CHAPTER 3 THE INDEX THEOREM Introduction In this the third chapter, we complete the proof of the index theorem for the four classical elliptic complexes. We give a proof of the Aityah-Singer theorem in general based on the Chern isomorphism between K-theory and cohomology (which is not proved). Our approach is to use the results of the first chapter to show there exists a suitable formula with the appropriate functorial properties. The results of the second chapter imply it must be a characteristic class. The normalizing constants are then determined using the method of universal examples. In section 3.1, we define the twisted signature complex and prove the Hirzebruch signature theorem. We shall postpone until section 3.4 the determination of all the normalizing constants if we take coefficients in an auxilary bundle. In section 3.2, we introduce spinors as a means of connect- ing the de Rham, signature and Dolbeault complexes. In section 3.3, we discuss the obstruction to putting a spin structure on a real vector bundle in terms of Stieffel-Whitney classes. We compute the characteristic classes of spin bundles. In section 3.4, we discuss the spin complex and the Â genus. In sec- tion 3.5, we use the spin complex together with the spin_{c} representation to discuss the Dolbeault complex and to prove the Riemann-Roch theo- rem for almost complex manifolds. In sections 3.6 and 3.7 we give another treatment of the Riemann-Roch theorem based on a direct approach for Kaehler manifolds. For Kaehler manifolds, the integrands arising from the heat equation can be studied directly using an invariant characterization of the Chern forms similar to that obtained for the Euler form. These two subsections may be deleted by a reader not interested in Kaehler geometry. In section 3.8, we give the preliminaries we shall need to prove the Atiyah- Singer index theorem in general. The only technical tool we will use which we do not prove is the Chern isomorphism between rational cohomology and K-theory. We give a discussion of Bott periodicity using Clifford algebras. In section 3.9, we show that the index can be treated as a formula in rational K-theory. We use constructions based on Clifford algebras to determine the normalizing constants involved. For these two subsections, some familarity with K-theory is helpful, but not essential. Theorems 3.1.4 and 3.6.10 were also derived by V. K. Patodi using a complicated cancellation argument as a replacement of the invariance the- ory presented in Chapter 2. A similar although less detailed discussion may also be found in the paper of Atiyah, Bott and Patodi.

3.1. The Hirzebruch Signature Formula.

The signature complex is best described using Clifford algebras as these provide a unified framework in which to discuss (and avoid) many of the ± signs which arise in dealing with the exterior algebra directly. The reader will note that we are choosing the opposite sign convention for our discussion of Clifford algebras from that adopted in the example of Lemma 2.1.5. This change in sign convention is caused by the √

−1 present in

discussing the symbol of a first order operator.

Let V be a real vector with a positive definite inner product. The Clifford algebra CLIF(V) is the universal algebra generated by V subject to the relations

v ∗ v + (v, v) = 0 for v ∈ V.

If the {e_i} are an orthonormal basis for V and if I = {1 ≤ i_1 < ⋯ < i_p ≤ dim(V)} then e_i ∗ e_j + e_j ∗ e_i = −2δ_{ij} is the Kronecker symbol and

e_I = e_{i1} ∗ ⋯ ∗ e_{ip}

is an element of the Clifford algebra. CLIF(V) inherits a natural inner product from V and the {e_I} form an orthonormal basis for V.

If Λ(V) denotes the exterior algebra of V and if END(Λ(V)) is the algebra of linear endomorphisms of Λ(V), there is a natural representation of CLIF(V) into END(Λ(V)) given by Clifford multiplication. Let ext: V → END(Λ(V)) be exterior multiplication on the left and let int(v) be interior multiplication, the adjoint. For example:

ext(e_1)(e_{i1} ∧ ⋯ ∧ e_{ip}) = { e_1 ∧ e_{i1} ∧ ⋯ ∧ e_{ip} if i_1 > 1 0 if i_1 = 1

int(e_1)(e_{i1} ∧ ⋯ ∧ e_{ip}) = { e_{i2} ∧ ⋯ ∧ e_{ip} if i_1 = 1 0 if i_1 > 1.

We define c(v) = ext(v) − int(v): V → END(Λ(V)).

It is immediate from the definition that

c(v)^2 = −(ext(v) int(v) + int(v) ext(v)) = −|v|^2I

so that c extends to define an algebra morphism

c: CLIF(V) → END(Λ(V)).

Furthermore:

c(e_I)1 = e_{i1} ∧ ⋯ ∧ e_{ip}

Hirzebruch Signature Formula 149

so the map w ↦ c(w)1 defines a vector space isomorphism (which is not, of course an algebra morphism) between CLIF(V) and Λ(V). Relative to an orthonormal frame, we simply replace Clifford multiplication by exterior multiplication. Since these constructions are all independent of the basis chosen, they extend to the case in which V is a real vector bundle over M with a fiber metric which is positive definite. We define CLIF(V), Λ(V), and c: CLIF(V) → END(Λ(V)) as above. Since we only want to deal with complex bundles, we tensor with C at the end to enable us to view these bundles as being complex. We emphasize, however, that the underlying constructions are all real. Clifford algebras provide a convenient way to describe both the de Rham and the signature complexes. Let (d + δ): C^{∞}(Λ(T^{∗}M)) → C^{∞}(Λ(T^{∗}M)) be exterior differentiation plus its adjoint as discussed earlier. The leading symbol of (d+δ) is √−1(ext(ξ)−int(ξ)) = √−1c(ξ). We use the following diagram to define an operator A; let ∇ be covariant differentiation. Then: A: C^{∞}(Λ(T^{∗}M)) ^{∇} → C^{∞}(T^{∗}M ⊗ Λ(T^{∗}M)) ^{c} → C^{∞}(Λ(T^{∗}M)). A is invariantly defined. If {e_{i}} is a local orthonormal frame for T^{∗}(M) which we identify with T (M), then: A(ω) = _{∑ i}(ext(e_{i}) − int(e_{i}))∇_{ei}(ω). Since the leading symbol of A is √−1c(ξ), these two operators have the same leading symbol so (d+δ)−A = A_{0} is an invariantly defined 0^{th} order operator. Relative to a coordinate frame, we can express A_{0} as a linear combination of the 1-jets of the metric with coefficients which are smooth in the {g_{ij}} variables. Given any point x_{0}, we can always choose a frame so the g_{ij/k} variables vanish at x_{0} so A_{0}(x_{0}) = 0 so A_{0} ≡ 0. This proves A = (d + δ) is defined by this diagram which gives a convenient way of describing the operator (d + δ) in terms of Clifford multiplication. This trick will be useful in what follows. If A and B are natural first order differential operators with the same leading symbol, then A = B since A − B is a 0^{th} order operator which is linear in the 1-jets of the metric. This trick does not work in the holomorphic category unless we impose the additional hypothesis that M is Kaehler. This makes the study of the Riemann-Roch theorem more complicated as we shall see later since there are many natural operators with the same leading symbol. We let α ∈ END(Λ(T^{∗}M)) be defined by: α(ω_{p}) = (−1)^{p}ω_{p} for ω_{p} ∈ Λ^{p}(T^{∗}M).

It is immediate that ext(ξ)α = −α ext(ξ)  and  int(ξ)α = −α int(ξ) so that −α(d+δ)α and (d+δ) have the same leading symbol. This implies α(d + δ) = −(d + δ)α. We decompose Λ(T^{∗}M) = Λ^{e}(T^{∗}M) ⊕ Λ^{o}(T^{∗}M) into the differential forms of even and odd degree. This decomposes Λ(T^{∗}M) into the ±1 eigenspaces of α. Since (d + δ) anti-commutes with α, we decompose (d + δ)_{e,o}: C^{∞}(Λ^{e,o}(T^{∗}M)) → C^{∞}(Λ^{e,o}(T^{∗}M)) where the adjoint of (d+δ)_{e} is (d+δ)_{o}. This is, of course, just the de Rham complex, and the index of this elliptic operator is χ(M). If dim(M) = m is even and if M is oriented, there is another natural endomorphism τ ∈ END(Λ(T^{∗}M)). It can be used to define an elliptic complex over M called the signature complex in just the same way that the de Rham complex was defined. Let dvol ∈ Λ^{m}(T^{∗}M) be the volume form. If {e_{i}} is an oriented local orthonormal frame for T^{∗}M, then dvol = e_{1} ∧ ⋯ ∧ e_{m}. We can also regard dvol = e_{1} ∗ ⋯ ∗ e_{m} ∈ CLIF(T^{∗}M) and we define τ = (√−1)^{m/2}c(dvol) = (√−1)^{m/2}c(e_{1}) … c(e_{m}). We compute: τ 2 = (−1)^{m/2}c(e_{1} ∗ ⋯ ∗ e_{m} ∗ e_{1} ∗ ⋯ ∗ e_{m})^{= (−1)m/2(−1)m+(m−1)+···+1} = (−1)^{(m+(m+1)m)/2} = 1. Because m is even, c(ξ)τ = −τc(ξ) so τ anti-commutes with the symbol of (d + δ). If we decompose Λ(T^{∗}M) = Λ^{+}(T^{∗}M) ⊕ Λ^{−}(T^{∗}M) into the ±1 eigenvalues of τ, then (d + δ) decomposes to define: (d + δ)_{±}: C^{∞}(Λ^{±}(T^{∗}M)) → C^{∞}(Λ^{∓}(T^{∗}M)) where the adjoint of (d + δ)_{+} is (d + δ)_{−}. We define: signature(M) = index(d + δ)+ to be the signature of M. (This is also often refered to as the index of M, but we shall not use this notation as it might be a source of some confusion).

We decompose the Laplacian Δ = Δ^{+} ⊕ Δ^{−} so signature(M) = dim N(Δ^{+}) − dim N(Δ^{−}). Let a_{s n}(x, orn) = a_{n}(x, Δ^{+}) − a_{n}(x, Δ^{−}) be the invariants of the heat equa- tion; they depend on the orientation orn chosen. Although we have com- plexified the bundles Λ(T^{∗}M) and CLIF(T^{∗}M), the operator (d+δ) is real. If m ≡ 2 (4), then τ is pure imaginary. Complex conjugation defines an isomorphism Λ^{+}(T^{∗}M) ^{≃} → Λ^{−}(T^{∗}M)  and Δ^{+} ^{≃} → Δ^{−}. This implies signature(M) = 0 and a_{s n}(x, orn) = 0 in this case. We can get a non-zero index if m ≡ 2 (4) if we take coefficients in some auxiliary bundle as we shall discuss shortly. If m ≡ 0 (4), then τ is a real endomorphism. In general, we compute: τ(e_{1} ∧ ⋯ ∧ e_{p}) = (√−1)^{m/2}e_{1} ∗ ⋯ ∗ e_{m} ∗ e_{1} ∗ ⋯ ∗ ep = (√−1)^{m/2}(−1)^{p(p−1)/2} e_{p+1} ∧ ⋯ ∧ e_{m}. If “∗” is the Hodge star operator discussed in the first sections, then τ_{p} = (√−1)^{m/2+p(p−1)} ∗p acting on p-forms. The spaces Λ^{p}(T^{∗}M)⊕Λ^{m−p}(T^{∗}M) are invariant under τ. If p ≠ m − p there is a natural isomorphism Λ^{p}(T^{∗}M) ^{≈} → (Λ^{p}(T^{∗}M) ⊕ Λ^{m−p}(T^{∗}M))^{±}  by ω_{p} ↦ _{1 2} (ω_{p} ± ω_{p}). This induces a natural isomorphism from Δ^{p} ^{≃} → Δ^{±} on (Λ^{p}(T^{∗}M) ⊕ Λ^{m−p}(T^{∗}M))± so these terms all cancel off in the alternating sum and the only contribution is made in the middle dimension p = m − p. If m = 4k and p = 2k then τ = ∗. We decompose N(Δ_{p}) = N(Δ_{+ p}) ⊕ N(Δ_{− p}) so signature(M) = dim N(Δ_{+ p}) − dim N(Δ_{− p}). There is a natural symmetric bilinear form on H^{2k}(T^{∗}M; C) = N(Δ_{p}) defined by I(α_{1}, α_{2}) = ∫_{M} α_{1} ∧ α_{2}. If we use the de Rham isomorphism to identify de Rham and simplicial cohomology, then this bilinear form is just the evaluation of the cup product

152 3.1. Hirzebruch Signature Formula of two cohomology classes on the top dimensional cycle. This shows I can be defined in purely topological terms. The index of a real quadratic form is just the number of +1 eigenvalues minus the number of −1 eigenvalues when it is diagonalized over R. Since I(α, β) = ∫_{M} α ∧ β = ∫_{M} (α, ∗β) dvol = (α, ∗β)_{L2} , we see index(I) = dim{+1 eigenspace of ∗ on H^{p}} − dim{−1 eigenspace of ∗ on H^{p}} = dim N(Δ^{+}) − dim N(Δ^{−}) = signature(M). This gives a purely topological definition of the signature of M in terms of cup product. We note that if we reverse the orientation of M, then the signature changes sign. Example: Let M = CP_{2k} be complex projective space. Let x ∈ H^{2}(M; C) be the generator. Since x^{k} is the generator of H^{2k}(M; C) and since x^{k} ∧ x^{k} = x^{2k} is the generator of H^{4k}(M; C), we conclude that ∗x^{k} = x^{k}. dim N(Δ_{+ 2k}) = 1 and dim N(Δ_{− 2k}) = 0 so signature(CP_{2k}) = 1. An important tool in the study of the de Rham complex was its multi- plicative properties under products. Let M_{i} be oriented even dimensional manifolds and let M = M_{1} ×M_{2} with the induced orientation. Decompose: Λ(T^{∗}M) = Λ(T^{∗}M_{1}) ⊗ Λ(T^{∗}M_{2}) CLIF(T^{∗}M) = CLIF(T^{∗}M_{1}) ⊗ CLIF(T^{∗}M_{2}) as graded non-commutative algebras—i.e., (ω_{1} ⊗ ω_{2}) ∘ (ω_{′1} ⊗ ω_{′2}) = (−1)^{deg ω_{2}·deg ω′1} (ω_{1} ∘ ω_{′1}) ⊗ (ω_{2} ∘ ω_{′2}) for ∘ = either ∧ or ∗. Relative to this decomposition, we have: τ = τ_{1} ⊗ τ_{2} where the τ_{i} commute. This implies that: Λ^{+}(M) = Λ^{+}(T^{∗}M_{1}) ⊗ Λ^{+}(T^{∗}M_{2}) ⊕ Λ^{−}(T^{∗}M_{1}) ⊗ Λ^{−}(T^{∗}M_{2}) Λ^{−}(M) = Λ^{−}(T^{∗}M_{1}) ⊗ Λ^{+}(T^{∗}M_{2}) ⊕ Λ^{+}(T^{∗}M_{1}) ⊗ Λ^{−}(T^{∗}M_{2}) N(Δ^{+}) = N(Δ_{+ 1}) ⊗ N(Δ_{+ 2}) ⊕ N(Δ_{− 1}) ⊗ N(Δ_{− 2}) N(Δ^{−}) = N(Δ_{− 1}) ⊗ N(Δ_{+ 2}) ⊕ N(Δ_{+ 1}) ⊗ N(Δ_{− 2}) signature(M) = signature(M_{1}) signature(M_{2}).

Example: Let ρ be a partition of k = i_{1} + ⋯ + i_{j} and M_{r ρ} = CP_{2i1} × ⋯ × CP_{2ij}. Then signature(M_{r ρ}) = 1. Therefore if L_{k} is the Hirzebruch L-polynomial,  signature(M_{r ρ}) = ∫_{M^{p} ρ} L_{k}(T (M_{r ρ})) by Lemma 2.3.5. We can now begin the proof of the Hirzebruch signature theorem. We shall use the same argument as we used to prove the Gauss-Bonnet theorem with suitable modifications. Let m = 4k and let a_{s n}(x, orn) = a_{n}(x, Δ^{+}) − a_{n}(x, Δ^{−}) be the invariants of the heat equation. By Lemma 1.7.6: ∫^{M} a^{s n}(x, orn) = { 0  if n ≠ m signature(M)  if n = m so this gives a local formula for signature(M). We can express τ functorially in terms of the metric tensor. We can find functorial local frames for Λ^{±} relative to any oriented coordinate system in terms of the coordinate frames for Λ(T^{∗}M). Relative to such a frame, we express the symbol of Δ^{±} functorially in terms of the metric. The leading symbol is |ξ|^{2}I; the first order symbol is linear in the 1-jets of the metric with coefficients which depend smoothly on the {g_{ij}} variables; the 0^{th} order symbol is linear in the 2-jets of the metric and quadratic in the 1-jets of the metric with coefficients which depend smoothly on the {g_{ij}} variables. By Lemma 2.4.2, we conclude a_{s n}(x, orn) is homogeneous of order n in the jets of the metric. It is worth noting that if we replace the metric G by c^{2}G for c > 0, then the spaces Λ^{±} are not invariant. On Λ^{p} we have: τ(c^{2}G)(ω_{p}) = c^{2p−m}τ(G)(ω_{p}). However, in the middle dimension we have τ is invariant as are the spaces Λ_{± p} for 2p = m. Clearly Δ_{± p}(c^{2}G) = c^{−2}Δ_{± p}(G). Since a_{s n}(x, orn) only depends on the middle dimension, this provides another proof that a_{s n} is homogeneous of order n in the derivatives of the metric since a_{s n}(x, orn)(c^{2}G) = a_{n}(x, c^{−2}Δ_{+ p}) − a_{n}(x, c^{−2}Δ_{− p}) = c^{−n}a_{n}(x, Δ_{+ p}) − c^{−n}a_{n}(x, Δ_{− p}) = c^{−n}a_{s n}(x, orn)(G). If we reverse the orientation, we interchange the roles of Δ^{+} and Δ^{−} so a^{s} n changes sign if we reverse the orientation. This implies a_{s n} can be regarded as an invariantly defined m-form; a_{s n}(x) = a_{s n}(x, orn) dvol ∈ P_{m,n,m}.

154 3.1. Hirzebruch Signature Formula Theorem 3.1.1. Let a_{s n} = {a_{n}(x, Δ^{+})−a_{n}(x, Δ^{−})} dvol ∈ P_{m,n,m} then: (a) a_{s n} = 0 if either m ≡ 2 (4) or if n < m. (b) If m = 4k, then a_{s 4k} = L_{k} is the Hirzebruch polynomial so signature(M) = ∫_{M} L_{k}. (Hirzebruch Signature Theorem). Proof: We already noted that Δ^{+} is naturally isomorphic to Δ^{−} if m ≡ 2 (4) so a_{s n} = 0 in that case. Lemma 2.5.6 implies a_{s n} = 0 for n < m. If m = 4k, then a_{s m} is a characteristic form of T (M) by Lemma 2.5.6. We know ∫_{M^{r} ρ} a_{s m} = signature(M_{r ρ}) = 1 = ∫_{M^{r} ρ} Lk so Lemma 2.3.4 implies a_{s m} = L_{k}. Since signature(M) = ∫_{M} a_{s m} for any manifold M, we conclude signature(M) = ∫_{M} L_{k} in general which com- pletes the proof of (b). If ω ∈ Λ(T^{∗}M), we define ∫_{M} ω = ∫_{M} ω_{m} of the top degree form. With this notational convention, we can also express signature(M) = ∫_{M} L which is a common form in which the Hirzebruch signature theorem ap- pears. It is worth making a few remarks about the proof of this result. Just as in the case of the de Rham complex, the heat equation furnishes us with the a priori local formula for the signature of M. The invariance theory of the second chapter identifies this local formula as a characteristic class. We evaluate this local formula on a sufficient number of classifying examples to determine the normalizing constants to prove a_{s m} = L_{k}. There are a great many consequences of this theorem and of the Gauss- Bonnet theorem. We present just a few to illustrate some of the applica- tions: Corrolary 3.1.2. (a) Let F → M_{1} → M_{2} be a finite covering projection. Then χ(M_{1}) = χ(M_{2})|F |. If M_{2} is orientable, then M_{1} is orientable and we give it the natu- ral orientation inherited from M_{2}. Then signature(M_{1})=signature(M_{2})|F |. (b) If M_{1} and M_{2} are manifolds of dimension m, we let M_{1} # M_{2} be the connected sum. This is defined by punching out disks in both manifolds and gluing along the common resulting boundaries. Then χ(M_{1} # M_{2}) + χ(S^{m}) = χ(M_{1}) + χ(M_{2}). If M_{1} and M_{2} are oriented by some orientation

Hirzebruch Signature Formula 155

on M_{1} # M_{2} then signature(M_{1} # M_{2}) = signature(M_{1}) + signature(M_{2}), if m ≡ 0 (4). Proof: (a) is an immediate consequence of the fact we have local formulas for the Euler characteristic and for the signature. To prove (b), we note that the two disks we are punching out glue together to form a sphere. We use the additivity of local formulas to prove the assertion about X. The second assertion follows similarly if we note signature(S^{m}) = 0. This corollary has topological consequences. Again, we present just one to illustrate the methods involved: Corollary 3.1.3. Let F → CP_{2j} → M be a finite covering. Then |F | = 1 and M = CP_{2j}. Proof: χ(CP_{2j}) = 2j + 1 so as 2j + 1 = |F |χ(M), we conclude |F | must be odd. Therefore, this covering projection is orientation preserving so M is orientable. The identity 1 = signature(CP_{2j}) = |F | signature(M) implies |F | = 1 and completes the proof. If m ≡ 2 (4), the signature complex does not give a non-trivial index. We twist by taking coefficients in an auxiliary complex vector bundle V to get a non-trivial index problem in any even dimension m. Let V be a smooth complex vector bundle of dimension v equipped with a Riemannian connection ∇. We take the Levi-Civita connection on T^{∗}(M) and on Λ(T^{∗}M) and let ∇ be the tensor product connection on Λ(T^{∗}M)⊗V. We define the operator (d + δ)_{V} on C^{∞}(Λ(T^{∗}M) ⊗ V) using the diagram: (d + δ)_{V}: C^{∞}(Λ(T^{∗}M) ⊗ V) ^{∇} → C(T^{∗}M ⊗ Λ(T^{∗}M) ⊗ V)^{c⊗1} ⟶ C^{∞}(Λ(T^{∗}M) ⊗ V). We have already noted that if V = 1 is the trivial bundle with flat connec- tion, then the resulting operator is (d + δ). We define τ_{V} = τ ⊗ 1, then a similar argument to that given for the signature complex shows τ_{2 V} = 1 and τ_{V} anti-commutes with (d + δ)_{V}. The ±1 eigenspaces of τ_{V} are Λ^{±}(T^{∗}M) ⊗ V and the twisted signature complex is defined by the diagram: (d + δ)_{± V}: C^{∞}(Λ^{±}(T^{∗}M) ⊗ V) → C^{∞}(Λ^{∓}(T^{∗}M) ⊗ V) where as before (d + δ)_{− V} is the adjoint of (d + δ)_{+ V}. We let Δ_{± V} be the associated Laplacians and define: signature(M, V) = index((d + δ)_{+ V}) = dim N(Δ_{+ V}) − dim N(Δ_{− V}) a_{s n}(x, V) = {a^{n}(x, Δ^{+ V}) − a^{n}(x, Δ^{− V})} dvol ∈ Λm ∫_{M} a_{s n}(x, V) = signature(M, V).

156 3.1. Hirzebruch Signature Formula The invariance of the index under homotopies shows signature(M, V) is independent of the metric on M, of the fiber metric on V, and of the Riemannian connection on V. If we do not choose a Riemannian connection on V, we can still compute signature(M, V) = index((d + δ)_{+ V}), but then (d + δ)_{− V} is not the adjoint of (d + δ)_{+ V} and a_{s n} is not an invariantly defined m form. Relative to a functorial coordinate frame for Λ^{±}, the leading symbol of Δ^{±} is |ξ|^{2}I. The first order symbol is linear in the 1-jets of the metric and the connection form on V. The 0^{th} order symbol is linear in the 2-jets of the metric and the connection form and quadratic in the 1-jets of the metric and the connection form. Thus a_{s n}(X, V) ∈ R_{m,n,m,v}. Theorem 2.6.1 implies a_{s n} = 0 for n < m while a_{s m} is a characteristic form of T (M) and of V. If m = 2 and if v = 1, then R_{2,2,2,1} is one dimensional and is spanned by the first Chern class c_{1}(V) = ch(V) = _{i 2π} Ω. Consequently a_{s 2} = cc_{1} in this case. We shall show later that this normalizing constant c = 2—i.e., Lemma 3.1.4. Let m = 2 and let V be a line bundle over M_{2}. Then: signature(M, V) = 2 ∫_{M} c_{1}(L). We postpone the proof of this lemma until later in this chapter. With this normalizing constant established, we can compute a formula for signature(M, V) in general: Theorem 3.1.5. Let L be the total L-polynomial and let ch(V) be the Chern character. Then: (a) a_{s n}(x, V) = 0 for n < m. (b) a_{s m}(x, V) = ∑_{4s+2t=m} L_{s}(T M) ∧ 2^{t}ch_{t}(V) so that: signature(M, V) = _{∑ 4s+2t=m} ∫_{M} L_{s}(T M) ∧ 2^{t}ch_{t}(V). The factors of 2^{t} are perhaps a bit mysterious at this point. They arise from the normalizing constant of Lemma 3.1.4 and will be explained when we discuss the spin and Dolbeault complexes. Proof: We have already proved (a). We know a_{s m}(x, V) is a characteristic form which integrates to signature(M, V) so it suffices to verify the formula of (b). If V_{1} and V_{2} are bundles, we let V = V_{1} ⊕ V_{2} with the direct sum connection. Since Δ_{± V} = Δ_{± V1} ⊕ Δ_{± V2} we conclude signature(M, V_{1} ⊕ V_{2}) = signature(M, V_{1}) + signature(M, V_{2}).

Since the integrals are additive, we apply the uniqueness of Lemma 2.6.3 to conclude the local formulas must be additive. This also follows from Lemma 1.7.5 so that: a_{s n}(x, V_{1} ⊕ V_{2}) = a_{s n}(x, V_{1}) + a_{s n}(x, V_{2}). Let {P_{ρ}}_{|ρ|=s} be the basis for P_{m,4s,4s} and expand: a_{s m}(x, V) = _{∑ 4|ρ|+2t=m} P_{ρ} ∧ Q_{m,t,v,ρ}  for Q_{m,t,v,ρ} ∈ Qm,2t,2t,v a characteristic form of V. Then the additivity under direct sum implies: Q_{m,t,v,ρ}(V_{1} ⊕ V_{2}) = Q_{m,t,v_{1},ρ}(V_{1}) + Q_{m,t,v_{2},ρ}(V_{2}). If v = 1, then Q_{m,t,1,ρ}(V_{1}) = c · c_{1}(V)^{t} since Q_{m,2t,2t,1} is one dimensional. If A is diagonal matrix, then the additivity implies: Q_{m,t,v,ρ}(A) = Q_{m,t,v,ρ}(λ) = c ·_{∑ j} λ_{tj} = c · ch_{t}(A). Since Q is determined by its values on diagonal matrices, we conclude: Q_{m,t,v,ρ}(V) = c(m, t, ρ)ch_{t}(V) where the normalizing constant does not depend on the dimension v. There- fore, we expand a_{s m} in terms of ch_{t}(V) to express: a_{s m}(x, V) = _{∑ 4s+2t=m} P_{m,s} ∧ 2^{t}ch_{t}(V)  for P_{m,s} ∈ P_{m,4s,4s} . We complete the proof of the theorem by identifying P_{m,s} = L_{s}; we have reduced the proof of the theorem to the case v = 1. We proceed by induction on m; Lemma 3.1.4 establishes this theorem if m = 2. Suppose m ≡ 0 (4). If we take V to be the trivial bundle, then if 4k = m,  a_{s m}(x, 1) = L_{k} = Pm,k follows from Theorem 3.1.1. We may therefore assume 4s < m in comput- ing P_{m,s}. Let M = M_{1} × S^{2} and let V = V_{1} ⊗ V_{2} where V_{1} is a line bundle over M_{1} and where V_{2} is a line bundle over S^{2} so ∫_{S 2} c_{1}(V_{2}) = 1. (We constructed such a line bundle in section 2.1 using Clifford matrices). We take the product connection on V_{1} ⊗ V_{2} and decompose: Λ^{+}(V) = Λ^{+}(V_{1}) ⊗ Λ^{+}(V_{2}) ⊕ Λ^{−}(V_{1}) ⊗ Λ^{−}(V_{2}) Λ^{−}(V) = Λ^{−}(V_{1}) ⊗ Λ^{+}(V_{2}) ⊕ Λ^{+}(V_{1}) ⊗ Λ^{−}(V_{2}).

158 3.1. Hirzebruch Signature Formula A similar decomposition of the Laplacians yields: signature(M, V) = signature(M_{1}, V_{1}) signature(M_{2}, V_{2}) = 2 signature(M_{1}, V_{1}) by Lemma 3.1.4. Since the signatures are multiplicative, the local formulas are multiplicative by the uniqueness assertion of Lemma 2.6.3. This also follows by using Lemma 1.7.5 that a_{s m}(x, V) = _{∑ p+q=m} a_{s p}(x_{1}, V_{1})a_{s q}(x_{2}, V_{2}) and the fact a_{p} = 0 for p < m_{1} and a_{q} = 0 for q < m_{2}. Thus we conclude: a_{s m}(x, V) = a_{s m1} (x_{1}, V_{1})a_{s m2} (x_{2}, V_{2}) where, of course, m_{2} = 2 and m_{1} = m − 2. We use the identity: ch(V_{1} ⊗ V_{2}) = ch(V_{1})ch(V_{2}) to conclude therefore: signature(M_{1}, V_{1}) = 1 2 signature(M, V) = 1 2{_{∑ 4s+2t=m−2} ∫_{MI} P_{m,s} ∧ 2^{t}ch_{t}(V_{1})} ∫_{M2} 2ch_{1}(V_{2}) = _{∑ 4s+2t=m−2} ∫_{M1} P_{m,s} ∧ 2^{t}ch_{t}(V_{1}). We apply the uniqueness assertion of Lemma 2.6.3 to conclude P_{m,s} = P_{m−2,s} for 4s ≤ m − 2. Since by induction, P_{m−2,s} = L_{s} this completes the proof of the theorem. We note that the formula is non-zero, so Lemma 2.6.3 implies that in any even dimension m, there always exist (M, V) so signature(M, V) ≠ 0. In fact, much more is true. Given any orientable manifold M, we can find V over M so signature(M, V) ≠ 0 if dim(M) is even. Since the proof of this assertion relies on the fact ch: K(M) ⊗ Q ^{≃} → H^{2∗}(M; Q) we postpone a discussion of this fact until we discuss the index theorem in general.

3.2. Spinors and their Representations. To define the signature complex, we needed to orient the manifold M. For any Riemannian manifold, by restricting to local orthonormal frames, we can always assume the transition functions of T (M) are maps g_{αβ}: U_{α} ∩ U_{β} → O(m). If M is oriented, by restricting to local orthonormal frames, we can choose the transition functions of T (M) to lie in SO(m) and to reduce the structure group from GL(m, R) to SO(m). The signature com- plex results from the represntation Λ^{±} of SO(m); it cannot be defined in terms of GL(m, R) or O(m). By contrast, the de Rham complex results from the representation Λ^{e,o} which is a representation of GL(m, R) so the de Rham complex is defined for non-orientable manifolds. To define the spin complex, which is in some sense a more fundamental elliptic complex than is either the de Rham or signature complex, we will have to lift the transition functions from SO(m) to SPIN(m). Just as every manifold is not orientable, in a similar fashion there is an obstruction to defining a spin structure. If m ≥ 3, then π_{1}(SO(m)) = Z_{2}. Abstractly, we define SPIN(m) to be the universal cover of SO(m). (If m = 2, then SO(2) = S^{1} and we let SPIN(2) = S^{1} with the natural double cover Z_{2} → SPIN(2) → SO(2) given by θ ↦ 2θ). To discuss the representations of SPIN(m), it is convenient to obtain a more concrete representation of SPIN(m) in terms of Clifford algebras. Let V be a real vector space of dimension v ≡ 0 (2). Let ⨂ V = R ⊕ V ⊕ (V ⊗ V) ⊕ ⋯ ⊕ ⨂^{k} V ⊕ ⋯ be the complete tensor algebra of V. We assume V is equipped with a symmetric positive definite bilinear form. Let I be the two-sided ideal of ⨂ V generated by {v ⊗ v + |v|^{2}}_{v∈V}, then the real Clifford algebra CLIF(V) = ⨂ V mod I. (Of course, we will always construct the corre- sponding complex Clifford algebra by tensoring CLIF(V) with the complex numbers). There is a natural transpose defined on ⨂ V by: (v_{1} ⊗ ⋯ ⊗ v_{k})^{t} = v_{k} ⊗ ⋯ ⊗ v_{1}. Since this preserves the ideal I, it extends to CLIF(V). If {e_{1},…, e_{v}} are the orthonormal basis for V, then e_{i} ∗ e_{j} + e_{j} ∗ e_{i} = −2δ_{ij} and (e_{i1} ∗ ⋯ ∗ e_{ip})^{t} = (−1)^{p(p−1)/2} e_{i1} ∗ ⋯ ∗ e_{ip}. If V is oriented, we let {e_{i}} be an oriented orthonormal basis and define: τ = (√−1)^{v/2}e_{1} ∗ ⋯ ∗ e_{v}.

We already computed that: τ 2 = (−1)^{v/2}e_{1} ∗ ⋯ ∗ e_{v} ∗ e_{1} ∗ … ev = (−1)^{v(v−1)/2} e_{1} ∗ ⋯ ∗ e_{v} ∗ e_{1} ∗ ⋯ ∗ ev = e_{1} ∗ ⋯ ∗ e_{v} ∗ e_{v} ∗ ⋯ ∗ e_{1} = (−1)^{v} = 1. We let SPIN(V) be the set of all w ∈ CLIF(V) such that w can be decomposed as a formal product w = v_{1} ∗ ⋯ ∗ v_{2j} for some j where the v_{i} ∈ V are elements of length 1. It is clear that w^{t} is such an element and that ww^{t} = 1 so that SPIN(V) forms a group under Clifford multiplication. We define ρ(w)x = wxw^{t}  for x ∈ CLIF, w ∈ CLIF(V). For example, if v_{1} = e_{1} is the first element of our orthonormal basis, then: e^{1}e^{i}e^{1} = { −e1  i = 1 ei  i ≠ 1. The natural inclusion of V in ⨂ V induces an inclusion of V in CLIF(V). ρ(e_{1}) preserves V and is reflection in the hyperplane defined by e_{1}. If w ∈ SPIN(V), then ρ(w): V → V is a product of an even number of hyperplane reflections. It is therefore in SO(V) so ρ: SPIN(V) → SO(V) is a group homomorphism. Since any orthogonal transformation of determi- nant one can be decomposed as a product of an even number of hyperplane reflections, ρ is subjective. If w ∈ CLIF(V) is such that wvw^{t} = v all v ∈ V and ww^{t} = 1 then wv = vw for all v ∈ V so w must be in the center of CLIF(V). Lemma 3.2.1. If dim V = v is even, then the center of CLIF(V) is one dimensional and consists of the scalars. Proof: Let {e_{i}} be an orthonormal basis for V and let {e_{I}} be the cor- responding orthonormal basis for CLIF(V). We compute: e_{i} ∗ e_{i1} ∗ ⋯ ∗ e_{ip} ∗ e_{i} = e_{i1} ∗ ⋯ ∗ e_{ip}  for p ≥ 1, if (a) p is even and i is one of the i_{j} or (b) p is odd and i is not one of the i_{j}. Thus given I we can choose i so e_{i} ∗ e_{I} = −e_{I} ∗ e_{i}. Thus

And their Representations 161

e_{i} ∗ (∑_{I} c_{I}e_{I}) = ∑_{I} c_{I}e_{I} ∗ e_{i} for all i implies c_{I} = 0 for |I| > 0 which completes the proof. (We note that this lemma fails if dim V is odd since the center in that case consists of the elements a+be_{1} ∗ ⋯∗e_{v} and the center is two dimensional). If ρ(w) = 1 and w ∈ SPIN(V), this implies w is scalar so w = ±1. By considering the arc in SPIN(V) given by w(θ) = ((cos θ)e_{1} + (sin θ)e_{2}) ∗ (−(cos θ)e_{1} + (sin θ)e_{2}), we note w(0) = 1 and w(_{π 2} ) = −1 so SPIN(V) is connected. This proves we have an exact sequence of groups in the form: Z_{2} → SPIN(V) → SO(V) and shows that SPIN(V) is the universal cover of SO(V) for v > 2. We note that is it possible to define SPIN(V) using Clifford algebras even if v is odd. Since the center of CLIF(V) is two dimensional for v odd, more care must be used with the relevant signs which arise. As we shall not need that case, we refer to Atiyah-Bott-Shapiro (see bibliography) for further details. SPIN(V) acts on CLIF(V) from the left. This is an orthogonal action. Lemma 3.2.2. Let dim V = 2v_{1}. We complexify CLIF(V). As a left SPIN(V) module, this is not irreducible. We can decompose CLIF(V) as a direct sum of left SPIN(V) modules in the form CLIF(V) = 2^{v1} Δ. This representation is called the spin representation. It is not irreducible but further decomposes in the form Δ = Δ^{+} ⊕ Δ^{−}. If we orient V and let τ be the orientation form discussed earlier, then left multiplication by τ is ±1 on Δ^{±} so these are inequivalent representations. They are irreducible and act on a representation space of dimension 2v^{1}−1 and are called the half-spin representations. Proof: Fix an oriented orthonormal basis {e_{i}} for V and define: α_{1} = √−1 e_{1}e_{2}, α_{2} = √−1 e_{3}e_{4}, …, α_{v1} = √−1 e_{v−1}ev as elements of CLIF(V). It is immediate that τ = α_{1} … α_{v1} and: α_{2 i} = 1 and α_{i}α_{j} = α_{j}α_{i}. We let the {α_{i}} act on CLIF(V) from the right and decompose CLIF(V) into the 2^{v1} simultaneous eigenspaces of this action. Since right and left multiplication commute, each eigenspace is invariant as a left SPIN(V)

162 3.2. Spinors module. Each eigenspace corresponds to one of the 2^{v1} possible sequences of + and − signs. Let ε be such a string and let Δ_{ε} be the corresponding representation space. Since e_{1}α_{1} = −α_{1}e_{1} and e_{1}α_{i} = α_{i}e_{1} for i > 1, multiplication on the right by e_{1} transforms Δ_{ε} → Δ_{ε′}, where ε(1) = −ε′(1) and ε(i) = ε′(i) for i > 1. Since this map commutes with left multiplication by SPIN(V), we see these two representations are isomorphic. A similar argument on the other indices shows all the representation spaces Δ_{ε} are equivalent so we may decompose CLIF(V) = 2^{v1} Δ as a direct sum of 2^{v1} equivalent repre- sentation spaces Δ. Since dim(CLIF (V)) = 2^{v} = 4^{v1} , the representation space Δ has dimension 2^{v1} . We decompose Δ into ±1 eigenspaces under the action of τ to define Δ^{±}. Since e_{1} ∗ ⋯ ∗ e_{v} is in the center of SPIN(V), these spaces are invariant under the left action of SPIN(V). We note that elements of V anti-commute with τ so Clifford multiplication on the left defines a map: cl: V ⊗ Δ^{±} → Δ∓ so both of the half-spin representations have the same dimension. They are clearly inequivalent. We leave the proof that they are irreducible to the reader as we shall not need this fact. We shall use the notation Δ^{±} to denote both the representations and the corresponding representation spaces. Form the construction we gave, it is clear the CLIF(V) acts on the left to preserve the space Δ so Δ is a representation space for the left action by the whole Clifford algebra. Since vτ = −τv for v ∈ V, Clifford multiplication on the left by an element of V interchanges Δ^{+} and Δ^{−}. Lemma 3.2.3. There is a natural map given by Clifford multiplication of V ⊗ Δ^{±} → Δ^{∓}. This map induces a map on the representations involved: ρ ⊗ Δ^{±} ↦ Δ^{∓}. If this map is denoted by v ∗ w then v ∗ v ∗ w = −|v|^{2}w. Proof: We already checked the map on the spaces. We check: wvw^{t} ⊗ wx ↦ wvw^{t}wx = wvx to see that the map preserves the relevant representation. We emphasize that Δ^{±} are complex representations since we must complexify CLIF(V) to define these representation spaces (ww^{t} = 1 for w ∈ SPIN). There is a natural map ρ: SPIN(V) → SO(V). There are natural repre- sentations Λ, Λ^{±}, Λ^{e,o} of SO(V) on the subspaces of Λ(V) = CLIF(V). We use ρ to extend these representations of SPIN(V) as well. They are related to the half-spin representations as follows: Lemma 3.2.4. (a) Λ = Δ ⊗ Δ. (b) (Λ^{+} − Λ^{−}) = (Δ^{+} − Δ^{−}) ⊗ (Δ^{+} + Δ^{−}).

(c) (Λ^{e} − Λ^{o}) = (Δ^{+} − Δ^{−}) ⊗ (Δ^{+} − Δ^{−})(−1)^{v/2}. These identities are to be understood formally (in the sense of K-theory). They are shorthand for the identities: Λ^{+} = (Δ^{+} ⊗ Δ^{+}) ⊕ (Δ^{+} ⊗ Δ^{−}) and Δ^{−} = (Δ^{−} ⊗ Δ^{+}) ⊕ (Δ^{−} ⊗ Δ^{−}) and so forth. Proof: If we let SPIN(V) act on CLIF(V) by right multiplication by w^{t} = w^{−1}, then this representation is equivalent to left multiplication so we get the same decomposition of CLIF(V) = 2^{v1} Δ as right spin modules. Since ρ(w)x = wxw^{t}, it is clear that Λ = Δ⊗Δ where one factor is viewed as a left and the other as a right spin module. Since Λ^{±} is the decomposition of SPIN(V) under the action of τ from the left, (b) is immediate. (c) follows similarly once the appropriate signs are taken into consideration; w ∈ CLIF(V)^{even} if and only if τwτ^{t} = (−1)^{v/2}w which proves (c). It is helpful to illustrate this for the case dim V = 2. Let {e_{1}, e_{2}} be an oriented orthonormal basis for V. We compute that: ((cos α)e_{1} + (sin α)e_{2})((cos β)e1 + (sin β)e^{2}) = (− cos α cos β − sin α sin β) + (cos α sin β − sin α cos β)e_{1}e2 so elements of the form cos γ+(sin γ)e_{1}e_{2} belong to SPIN(V). We compute: (cos α + (sin α)e_{1}e_{2})(cos β + (sin β)e_{1}e_{2}) = cos(α + β) + sin(α + β)e_{1}e2 so spin(V) is the set of all elements of this form and is naturally isomorphic to the circle S^{1} = [0, 2π] with the endpoints identified. We compute that ρ(w)(e_{1}) = (cos θ + (sin θ)e^{1}e^{2})e^{1}(cos θ + (sin θ)e^{2}e^{1}) = (cos^{2} θ − sin^{2} θ)e_{1} + 2(cos θ sin θ)e2 = cos(2θ)e_{1} + sin(2θ)e2 ρ(w)(e_{2}) = cos(2θ)e_{2} − sin(2θ)e1 so the map ρ: S^{1} → S^{1} is the double cover θ ↦ 2θ. We construct the one dimensional subspaces V_{i} generated by the ele- ments: v1 = 1 + τ, τv1 = v_{1}, v_{1}τ = v_{1},  v2 = 1 − τ, τv2 = −v_{2}, v_{2}τ = −v_{2},  v3 = (1 + τ)e_{1}, τv3 = v_{3}, v_{3}τ = −v_{3},  v4 = (1 − τ)e_{1}, τv4 = −v_{4}, v_{4}τ = v_{4}.

When we decompose CLIF(V) under the left action of SPIN(V), V_{1} ≃ V_{3} ≃ Δ^{+}  and V_{2} ≃ V_{4} ≃ Δ^{−}. When we decompose CLIF(V) under the right action of SPIN(V), and replace τ by −τ = τ^{t} acting on the right: V_{2} ≃ V_{3} ≃ Δ^{+}  and V_{1} ≃ V_{4} ≃ Δ^{−}. From this it follows that as SO(V) modules we have: V^{1} = Δ^{+} ⊗ Δ^{−}, V^{2} = Δ^{−} ⊗ Δ^{+}, V^{3} = Δ^{+} ⊗ Δ^{+}, V^{4} = Δ^{−} ⊗ Δ− from which it is immediate that: Λe = V_{1} ⊕ V2 = Δ^{+} ⊗ Δ^{−} ⊕ Δ^{−} ⊗ Δ+ Λo = V_{3} ⊕ V4 = Δ^{+} ⊗ Δ^{+} ⊕ Δ^{−} ⊗ Δ− Λ+ = V_{1} ⊕ V3 = Δ^{+} ⊗ Δ^{−} ⊕ Δ^{+} ⊗ Δ+ Λ− = V_{2} ⊕ V4 = Δ^{−} ⊗ Δ^{+} ⊕ Δ^{−} ⊗ Δ^{−}. If V is a one-dimensional complex vector space, we let V_{r} be the under- lying real vector space. This defines a natural inclusion of U(1) → SO(2). If we let J : V → V be complex multiplication by √−1 then J(e_{1}) = e2 and J(e_{2}) = −e_{1} so J is equivalent to Clifford multiplication by e_{1}e_{2} on the left. We define a complex linear map from V to Δ^{−} ⊗ Δ^{−} by: T (v) = v − iJ(v) ∈ V4 by computing T (e_{1}) = (e_{1} − ie_{2}) = (1 − ie_{1}e_{2})(e_{1}) and T (e_{2}) = e_{2} + ie_{1} = (i)(e_{1} − ie_{2}). Lemma 3.2.5. Let U(1) be identified with SO(2) in the usual manner. If V is the underlying complex 1-dimensional space corresponding to V_{r} then V ≃ Δ^{−} ⊗ Δ^{−} and V^{∗} ≃ Δ^{+} ⊗ Δ^{+} as representation spaces of SPIN(2). Proof: We have already verified the first assertion. The second follows from the fact that V^{∗} was made into a complex space using the map −J instead of J on V_{r}. This takes us into V_{3} instead of into V_{4}. It is also clear that Δ^{−} = (Δ^{+})^{∗} if dim V = 2. Of course, all these statements are to be interpreted as statements about representations since they are trivial as statements about vector spaces (since any vector spaces of the same dimension are isomorphic).

3.3. Spin Structures on Vector Bundles. We wish to apply the constructions of 3.2 to vector bundles over mani- folds. We first review some facts regarding principal bundles and Stieffel- Whitney classes which we shall need. Principal bundles are an extremely convenient bookkeeping device. If G is a Lie group, a principal G-bundle is a fiber space π: P_{G} → M with fiber G such that the transition functions are elements of G acting on G by left multiplication in the group. Since left and right multiplication commute, we can define a right action of G on P_{G} which is fiber preserving. For example, let SO(2k) and SPIN(2k) be the groups defined by R^{2k} with the cannonical inner product. Let V be an oriented Riemannian vector bundle of dimension 2k over M and let P_{SO} be the bundle of oriented frames of V. P_{SO} is an SO(2k) bundle and the natural action of SO(2k) from the right which sends an oriented orthonormal frame s = (s_{1},…, s_{2k}) to s · g = (s_{′1},…, s_{′2k}) is defined by: s_{′i} = s_{1}g_{1,i} + ⋯ + s_{2k}g_{2k,i}. The fiber of P_{SO} is SO(V_{x}) where V_{x} is the fiber of V over the point x. This isomorphism is not natural but depends upon the choice of a basis. It is possible to define the theory of characteristic classes using principal bundles rather than vector bundles. In this approach, a connection is a splitting of T (P_{G}) into vertical and horizontal subspaces in an equivariant fashion. The curvature becomes a Lie algebra valued endomorphism of T (P_{G}) which is equivariant under the right action of the group. We refer to Euguchi, Gilkey, Hanson for further details. Let {U_{α}} be a cover of M so V is trivial over U_{α} and let sα be local oriented orthonormal frames over U_{α}. On the overlap, we express sα = g_{αβ}sβ where g_{αβ}: U_{α} ∩ U_{β} → SO(2k). These satisfy the cocycle condition: g_{αβ}g_{βγ}g_{γα} = I  and g_{αα} = I. The principal bundle P_{SO} of oriented orthonormal frames has transition functions g_{αβ} acting on SO(2k) from the left. A spin structure on V is a lifting of the transition functions to SPIN(2k) preserving the cocyle condition. If the lifting is denoted by g′, then we assume: ρ(g_{′αβ}) = g_{αβ} ,  g_{′αβ}g_{′βγ}g_{′γα} = I ,  and g_{′αα} = I. This is equivalent to constructing a principal SPIN(2k) bundle P_{SPIN} to- gether with a double covering map ρ: P_{SPIN} → P_{SO} which preserves the group action—i.e.,  ρ(x · g′) = ρ(x) · ρ(g′).

166 3.3. Spin Structures The transition functions of P_{SPIN} are just the g_{′αβ} acting on the left. Attempting to find a spin structure on V is equivalent to taking a square root in a certain sense which we will make clear later. There is an ob- struction to defining a spin structure which is similar to that to defining an orientation on V. These obstructions are Z_{2} characteristic classes called the Stieffel-Whitney classes and are most easily defined in terms of Čech cohomology. To understand these obstructions better, we review the con- struction briefly. We fix a Riemannian structure on T (M) to define a notion of distance. Geodesics on M are curves which locally minimize distance. A set U is said to be geodesically convex if (a) given x, y ∈ U there exists a unique geodesic in M joining x to y with d(x, y) = length(γ) and (b) if γ is any such geodesic then γ is actually contained in U. It is immediate that the intersection of geodesically convex sets is again geodesically convex and that every geodesically convex set is contractible. It is a basic theorem of Riemannian geometry that there exist open covers of M by geodesically convex sets. A cover {U_{α}} is said to be simple if the intersection of any number of sets of the cover is either empty or is contractible. A cover of M by open geodesically convex sets is a simple cover. We fix such a simple cover hence forth. Since U_{α} is contractible, any vector bundle over M is trivial over U_{α}. Let Z_{2} be the multiplicative group {±1}. A Čech j-cochain is a function f(α_{0},…, α_{j}) ∈ Z_{2} defined for j + 1-tuples of indices where U_{α0} ∩ ⋯ ∩ U_{αj} ≠ ∅ which is totally symmetric—i.e., f(α_{σ(0)},…, α_{σ(j)}) = f(α_{0},…, α_{j}) for any permutation σ. If C^{j}(M; Z_{2}) denotes the multiplicative group of all such functions, the coboundary δ: C^{j}(M, Z_{2}) → C^{j+1}(M; Z_{2}) is defined by: (δf)(α_{0},…, α_{j+1}) = _{j+1 ∏ i=0} f(α_{0},…, αi,…, α_{j+1}). The multiplicative identity of C^{j}(M; Z_{2}) is the function 1 and it is an easy combinatorial exercise that δ^{2}f = 1. For example, if f_{0} ∈ C^{0}(M; Z_{2}) and if f_{1} ∈ C^{1}(M; Z_{2}), then: δ(f_{0})(α_{0}, α_{1}) = f(α_{1})f(α_{0}) δ(f_{1})(α_{0}, α_{1}, α_{2}) = f(α_{1}, α_{2})f(α_{0}, α_{1})f(α_{o}, α_{2}). Z_{2} is a particularly simple coefficient group to work with since every ele- ment is its own inverse; in defining the Čech cohomology with coefficients

On Vector Bundles 167

in other abelian groups, more care must be taken with the signs which arise. Let H^{j}(M; Z_{2}) = N(δ_{j})/ R(δ_{j−1}) be the cohomology group; it is an easy exercise to show these groups are independent of the particular simple cover chosen. There is a ring structure on H^{∗}(M; Z_{2}), but we shall only use the “additive” group structure (which we shall write multiplicatively). Let V be a real vector bundle, not necessarily orientable. Since the Uα are contractible, V is trivial over the U and we can find a local orthonormal frame sα for V over U_{α}. We let sα = g_{αβ}sβ and define the 1-cochain: f(αβ) = det(g_{αβ}) = ±1. This is well defined since U_{α} ∩ U_{β} is contractible and hence connected. Since f(α, β) = f(β, α), this defines an element of C^{1}(M; Z_{2}). Since the {g_{αβ}} satisfy the cocycle condition, we compute: δf(α, β, γ) = det(g_{αβ}g_{βγ}g_{γα}) = det(I) = 1 so δ(f) = 1 and f defines an element of H^{1}(M; Z_{2}). If we replace sα by s_{′α} = h_{α}sα the new transition functions become g_{′αβ} = h_{α}g_{αβ}h_{−1 β} so if f_{0}(α) = det(h_{α}), f′(α, β) = det(h_{α}g_{αβ}h_{−1 β}) = det(h_{α})f(α, β) det(h_{β}) = δ(f_{0})f and f changes by a coboundary. This proves the element in cohomology defined by f is independent of the particular frame chosen and we shall denote this element by w_{1}(V) ∈ H^{1}(M; Z_{2}). If V is orientable, we can choose frames so det(g_{αβ}) = 1 and thus w_{1}(V) = 1 represents the trivial element in cohomology. Conversely, if w_{1}(V) is trivial, then f = δf_{0}. If we choose h_{α} so det(h_{α}) = f_{0}(α), then the new frames s_{′α} = h_{α}sα will have transition functions with det(g_{′αβ}) = 1 and define an orientation of V. Thus V is orientable if and only if w_{1}(V) is trivial and w_{1}(V), which is called the first Stieffel-Whitney class, measures the obstruction to orientability. If V is orientable, we restrict henceforth to oriented frames. Let dim V = 2k be even and let g_{αβ} ∈ SO(2k) be the transition functions. We choose any lifting g̃αβ to SPIN(2k) so that: ρ(g̃αβ) = g_{αβ} and g̃αβg̃βα = I; since the U_{α} are contractible such lifts always exist. We have g_{αβ}g_{βγ}gγα = I so ρ(g̃αβg̃βγg̃γα) = I and hence g̃αβg̃βγg̃γα = ±I = f(α, β, γ)I where f(α, β, γ) ∈ Z_{2}. V admits a spin structure if and only if we can choose the lifting so f(α, β, γ) = 1. It is an easy combinatorial exercise to show that f is symmetric and that δf = 0. Furthermore, if we change the choice of the sα or change the choice of lifts, then f changes by a coboundary. This implies f defines an element w_{2}(V) ∈ H^{2}(M; Z_{2}) independent of the choices made. w_{2} is called the second Stieffel-Whitney class and is trivial if and only if V admits a spin structure.

We suppose w_{1}(V) = w_{2}(V) = 1 are trivial so V is orientable and admits a spin structure. Just as there are two orientations which can be chosen on V, there can be several possible inequivalent spin structures (i.e., several non-isomorphic principal bundles P_{SPIN} ). It is not difficult to see that inequivalent spin structures are parametrized by representations of the fundamental group π_{1}(M) → Z_{2} just as inequivalent orientations are parametrized by maps of the components of M into Z_{2}.

To illustrate the existence and non-existence of spin structures on bundles, we take M = S^{2}. S^{2} = CP_{1} is the Riemann sphere. There is a natural projection from C^2 − 0 to S^2 given by sending (x, w) ↦ z/w; S^2 is obtained by identifying (z, w) = (λz, λw) for (z, w) ≠ (0, 0) and λ ≠ 0. There are two natural charts for S^{2}:

U_{1} = {z : |z| ≤ 1} and U_{2} = {z : |z| ≥ 1} ∪ {∞}

where w = 1/z gives coordinates on U_{2}.

CP_{1} is the set of lines in C^{2}; we let L be the natural line bundle over S^{2}; this is also refered to as the tautological line bundle. We have natural sections to L over U_{i} defined by:

s_{1} = (z, 1) over U_{1} and s_{2} = (1, w) over U_{2}.

these are related by the transition function:

s_{1} = zs2

so on U_{1} ∩ U_{2} we have g_{12} = e^{iθ}. The double cover SPIN(2) → SO(2) is defined by θ ↦ 2θ so this bundle does not have a spin structure since this transition function cannot be lifted to SPIN(2).

This cover is not a simple cover of S^{2} so we construct the cover given in the following

V4

V2

V1
V3

[figure]

(where we should “fatten up” the picture to have an open cover). If we have sections si to L over V_{i}, then we can choose the transition functions so sj = e^{iθ}s4 for 1 ≤ j ≤ 3 and s1 = s2 = s3. We let 0 ≤ θ ≤ 2π/3 parametrize V_{1} ∩ V4 2π/3 ≤ θ ≤ 4π/3 parametrize V_{2} ∩ V4 4π/3 ≤ θ ≤ 2π  parametrize V_{3} ∩ V4 and define:  ^{g̃14} = e^{iθ/2} , ^{g̃24} = e^{iθ/2} , ^{g̃34} = eiθ/2 and all the g̃jk = 1 for 1 ≤ j, k ≤ 3 then we compute: g̃13g̃34g̃41 = e^{0}e^{iπ} = −1 so the cochain defining w_{2} satisfies: f(1, 3, 4) = −1 with f(i, j, k) = 1 if i or j or k=2. We compute this is non-trivial in cohomology as follows: suppose f = δf_{1}. Then:  1 = f(1, 2, 3) = f_{1}(1, 2)f_{1}(1, 3)f_{1}(2, 3) 1 = f(1, 2, 4) = f_{1}(1, 2)f_{1}(1, 4)f_{1}(2, 4) 1 = f(2, 3, 4) = f_{1}(2, 3)f_{1}(2, 4)f_{1}(3, 4) and multiplying together: 1 = f_{1}(1, 2)^{2}f_{1}(2, 3)^{2}f_{1}(2, 4)^{2}f_{1}(1, 4)f_{1}(1, 3)f_{1}(3, 4) = f_{1}(1, 4)f_{1}(1, 3)f_{1}(3, 4) = f(1, 3, 4) = −1 which is a contradiction. Thus we have computed combinatorially that w_{2} ≠ 1 is non-trivial. Next we let V = T (M) be the real tangent bundle. We identify U(1) with SO(2) to identify T (M) with T_{C}(M). Since w = 1/z we have: d dz = dw dz d dw = −z^{−2} d dw so the transition function on the overlap is −e^{−2iθ}. The minus sign can be eliminated by using − d dw instead of d dw as a section over U_{2} to make the transition function be e^{−2iθ}. Since the double cover of SPIN(2) → SO(2) is given by θ ↦ 2θ, this transition function lifts and T (M) has a spin structure. Since S^{2} is simply connected, the spin structure is unique. This

also proves T_C(M) = L^∗⊗L^∗ since the two bundles have the same transition functions.

There is a natural inclusion of SPIN(V) and SPIN(W) into subgroups of SPIN(V ⊕ W) which commute. This induces a map from SPIN(V) × SPIN(W) → SPIN(V ⊕ W). Using this map, it is easy to compute w_2(V ⊕ W) = w_2(V)w_2(W). To define w_2(V), we needed to choose a fixed orientation of V. It is not difficult to show that w_2(V) is independent of the orientation chosen and of the fiber metric on V. (Our definition does depend upon the fact that V is orientable. Although it is possible to define w_2(V) more generally, the identity w_2(V ⊕ W) = w_2(V)w_2(W) fails if V and W are not orientable in general). We summarize the properties of w_2 we have derived:

Lemma 3.3.1. Let V be a real oriented vector bundle and let w_2(V) ∈ H^2(M; Z_2) be the second Stieffel-Whitney class. Then:

(a) w_2(V) = 1 represents the trivial cohomology class if and only if V admits a spin structure.

(b) w_2(V ⊕ W) = w_2(V)w_2(W).

(c) If L is the tautological bundle over S^2 then w_2(L) is non-trivial.

We emphasize that (b) is written in multiplicative notation since we have chosen the multiplicative version of Z_2. w_2 is also functorial under pull-backs, but as we have not defined the Čech cohomology as a functor, we shall not discuss this property. We also note that the Stieffel-Whitney classes can be defined in general; w(V) = 1 + w_1(V) + ⋯ ∈ H^∗(M; Z_2) is defined for any real vector bundle and has many of the same properties that the Chern class has for complex bundles.

We can use Lemma 3.3.1 to obtain some other results on the existence of spin structures:

Lemma 3.3.2.

(a) If M = S^m then any bundle over S^m admits a spin structure for m ≠ 2.

(b) If M = CP_k and if L is the tautological bundle over M, then L does not admit a spin structure.

(c) If M = CP_k and if V = T (M) is the tangent space, then V admits a spin structure if and only if k is odd.

(d) If M = QP_k is quaternionic projective space, then any bundle over M admits a spin structure.

Proof: H^2(M; Z_2) = {1} in (a) and (d) so w_2 must represent the trivial element. To prove (b), we suppose L admits a spin structure. S^2 is embedded in CP_k for k ≥ 2 and the restriction of L to S^2 is the tautological bundle over S^2. This would imply L admits a spin structure over S^2 which is false. Finally, we use the representation:

T_c(CP_j) ⊕ 1 = L^∗ ⊕ ⋯ ⊕ L^∗

On Vector Bundles 171

so that T (CP^{j}) ⊕ 1^{2} = (L_{∗ r}) ⊕ ⋯ ⊕ (L_{∗ r}) j+1 times where L_{∗ r} denotes the real vector bundle obtained from L^{∗} by forgetting the complex structure. Then w_{2}(L_{∗ r}) = w_{2}(L_{r}) since these bundles are isomorphic as real bundles. Furthermore: w^{2}(T (CP^{j})) = w^{2}(L^{r})j+1 and this is the trivial class if j + 1 is even (i.e., j is odd) while it is w_{2}(L_{r}) and non-trivial if j + 1 is odd (i.e., j is even). Let M be a manifold and let V be a real vector bundle over M. If V admits a spin structure, we shall say V is spin and not worry for the moment about the lack of uniqueness of the spin structure. This will not become important until we discuss Lefschetz fixed point formulas. If V is spin, we define the bundles Δ^{±}(V) to have transition functions Δ^{±}(g̃αβ) where we apply the representation Δ^{±} to the lifted transition functions. Alternatively, if P_{SPIN} is the principal spin bundle defining the spin structure, we define:Δ^{±}(V) = P^{SPIN} ⊗^{Δ±} Δ± where the tensor product across a representation is defined to be P_{SPIN} × Δ^{±} module the identification (p · g) × z = p × (Δ^{±}(g)z) for p ∈ P_{SPIN} , g ∈ P_{SPIN} (2k), z ∈ Δ^{±}. (The slight confusion of notation is caused by our convention of using the same symbol for the representation and the representation space.) Let b_{±} be a fixed unitary frame for Δ^{±}. If s is a local oriented orthonor- mal frame for V, we let s̃i be the two lifts of s ∈ P_{SO} to P_{SPIN} . s̃1 = −s̃2 but there is no natural way to distinguish these lifts, although the pair is cannonically defined. If ∇ is a Riemannian connection on V, let ∇_{s} = ωs be the connection 1-form. ω is a skew symmetric matrix of 1-forms and is a 1-form valued element of the Lie algebra of SO(2k). Since the Lie alge- bra of SO(2k) and SPIN(2k) coincide, we can also regard ω as an element which is 1-form valued of the Lie algebra of SPIN(2k) and let Δ^{±}(ω) act on Δ^{±}(V). We define bases s̃i ⊗ b_{±} for Δ^{±}(V) and define: ∇(s̃i ⊗ b_{±}) = s̃i ⊗ Δ^{±}(ω)b± to define a natural connection on Δ^{±}(V). Since the same connection is defined whether s̃1 or s̃2 = −s̃1 is chosen, the Z_{2} ambiguity is irrelevant and ∇ is well defined. Lemma 3.2.4 and 3.2.5 extend to:

Lemma 3.3.3. Let V be a real oriented Riemannian vector bundle of even fiber dimension v. Suppose V admits a spin structure and let Δ^{±}(V) be the half-spin bundles. Then: (a) Λ(V) ≃ Δ(V) ⊗ Δ(V). (b) (Λ^{+} − Λ^{−})(V) ≃ (Δ^{+} − Δ^{−}) ⊗ (Δ^{+} − Δ^{−})(V). (c) (Λ^{e} − Λ^{o})(V) ≃ (−1)^{v/2}(Δ^{+} − Δ^{−}) ⊗ (Δ^{+} − Δ^{−})(V). (d) If v = 2 and if V is the underlying real bundle of a complex bundle Vc then V_{c} ≃ Δ^{−} ⊗ Δ^{−} and V_{∗ c} ≃ Δ^{+} ⊗ Δ^{+}. (e) If ∇ is a Riemannian connection on V and if we extend ∇ to Δ^{±}(V), then the isomorphisms given are unitary isomorphisms which preserve ∇. Spinors are multiplicative with respect to products. It is immediate from the definitions we have given that: Lemma 3.3.4. Let V_{i} be real oriented Riemannian vector bundles of even fiber dimensions v_{i} with given spin structures. Let V_{1} ⊕V_{2} have the natural orientation and spin structure. Then: (Δ^{+} − Δ^{−})(V_{1} ⊕ V_{2}) ≃ {(Δ^{+} − Δ^{−})(V_{1})} ⊗ {(Δ^{+} − Δ^{−})(V_{2})}. If ∇_{i} are Riemannian connections on V_{i} and if we define the natural in- duced connections on these bundles, then the isomorphism is unitary and preserves the connections. A spin structure always exists locally since the obstruction to a spin structure is global. Given any real oriented Riemannian bundle V of even fiber dimension with a fixed Riemannian connection ∇, we define Λ(V) and Δ(V) = Δ^{+}(V) ⊕ Δ^{−}(V). With the natural metrics and connections, we relate the connection 1-forms and curvatures of these 3 bundles: Lemma 3.3.5. Let {e_{i}} be a local oriented orthonormal frame for V. We let ∇e_{j} = ω_{jk}s_{k} represent the connection 1-form and Ωe_{j} = Ω_{jk}e_{k} be the curvature matrix for Ω_{jk} = dω_{jk} − ω_{jl} ∧ ω_{lk}. Let Λe and Δe denote the natural orthonormal frames on Λ(V) and Δ(V). Relative to these frames we compute the connection 1-forms and curvature matrices of Λ(V) and Δ(V) by: ωΛ = ω_{jk} ext(e_{k}) int(e_{j})  Ω_{Λ} = Ω_{jk} ext(e_{k}) int(e_{j}) (a)  ωΔ = _{1 4} ω_{jk}e_{j} ∗ ek  Ω_{Δ} = _{1 4} Ω_{jk}e_{j} ∗ e_{k}. (b) Proof: We sum over repeated indices in these expressions. We note (a) is true by definition on Λ^{1}(V) = V. Since both ω and Ω extend to act as derivations on the exterior algebra, this implies (a) is true on forms of all degree. Let so(n) = {A ∈ n×n real matrices with A+A^{t} = 0} be the Lie algebra of SO(n). This is also the Lie algebra of SPIN(n) so we must identify this

with an appropriate subset of the Clifford algebra in order to prove (b). We choose a representative element of so(n) and define Ae_{1} = e_{2}, Ae_{2} = −e_{1}, Ae_{j} = 0  for j > 2; i.e.,  A_{12} = 1, A_{21} = −1, A_{jk} = 0  otherwise. If we let g(t) ∈ SO(n) be defined by g(t)e1 = (cos t)e_{1} + (sin t)e_{2}, g(t)e2 = (cos t)e_{2} − (sin t)e_{1}, g(t)ej = e_{j}  j > 2, then g(0) = I and g′(0) = A. We must lift g(t) from SO(n) to SPIN(n). Define: h(t) = (cos(t/4)e^{1} + sin(t/4)e^{2})(− cos(t/4)e^{1} + sin(t/4)e^{2}) = cos(t/2) + sin(t/2)e_{1}e_{2} ∈ SPIN(n) Then ρ(h) ∈ SO(n) is defined by: ρ(h)e^{j} = (cos(t/2) + sin(t/2)e^{1}e^{2})e^{j}(cos(t/2) − sin(t/2)e^{1}e^{2}) so that ρ(h)e_{j} = e_{j} for j > 2. We compute: ρ(h)e1 = (cos(t/2)e^{1} + sin(t/2)e^{2})(cos(t/2) − sin(t/2)e^{1}e^{2}) = (cos^{2}(t/2) − sin^{2}(t/2))e_{1} + 2 sin(t/2) cos(t/2)e2 ρ(h)e2 = (cos(t/2)e^{2} − sin(t/2)e^{1})(cos(t/2) − sin(t/2)e^{1}e^{2}) = (cos^{2}(t/2) − sin^{2}(t/2))e_{2} − 2 sin(t/2) cos(t/2)e1 so that ρ(h) = g. This gives the desired lift from SO(n) to SPIN(n). We differentiate to get h′(0) = 1 2e_{1}e_{2} = 1 4A_{jk}e_{j} ∗ e_{k}. This gives the lift of a matrix in this particular form. Since the whole Lie algebra is generated by elements of this form, it proves that the lift of A_{jk} in general is given by _{1 4} A_{jk}e_{j} ∗ e_{k}. Since SPIN(n) acts on the Clifford algebra by Clifford multiplication on the left, this gives the action of the curvature and connection 1-form on the spin representations and completes the proof.

We can define ch(Δ^{±}(V)) as SO characteristic forms. They can be ex- pressed in terms of the Pontrjagin and Euler forms. We could use the explicit representation given by Lemma 3.3.5 to compute these forms; it is most easy, however, to compute in terms of generating functions. We introduce formal variables {x_{j}} for 1 ≤ j ≤ (dim V)/2 so that p(V) = _{∏ j}(1 + x_{2 j})  and e(V) = _{∏ j} x_{j}. All these computations are really induced from the corresponding matrix identities and the {x_{j}} arise from putting a skew-symmetric matrix A in block form. Lemma 3.3.6. Let V be an oriented real Riemannian vector bundle of dimension n ≡ 0 (2). Let ch(Δ^{±}(V)) be a real characteristic form; these are well defined even if V does not admit a global spin structure. Then: (a) ch(Δ^{+}(V)) + ch(Δ^{−}(V)) = ch(Δ(V)) = _{∏ j}{e^{x_{j}/2} + e^{−x_{j}/2}}. (b)(−1)^{n/2}{ch(Δ^{+}(V)) − ch(Δ^{−}(V))} = _{∏ j}{e^{x_{j}/2} − e^{−x_{j}/2}} = e(V)(1 + higher order terms). Proof: This is an identity among invariant polynomials in the Lie algebra so(n). We may therefore restrict to elements of the Lie algebra which split in block diagonal form. Using the multiplicative properties of the Chern character and Lemma 3.3.4, it suffices to prove this lemma for the special case that n = 2 so Δ^{±}(V) are complex line bundles. Let V_{c} be a complex line bundle and let V be the underlying real bundle. Then x_{1} = x = c_{1}(V_{c}) = e(V). Since: V^{c} = Δ^{−} ⊗ Δ^{−}  and V^{∗ c} = Δ^{+} ⊗ Δ+ we conclude: x = 2c_{1}(Δ^{−})  and  − x = 2c_{1}(Δ^{+}) which shows (a) and the first part of (b). We expand: e^{x/2} − e^{−x/2} = x + 1 24x^{3} + ⋯ to see ch(Δ^{−}) − ch(Δ^{+}) = e(V)(1 + _{1 24} p_{1}(V) + ⋯) to complete the proof of (b).

There is one final calculation in characteristic classes which will prove helpful. We defined L and Â by the generating functions: L(x) = ^{∏ j} xj tanh xj = ^{∏ j} x^{j} e^{xj} + e−xj e^{xj} − e−xĵ A(x) = _{∏ j}  xj sinh(x_{j}/2) = _{∏ j} _{xj ex^{j}/2 − e−x^{j}/2} . Lemma 3.3.7. Let V be an oriented Riemannian real vector bundle of dimension n = 2n_{0}. If we compute the component of the differential form which is in Λ^{n}(T^{∗}M) then: {L(V))}_{n} = {ch(Δ(V)) ∧ Â(V)}_{n}. Proof: It suffices to compute the component of the corresponding sym- metric functions which are homogeneous of order n_{0}. If we replace x_{j} by xj/2 then:  {L(x_{j})}n0 = {2^{n0} L(x^{j}/2)}n0 = {^{∏ j} x^{j} tanh(x^{j}/2)}n0 = ^{∏ j} x^{j} ^{ex/2 + e−x/2 ex^{j}/2 − e−x^{j}/2} n0 = {ch(Δ(V)) ∧ Â(V)}n0 which completes the proof.

3.4. The Spin Complex. We shall use the spin complex chiefly as a formal construction to link the de Rham, signature, and Dolbeault complexes. Let M be a Riemannian manifold of even dimension m. Let T (M) be the real tangent space. We assume that M is orientable and that T (M) admits a spin structure. We let Δ^{±}(M) be the half-spin representations. There is a natural map given by Lemma 3.2.3 from the representations of T^{∗}(M) → HOM(Δ^{±}(M), Δ^{∓}(M)) which we will call c(ξ) (since it is essentially Clifford multiplication) such that c(ξ)^{2} = −|ξ|^{2}I. We extend the Levi-Civita connection to act naturally on these bundles and define the spin complex by the diagram: A^{±}: C^{∞}(Δ^{±}(M)) ^{∇} → C^{∞}(T^{∗}M ⊗ Δ^{±}(M)) ^{c} → C^{∞}(Δ^{∓}(M)) to be the operator with leading symbol c. (A^{+})^{∗} = A^{−} and A^{±} is elliptic since c(ξ)^{2} = −|ξ|^{2}I; this operator is called the Dirac operator. Let V be a complex bundle with a Riemannian connection ∇. We define the spin complex with coefficients in ∇ by using the diagram: A_{± V}: C^{∞}(Δ^{±}(M) ⊗ V) → C^{∞}(T^{∗}M ⊗ Δ^{±}(M) ⊗ V)^{c⊗1} ⟶ C^{∞}(Δ^{∓}(M) ⊗ V). This is completely analogous to the signature complex with coefficients in V. We define: index(V, spin) = index(A_{+ V}); a priori this depends on the particular spin structure chosen on V, but we shall show shortly that it does not depend on the particular spin structure, although it does depend on the orientation. We let a_{spin n} (x, V) denote the invariants of the heat equation. If we reverse the orientation of M, we interchange the roles of Δ^{+} and of Δ^{−} so a_{spin n} changes sign. This implies a_{spin n} can be regarded as an invariantly defined m-form since the scalar invariant changes sign if we reverse the orientation. Let X be an oriented coordinate system. We apply the Gramm-Schmidt process to the coordinate frame to construct a functorial orthonormal frame s (X) for T (M) = T^{∗}(M). We lift this to define two local sections si(X) to the principal bundle P_{SPIN} with s1(X) = −s2(X). There is, of course, no cannonical way to prefer one over the other, but the pair is invariantly defined. Let b^{±} be fixed bases for the representation space Δ^{±} and let

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b_{± i}(X) = si(X) ⊗ b^{±} provide local frames for Δ^{±}(M). If we fix i = 1 or i = 2, the symbol of the spin complex with coefficients in V can be functorially expressed in terms of the 1-jets of the metric on M and in terms of the connection 1-form on V. The leading symbol is given by Clifford multiplication; the 0^{th} order term is linear in the 1-jets of the metric on M and in the connection 1-form on V with coefficients which are smooth functions of the metric on M. If we replace b_{± 1} by b_{± 2} = −b_{± 1} then the local representation of the symbol is unchanged since multiplication by −1 commutes with differential operators. Thus we may regard a_{spin n} (x, V) ∈ R_{m,n,m,dim V} as an invariantly defined polynomial which is homogeneous of order n in the jets of the metric and of the connection form on V which is m-form valued. This interpretation defines a_{spin n} (x, V) even if the base manifold M is not spin. We can always define the spin complex locally as the Z_{2} indeter- minacy in the choice of a spin structure will not affect the symbol of the operator. Of course, ∫_{M} a_{spin m} (x, V) can only be given the interpretation of index(V, spin) if M admits a spin structure. In particular, if this integral is not an integer, M cannot admit a spin structure. a_{spin n} (x, V) is a local invariant which is not affected by the particular global spin structure chosen. Thus index(V, spin) is independent of the particular spin structure chosen. We use exactly the same arguments based on the theorem of the second chapter and the multiplicative nature of the twisted spin complex as were used to prove the Hirzebruch signature theorem to establish: Lemma 3.4.1. (a) a_{spin n} = 0 if n < m and a_{spin m} is a characteristic form of T (M) and of V. (b) ∫_{M} a_{spin m} (x, V) = index(V, spin). (c) There exists a characteristic form Â′ of T (M) in the form: Â′ = 1 + Â1 + ⋯ which does not depend on the dimension of M together with a universal constant c such that a_{spin m} = _{∑ 4s+2t=m} Â′s ∧ c^{t}ch(V)_{t}. We use the notation Â′ since we have not yet shown it is the A-roof genus defined earlier. In proving the formula splits into this form, we do not rely on the uniqueness property of the second chapter, but rather on the multiplicative properties of the invariants of the heat equation discussed in the first chapter. The spin complex has an intimate relation with both the de Rham and signature complexes:

178 3.4. The Spin Complex Lemma 3.4.2. Let U be an open contractible subset of M. Over U, we define the signature, de Rham, and spin complexes. Then: (a) (Λ^{e} − Λ^{o}) ⊗ V ≃ (−1)^{m/2}(Δ^{+} − Δ^{−}) ⊗ (Δ^{+} − Δ^{−}) ⊗ V. (b) (Λ^{+} − Λ^{−}) ⊗ V ≃ (Δ^{+} − Δ^{−}) ⊗ (Δ^{+} + Δ^{−}) ⊗ V. (c) These two isomorphisms preserve the unitary structures and the con- nections. They also commute with Clifford multiplication on the left. (d) The natural operators on these complexes agree under this isomor- phism. Proof: (a)–(c) follow from previous results. The natural operators on these complexes have the same leading symbol and therefore must be the same since they are natural first order operators. This proves (d). We apply this lemma in dimension m = 2 with V = the trivial bundle and M = S^{2}: χ(S^{2}) = 2 = index(d + δ) = index(Δ^{−} − Δ^{+}, spin) = ∫_{S 2} c{c_{1}(Δ^{−}) − c_{1}(Δ^{+})} = c ∫_{S 2} e(T M) = 2c by Lemmas 2.3.1, 3.3.5, and 3.3.4. This establishes that the normalizing constant of Lemma 3.4.1 must be 1 so that: a_{spin m} = _{∑ 4s+2t=m} Â′ ∧ ch_{t}(V). We apply this lemma to the twisted signature complex in dimension m = 2 with V a non-trivial line bundle over S^{2} to conclude: signature(S^{2}, V) = index((Δ^{+} ⊕ Δ^{−}) ⊗ V, spin) = ∫_{S 2} ch((Δ^{+} ⊕ Δ^{−}) ⊗ V) = ∫_{S 2} 2 · c_{1}(V). This shows that the normalizing constant of Lemma 3.1.4 for the twisted signature complex is 2 and completes the proof of Lemma 3.1.4. This there- fore completes the proof of the Hirzebruch signature theorem in general. The de Rham complex with coeffiecients in V is defined by the diagram: C^{∞}(Λ^{e,o}(M) ⊗ V) ^{∇} → C^{∞}(T^{∗}M ⊗ Λ^{e,o}(M) ⊗ V) ^{c⊗1} ⟶ C^{∞}(Λ^{o,e}(M) ⊗ V) and we shall denote the operator by (d + δ)_{e V}. The relations given by Lemma 3.4.2 give rise to relations among the local formulas: a_{n}(x, (d + δ)_{e V}) = a_{n}(x, A_{+ (−1)^{m/2}(Δ^{+}−Δ^{−})⊗V} ) a_{n}(x, (d + δ)_{+ V}) = a_{n}(x, A_{+ (Δ^{+}−Δ^{−})⊗V} )

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where (d + δ)_{+ V} is the operator of the twisted signature complex. These relations are well defined regardless of whether or not M admits a spin structrue on T (M). We deal first with the de Rham complex. Using Lemma 3.4.1 and the fact that the normalizing constant c is 1, we conclude: a_{n}(x, (d + δ)_{e V}) = Â′ ∧ ch((−1)^{m/2}(Δ^{+} − Δ^{−}))) ∧ ch(V). Since ch((−1)^{m/2}(Δ^{+} −Δ^{−})) = e(M) is already a top dimensional form by Lemma 3.3.5(b), we conclude a_{n}(x, (d + δ)_{e V}) = (dim V)e(M). This proves: Theorem 3.4.3. Let (d + δ)_{e V} be the de Rham complex with coefficients in the bundle V and let a_{n}(x, (d + δ)_{e V}) be the invariants of the heat equa- tion. Then: (a) a_{n}(x, (d + δ)_{e V}) = 0 if n < m. (b) a_{m}(x, (d+δ)_{e V}) = (dim V)e(M) where e(M) is the Euler form of T (M). (c) index((d + δ)_{e V}) = (dim V)χ(M) = ∫_{m}(dim V)e(M). This shows that no information about V (except its dimension) is ob- tained by considering the de Rham complex with coefficients in V and it is for this reason we did not introduce this complex earlier. This gives a second proof of the Gauss-Bonnet theorem independent of the proof we gave earlier. Next, we study the signature complex in order to compute Â. Using Lemma 3.4.1 with the bundle V = 1, we conclude for m = 4k, a_{m}(x, (d + δ)_{+ V}) = {Â′ ∧ ch(Δ)}_{m}. Using Theorem 3.1.1 and Lemma 3.3.6, we compute therefore: L_{k} = {ch(Δ) ∧ Â}_{m} = {ch(Δ) ∧ Â′}_{m} = a_{m}(x, (d + δ)_{+ V}) so as the Chern character is formally invertible; Â = Â′ is given by the generating function z_{j}/ sinh(z_{j}/2) by Lemma 3.3.7. We can now improve Lemma 3.4.1 and determine all the relevant normalizing constants. Theorem 3.4.4. Let T (M) admit a spin structure, then: (a) a_{spin n} = 0 if n < m. (b) a_{spin m} (x, V) = ∑_{4s+2t=m} Âs ∧ ch_{t}(V). (c) index(V, spin) = ∫^{M} a^{spin m} (x, V).

3.5. The Riemann-Roch Theorem For Almost Complex Manifolds. So far, we have discussed three of the four classical elliptic complexes, the de Rham, the signature, and the spin complexes. In this subsection, we define the Dolbeault complex for an almost complex manifold and relate it to the spin complex. Let M be a Riemannian manifold of dimension m = 2n. An almost complex structure on M is a linear map J: T (M) → T (M) with J^{2} = −1. The Riemannian metric G is unitary if G(X, Y) = G(JX, JY) for all X, Y ∈ T (M); we can always construct unitary metrics by averaging over the action of J. Henceforth we assume G is unitary. We extend G to be Hermitian on T (M) ⊗ C, T^{∗}(M) ⊗ C, and Λ(M) ⊗ C. Since J^{2} = −1, we decompose T (M) ⊗ C = T′(M) ⊕ T′′(M) into the ±i eigenspaces of J. This direct sum is orthogonal with respect to the metric G. Let Λ^{1,0}(M) and Λ^{0,1}(M) be the dual spaces in T^{∗}(M) ⊗ C to T′ and T′′. We choose a local frame {e_{j}} for T (M) so that J(e_{j}) = ej+n for 1 ≤ i ≤ n. Let {e^{j}} be the corresponding dual frame for T^{∗}(M). Then: T′(M) = span_{C}{e_{j} − ie_{j+n}}^{n} j=1  T′′(M) = span_{C}{e_{j} + ie_{j+n}}^{n} j=1 Λ^{1,0}(M) = span_{C}{e^{j} + ie^{j+n}}^{n} j=1  Λ^{0,1}(M) = span_{C}{e^{j} − ie^{j+n}}_{n j=1}. These four vector bundles are all complex vector bundles over M. The metric gives rise to natural isomorphisms T′(M) ≃ Λ^{0,1}(M) and T′′(M) = Λ^{1,0}(M). We will use this isomorphism to identify e_{j} with e^{j} for much of what follows. If we forget the complex structure on T′(M), then the underlying real vector bundle is naturally isomorphic to T (M). Complex multiplication by i on T′(M) is equivalent to the endomorphism J under this identification. Thus we may regard J as giving a complex structure to T (M). The decomposition: T^{∗}(M) ⊗ C = Λ^{1,0}(M) ⊕ Λ^{0,1}(M) gives rise to a decomposition: Λ(T^{∗}M) ⊗ C = _{⨁ p,q} Λ^{p,q}(M) for  Λ^{p,q}(M) = Λ^{p}(Λ^{1,0}(M)) ⊗ Λ^{q}(Λ^{0,1}(M)). Each of the bundles Λ^{p,q} is a complex bundle over M and this decomposi- tion of Λ(T^{∗}M)⊗C is orthogonal. Henceforth we will denote these bundles by T′, T′′, and Λ^{p,q} when no confusion will arise.

If M is a holomorphic manifold, we let z = (z_{1},…, z_{n}) be a local holo- morphic coordinate chart. We expand z_{j} = x_{j} + iy_{j} and define: ∂ ∂zj = 1 2 ( ∂ ∂xj − i ∂ ∂yj ) ,  ∂ ∂z̄j = 1 2 ( ∂ ∂xj + i ∂ ∂yj ) dzj = dx_{j} + idy_{j},  dz̄j = dx_{j} − idy_{j}. We define: ∂(f) = _{∑ j} ∂f ∂zj dz_{j}  and  ¯∂(f) = _{∑ j} ∂f ∂z̄j dz̄j. The Cauchy-Riemann equations imply that a function f is holomorphic if and only if ¯∂(f) = 0. If w = (w_{1},…, w_{n}) is another holomorphic coordinate system, then: ∂ ∂wj = _{∑ k} ∂zk ∂wj ∂ ∂zk ,  ∂ ∂w̄j = ^{∑ k} ∂z̄k ∂w̄j ∂ ∂z̄k dwj = _{∑ k} ∂wj ∂zk dz_{k},  dw̄j = ∂w̄j ∂z̄k dz̄k. We define:^{T′(M) = span { ∂ ∂zj }_{n j=1} ,  T′′(M) = span { ∂ ∂z̄j }n j=1} Λ^{1,0}(M) = span{dz_{j}}_{n j=1},  Λ^{0,1}(M) = span{dz̄j}^{n} j=1 then these complex bundles are invariantly defined independent of the choice of the coordinate system. We also note ∂: C^{∞}(M) → C^{∞}(Λ^{1,0}(M))  and  ¯∂: C^{∞}(M) → C^{∞}(Λ^{0,1}(M)) are invariantly defined and decompose d = ∂ + ¯∂. There is a natural isomorphism of T′(M) with T (M) as real bundles in this example and we let J be complex multiplication by i under this isomorphism. Equivalently: J ( ∂ ∂xj ) = ( ∂ ∂yj )  and J ( ∂ ∂yj ) = − ∂ ∂xj . T′(M) = T_{c}(M) is the complex tangent bundle in this example; we shall reserve the notation T_{c}(M) for the holomorphic case. Not every almost complex structure arises from a complex structure; there is an integrability condition. If J is an almost complex structure, we

decompose the action of exterior differentiation d on C^{∞}(Λ) with respect to the bigrading (p, q) to define: ∂: C^{∞}(Λ^{p,q}) → C^{∞}(Λ^{p+1,q})  and  ¯∂: C^{∞}(Λ^{p,q}) → C^{∞}(Λ^{p,q+1} ). Theorem 3.5.1 (Nirenberg-Neulander). The following are equiv- alent and define the notion of an integrable almost complex structure: (a) The almost complex structure J arises from a holomorphic structure on M. (b) d = ∂ + ¯∂. (c) ¯∂ ¯∂ = 0. (d) T′(M) is integrable—i.e., given X, Y ∈ C^{∞}(T′(M)), then the Lie bracket [X, Y] ∈ C^{∞}(T′(M)) where we extend [ , ] to complex vector fields in the obvious fashion. Proof: This is a fairly deep result and we shall not give complete details. It is worth, however, giving a partial proof of some of the implications to illustrate the concepts we will be working with. Suppose first M is holomorphic and let {z_{j}} be local holomorphic coordinates on M. Define: dz^{I} = dz^{i1} ∧ ⋯ ∧ dz^{ip}  and dz̄J = dz̄j1 ∧ ⋯ ∧ dz̄jq then the collection {dz^{I} ∧ dz̄J } gives a local frame for Λ^{p,q} which is closed. If ω ∈ C^{∞}(Λ^{p,q}), we decompose ω = ∑ f_{I,J} dz^{I} ∧ dz̄J and compute: dω = d{∑ f_{I,J} dz^{I} ∧ dz̄J } = ∑ df_{I,J} ∧ dz^{I} ∧ dz̄J . On functions, we decompose d = ∂ + ¯∂. Thus dω ∈ C^{∞}(Λ^{p+1,q} ⊕ Λ^{p,q+1} ) has no other components. Therefore dω = ∂ω + ¯∂ω so (a) implies (b). We use the identity d^{2} = 0 to compute (∂ + ¯∂)^{2} = (∂^{2}) + (∂ ¯∂ + ¯∂∂) + ( ¯∂2) = 0. Using the bigrading and decomposing this we conclude (∂^{2}) = (∂ ¯∂ + ¯∂∂) = ( ¯∂2) = 0 so (b) implies (c). Conversely, suppose that ¯∂ ¯∂ = 0 on C^{∞}(M). We must show d: C^{∞}(Λ^{p,q}) → C^{∞}(Λ^{p+1,q} ⊕ Λ^{p,q+1} ) has no other components. Let {e_{j}} be a local frame for Λ^{1,0} and let {ēj} be the corresponding local frame for Λ^{0,1}. We decompose: dej = ∂e_{j} + ¯∂e_{j} + Aj  for A^{j} ∈ Λ0,2 dēj = ¯∂ēj + ∂ēj + Āj  for Āj ∈ Λ^{2,0}. Then we compute: d(∑ f_{j}e_{j}) = ∑ df_{j} ∧ e_{j} + f_{j}dej = ∑{∂f_{j} ∧ e_{j} + ¯∂f_{j} ∧ e_{j} + f_{j}∂e_{j} + f_{j} ¯∂e_{j} + f_{j}A_{j}} = ∂(∑ f^{j}e^{j}) + ¯∂(∑ f^{j}e^{j}) + ∑ f^{j}A^{j}.

Similarly d(∑ f_{j}ēj) = ∂(∑ f_{j}ēj) + ¯∂(∑ f_{j}ēj) + ∑ f_{j}Āj. If A and Ā denote the 0^{th} order operators mapping Λ^{1,0} → Λ^{0,2} and Λ^{0,1} → Λ^{2,0} then we compute: d(ω) = ∂(ω) + ¯∂(ω) + Aω  for ω ∈ C^{∞}(Λ^{1,0}) d(ω̄) = ∂(ω̄) + ¯∂(ω̄) + Āω̄  for ω̄ ∈ C^{∞}(Λ^{0,1}). Let f ∈ C^{∞}(M) and compute: 0 = d^{2}f = d(∂f + ¯∂f) = (∂^{2} + Ā ¯∂)f + (∂ ¯∂ + ¯∂∂)f + ( ¯∂2 + A∂)f where we have decomposed the sum using the bigrading. This implies that ( ¯∂2 +A∂)f = 0 so A∂f = 0. Since {∂f} spans Λ^{1,0}, this implies A = Ā = 0 since A is a 0^{th} order operator. Thus de_{j} = ∂e_{j} + ¯∂e_{j} and dēj = ∂ēj + ¯∂ēj. We compute d(e^{I} ∧ ēJ) has only (p + 1, q) and (p, q + 1) components so d = ∂ + ¯∂. Thus (b) and (c) are equivalent. It is immediate from the definition that X ∈ T′(M) if and only if ω(X) = 0 for all ω ∈ Λ^{0,1}. If X, Y ∈ C^{∞}(T^{∗}M), then Cartan’s identity implies: dω(X, Y) = X(ωY) − Y (ωX) − ω([X, Y]) = −ω([X, Y]). If (b) is true then dω has no component in Λ^{2,0} so dω(X, Y) = 0 which implies ω([X, Y]) = 0 which implies [X, Y] ∈ C^{∞}(T′M) which implies (d). Conversly, if (d) is true, then dω(X, Y) = 0 so dω has no component in Λ^{2,0} so d = ∂ + ¯∂ on Λ^{0,1}. By taking conjugates, d = ∂ + ¯∂ on Λ^{1,0} as well which implies as noted above that d = ∂ + ¯∂ in general which implies (c). We have proved that (b)–(d) are equivalent and that (a) implies (b). The hard part of the theorem is showing (b) implies (a). We shall not give this proof as it is quite lengthy and as we shall not need this implication of the theorem. As part of the previous proof, we computed that d−(∂+ ¯∂) is a 0^{th} order operator (which vanishes if and only if M is holomorphic). We now compute the symbol of both ∂ and ¯∂. We use the metric to identify T (M) = T^{∗}(M). Let {e_{j}} be a local orthonormal frame for T (M) such that J(e_{j}) = ej+n for 1 ≤ j ≤ n. We extend ext and int to be complex linear maps from T^{∗}(M) ⊗ C → END(Λ(T^{∗}M) ⊗ C). Lemma 3.5.2. Let ∂ and ¯∂ be defined as before and let δ′ and δ′′ be the formal adjoints. Then:

184 3.5. The Riemann-Roch Theorem (a) ∂: C^{∞}(Λ^{p,q}) → C^{∞}(Λ^{p+1,q}) and σ_{L}(∂)(x, ξ) = i2 _{∑ j≤n}(ξ_{j} − iξ_{j+n}) ext(e_{j} + ie_{j+n}) ¯∂: C^{∞}(Λ^{p,q}) → C^{∞}(Λ^{p,q+1} ) and σ_{L}( ¯∂)(x, ξ) = i2 _{∑ j≤n}(ξ_{j} + iξ_{j+n}) ext(e_{j} − ie_{j+n}) δ′: C^{∞}(Λ^{p,q}) → C^{∞}(Λ^{p−1,q}) and σ_{L}(δ′)(x, ξ) = − i2 _{∑ j≤n}(ξ_{j} + iξ_{j+n}) int(e_{j} − ie_{j+n}) δ′′: C^{∞}(Λ^{p,q}) → C^{∞}(Λ^{p,q−1}) and σ_{L}(δ′′)(x, ξ) = − i2 _{∑ j≤n}(ξ_{j} − iξ_{j+n}) int(e_{j} + ie_{j+n}). (b) If Δ_{′c} = (∂ + δ′)^{2} and Δ_{′′ c} = ( ¯∂ + δ′′)^{2} then these are elliptic on Λ with σ_{L}(Δ_{′c}) = σ_{l}(Δ_{′′ c}) = _{1 2} |ξ|^{2}I. Proof: We know σ_{L}(d)(x, ξ) = i ∑_{j} ξ_{j} ext(e_{j}). We define A(ξ) = 1 2 _{∑ j≤n}(ξ_{j+n}) ext(e_{j} + ie_{j+n}) then A(ξ): Λ^{p,q} → Λ^{p+1,q} and Ā(ξ): Λ^{p,q} → Λ^{p,q+1} . Since iA(ξ) + iĀ(ξ) = σ_{L}(d), we conclude that iA and iĀ represent the decomposition of σ_{L}(d) under the bigrading and thus define the symbols of ∂ and ¯∂. The symbol of the adjoint is the adjoint of the symbol and this proves (a). (b) is an immediate consequence of (a). If M is holomorphic, the Dolbeault complex is the complex { ¯∂, Λ^{0,q}} and the index of this complex is called the arithmetic genus of M. If M is not holomorphic, but only has an almost complex structure, then ¯∂2 ≠ 0 so we can not define the arithmetic genus in this way. Instead, we use a trick called “rolling up” the complex. We define: Λ^{0,+} = ^{⨁ q} Λ^{0,2q} and Λ^{0,−} = ^{⨁ q} Λ0,2q+1 to define a Z_{2} grading on the Dolbeault bundles. (These are also often denoted by Λ^{0,even} and Λ^{0,odd} in the literature). We consider the two term elliptic complex: ( ¯∂ + δ′′)_{±}: C^{∞}(Λ^{0,±}) → C^{∞}(Λ^{0,∓})

For Almost Complex Manifolds 185

and define the arithmetic genus of M to be the index of ( ¯∂ + δ′′)_{+}. The adjoint of ( ¯∂ + δ′′)_{+} is ( ¯∂ + δ′′)_{−} and the associated Laplacian is just Δ_{′′ c} restricted to Λ^{0,±} so this is an elliptic complex. If M is holomorphic, then the index of this elliptic complex is equal to the index of the complex ( ¯∂, Λ^{0,q}) by the Hodge decomposition theorem. We define: c′(ξ) = (√2)^{−1} ∑ j≤n{(ξ_{j} + iξ_{j+n}) ext(e_{j} − ie_{j+n}) − (ξ_{j} − iξ_{j+n}) int(e_{j} + ie_{j+n})} then it is immediate that: c′(ξ)c′(ξ) = −|ξ|^{2} and σ_{L}( ¯∂ + δ′′) = ic′(ξ)/√2. Let ∇ be a connection on Λ^{0,±}, then we define the operator A^{±}(∇) by the diagram:A^{±}(∇): C^{∞}(Λ^{0,±}) ^{∇} → C^{∞}(T^{∗}M ⊗ Λ^{0,±}) ^{c′/√2} ⟶ C^{∞}(Λ^{0,∓}) This will have the same leading symbol as ( ¯∂ + δ′′)_{±}. There exists a unique connection so A^{±}(∇) = ( ¯∂ + δ′′)_{±} but as the index is constant under lower order perturbations, we shall not need this fact as the index of A^{+}(∇) = index( ¯∂ + δ′′)_{+} for any ∇. We shall return to this point in the next subsection. We now let V be an arbitrary coefficient bundle with a connection ∇ and define the Dolbeault complex with coefficients in V using the diagram: A^{±}: C^{∞}(Λ^{0,±} ⊗ V) ^{∇} → C^{∞}(T^{∗}M ⊗ Λ^{0,±} ⊗ V) ^{c′/√2⊗1} ⟶ C^{∞}(Λ^{0,∓} ⊗ V) and we define index(V, Dolbeault) to be the index of this elliptic complex. The index is independent of the connections chosen, of the fiber metrics chosen, and is constant under perturbations of the almost complex struc- ture. If M is holomorphic and if V is a holomorphic vector bundle, then we can extend ¯∂: C^{∞}(Λ^{0,q} ⊗ V) → C^{∞}(Λ^{0,q+1} ⊗ V) with ¯∂ ¯∂ = 0. Exactly as was true for the arithmetic genus, the index of this elliptic complex is equal to the index of the rolled up elliptic complex so our definitions generalize the usual definitions from the holomorphic category to the almost complex category. We will compute a formula for index(V, Dolbeault) using the spin com- plex. There is a natural inclusion from U(_{m 2} ) = U(n) into SO(m), but this does not lift in general to SPIN(m). We saw earlier that T (CP_{k}) does not admit a spin structure if k is even, even though it does admit a unitary structure. We define SPIN_c(m) = SPIN(m) × S^1/Z_2 where we choose the Z_2 identification (g, λ) = (−g, −λ). There is a natural map ρ_c: SPIN_c(m) → SO(m) × S^1 induced by the map which sends (g, λ) ↦ (ρ(g), λ^2). This map is a Z_2 double cover and is a group homomorphism.

There is a natural map of U(m/2) into SO(m) × S^1 defined by sending g ↦ (g, det(g)). The interesting thing is that this inclusion does lift; we can define f: U(m/2) → SPIN_c(m) so the following diagram commutes:

↗

SPIN_c(m)

f ρ_c = ρ × λ^2

U(m/2) 1 × det ⟶ SO(m) × S^1

We define the lifting as follows. For U ∈ U(m/2), we choose a unitary basis {e_j}^n_{j=1} so that U(e_j) = λ_je_j. We define e_{j+n} = ie_j so {e_j}^{2n}_{j=1} is an orthogonal basis for R^m = C^n. Express λ_j = e^{iθj} and define:

f(U) =

n ∏

j=1

{cos(θ_j/2) + sin(θ_j/2)e_je_{j+n}} ×

n ∏

j=1

e^{iθj/2} ∈ SPIN_c(m).

We note first that e_je_{j+n} is an invariant of the one-dimensional complex subspace spanned by e_j and does not change if we replace e_j by ze_j for |z| = 1. Since all the factors commute, the order in which the eigenvalues is taken does not affect the product. If there is a multiple eigenvalue, this product is independent of the particular basis which is chosen. Finally, if we replace θ_j by θ_j + 2π then both the first product and the second product change sign. Since (g, λ) = (−g, −λ) in SPIN_c(m), this element is invariantly defined. It is clear that f(I) = I and that f is continuous. It is easily verified that ρ_cf(U) = i(U) × det(U) where i(U) denotes the matrix U viewed as an element of SO(m) where we have forgotten the complex structure on C^n. This proves that f is a group homomorphism near the identity and consequently f is a group homomorphism in general. ρ_c is a covering projection.

The cannonical bundle K is given by:

K = Λ^{n,0} = Λ^n(T^′M)^∗

so K^{∗} = Λ^{n}(T′M) is the bundle with transition functions det(U_{αβ}) where the U_{αβ} are the transitions for T′M. If we have a spin structure on M, we can split:  P_{SPIN c} = P_{SPIN} × P_{S 1}/Z2 where P_{S 1} represents a line bundle L_{1} over M. From this description, it is clear K^{∗} = L_{1} ⊗ L_{1}. Conversely, if we can take the square root of K (or equivalently of K^{∗}), then M admits a spin structure so the obstruction to constructing a spin structure on an almost complex manifold is the obstruction to finding a square root of the cannonical bundle. Let V = C^{n} = R^{m} with the natural structures. We extend Δ^{±} to representations Δ^{±} of SPIN_{c}(m) in the natural way. We relate this repre- sentation to the Dolbeault representation as follows: Lemma 3.5.3. There is a natural isomorphism between Λ^{0,±} and Δ^{±} c which defines an equivalence of these two representations of U(_{m 2} ). Under this isomorphism, the action of V by Clifford multiplication on the left is preserved. Proof: Let {e_{j}} be an orthonormal basis for R^{m} with Je_{j} = e_{j+n} for 1 ≤ j ≤ n. Define: α_{j} = ie_{j}e_{j+n},  β_{j} = e_{j} + ie_{j+n},  β̄j = e_{j} − ie_{j+n}  for 1 ≤ j ≤ n. We compute: β_{j}α_{j} = −β_{j} and β_{j}α_{k} = α_{k}β_{j}  for k ≠ j. We define γ = β_{1} … β_{n} then γ spans Λ^{n,0} and γα_{j} = −γ, 1 ≤ j ≤ n. We define: Λ^{0,∗} = _{⨁ q} Λ^{0,q}  and Λ^{n,∗} = _{⨁ q} Λ^{n,q} = Λ^{0,∗}γ. Since dim(Λ^{0,∗}) = 2^{n}, we conclude that Λ^{0,∗}γ is the simultaneous −1 eigenspace for all the α_{j} and that therefore: Δ_{c} = Λ^{0,∗}γ as a left representation space for SPIN(m). Again, we compute: α_{j}β_{j} = β_{j}, α_{j}β̄j = −β̄j, α_{j}β_{k} = β_{k}α_{j}, α_{j}β̄k = β̄kα_{j}  for j ≠ k so that if x ∈ Λ^{0,q}γ then since τ = α_{1} … α_{n} we have τx = (−1)^{q}x so that Δ_{± c} = Λ^{0,±}γ.

We now study the induced representation of U(n). Let g = (cos(θ/2) + sin(θ/2)e_{1}Je_{1}) · e^{iθ/2} ∈ SPIN_{c}(m). We compute: gβ1 = {cos(θ/2) − i sin(θ/2)}e^{iθ/2}β_{1} = β1 gβ̄1 = {cos(θ/2) + i sin(θ/2)}e^{iθ/2}β̄1 = e^{iθ}β̄1 gβk = β_{k}g  and gβ̄k = β̄kg for k > 1. Consequently:  gβ̄jγ = { β̄jγ  if j_{1} > 1 e^{iθ}β̄jγ  if j^{1} = 1. A similar computation goes for all the other indices and thus we compute that if Ue_{j} = e^{iθj} e_{j} is unitary that: f(U)β̄J γ = e^{iθj1} … e^{iθjq} β̄J γ which is of course the natural represenation of U(n) on Λ^{0,q}. Finally we compare the two actions of V by Clifford multiplication. We assume without loss of generality that ξ = (1, 0,…, 0) so we must study Clifford multiplication by e_{1} on Δ_{c} and {ext(e_{1} − ie_{1+n}) − int(e_{1} + ie_{1+n})}/√2 = c′(e_{1}) on Λ^{0,∗}. We compute: e_{1}β1 = −1 + ie_{1}e_{2} = (e_{1} − ie_{2})(e_{1} + ie_{2})/2 = β̄1β_{1}/2, e_{1}β̄1β1 = −2β_{1}, e_{1}β̄k = −β̄ke_{1}  for k > 1. From this it follows immediately that: e^{1}β̄J γ = (−1)^{q} { β̄1β̄J γ/2  if j_{1} > 1 2β̄J ′γ  where J′ = {j^{2},…, j^{q}} if j^{1} = 1. Similarly, we compute: c′(e^{1})β̄J = { β̄1β̄J /√2  if j_{1} > 1 −√2β̄J ′  for J′ = {j^{2},…, j^{q}} if j^{1} = 1. From these equations, it is immediate that if we define T (β̄J) = β̄J γ, then T will not preserve Clifford multiplication. We let a(q) be a sequence of non-zero constants and define T: Λ^{0,±} → Δ_{± c} by: T (β̄J) = a(|J|)β̄J γ.

For Almost Complex Manifolds 189

Since the spaces Λ^{0,q} are U(n) invariant, this defines an equivalence between these two representations of U(n). T will induce an equivalence between Clifford multiplication if and only if we have the relations: (−1)^{q}a(q)/2 = a(q + 1)/√2 and √2a(q − 1) = (−1)^{q} · 2 · a(q). These give rise to the inductive relations: a(q + 1) = (−1)^{q}a(q)/√2 and a(q) = (−1)^{q−1}a(q − 1)/√2. These relations are consistent and we set a(q) = (√2)^{−q}(−1)^{q(q−1)/2} to define the equivalence T and complete the proof of the Lemma. From this lemma, we conclude: Lemma 3.5.4. There is a natural isomorphism of elliptic complexes (Δ_{+ c} − Δ_{− c}) ⊗ V ≃ (Λ^{0,+} − Λ^{0,−}) ⊗ V which takes the operator of the SPIN_{c} complex to an operator which has the same leading symbol as the Dolbeault complex (and thus has the same index). Furthermore, we can represent the SPIN_{c} complex locally in terms of the SPIN complex in the form: (Δ_{+ c} − Δ_{− c}) ⊗ V = (Δ^{+} − Δ^{−}) ⊗ L_{1} ⊗ V where L_{1} is a local square root of Λ^{n}(T′M). We use this sequence of isomorphisms to define an operator on the Dol- beault complex with the same leading symbol as the operator ( ¯∂ + ∂′′) which is locally isomorphic to the natural operator of the SPIN complex. The Z_{2} ambiguity in the definition of L_{1} does not affect this construction. (This is equivalent to choosing an appropriate connection called the spin connection on Λ^{0,±}.) We can compute index(V, Dolbeault) using this op- erator. The local invariants of the heat equation for this operator are the local invariants of the twisted spin complex and therefore arguing exactly as we did for the signature complex, we compute: index(V, Dolbeault) = ∫_{M} Â(T M) ∧ ch(L_{1}) ∧ ch(V) where ch(L_{1}) is to be understood as a complex characteristic class of T′(M). Theorem 3.5.5 (Riemann-Roch). Let T d(T′M) be the Todd class defined earlier by the generating function T d(A) = ∏_{ν} x_{ν}/(1−e^{−xν}). Then index(V, Dolbeault) = ∫_{M} T d(T′M) ∧ ch(V). Proof: We must simply identify Â(T M) ∧ ch(L_{1}) with T d(T′M). We perform a computation in characteristic classes using the splitting principal.

We formally decompose T^′M = L̃_1 ⊕ ⋯ ⊕ L̃_n as the direct sum of line bundles. Then Λ^n(T^′M) = L̃_1 ⊗ ⋯ ⊗ L̃_n so c_1(Λ^nT^′M) = c_1(L̃_1) + ⋯ + c_1(L̃_n) = x_1 + ⋯ + x_n. Since L_1 ⊗ L_1 = Λ^nT^′M, we conclude c_1(L_1) = 1/2(x_1 + ⋯ + x_n) so that:

ch(L_1) = ∏

ν

e^{xν /2}.

Therefore:

Â(M) ∧ ch(L_1) = ∏

ν

x_ν (e^{xν /2}) / (e^{xν /2} − e^{−xν /2})

= ∏

ν

(x_ν) / (1 − e^{−xν}) = T d(T^′M).

Of course, this procedure is only valid if the given bundle does in fact split as the direct sum of line bundles. We can make this procedure a correct way of calculating characteristic classes by using flag manifolds or by actually calculating on the group representaton and using the fact that the diagonizable matrices are dense.

It is worth giving another proof of the Riemann-Roch formula to ensure that we have not made a mistake of sign somewhere in all our calculations. Using exactly the same multiplicative considerations as we used earlier and using the qualitative form of the formala for index(V, Dolbeault) given by the SPIN_c complex, it is immediate that there is some formula of the form:

index(V, Dolbeault) = ∫_N ∑

s+t=n

T d^′_s(T^′M) ∧ c^tch_t(V)

where c is some universal constant to be determined and where T d^′_s is some characteristic form of T^′M.

If M = CP_j, then we shall show in Lemma 3.6.8 that the arithmetic genus of CP_j is 1. Using the multiplicative property of the Dolbeault complex, we conclude the arithmetic genus of CP_{j1} × ⋯ × CP_{jk} is 1 as well. If M^c_ρ are the manifolds of Lemma 2.3.4 then if we take V = 1 we conclude:

1 = index(1, Dolbeault) = arithmetic genus of CP_{j1} × ⋯ × CP_{jk}

= ∫_{M c}

ρ

T d^′(T^′M).

We verified in Lemma 2.3.5 that T d(T^′M) also has this property so by the uniqueness assertion of Lemma 2.3.4 we conclude T d = T d^′. We take m = 2 and decompose:

Λ^0 ⊕ Λ^2 = Λ^{0,0} ⊕ Λ^{1,1} and Λ^1 = Λ^{1,0} ⊕ Λ^{0,1}

so that  Λ^{e} − Λ^{o} = (Λ^{0,0} − Λ^{0,1}) ⊗ (Λ^{0,0} − Λ^{1,0}). In dimension 2 (and more generally if M is Kaehler), it is an easy exercise to compute that Δ = 2( ¯∂δ′′ + δ′′ ¯∂) so that the harmonic spaces are the same therefore: χ(M) = index(Λ^{0,0}, Dolbeault) − index(Λ^{1,0}, Dolbeault) = −c ∫_{M} ch(Λ^{1,0}) = c ∫_{M} ch(T′M) = c ∫_{M} e(M). The Gauss-Bonnet theorem (or the normalization of Lemma 2.3.3) implies that the normalizing constant c = 1 and gives another equivalent proof of the Riemann-Roch formula. There are many applications of the Riemann-Roch theorem. We present a few in dimension 4 to illustrate some of the techniques involved. If dim M = 4, then we showed earlier that: c_{2}(T′M) = e_{2}(T M)  and (2c_{2} − c_{2 1})(T′M) = p_{1}(T′M). The Riemann-Roch formula expresses the arithmetic genus of M in terms of c_{2} and c_{2 1}. Consequently, there is a formula: arithmetic genus = a_{1} ∫_{M} e_{2}(T M) + a_{2} ∫_{M} p_{1}(T M)/3 = a_{1}χ(M) + a_{2} signature(M) where a_{1} and a_{2} are universal constants. If we consider the manifolds S^{2} × S^{2} and CP_{2} we derive the equations: 1 = a_{1} · 4 + a_{2} · 0  and 1 = a_{1} · 3 + a_{2} · 1 so that a_{1} = a_{2} = 1/4 which proves: Lemma 3.5.6. If M is an almost complex manifold of real dimension 4, then:  arithmetic genus(M) = {χ(M) + signature(M)}/4. Since the arithmetic genus is always an integer, we can use this result to obtain some non-integrability results: Corollary 3.5.7. The following manifolds do not admit almost com- plex structures: (a) S^{4} (the four dimensional sphere). (b) CP_{2} with the reversed orientation.

192 3.5. The Riemann-Roch Theorem

(c) M_{1} # M_{2} where the M_{i} are 4-dimensional manifolds admitting almost complex structures (# denotes connected sum). Proof: We assume the contrary in each of these cases and attempt to compute the arithmetic genus: a.g.(S^{4}) = ^{1 4} (2 + 0) = ^{1} 2 a.g.(−CP_{2}) = ^{1 4} (3 − 1) = ^{1} 2 a.g.(M_{1} # M_{2}) = _{1 4} (χ(M_{1} # dM_{2}) + sign(M_{1} # M_{2}) = _{1 4} (χ(M_{1}) + χ(M_{2}) − 2 + sign(M_{1}) + sign(M_{2})) = a.g.(M_{1}) + a.g.(M_{2}) − _{1 2} . In none of these examples is the arithmetic genus an integer which shows the impossibility of constructing the desired almost complex structure.

3.6. A Review of Kaehler Geometry. In the previous subsection, we proved the Riemann-Roch theorem using the SPIN_{c} complex. This was an essential step in the proof even if the manifold was holomorphic. Let M be holomorphic and let the coefficient bundle V be holomorphic (i.e., the transition functions are holomorphic). We can define the Dol- beault complex directly by defining: ¯∂V : C^{∞}(Λ^{0,q} ⊗ V) → C^{∞}(Λ^{0,q+1} ⊗ V) by ¯∂V (ω ⊗ s) = ¯∂ω ⊗ s where s is a local holomorphic section to V. It is immediate ¯∂V ¯∂V = 0 and that this is an elliptic complex; the index of this elliptic complex is just index(V, Dolbeault) as defined previously. We let a_{n}(x, V, Dolbeault) be the invariant of the heat equation for this elliptic complex; we use the notation a_{j}(x, Dolbeault) when V is the trivial bundle. Then: Remark 3.6.1. Let m = 2n > 2. Then there exists a unitary Rieman- nian metric on the m-torus (with its usual complex structure) and a point x such that: (a) a_{j}(x, Dolbeault) ≠ 0 for j even and j ≥ n, (b) a_{m}(x, Dolbeault) ≠ Td_{n} where both are viewed as scalar invariants. The proof of this is quite long and combinatorial and is explained else- where; we simply present the result to demonstrate that it is not in general possible to prove the Riemann-Roch theorem directly by heat equation methods. The difficulty is that the metric and the complex structure do not fit together properly. Choose local holomorphic coordinates z = (z_{1},…, z_{n}) and extend the metric G to be Hermitian on T (M) ⊗ C so that T′ and T ′′ are orthogonal. We define:g_{jk̄} = G(∂/∂z_{j}, ∂/∂z_{k}) then the matrix g_{jk̄} is a positive definite Hermitian matrix which deter- mines the original metric on T (M): ds2 = 2^{∑ j,k} g^{jk̄}dz^{j} · dz̄k G(∂/∂x_{j}, ∂/∂x_{k}) = G(∂/∂y_{j}, ∂/∂y_{k}) = (g_{jk̄} + g_{kj̄}) G(∂/∂x_{j}, ∂/∂y_{k}) = −G(∂/∂y_{j}, ∂/∂x_{k}) = 1i (g_{jk̄} − g_{kj̄}) We use this tensor to define the Kaehler 2-form: Ω = i_{∑ j,k} g_{jk̄}dz^{j} ∧ dz̄k.

This is a real 2-form which is defined by the identity: Ω[X, Y] = −G(X, JY). (This is a slightly different sign convention from that sometimes followed.) The manifold M is said to be Kaehler if Ω is closed and the heat equation gives a direct proof of the Riemann-Roch theorem for Kaehler manifolds. We introduce variables:^{gjk̄/l  and gjk̄/l̄} for the jets of the metric. On a Riemannian manifold, we can always find a coordinate system in which all the 1-jets of the metric vanish at a point; this concept generalizes as follows: Lemma 3.6.2. Let M be a holomorphic manifold and let G be a unitary metric on M. The following statements are equivalent: (a) The metric is Kaeher (i.e., dΩ = 0). (b) For every z_{0} ∈ M there is a holomorphic coordinate system Z centred at z_{0} so that g_{jk̄}(Z, G)(z_{0}) = δ_{jk} and g_{jk̄/l}(Z, G)(z_{0}) = 0. (c) For every z_{0} ∈ M there is a holomorphic coordinate system Z centred at z_{0} so g_{jk̄}(Z, G)(z_{0}) = δ_{jk} and so that all the 1-jets of the metric vanish at z_{0}. Proof: We suppose first the metric is Kaehler and compute: dΩ = i_{∑ j,k}{g_{jk̄/l}dz^{l} ∧ dz^{j} ∧ dz̄k + g_{jk̄/l̄}dz̄l ∧ dz^{j} ∧ dz̄k}. Thus Kaehler is equivalent to the conditions: g_{jk̄/l} − g_{lk̄/j} = g_{jk̄/l̄}− g_{jl̄/k̄} = 0. By making a linear change of coordinates, we can assume that the holo- morphic coordinate system is chosen to be orthogonal at the center z_{0}. We let  z_{′j} = z_{j} + ∑ c_{jkl}z_{k}z_{l}  (where c_{jkl} = c_{jlk}) and compute therefore: dz′j = dz_{j} + 2 ∑ c_{jkl}z_{l}dzk ∂/∂z′j = ∂/∂z_{j} − 2 ∑ c_{k jl}z_{l}∂/∂z_{k} + O(z^{2}) g′j_{}k̄ = g_{jk̄} − 2 ∑ c_{k jl}z_{l} + terms in z̄ + O(|z|^{2})^{g′jk̄/l} = g_{jk̄/l} − 2c_{k jl} at z_{0}.

Kaehler Geometry 195

We define c_{k jl} = _{1 2} g_{jk̄/l} and observe that the Kaehler condition shows this is symmetric in the indices (j, l) so (a) implies (b). To show (b) implies (c) we simply note that g_{jk̄/l̄} = gkj̄ /l = 0 at z_{0}. To prove (c) implies (a) we observe dΩ(z_{0}) = 0 so since z_{0} was arbitrary, dΩ ≡ 0. If M is Kaehler, then δΩ is linear in the 1-jets of the metric when we compute with respect to a holomorphic coordinate system. This implies that δΩ = 0 so Ω is harmonic. We compute that Ω^{k} is also harmonic for 1 ≤ k ≤ n and that Ω^{n} = c · dvol is a multiple of the volume form (and in particular is non-zero). This implies that if x is the element in H^{2}(M; C) defined by Ω using the Hodge decomposition theorem, then 1, x,…, xn all represent non-zero elements in the cohomology ring of M. This can be used to show that there are topological obstructions to constructing Kaehler metrics: Remark 3.6.3. If M = S^{1} × S^{m−1} for m even and m ≥ 4 then this admits a holomorphic structure. No holomorphic structure on M admits a Kaehler metric. Proof: Since H^{2}(M; C) = 0 it is clear M cannot admit a Kaehler metric. We construct holomorphic structures on M as follows: let λ ∈ C with |λ| > 1 and let M_{λ} = {C^{n} − 0}/λ (where we identify z and w if z = λ^{k}w for some k ∈ Z). The M_{λ} are all topologically S^{1} × S^{m−1}; for example if λ ∈ R and we introduce spherical coordinates (r, θ) on C^{n} = R^{m} then we are identifying (r, θ) and (λ^{k}r, θ). If we let t = log r, then we are identifying t with t + k log λ so the manifold is just [1, log λ] × S^{m−1} where we identify the endpoints of the interval. The topological identification of M_{λ} for other λ is similar. We now turn to the problem of constructing a Kaehler metric on CP_{n}. Let L be the tautological line bundle over CP_{n} and let x = −c_{1}(L) ∈ Λ^{1,1}(CP_{n}) be the generator of H^{2}(CP_{n}; Z) discussed earlier. We expand x in local coordinates in the form: x = i 2π ^{∑ j,k} g^{jk̄}dz^{j} ∧ dz̄k and define G(∂/∂z_{k}, ∂/∂z_{k}) = g_{jk̄}. This gives an invariantly defined form called the Fubini-Study metric on T_{c}(CP_{n}). We will show G is positive definite and defines a unitary metric on the real tangent space. Since the Kaehler form of G, Ω = 2πx, is a multiple of x, we conclude dΩ = 0 so G will be a Kaehler metric. The 2-form x is invariant under the action of U(n+1). Since U(n+1) acts transitively on CP_{n}, it suffices to show G is positive definite and symmetric at a single point. Using the notation of section 2.3, let U_{n} = {z ∈ CP_{n} : z_{n+1}(z) ≠ 0}. We identify U_{n} with C^{n} by identifying z ∈ C^{n} with the line 196 3.6. Kaehler Geometry in C^{n+1} through the point (z, 1). Then: x = i 2π ∂ ¯∂ log(1 + |z|^{2}) so at the center z = 0 we compute that: x = i 2π _{∑ j} dz^{j} ∧ dz̄j so that g_{jk̄} = δ_{jk̄}. If G′ is any other unitary metric on CP_{n} which is invariant under the action of U(n + 1), then G′ is determined by its value on T_{c}(CP_{n}) at a single point. Since U(n) preserves the origin of U_{n} = C^{n}, we conclude G′ = cG for some c > 0. This proves: Lemma 3.6.4. (a) There exists a unitary metric G on CP_{n} which is invariant under the action of U(n + 1) such that the Kaehler form of G is given by Ω = 2πx where x = −c_{1}(L) (L is the tautological bundle over CP_{n}). G is a Kaehler metric. (b) ∫_{CPn} x^{n} = 1. (c) If G′ is any other unitary metric on CP_{n} which is invariant under the action of U(n + 1) then G′ = cG for some constant c > 0. A holomorphic manifold M is said to be Hodge if it admits a Kaehler metric G such that the Kaehler form Ω has the form Ω = c_{1}(L) for some holomorphic line bundle L over M. Lemma 3.6.4 shows CP_{n} is a Hodge manifold. Any submanifold of a Hodge manifold is again Hodge where the metric is just the restriction of the given Hodge metric. Thus every algebraic variety is Hodge. The somewhat amazing fact is that the converse is true: Remark 3.6.5. A holomorphic manifold M is an algebraic variety (i.e., is holomorphically equivalent to a manifold defined by algebraic equations in CP_{n} for some n) if and only if it admits a Hodge metric. We shall not prove this remark but simply include it for the sake of com- pleteness. We also note in passing that there are many Kaehler manifolds which are not Hodge. The Riemann period relations give obstructions on complex tori to those tori being algebraic. We now return to our study of Kaehler geometry. We wish to relate thē ∂ cohomology of M to the ordinary cohomology. Define: H^{p,q}(M) = ker ¯∂/ im ¯∂ in bi-degree (p, q). Since the Dolbeault complex is elliptic, these groups are finite dimensional using the Hodge decomposition theorem.

Lemma 3.6.6. Let M be Kaehler and let Ω = i ∑ g_{jk̄}dz^{j} ∧ dz̄k be the Kaehler form. Let d = ∂ + ¯∂ and let δ = δ′ + δ′′. Then we have the identities: (a) ¯∂ int(Ω) − int(Ω) ¯∂ = iδ′. (b) ¯∂δ′ + δ′ ¯∂ = ∂δ′′ + δ′′∂ = 0. (c) dδ + δd = 2(∂δ′ + δ′∂) = 2( ¯∂δ′′ + δ′′ ¯∂). (d) We can decompose the de Rham cohomology in terms of the Dolbeault cohomology: H^{n}(M; C) = _{⨁ p+q=n} H^{p,q}(M; C). (e) There are isomorphisms H^{p,q} ≃ H^{q,p} ≃ H^{n−p,n−q} ≃ H^{n−q,n−p}. Proof: We first prove (a). We define: A = ¯∂ int(Ω) − int(Ω) ¯∂ − iδ′. If we can show that A is a 0^{th} order operator, then A will be a functori- ally defined endomorphism linear in the 1-jets of the metric. This implies A = 0 by Lemma 3.6.2. Thus we must check σ_{L}(A) = 0. We choose an orthonormal basis {e_{j}} for T (M) = T^{∗}(M) so Je_{j} = e_{j+n}, 1 ≤ j ≤ n. By Lemma 3.5.2, it suffices to show that: ext(e_{j} − ie_{j+n}) int(Ω) − int(Ω) ext(e_{j} − ie_{j+n}) = −i · int(e_{j} − ie_{j+n}) for 1 ≤ j ≤ n. Since g_{jk̄} = _{1 2} δ_{jk̄}, Ω is given by Ω = i2 _{∑ j}(e_{j} + ie_{j+n}) ∧ (e_{j} − ie_{j+n}) = _{∑ j} e_{j} ∧ e_{j+n}. The commutation relations: int(e_{k}) ext(e_{l}) + ext(e_{l}) int(e_{k}) = δk,l imply that int(e_{k} ∧ e_{k+n}) = − int(e_{k}) int(e_{k+n}) commutes with ext(e_{j} − ie_{j+n}) for k ≠ j so these terms disappear from the commutator. We must show: ext(e_{j} − ie_{j+n}) int(e_{j} ∧ e_{j+n}) − int(e_{j} ∧ e_{j+n}) ext(e_{j} − ie_{j+n}) = −i · int(e_{j} − ie_{j+n}). This is an immediate consequence of the previous commutation relations for int and ext together with the identity int(e_{j}) int(e_{k})+int(e_{k}) int(e_{j}) = 0. This proves (a).

From (a) we compute: ¯∂δ′ + δ′ ¯∂ = i(− ¯∂ int(Ω) ¯∂) + i( ¯∂ int(Ω) ¯∂) = 0 and by taking complex conjugates ∂δ′′ + δ′′∂ = 0. This shows (b). Thus Δ = (d+δ)^{2} = dδ +δd = {∂δ′ +δ′∂}+{ ¯∂δ′′ +δ′′∂} = {∂+δ′}^{2} +{ ¯∂ +δ′′}2 since all the cross terms cancel. We apply (a) again to compute: ∂ ¯∂ int(Ω) − ∂ int(Ω) ¯∂ = i∂δ ′ ¯∂ int(Ω)∂ − int(Ω) ¯∂∂ = iδ′∂ which yields the identity: i(∂δ′ + δ′∂) = ∂ ¯∂ int(Ω) − ∂ int(Ω) ¯∂ + ¯∂ int(Ω)∂ − int(Ω) ¯∂∂. Since Ω is real, we take complex conjugate to conclude: −i( ¯∂δ′′ + δ′′ ¯∂) = ¯∂∂ int(Ω) − ¯∂ int(Ω)∂ + ∂ int(Ω) ¯∂ − int(Ω)∂ ¯∂ = −i(∂δ′ + δ′∂). This shows ( ¯∂δ′′ + δ′′ ¯∂) = (∂δ′ + δ′∂) so Δ = 2( ¯∂δ′′ + δ′′ ¯∂) which proves (c). H^{n}(M; C) denotes the de Rham cohomology groups of M. The Hodge decomposition theorem identifies these groups with the null-space of (d+δ). Since N(d + δ) = N( ¯∂ + δ′′), this proves (d). Finally, taking complex conjugate and applying ∗ induces the isomorphisms: H^{p,q} ≃ H^{q,p} ≃ H^{n−q,n−p} ≃ H^{n−p,n−q} . In particular dim H^{1}(M; C) is even if M admits a Kaehler metric. This gives another proof that S^{1} × S^{2n−1} does not admit a Kaehler metric for n > 1. The duality operations of (e) can be extended to the Dolbeault com- plex with coefficients in a holomorphic bundle V. If V is holomorphic, the transition functions are holomorphic and hence commute with ¯∂. We define:  ¯∂: C^{∞}(Λ^{p,q} ⊗ V) → (Λ^{p,q+1} ⊗ V) by defining ¯∂(ω ⊗ s) = ¯∂ω ⊗ s relative to any local holomorphic frame for V. This complex is equivalent to the one defined in section 3.5 and defines cohomology classes: H^{p,q}(V) = ker ¯∂/ im ¯∂  on C^{∞}(Λ^{p,q} ⊗ V).

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It is immediate from the definition that: H^{p,q}(V) = H^{0,q}(Λ^{p,0} ⊗ V). We define index(V, ¯∂) = index(V, Dolbeault) = ∑(−1)^{q} dim H^{0,q}(V) to be the index of this elliptic complex. There is a natural duality (called Serre duality) which is conjugate linear̄ ∗: H^{p,q}(V) → H^{n−p,n−q} (V^{∗}), where V^{∗} is the dual bundle. We put a unitary fiber metric on V and define: T (s_{1})(s_{2}) = s_{2} · s1 so T: V → V^{∗} is conjugate linear. If ∗: Λ^{k} → Λ^{m−k} is the ordinary Hodge operator, then it extends to define ∗: Λ^{p,q} → Λ^{n−q,n−p}. The complex con- jugate operator defines ¯∗: Λ^{p,q} → Λ^{n−p,n−q}. We extend this to have coeffi- cients in V by defining: ¯∗(ω ⊗ s) = ¯∗ω ⊗ T s so ¯∗: Λ^{p,q} ⊗ V → Λ^{n−p,n−q} ⊗ V^{∗} is conjugate linear. Then: Lemma3.6.7. Let δ_{′′ V} be the adjoint of ¯∂V on C^{∞}(Λ^{p,∗} ⊗ V) with respect to a unitary metric on M and a Hermitian metric on V. Then (a) δ′′ = −¯∗ ¯∂V ¯∗. (b) ¯∗ induces a conjugate linear isomorphism between the groups H^{p,q}(V) and H^{n−p,n−q} (V^{∗}). Proof: It is important to note that this lemma, unlike the previous one, does not depend upon having a Kaehler metric. We suppose first V is holomorphically trivial and the metric on V is flat. Then Λ ⊗ V = Λ ⊗ 1k for some k and we may suppose k = 1 for the sake of simplicity. We noted δ = − ∗ d∗ = −¯∗d¯∗ in section 1.5. This decomposes δ = δ′ + δ′′ = −¯∗∂¯∗ − ¯∗ ¯∂¯∗. Since ¯∗: Λ^{p,q} → Λ^{n−p,n−q}, we conclude δ′ = −¯∗∂¯∗ and δ′′ = −¯∗ ¯∂¯∗. This completes the proof if V is trivial. More generally, we define A = δ_{′′ V} + ¯∗ ¯∂¯∗.

This must be linear in the 1-jets of the metric on V. We can always normalize the choice of local holomorphic frame so the 1-jets vanish at a basepoint z_{0} and thus A = 0 in general which proves (a). We identify H^{p,q}(V) with N( ¯∂V )∩N(δ_{′′ V}) using the Hodge decomposition theorem. Thus̄ ∗: H^{p,q}(V) → H^{n−p,n−q} (V^{∗}). Since ¯∗2 = ±1, this completes the proof. We defined x = −c_{1}(L) ∈ Λ^{1,1}(CP_{n}) for the standard metric on the tautological line bundle over CP_{n}. In section 2.3 we showed that x was harmonic and generates the cohomology ring of CP_{n}. Since x defines a Kaehler metric, called the Fubini-Study metric, Lemma 3.6.6 lets us de- compose H^{n}(CP_{n}; C) = ⨁_{p+q=n} H^{p,q}(CP_{n}). Since x^{k} ∈ H^{k,k} this shows: Lemma 3.6.8. Let CP_{n} be given the Fubini-Study metric. Let x = −c_{1}(L). Then H^{k,k}(M) ≃ C is generated by x^{k} for 1 ≤ k ≤ n. H^{p,q}(M) = 0 for p ≠ q. Thus in particular, index( ¯∂) = arithmetic genus = ∑(−1)^{k} dim H^{0,k} = 1  for CP_{n}. Since the arithmetic genus is multiplicative with respect to products, index( ¯∂) = 1  for CP_{n1} × ⋯ × CP_{nk}. We used this fact in the previous subsection to derive the normalizing constants in the Riemann-Roch formula. We can now present another proof of the Riemann-Roch theorem for Kaehler manifolds which is based directly on the Dolbeault complex and not on the SPIN_{c} complex. Let Z be a holomorphic coordinate system and let  g_{uv̄} = G(∂/∂z_{u}, ∂/∂z_{v}) represent the components of the metric tensor. We introduce additional variables: g_{uv̄/αβ̄} = d_{α z}d_{β̄ z}gu_{}v̄ for 1 ≤ u, v ≤ n, α = (α_{1},…, α_{n}), and β = (β_{1},…, β_{n}). These variables represent the formal derivatives of the metric. We must also consider the Dolbeault complex with coefficients in a holo- morphic vector bundle V. We choose a local fiber metric H for V. If s = (s_{1},…, s_{k}) is a local holomorphic frame for V, we introduce variables: h_{p,q̄} = H(s_{p}, s_{q}),  h_{p,q̄/αβ̄} = d_{α z}d_{β̄ z}h_{pq̄}  for 1 ≤ p, q ≤ k = dim V . If Z is a holomorphic coordinate system centered at z_{0} and if s is a local holomorphic frame, we normalize the choice so that: g_{uv̄}(Z, G)(z_{0}) = δ_{u,v}  and h_{pq̄}(s, H)(z_{0}) = δ_{p,q}.

The coordinate frame {∂/∂z_{u}} and the holomorphic frame {s_{p}} for V are orthogonal at z_{0}. Let R be the polynomial algebra in these variables. If P ∈ R we can evaluate P (Z, G, s, H)(z_{0}) once a holomorphic frame Z is chosen and a holomorphic frame s is chosen. We say P is invariant if P (Z, G, s, H)(z_{0}) = P (G, H)(z_{0}) is independent of the particular (Z, s) chosen; let R_{c n,k} denote the sub-algebra of all invariant polynomials. As before, we define: ord(g_{u,v̄/αβ̄}) = ord(s_{pq̄/αβ̄}) = |α| + |β|. We need only consider those variables of positive order as g_{u,v̄} = δ_{jv̄} and s_{p,q̄} = δ_{p,q} at z_{0}. If R_{c n,ν,k} denotes the subspace of invariant polynomials homogenous of order ν in the jets of the metrics (G, H), then there is a direct sum decomposition: R_{c n,k} = ⨁ R_{c n,ν,k} . In the real case in studying the Euler form, we considered the restriction map from manifolds of dimension m to manifolds of dimension m − 1. In the complex case, there is a natural restriction map^{r: Rc n,ν,k → Rc n−1,ν,k} which lowers the complex dimension by one (and the real dimension by two). Algebraically, r is defined as follows: let deg_{k}(g_{uv̄/αβ̄}) = δ_{k,u} + α(k) deg_{k}(h_{pq̄/αβ̄}) = α(k)  deg_{k̄}(g_{uv̄/αβ̄}) = δ_{k,v} + β(k) deg_{k̄}(h_{pq̄/αβ̄}) = β(k). We define: r(g_{uv̄/αβ̄}) = { guv̄/αβ̄  if deg_{n}(g_{uv̄/αβ̄}) + deg_{n̄}(g_{uv̄/αβ̄}) = 0 0  if deg^{n}(g^{uv̄/αβ̄}) + deg^{n̄}(g^{uv̄/αβ̄}) > 0 r(h_{pq̄/αβ̄}) = { hpq̄/αβ̄  if deg_{n}(h_{pq̄/αβ̄}) + deg_{n̄}(h_{pq̄/αβ̄}) = 0 0  if deg^{n}(h^{pq̄/αβ̄}) + deg^{n̄}(h^{pq̄/αβ̄}) > 0. This defines a map r: R_{n} → R_{n−1} which is an algebra morphism; we simply set to zero those variables which do not belong to R_{n−1}. It is immediate that r preserves both invariance and the order of a polynomial. Geometri- cally, we are considering manifolds of the form M^{n} = M^{n−1} × T_{2} where T_{2} is the flat two torus, just as in the real case we considered manifolds M^{m} = M^{m−1} × S^{1}. Let R_{ch n,p,k} be the space of p-forms generated by the Chern forms of T_{c}(M) and by the Chern forms of V. We take the holomorphic connection defined by the metrics G and H on T_{c}(M) and on V. If P ∈ R_{ch n,m,k} then ∗P is a scalar invariant. Since P vanishes on product metrics of the form M^{n−1} × T_{2} which are flat in one holomorphic direction, r(∗P) = 0. This is the axiomatic characterization of the Chern forms which we shall need.

202 3.6. Kaehler Geometry Theorem 3.6.9. Let the complex dimension of M be n and let the fiber dimension of V be k. Let P ∈ R_{c n,ν,k} and suppose r(P) = 0. Then: (a) If ν < 2n then P (G, H) = 0 for all Kaehler metrics G. (b) If ν = 2n then there exists a unique Q ∈ R_{ch n,m,k} so P (G, H) = ∗Q(G, H) for all Kaehler metrics G. As the proof of this theorem is somewhat technical, we shall postpone the proof until section 3.7. This theorem is false if we don’t restrict to Kaehler metrics. There is a suitable generalization to form valued invariants. We can apply Theorem 3.6.9 to give a second proof of the Riemann-Roch the- orem: Theorem 3.6.10. Let V be a holomorphic bundle over M with fiber metric H and let G be a Kaehler metric on M. Let ¯∂V : C^{∞}(Λ^{0,q} ⊗ V) → C^{∞}(Λ^{0,q+1} ⊗ V) denote the Dolbeault complex with coefficients in V and let a_{ν}(x, ¯∂V ) be the invariants of the heat equation. a_{ν} ∈ R_{c n,ν,k} and ∫^{M} a^{ν}(x, ¯∂V ) dvol(x) = { 0 if ν ≠ 2n index( ¯∂V ) if ν = 2n. Then (a) a_{ν}(x, ¯∂V ) = 0 for ν < 2n. (b) a_{2n}(x, ¯∂V ) = ∗{Td(T_{c}M) ∧ ch(V)}_{m}. Proof: We note this implies the Riemann-Roch formula. By remark 3.6.1, we note this theorem is false in general if the metric is not assumed to be Kaehler. The first assertions of the theorem (including the homogene- ity) follow from the results of Chapter 1, so it suffices to prove (a) and (b). The Dolbeault complex is multiplicative under products M = M^{n−1} × T_{2}. If we take the product metric and assume V is the pull-back of a bundle over M_{n−1} then the natural decomposition: Λ^{0,q} ⊗ V ≃ Λ^{0,q}(M^{n−1}) ⊗ V ⊕ Λ^{0,q−1} (M^{n−1}) ⊗ V shows that r(a_{ν}) = 0. By Theorem 3.6.9, this shows a_{ν} = 0 for ν < 2n proving (a). In the limiting case ν = 2n, we conclude ∗a_{2n} is a characteristic 2n form. We first suppose m = 1. Any one dimensional complex manifold is Kaehler. Thenindex( ¯∂V ) = a_{1} ∫_{M} c_{1}(T_{c}(M)) + a_{2} ∫_{M} c_{1}(V) where a_{1} and a_{2} are universal constants to be determined. Lemma 3.6.6 im- plies that dim H^{0,0} = dim H^{0} = 1 = dim H^{2} = dim H^{1,1} while dim H^{1,0} = dim H^{0,1} = g where g is the genus of the manifold. Consequently: χ(M) = 2 − 2g = 2 index( ¯∂).

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We specialize to the case M = S^{2} and V = 1 is the trivial bundle. Then g = 0 and 1 = index( ¯∂V ) = a_{1} ∫_{M} c_{1}(T_{c}(M)) = 2a1 so a_{1} = _{1 2} . We now take the line bundle V = Λ^{1,0} so index( ¯∂V ) = g − 1 = −1 = 1 + a_{2} ∫_{M} c_{1}(T_{∗ c}) = 1 − 2a2 so that a_{2} = 1. Thus the Riemann-Roch formula in dimension 1 becomes: index( ¯∂V ) = 1 2 ∫_{M} c_{1}(T_{c}(M)) + ∫_{M} c_{1}(V). We now use the additive nature of the Dolbeault complex with respect to V and the multiplicative nature with respect to products M_{1} × M_{2} to conclude that the characteristic m-form ∗a_{m} must have the form: {T_{′ d}(T_{c}M) ∧ ch(V)} where we use the fact the normalizing constant for c_{1}(V) is 1 if m = 1. We now use Lemma 2.3.5(a) together with the fact that index( ¯∂V ) = 1 for products of complex projective spaces to show Td′ = Td. This completes the proof.

3.7. An Axiomatic Characterization Of the Characteristic Forms for Holomorphic Manifolds with Kaehler Metrics. In subsection 3.6 we gave a proof of the Riemann-Roch formula for Kaehler manifolds based on Theorem 3.6.9 which gave an axiomatic char- acterization of the Chern forms. This subsection will be devoted to giving the proof of Theorem 3.6.9. The proof is somewhat long and technical so we break it up into a number of steps to describe the various ideas which are involved. We introduce the notation:^{gu_{0}v_{0}/u_{1}...u_{j}v̄1...v̄k} for the jets of the metric on M. Indices (u, v, w) will refer to T_{c}(M) and will run from 1 thru n = _{m 2} . We use (ū, v̄, w̄) for anti-holomorphic indices. The symbol “∗” will refer to indices which are not of interest in some particular argument. We let A_{0} denote a generic monomial. The Kaehler condition is simply the identity: g_{uv̄/w} = g_{wv̄/u} and g_{uv̄/w̄} = g_{uw̄/v̄}. We introduce new variables: g(u_{0},…, u_{j}; v̄0,…, v̄k) = g_{u_{0}v_{0}/u_{1}...u_{j}v̄1...v̄k} . If we differentiate the Kaehler identity, then we conclude g(u; v) is sym- metric in u = (u_{0},…, u_{j}) and v = (v_{0},…, v_{j}). Consequently, we may also use the multi-index notation g(α; β̄) to denote these variables. We define: ord(g(α; β̄)) = |α| + |β| − 2, deg_{u} g(α; β̄) = α(u), deg_{ū} g(α; β̄) = β(u). We already noted in Lemma 3.6.2 that it was possible to normalize the coordinates so g_{uv̄/w}(z_{0}) = g_{uv̄/w̄}(z_{0}) = 0. The next lemma will permit us to normalize coordinates to arbitrarily high order modulo the action of the unitary group: Lemma 3.7.1. Let G be a Kaehler metric and let z_{0} ∈ M. Let ν ≥ 2 be given. Then there exists a holomorphic coordinate system Z centered at z_{0} so that (a) g_{uv̄}(Z, G)(z_{0}) = δ_{uv}. (b) g_{uv̄/α}(Z, G)(z_{0}) = g_{uv̄/ᾱ}(Z, G)(z_{0}) = 0 for 1 ≤ |α| ≤ ν − 1. (c) Z is unique modulo the action of the unitary group U(n) and modulo coordinate transformations of order ν + 1 in z. Proof: We proceed by induction. It is clear U(n) preserves such coor- dinate systems. The case ν = 2 is just Lemma 3.6.2 and the unique- ness is clear. We now consider W given which satisfies (a) and (b) for

Characteristic Forms 205

1 ≤ |α| ≤ ν − 2 and define a new coordinate system Z by setting: w^{u} = z^{u} +^{∑ |α|=ν} C^{u,α}zα where the constants C_{u,α} remain to be determined. Since this is the identity transformation up to order ν, the ν − 2 jets of the metric are unchanged so conditions (a) and (b) are preserved for |β| < ν − 1. We compute: ∂ ∂zu = ∂ ∂wu +^{∑ v,|α|=ν} C^{v,α}α(u)z^{αu} ∂ ∂wv where α_{u} is the multi-index defined by the identity z_{u}z^{αu} = z^{α} (and which is undefined if α(u) = 0.) This implies immediately that: g_{uv̄}(Z, G) = g_{uv̄}(W, G) +_{∑ |α|=n} C_{v,α}α(u)z^{αu} + terms in z̄ + O(z^{ν+1} ). The ∂/∂z_{u} and ∂/∂w_{u} agree to first order. Consequently: g_{uv̄/αu}(Z, G) = g_{uv̄/αu}(W, G) + α! C_{v,α} + O(z, z̄). Therefore the symmetric derivatives are given by: g(α; v̄)(Z, G)(z_{0}) = g(α; v̄)(W, G)(z_{0}) + α! Cv,α The identity g(α; v̄)(Z, G)(z_{0}) = 0 determines the C_{v,α} uniquely. We take complex conjugate to conclude g(v, ᾱ)(Z, G)(z_{0}) = 0 as well. This permits us to choose the coordinate system so all the purely holo- morphic and anti-holomorphic derivatives vanish at a single point. We restrict henceforth to variables g(α; β̄) so that |α| ≥ 2, |β| ≥ 2. We introduced the notation h_{pq̄/αβ̄} for the jets of the metric H on the auxilary coefficient bundle V. Lemma 3.7.2. Let H be a fiber metric on a holomorphic bundle V. Then given v ≥ 1 there exists a holomorphic frame s near z_{0} for V so that: (a) h_{pq̄}(s, H)(z_{0}) = δ_{pq}. (b) h_{pq̄/α}(s, H)(z_{0}) = h_{pq̄/ᾱ}(s, H)(z_{0}) = 0 for 1 ≤ |α| ≤ ν. (c) The choice of s is unique modulo the action of the unitary group U(k) (where k is the fiber dimension of V) and modulo transformations of order ν + 1. Proof: If ν = 0, we make a linear change to assume (a). We proceed by induction assuming s′ chosen for ν − 1. We define: s^{p} = s^{′p} +_{∑ q, |α|=ν} C^{q,α}s^{′q}z^{α}.

We adopt the notational conventions that indices (p, q) run from 1 through k and index a frame for V. We compute: h_{pq̄}(s, H) = h^{pq̄}(s′, H) +^{∑ |α|=ν} C^{q,α}zα + terms in z̄+ terms vanishing to order ν + 1 h_{pq̄/α}(x, H)(z_{0}) = h_{pq̄/α}(s′, H)(z_{0}) + α!C_{q,α}  for |α| = ν and derivatives of lower order are not disturbed. This determines the C_{q,α} uniquely so h_{pq̄/α}(s, H)(z_{0}) = 0 and taking complex conjugate yields h_{qp̄/ᾱ}(s, H)(z_{0}) = 0 which completes the proof of the lemma. Using these two normalizing lemmas, we restrict henceforth to polyno- mials in the variables {g(α; β̄), h_{pq̄/α_{1}β̄1} } where |α| ≥ 2, |β| ≥ 2, |α_{1}| ≥ 1, |β_{1}| ≥ 1. We also restrict to unitary transformations of the coordinate system and of the fiber of V. If P is a polynomial in these variables and if A is a monomial, let c(A, P) be the coefficient of A in P. A is a monomial of P if c(A, P) ≠ 0. Lemma 2.5.1 exploited invariance under the group SO(2) and was central to our axiomatic characterization of real Pontrjagin forms. The natural groups to study here are U(1), SU(2) and the coordinate permutations. If we set ∂/∂w_{1} = a∂/∂z_{1}, ∂/∂w̄1 = ā∂/∂z̄1 for aā = 1 and if we leave the other indices unchanged, then we compute: A(W, ∗) = a^{deg_{1}(A)}ā^{deg}^{¯1}(A)A(Z, ∗). If A is a monomial of an invariant polynomial P, then deg_{1}(A) = deg_{¯1}(A) follows from this identity and consequently deg_{u}(A) = deg_{ū}(A) for all 1 ≤ u ≤ n. This is the only conclusion which follows from U(1) invariance so we now study the group SU(2). We consider the coordinate transformation: ∂/∂w1 = a∂/∂z_{1} + b∂/∂z_{2}, ∂/∂w̄1 = ā∂/∂z̄1 + b̄∂/∂z̄2,  ∂/∂w2 = −b̄∂/∂z_{1} + ā∂/∂z_{2}, ∂/∂w̄2 = −b∂/∂z̄1 + a∂/∂z̄2, ∂/∂wu = ∂/∂z_{u}  for u > 2, ∂/∂w̄u = ∂/∂z̄u  for u > 2, aā + bb̄ = 1. We let j = deg_{1}(A) + deg_{2}(A) = deg_{¯1}(A) + deg_{¯2}(A) and expand A(W, ∗) = a^{j}ājA(Z, ∗) + a^{j−1}ājbA(1 → 2 or ¯2 → ¯1)(Z, ∗) + a^{j}āj−1b̄A(2 → 1 or ¯1 → ¯2)(Z, ∗) + other terms.

The notation “A(1 → 2 or ¯2 → ¯1)” indicates all the monomials (with mul- tiplicity) of this polynomial constructed by either changing a single index 1 → 2 or a single index ¯2 → ¯1. This plays the same role as the polyno- mial A^{(1)} of section 2.5 where we must now also consider holomorphic and anti-holomorphic indices. Let P be U(2) invariant. Without loss of generality, we may assume deg_{1}(A) + deg_{2}(A) = j is constant for all monomials A of P since this condition is U(2) invariant. (Of course, the use of the indices 1 and 2 is for notational convenience only as similar statements will hold true for any pairs of indices). We expand: P (W, ∗) =a^{j}ājP (Z, ∗) + a^{j−1}ājbP (1 → 2 or ¯2 → ¯1) + a^{j}āj−1b̄P (2 → 1 or ¯1 → ¯2) + O(b, b̄)^{2}. Since P is invariant, we conclude: P (1 → 2 or ¯2 → ¯1) = P (2 → 1 or ¯1 → ¯2) = 0. We can now study these relations. Let B be an arbitrary monomial and expand: B(1 → 2) = c_{0}A_{0} + ⋯ + c_{k}A_{k}  B(¯2 → ¯1) = −(d_{0}A_{′0} + ⋯ + d_{k}A_{′k}) where the c’s and d’s are positive integers with ∑ c_{ν} = deg_{1}(B) and ∑ d_{ν} = deg_{¯2}(B). Then it is immediate that the {A_{j}} and the {A_{′j}} denote disjoint collections of monomials since deg_{1}(A)_{ν} = deg_{1}(B) − 1 while deg_{1}(A)_{′ν} = deg_{1}(B). We compute: A_{ν}(2 → 1) = −c_{′ν}B + other terms  A_{′ν}(¯1 → ¯2) = d_{′ν}B + other terms where again the c’s and d’s are positive integers (related to certain multi- plicities). Since P (2 → 1 or ¯1 → ¯2) is zero, if P is invariant, we conclude an identity: c(B, P (2 → 1 or ¯1 → ¯2)) = _{∑ ν} −c_{′ν}c(A_{ν}, P) +_{∑ ν} d_{′ν}c(A_{′ν}, P) = 0. By varying the creating monomial B (and also interchanging the indices 1 and 2), we can construct many linear equations among the coefficients. We note that in practice B will never be a monomial of P since deg_{1}(B) ≠ deg_{¯1}(B). We use this principle to prove the following generalization of Lemmas 2.5.1 and 2.5.5:

208 3.7. Characteristic Forms Lemma 3.7.3. Let P be invariant under the action of U(2) and let A be a monomial of P. (a) If A = g(α; β̄)A_{′0}, then by changing only 1 and 2 indices and ¯1 and̄ 2 indices we can construct a new monomial A_{1} of P which has the form A_{1} = g(α_{1}; β̄1)A_{′′ 0} where α(1) + α(2) = α_{1}(1) and where α_{1}(2) = 0. (b) If A = h_{pq̄/αβ̄}A_{′0} then by changing only 1 and 2 indices and ¯1 and̄ 2 indices we can construct a new monomial A_{1} of P which has the form A_{1} = h_{pq̄/α_{1}β̄1} A_{′′ 0} where α(1) + α(2) = α_{1}(1) and where α_{1}(2) = 0. Proof: We prove (a) as the proof of (b) is the same. Choose A of this form so α(1) is maximal. If α(2) is zero, we are done. Suppose the contrary. Let B = g(α_{1}; β̄)A_{′0} where α_{1} = (α(1) + 1, α(2) − 1, α(3),…, α(n)). We expand:B(1 → 2) = α_{1}(1)A + monomials divisible by g(α_{1}; β̄) B(¯2 → ¯1) = monomials divisible by g(α_{1}; β̄1) for some β_{1}. Since A is a monomial of P, we use the principle described previously to conclude there is some monomial of P divisible by g(α_{1}; β̄1) for some β_{1}. This contradicts the maximality of α and shows α(2) = 0 completing the proof. We now begin the proof of Theorem 3.6.9. Let 0 ≠ P ∈ R_{c n,ν,k} be a scalar valued invariant homogeneous of order ν with r(P) = 0. Let A be a monomial of P. Decompose A = g(α_{1}; β̄1) … g(α_{r}; β̄r)h_{p_{1}q_{1}/α_{r+1}β̄r+1} … h_{p_{s}q̄s/α_{r+s}β̄r+s} . Let ℓ(A) = r + s be the length of A. We show ℓ(A) ≥ n as follows. Without changing (r, s, |α_{ν}|, |β_{ν}|) we can choose A in the same form so that α_{1}(u) = 0 for u > 1. We now fix the index 1 and apply Lemma 3.7.3 to the remaining indices to choose A so α_{2}(u) = 0 for ν > 2. We continue in this fashion to construct such an A so that deg_{u}(A) = 0 for u > r + s. Since deg_{n}(A) = deg_{n̄}(A) > 0, we conclude therefore that r + s ≥ n. We estimate:ν = ord(P) = _{∑ μ≤r}{|α_{μ}| + |β_{μ}| − 2} +_{∑ r<μ≤r+s}{|α_{μ}| + |β_{u}|} ≥ 2r + 2s ≥ 2n = m since ord(g(α, β)) ≥ 2 and ord(h_{pq̄/αβ̄}) ≥ 2. This proves ν ≥ m ≥ 2. Consequently, if ν < m we conclude P = 0 which proves the first assertion of Theorem 3.6. We assume henceforth that we are in the limiting case ν = m. In this case, all the inequalities must have been equalities. This implies r + s = n and |α_{μ}| + |β_{μ}| = 4 for μ ≤ r and |α_{μ}| + |β_{μ}| = 2 for r < μ ≤ s + r.

For Kaehler Manifolds 209

This holds true for every monomial of P since the construction of A_{1} from A involved the use of Lemma 3.7.3 and does not change any of the orders involved. By Lemma 3.7.1 and 3.7.2 we assumed all the purely holomorphic and purely anti-holomorphic derivatives vanished at z_{0} and consequently: |α_{μ}| = |β_{μ}| = 2 for μ ≤ r and |α_{μ}| = |β_{μ}| = 1 for r < μ ≤ s + r. Consequently, P is a polynomial in the {g(i_{1}i_{2}; j̄1j̄2)h_{pq̄/ij̄}} variables; it only involves the mixed 2-jets involved. We wish to choose a monomial of P in normal form to begin counting the number of possible such P. We begin this process with: Lemma 3.7.4. Let P satisfy the hypothesis of Theorem 3.6.8(b). Then there exists a monomial A of P which has the form: A = g(11; j̄1j̄′1) … g(tt; j̄tj̄′t)h^{p_{1}q_{1}/t+1,j̄t+1} … hp^{s}q^{s}/m,^{}j̄n where t + s = n. Proof: We let ∗ denote indices which are otherwise unspecified and let A_{0} be a generic monomial. We have deg_{n}(A) > 0 for every monomial A of P so deg_{u}(A) > 0 for every index u as well. We apply Lemma 3.7.3 to choose A of the form: A = g(11; j̄1j̄′1)A_{0}. Suppose r > 1. If A = g(11; ∗)g(11 : ∗)A_{0}, then we could argue as before using Lemma 3.7.3 that we could choose a monomial A of P so deg_{k}(A) = 0 for k > 1 + ℓ(A_{0}) = r + s − 1 = n − 1 which would be false. Thus A = g(11; ∗)g(jk; ∗)A_{0} where not both j and k are 1. We may apply a coordinate permutation to choose A in the form A = g(11; ∗)g(2j; ∗)A_{0}. If j ≥ 2 we can apply Lemma 3.7.3 to choose A = g(11; ∗)g(22; ∗)A_{0}. Otherwise we suppose A = g(11; ∗)g(12; ∗)A_{0}. We let B = g(11; ∗)g(22; ∗)A_{0} and compute: B(2 → 1) = A + terms divisible by g(11; ∗)g(22; ∗) B(¯1 → ¯2) = terms divisible by g(11; ∗)g(22; ∗) so that we conclude in any event we can choose A = g(11; ∗)g(22; ∗)A_{0}. We continue this argument with the remaining indices to construct: A = g(11; ∗)g(22; ∗) … g(tt; ∗)h_{p_{1}q̄1/u_{1}v̄1} … h_{p_{s}q̄s/u_{s}v̄s}. We have deg_{u}(A) ≠ 0 for all u. Consequently, the indices {t + 1,…, t + s} must appear among the indices {u_{1},…, u_{s}}. Since these two sets have s elements, they must coincide. Thus by rearranging the indices we can assume u_{ν} = ν + t which completes the proof. We note deg_{u}(A) = 2 for u ≤ t and deg_{u}(A) = 1 for u > t. This lemma does not control the anti-holomorphic indices, we further normalize the choice of A in the following:

Lemma 3.7.5. Let P satisfy the hypothesis of Theorem 3.6.8(b). Then there exists a monomial A of P which has the form:

A = g(11; j̄_1j̄^′_1) … g(tt; j̄_tj̄^′_t)h_{p1q̄1/t+1,t+1} … h_{psq̄s/nn̄}

for 1 ≤ j_μ, j^′_μ ≤ t.

Proof: If A has this form, then deg_j(A) = 1 for j > t. Therefore deg_j̄ (A) = 1 for j > t which implies 1 ≤ j_μ, j^′_μ ≤ t automatically. We say that j̄ touches itself in A if A is divisible by g(∗; j̄ j̄) for some ∗. We say that j touches j̄ in A if A is divisible by h_{pq/jj̄} for some (p, q). Choose A of the form given in Lemma 3.7.4 so the number of indices j > t which touch j̄ in A is maximal. Among all such A, choose A so the number of j̄ ≤ t̄ which touch themselves in A is maximal. Suppose A does not satisfy the conditions of the Lemma. Thus A must be divisible by h_{pq̄/uv̄} for u ≠ v and u > t. We suppose first v > t. Since deg_v̄(A) = deg_v(A) = 1, v does not touch v̄ in A. We let A = h_{pq̄/uv̄}A_0 and B = h_{pq̄/uū}A_0 then deg_v̄(B) = 0. We compute:

B(ū → v̄) = A + A_1 and B(v → u) = A_2

where A_1 is defined by interchanging ū and v̄ and where A_2 is defined by replacing both v and v̄ by u and ū in A. Thus deg_v(A)_2 = 0 so A_2 is not a monomial of A. Thus A_1 = h_{pq̄/uū}A^′_0 must be a monomial of P. One more index (namely u) touches its holomorphic conjugate in A_1 than in A. This contradicts the maximality of A and consequently A = h_{pq̄/uv̄}A^′_0 for u > t, v ≤ t. (This shows h_{p0q0/u0v̄0} does not divide A for any u_0 ≠ v_0 and v̄_0 > t̄.) We have deg_ū(A) = 1 so the anti-holomorphic index ū must appear somewhere in A. It cannot appear in a h variable and consequently A has the form A = g(∗; ūw̄)h_{pq̄/uv̄}A^{′′}_0. We define B = g(∗; w̄w̄)h_{pq̄/uv̄}A^{′′}_0 and compute:

B(w̄ → ū) = A + terms divisible by g(∗; w̄w̄)

B(u → w) = g(∗; w̄w̄)h_{pq̄/wv̄}A^{′′}_0.

Since g(∗; w̄w̄)h_{pq̄/wv̄}A^{′′}_0 does not have the index u, it cannot be a monomial of P. Thus terms divisible by g(∗; w̄w̄) must appear in P. We construct these terms by interchanging a w̄ and ū index in P so the maximality of indices u_0 touching ū_0 for u_0 > t is unchanged. Since w̄ does not touch w̄ in A, we are adding one additional index of this form which again contradicts the maximality of A. This final contradiction completes the proof.

This constructs a monomial A of P which has the form

A = A_0A_1 for { deg_u(A_0) = deg_ū(A_0) = 0, u > t, deg_u(A_1) = deg_ū(A_1) = 0, u ≤ t.

For Kaehler Manifolds 211

A_{0} involves only the derivatives of the metric g and A_{1} involves only the derivatives of h. We use this splitting in exactly the same way we used a similar splitting in the proof of Theorem 2.6.1 to reduce the proof of Theorem 3.6.9 to the following assertions: Lemma 3.7.6. Let P satisfy the hypothesis of Theorem 3.6.9(b). (a) If P is a polynomial in the {h_{pq̄/uv̄}} variables, then P = ∗Q for Q a Chern m form of V. (b) If P is a polynomial in the {g(u_{1}u_{2}; v̄1v̄2)} variables, then P = ∗Q for Q a Chern m form of T_{c}M. Proof: (a). If A is a monomial of P, then deg_{u}(A) = deg_{ū}(A) = 1 for all u so we can express ^{A = hp_{1}q_{1}/1ū1 … hp_{n}q_{n}/nūn} where the {u_{ν}} are a permutation of the indices i through n. We let A = h_{∗/1ū1} h_{∗/2ū2} A_{′0} and B = h_{∗/1ū1} h_{∗/2ū1} A_{′0}. Then: B(ū1 → ū2) = A + A1 B(u_{2} → u_{1}) = A2 for A_{1} = h_{∗/1ū2} h_{∗/2ū1} A′0 for deg_{u2} (A_{2}) = 0. Therefore A_{2} is not a monomial of P. Thus A_{1} is a monomial of P and furthermore c(A, P) + c(A_{1}, P) = 0. This implies when we interchange ū1 and ū2 that we change the sign of the coefficient involved. This implies immediately we can express P is terms of expressions: ∗(Ω_{V p_{1}q̄1} ∧ ⋯ ∧ Ω_{V p_{n}q̄n}) = _{∑ u,v} ∗(h_{p_{1}q̄1/u_{1}v̄1} du^{1} ∧ dv̄1 ⋯ ∧ h_{p_{n}q̄n/u_{n}v̄n} du^{n} ∧ dv^{n}) = n!_{∑ ρ} sign(ρ)h_{p_{1}q_{1}/1ρ̄(1)} … h_{p_{n}q_{n}/nρ̄(n)} + ⋯ where ρ is a permutation. This implies ∗P can be expressed as an invariant polynomial in terms of curvature which implies it must be a Chern form as previously computed. The remainder of this section is devoted to the proof of (b). Lemma 3.7.7. Let P satisfy the hypothesis of Lemma 3.7.6(b). Then we can choose a monomial A of P which has the form: A = g(11; ū1ū1) … g(nn; ūnūn). This gives a normal form for a monomial. Before proving Lemma 3.7.7, we use this lemma to complete the proof of Lemma 3.7.6(b). By making a 212 3.7. Characteristic Forms coordinate permutation if necessary we can assume A has either the form g(11; ¯1¯1)A_{′0} or g(11; ¯2¯2)A_{′0}. In the latter case, we continue inductively to express A = g(11; ¯2¯2)g(22; ¯3¯3) … g(u − 1, u − 1; ūū)g(uu; ¯1¯1)A_{′0} until the cycle closes. If we permit u = 1 in this decomposition, we can also include the first case. Since the indices 1 through u appear exactly twice in A they do not appear in A_{′0}. Thus we can continue to play the same game to decompose A into cycles. Clearly A is determined by the length of the cycles involved (up to coordinate permutations); the number of such classifying monomials is π(n), the number of partitions of n. This shows that the dimension of the space of polynomials P satisfying the hypothesis of Lemma 3.7.6(b) is ≤ π(n). Since there are exactly π(n) Chern forms the dimension must be exactly π(n) and every such P must be a Chern m form as claimed. We give an indirect proof to complete the proof of Lemma 3.7.7. Choose A of the form given by Lemma 3.7.5 so the number of anti-holomorphic indices which touch themselves is maximal. If every anti-holomorphic index touches itself, then A has the form of Lemma 3.7.7 and we are done. We suppose the contrary. Since every index appears exactly twice, every anti- holomorphic index which does not touch itself touches another index which also does not touch itself. Every holomorphic index touches only itself. We may choose the notation so A = g(∗; ¯1¯2)A_{′0}. Suppose first ¯1 does not touch̄ 2 in A_{′0}. Then we can assume A has the form: A = g(∗; ¯1¯2)g(∗; ¯1¯3)g(∗; ¯2k̄)A′0 where possibly k = 3 in this expression. The index 1 touches itself in A. The generic case will be: A = g(11; ∗)g(∗; ¯1¯2)g(∗; ¯1¯3)g(∗; ¯2k̄)A_{′0}. The other cases in which perhaps A = g(11; ¯1¯2) … or g(11, ¯1¯3) … or g(11; ¯2k̄) … are handled similarly. The holomorphic and anti-holomorphic indices do not interact. In exactly which variable they appear does not matter. This can also be expressed as a lemma in tensor algebras. We suppose k ≠ 3; we will never let the 2 and 3 variables interact so the case in which k = 3 is exactly analogous. Thus A has the form: A = g(11; ∗)g(∗; ¯1¯2)g(∗; ¯1¯3)g(∗; ¯2k̄)g(∗; ¯3j̄)A′0 where possibly j = k. Set B = (11; ∗)g(∗; ¯2¯2)g(∗; ¯1¯3)g(∗; ¯2k̄)g(∗; ¯3j̄)A′0 then: B(¯2 → ¯1) = 2A + A′′ B(1 → 2) = 2A1

for  A′′ = g(11; ∗)g(∗; ¯2¯2)g(∗; ¯1¯3)g(∗; ¯1k̄)g(∗; ¯3j̄)A′0 A1 = g(12; ∗)g(∗; ¯2¯2)g(∗; ¯1¯3)g(∗; ¯2k̄)g(∗; ¯3j̄)A_{′0}. A′′ is also a monomial of the form given by Lemma 3.7.5. Since one more anti-holomorphic index touches itself in A′′ then does in A, the maximality of A shows A′′ is not a monomial of P. Consequently A_{1} is a monomial of P. Set B_{1} = g(12; ∗)g(∗; ¯2¯2)g(∗; ¯3¯3)g(∗; ¯2k̄)g(∗; ¯3j̄)A_{′0} then:^{B_{1}(¯3 → ¯1) = 2A_{1} + A2 B_{1}(1 → 3) = A′′′} for  A2 = g(12; ∗)g(∗; ¯2¯2)g(∗; ¯3¯3)g(∗; ¯2k̄)g(∗; ¯1j̄)A′0 A′′′ = g(32; ∗)g(∗; ¯2¯2)g(∗; ¯3¯3)g(∗; ¯2k̄)g(∗; ¯3j̄)A_{′0}. However, deg_{1}(A′′′) = 0. Since r(P) = 0, A′′′ cannot be a monomial of P so A_{2} is a monomial of P. Finally we set B_{2} = g(11; ∗)g(∗; ¯2¯2)g(∗; ¯3¯3)g(∗; ¯2k̄)g(∗; ¯1j̄)A′0 so  B_{2}(1 → 2) = A2 B_{2}(¯2 → ¯1) = A_{3} + 2A4 for  A3 = g(11; ∗)g(∗; ¯2¯2)g(∗; ¯3¯3)g(∗; ¯1k̄)g(∗; ¯1j̄)A′0 A4 = g(11; ∗)g(∗; ¯1¯2)g(∗; ¯3¯3)g(∗; ¯2k̄)g(∗; ¯1j̄)A_{′0}. This implies either A_{3} or A_{4} is monomial of P. Both these have every holo- morphic index touching itself. Furthermore, one more anti-holomorphic index (namely ¯3) touches itself. This contradicts the maximality of A. In this argument it was very important that ¯2 ≠ ¯3 as we let the index ¯1 interact with each of these indices separately. Thus the final case we must consider is the case in which A has the form: A = g(11; ∗)g(22; ∗)g(∗; ¯1¯2)g(∗; ¯1¯2)A_{0}. So far we have not had to take into account multiplicities or signs in com- puting A(1 → 2) etc; we have been content to conclude certain coefficients are non-zero. In studying this case, we must be more careful in our analysis as the signs involved are crucial. We clear the previous notation and define: A1 = g(12; ∗)g(22; ∗)g(∗; ¯1¯2)g(∗; ¯2¯2)A0 A2 = g(12; ∗)g(22; ∗)g(∗; ¯2¯2)g(∗; ¯1¯2)A0 A3 = g(11; ∗)g(22; ∗)g(∗; ¯1¯1)g(∗; ¯2¯2)A0 A4 = g(22; ∗)g(22; ∗)g(∗; ¯2¯2)g(∗; ¯2¯2)A_{0}.

We note that A_{3} is not a monomial of P by the maximality of A. A4 is not a monomial of P as deg_{1}(A_{4}) = 0 and r(P) = 0. We let B = g(11; ∗)g(22; ∗)g(∗; ¯1¯2)g(∗; ¯2¯2)A_{0}. Then B(¯2 → ¯1) = 2A + A_{3},  B(1 → 2) = 2A1 so that A_{1} must be a monomial of P since A is a monomial of P and A_{4} is not. We now pay more careful attention to the multiplicities and signs: A(¯1 → ¯2) = B + ⋯  and A_{1}(2 → 1) = B + ⋯ However ¯1 → ¯2 introduces a b̄ while 2 → 1 introduces a −b̄ so using the argument discussed earlier we conclude not only c(A_{1}, P) ≠ 0 but that c(A, P) − c(A_{1}, P) = 0 so c(A_{1}, P) = c(A, P). A_{2} behaves similarly so the analogous argument using B_{1} = g(11; ∗)g(22; ∗)g(∗; ¯2¯2)g(∗; ¯1¯2) shows c(A_{2}, P) = c(A, P). We now study B_{2} = g(12; ∗)g(22; ∗)g(∗, ¯2¯2)g(∗; ¯2¯2) and compute: B_{2}(1 → 2) = A_{4}  B_{2}(¯2 → ¯1) = 2A_{1} + 2A2 A_{4} is not a monomial of P as noted above. We again pay careful attention to the signs: A_{1}(¯1 → ¯2) = B_{2}  and A_{2}(¯1 → ¯2) = B2 This implies c(A_{1}, P) + c(A_{2}, P) = 0. Since c(A_{1}, P) = c(A_{2}, P) = c(A, P) this implies 2c(A, P) = 0 so A was not a monomial of P. This final con- tradiction completes the proof. This proof was long and technical. However, it is not a theorem based on unitary invariance alone as the restriction axiom plays an important role in the development. We know of no proof of Theorem 3.6.9 which is based only on H. Weyl’s theorem; in the real case, the corresponding characterization of the Pontrjagin classes was based only on orthogonal invariance and we gave a proof based on H. Weyl’s theorem in that case.

3.8. The Chern Isomorphism and Bott Periodicity. In section 3.9 we shall discuss the Atiyah-Singer index theorem in general using the results of section 3.1. The index theorem gives a topological formula for the index of an arbitrary elliptic operator. Before begining the proof of that theorem, we must first review briefly the Bott periodicity theorem from the point of Clifford modules. We continue our consideration of the bundles over S^{n} constructed in the second chapter. Let GL(k, C) = { A : A is a k × k complex matrix with det(A) ≠ 0 } GL′(k, C) = { A ∈ GL(k, C) : det(A − it) ≠ 0 for all t ∈ R } U(k) = { A ∈ GL(k, C) : A · A^{∗} = I } S(k) = { A ∈ GL(k, C) : A^{2} = I and A = A^{∗} } S_{0}(k) = { A ∈ S(k) : Tr(A) = 0 }. We note that S_{0}(k) is empty if k is odd. U(k) is compact and is a deformation retract of GL(k, C); S(k) is compact and is a deformation retract of GL′(k, C). S_{0}(k) is one of the components of S(k). Let X be a finite simplicial complex. The suspension ΣX is defined by identifying X × {_{π 2} } to a single point N and X × {−_{π 2} } to single point S in the product X × [−_{π 2} , _{π 2} ]. Let D_{±}(X) denote the northern and southern “hemispheres” in the suspension; the intersection D_{+}(X) ∩ D_{−}(X) = X. [Reprinter's note: A figure depicting the suspension of X belongs here.] We note both D_{+}(X) and D_{−}(X) are contractible. Σ(X) is a finite sim- plicial complex. If S^{n} is the unit sphere in R^{n+1}, then Σ(S^{n}) = S^{n+1}. Finally, if W is a vector bundle over some base space Y, then we choose a fiber metric on W and let S(W) be the unit sphere bundle. Σ(W) is the fiberwise suspension of S(W) over Y. This can be identified with S(W ⊕1). It is beyond the scope of this book to develop in detail the theory of vector bundles so we shall simply state relevant facts as needed. We let Vect_{k}(X) denote the set of isomorphism classes of complex vector bundles 216 3.8. The Chern Isomorphism over X of fiber dimension k. We let Vect(X) = ⋃_{k} Vect_{k}(X) be the set of isomorphism classes of complex vector bundles over X of all fiber dimen- sions. We assume X is connected so that the dimension of a vector bundle over X is constant. There is a natural inclusion map Vect_{k}(X) → Vect_{k+1}(X) defined by sending V ↦ V ⊕ 1 where 1 denotes the trivial line bundle over X. Lemma 3.8.1. If 2k ≥ dim X then the map Vect_{k}(X) → Vect_{k+1}(X) is bijective. We let F (Vect(X)) be the free abelian group on these generators. We shall let (V) denote the element of K(X) defined by V ∈ Vect(X). K(X) is the quotient modulo the relation (V ⊕W) = (V)+(W) for V, W ∈ Vect(X). K(X) is an abelian group. The natural map dim: Vect(X) → Z^{+} extends to dim: K(X) → Z; K(X) is the kernel of this map. We shall say that an element of K(X) is a virtual bundle; K(X) is the subgroup of virtual bundles of virtual dimension zero. It is possible to define a kind of inverse in Vect(X): Lemma 3.8.2. Given V ∈ Vect(X), there exists W ∈ Vect(X) so V ⊕W = 1^{j} is isomorphic to a trivial bundle of dimension j = dim V + dim W. We combine these two lemmas to give a group structure to Vect_{k}(X) for 2k ≥ dim X. Given V, W ∈ Vect_{k}(X), then V ⊕W ∈ Vect_{2k}(X). By Lemma 3.8.1, the map Vect_{k}(X) → Vect_{2k}(X) defined by sending U ↦ U ⊕ 1^{k} is bijective. Thus there is a unique element we shall denote by V + W ∈ Vect_{k}(X) so that (V + W) ⊕ 1^{k} = V ⊕ W. It is immediate that this is an associative and commutative operation and that the trivial bundle 1k functions as the unit. We use Lemma 3.8.2 to construct an inverse. Given V ∈ Vect_{k}(X) there exists j and W ∈ Vect_{j}(X) so V ⊕ W = 1^{j+k}. We assume without loss of generality that j ≥ k. By Lemma 3.8.1, we choose W ∈ Vect_{k}(X) so that V ⊕W ⊕1^{j−k} = V ⊕W = 1^{j+k}. Then the bijectivity implies V ⊕ W = 1^{2k} so W is the inverse of V. This shows Vect_{k}(X) is a group under this operation. There is a natural map Vect_{k}(X) → K(X) defined by sending V ↦ (V) − (1^{k}). It is immediate that: (V + W) − 1k = (V + W) + (1^{k}) − (1^{2k}) = ((V + W) ⊕ 1^{k}) − (1^{2k}) = (V ⊕ W) − (1^{2k}) = (V) + (W) − (1^{2k}) = (V) − (1^{k}) + (W) − (1^{k}) so the map is a group homomorphism. K(X) is generated by elements of the form (V) − (W) for V, W ∈ Vect_{j} X for some j. If we choose W so W ⊕ W = 1^{v} then (V) − (W) = (V ⊕ W) − (1^{v}) so K(X) is generated by elements of the form (V) − (1^{v}) for V ∈ Vect_{v}(X). Again, by adding

trivial factors, we may assume v ≥ k so by Lemma 3.8.1 V = V^{1} ⊕ 1j−k and (V) − (1^{j}) = (V_{1}) − (1^{k}) for V_{1} ∈ Vect_{k}(X). This implies the map Vect_{k}(X) → K(X) is subjective. Finally, we note that that in fact K(X) is generated by Vect_{k}(X) subject to the relation (V) + (W) = (V ⊕ W) = (V + W) + (1^{k}) so that this map is injective and we may identify K(X) = Vect_{k}(X) and K(X) = Z ⊕ K(X) = Z ⊕ Vect_{k}(X) for any k such that 2k ≥ dim X. Tensor product defines a ring structure on K(X). We define (V)⊗(W) = (V ⊗ W) for V, W ∈ Vect(X). Since (V_{1} ⊕ V_{2}) ⊗ (W) = (V_{1} ⊗ W ⊕ V_{2} ⊗ W), this extends from F (Vect(X)) to define a ring structure on K(X) in which the trivial line bundle 1 functions as a multiplicative identity. K(X) is an ideal of K(X). K(X) is a Z-module. It is convenient to change the coefficient group and define:  K(X; C) = K(X) ⊗_{Z} C to permit complex coefficients. K(X; C) is the free C-vector space gen- erated by Vect(X) subject to the relations V ⊕ W = V + W. By using complex coefficients, we eliminate torsion which makes calculations much simpler. The Chern character is a morphism: ch: Vect(X) → H^{even} (X; C) = _{⨁ q} H^{2q}(X; C). We define ch using characteristic classes in the second section if X is a smooth manifold; it is possible to extend this definition using topological methods to more general topological settings. The identities: ch(V ⊕ W) = ch(V) + ch(W)  and  ch(V ⊗ W) = ch(V) ch(W) imply that we can extend: ch: K(X) → H^{even} (X; C) to be a ring homomorphism. We tensor this Z-linear map with C to get ch: K(X; C) → H^{even} (X; C). Lemma 3.8.3 (Chern isomorphism). ch: K(X; C) → H^{even} (X; C) is a ring isomorphism. If Heven (X; C) = ⨁_{q>o} H^{2q}(X; C) is the reduced even dimensional co- homology, then ch: K(X; C) → Heven (X; C) is a ring isomorphism. For this reason, K(X) is often refered to as reduced K-theory. We emphasize that in this isomorphism we are ignoring torsion and that torsion is crucial

218 3.8. The Chern Isomorphism to understanding K-theory in general. Fortunately, the index is Z-valued and we can ignore torsion in K-theory for an understanding of the index theorem. We now return to studying the relation between K(X) and K(ΣX). We shall let [X, Y] denote the set of homotopy classes of maps from X to Y. We shall always assume that X and Y are equipped with base points and that all maps are basepoint preserving. We fix 2k ≥ dim X and let f: X → S_{0}(2k). Since f(x) is self-adjoint, and f(x)^{2} = I, the eigenvalues of f(x) are ±1. Since Tr f(x) = 0, each eigenvalue appears with multiplicity k. We let Π^{±}(f) be the bundles over X which are sub-bundles of X × 1k so that the fiber of Π_{±}(f) at x is just the ±1 eigenspace of f(x). If we define π_{±}(f)(x) = _{1 2} (1 ± f(x)) then these are projections of constant rank k with range Π_{±}(f). If f and f_{1} are homotopic maps, then they determine isomorphic vector bundles. Thus the assignment f ↦ Π_{+}(f) ∈ Vect_{k}(X) defines a map Π_{+}: [X, S_{0}(2k)] → Vect_{k}(X). Lemma 3.8.4. The natural map [X, S_{0}(2k)] → Vect_{k}(X) is bijective for 2k ≥ dim X. Proof: Given V ∈ Vect_{k}(X) we choose W ∈ Vect_{k}(X) so V ⊕ W = 1^{2k}. We choose fiber metrics on V and on W and make this sum orthogonal. By applying the Gram-Schmidt process to the given global frame, we can assume that there is an orthonormal global frame and consequently that V ⊕ W = 1^{2k} is an orthogonal direct sum. We let π_{+}(x) be orthogonal projection on the fiber V_{x} of 1^{2k} and f(x) = 2π_{+}(x) − I. Then it is immediate that f: X → S_{0}(2k) and π_{+}(f) = V. This proves the map is subjective. The injectivity comes from the same sorts of considerations as were used to prove Lemma 3.8.1 and is therefore omitted. If f: X → GL′(k, C), we can extend the definition to let Π_{±}(f)(x) be the span of the generalized eigenvectors of f corresponding to eigenval- ues with positive/negative real part. Since there are no purely imaginary eigenvalues, Π_{±}(f) has constant rank and gives a vector bundle over X. In a similar fashion, we can classify Vect_{k} ΣX = [X, U(k)]. Since U(k) is a deformation retract of GL(k, C), we identify [X, U(k)] = [X, GL(k, C)]. If g: X → GL(k, C), we use g as a clutching function to define a bundle over ΣX. Over D_{±}(X), we take the bundles D_{±}(X) × C^{k}. D_{+}(X) ∩ D_{−}(X) = X. On the overlap, we identify (x, z)_{+} = (x, z′)_{−} if z · g(x) = z′. If we let s^{+} and s^{−} be the usual frames for C^{k} over D_{±} then ∑ z_{+ i}g_{ij}s_{+ j} = ∑ z_{− i}s^{−} i so that we identify the frames using the identity s^{−} = gs^{+}. We denote this bundle by V_{g}. Homotopic maps define isomorphic bundles so we have a map [X, GL(k, C)] → Vect_{k} ΣX. Conversely, given a vector bundle V over ΣX we can always choose local trivializations for V over

D_{±}(X) since these spaces are contractible. The transition function s^{−} = gs^{+} gives a map g: X → GL(k, C). It is convenient to assume X has a base point x_{0} and to choose s^{−} = s^{+} at x_{0}. Thus g(x_{0}) = I. This shows the map [X, GL(k, C)] → Vect_{k} ΣX is surjective. If we had chosen different trivializations s̃+ = h^{+}s^{+} and s̃− = h^{−}s^{−} then we would have obtained a new clutching function g̃ = h^{−}g(h^{+})^{−1}. Since h^{±} are defined on contractible sets, they are null homotopic so g̃ is homotopic to g. This proves: Lemma 3.8.5. The map [X, GL(k, C)] = [X, U(k)] → Vect_{k}(ΣX), given by associating to a map g the bundle defined by the clutching function g, is bijective. It is always somewhat confusing to try to work directly with this defini- tion. It is always a temptation to confuse the roles of g and of its inverse as well as the transposes involved. There is another definition which avoids this difficulty and which will be very useful in computing specific examples. If g: X → GL(k, C), we shall let g(x)z denote matrix multiplication. We shall regard C^{k} as consisting of column vectors and let g act as a matrix from the left. This is, of course, the opposite convention from that used previously. Let θ ∈ [−_{π 2} , _{π 2} ] be the suspension parameter. We define: Σ: [X, GL(k, C)] → [ΣX, GL′(2k, C)] Σ: [X, GL′(k, C)] → [ΣX, GL(k, C)] by  Σg(x, θ) = ( (sin θ)Ik (cos θ)g^{∗}(x) (cos θ)g(x) −(sin θ)Ik) Σf(x, θ) = (cos θ)f(x) − i(sin θ)I_{k}. We check that Σ has the desired ranges as follows. If g: X → GL(k, C), then it is immediate that Σg is self-adjoint. We compute: (Σg)^{2} = ( (sin^{2} θ)I_{k} + (cos^{2} θ)g^{∗}g  0 0  (sin^{2} θ)I_{k} + (cos^{2} θ)gg∗ ) . This is non-singular since g is invertible. Therefore Σg is invertible. If θ = π/2, then Σg = ( {I 0 \atop 0 −I} ) is independent of x. If g is unitary, Σg ∈ S_{0}(2k). If f ∈ GL′(k, C), then f has no purely imaginary eigenvalues so Σf is non-singular.

Lemma 3.8.6. Let g: X → GL(k, C) and construct the bundle Π_+(Σg) over ΣX. Then this bundle is defined by the clutching function g.

Proof: We may replace g by a homotopic map without changing the isomorphism class of the bundle Π_+(Σg). Consequently, we may assume without loss of generality that g: X → U(k) so g^∗g = gg^∗ = I_k. Consequently (Σg)^2 = I_{2k} and π_+(Σg) = 1/2(I_{2k} + Σg). If z ∈ C^k, then:

π_+(Σg)(x, θ)( z 0) = 1 / (2() z + (sin θ)z (cos θ)g(x)z) .

This projection from C^k to Π_+(Σg) is non-singular away from the south pole S and can be used to give a trivialization of Π_+(Σg) on ΣX − S.

From this description, it is clear that Π_+(Σg) is spanned by vectors of

the form 1 2( (1 + (sin θ))z (cos θ)g(x)z )

away from the south pole. At the south pole,

Π_+(Σg) consists of all vectors of the form ( 0 w)

. Consequently, projection

on the second factor π_+(S): ( a b)

→ ( 0 b)

is non-singular away from the

north pole N and gives a trivialization of Π_+(Σg) on ΣX−N. We restrict to the equator X and compute the composite of these two maps to determine the clutching function: (

z

0) ↦ 1 / (2()

z

g(x)z) ↦ 1 / (2()

0

g(x)z) .

The function g(x)/2 is homotopic to g which completes the proof.

This is a very concrete description of the bundle defined by the clutching function g. In the examples we shall be considering, it will come equipped with a natural connection which will make computing characteristic classes much easier.

We now compute the double suspension. Fix f: X → S_0(2k) and g: X → GL(k, C).

Σ^2f(x, θ, ϕ)

= (

sin ϕ cos ϕ{(cos θ)f^∗ + i sin θ}

cos ϕ{(cos θ)f(x) − i sin θ}  − sin ϕ  )

Σ^2g(x, θ, ϕ)

= ( cos ϕ sin θ − i sin ϕ cos ϕ cos θg^∗(x)

cos ϕ cos θg(x) − cos ϕ cos θ − i sin ϕ )

We let U(∞) be the direct limit of the inclusions U(k) → U(k + 1) ⋯ and S_{0}(∞) be the direct limit of the inclusions S_{0}(2k) → S_{0}(2k + 2) → · · ·. Then we have identified: K(X) = [X, S_{0}(∞)]  and K(ΣX) = [ΣX, S_{0}(∞)] = [S, U(∞)] We can now state: Theorem 3.8.7 (Bott periodicity). The map Σ^{2}: K(X) = [X, S_{0}(∞)] → K(Σ^{2}X) = [Σ^{2}X, S_{0}(∞)] induces a ring isomorphism. Similarly, the map Σ^{2}: K(ΣX) = [X, U(∞)] → K(Σ^{2}X) = [Σ^{2}X, U(∞)] is a ring isomorphism. We note that [X, U(∞)] inherits a natural additive structure from the group structure on U(∞) by letting g ⊕ g′ be the direct sum of these two maps. This group structure is compatible with the additive structure on K(ΣX) since the clutching function of the direct sum is the direct sum of the clutching functions. Similarly, we can put a ring structure on [X, U(∞)] using tensor products to be compatible with the ring structure on K(ΣX). The Chern character identifies K(X; C) with Heven (X; C). We may identify Σ^{2}X with a certain quotient of X × S^{2}. Bott periodicity in this context becomes the assertion K(X × S^{2}) = K(X) ⊗ K(S^{2}) which is the Kunneth formula in cohomology. We now consider the case of a sphere X = S^{n}. The unitary group U(2) decomposes as U(2) = U(1)×SU(2) = S^{1} ×S^{3} topologically so π_{1}(U(2)) = Z, π_{2}(U(2)) = 0 and π_{3}(U(2)) = Z. This implies K(S^{1}) = K(S^{3}) = 0 and K(S^{2}) = K(S^{4}) = Z. Using Bott periodicity, we know more generally that: Lemma 3.8.8 (Bott periodicity). If n is odd, then K(S^{n}) ≃ π_{n−1}(U(∞)) = 0. If n is even, then  K(S^{n}) ≃ π_{n−1}(U(∞)) = Z. It is useful to construct explicit generators of these groups. Let x = (x_{0},…, x_{n}) ∈ S^{n}, let y = (x, x_{n+1}) ∈ S^{n+1}, and let z = (y, x_{n+2}) ∈ S^{n+2}. If f: S^{n} → GL′(k, C) and g: X → GL(k, C) we extend these to

be homogeneous of degree 1 with values in the k × k matrices. Then we compute:  Σf(y) = f(x) − ixn+1 Σ^{2}f(z) = ( xn+2  f^{∗}(x) + ixn+1_{f(x) − ixn+1  −xn+2} ) Σg(y) = ( xn+1 g^{∗}(x)_{g(x) −xn+1} ) Σ^{2}g(z) = ( {x_{n+1} − ixn+2  g^{∗}(x) \atop g(x)  −xn+1 − ixn+2} ) . If we suppose that f is self-adjoint, then we can express: Σ^{2}f(z) = x_{n+2} ( {1 0 \atop 0 −1} ) ⊗ I_{k} + x_{n+1} ( {0 i \atop −i 0} ) ⊗ I_{k} + ( {0 1 \atop 1 0} ) ⊗ f(x). We can now construct generators for π_{n−1} U(∞) and K(S^{n}) using Clif- ford algebras. Let g(x_{0}, x_{1}) = x_{0} − ix_{1} generate π_{1}(S^{1}) = Z then Σg(x_{0}, x_{1}, x_{2}) = x_{0}e_{0} + x_{1}e_{1} + x_{2}e2 for  e_{0} = ( {0 1 \atop 1 0} ) ,  e_{1} = ( {0 i \atop −i 0} ) ,  e_{2} = ( {1 0 \atop 0 −1} ) . The \{e_{j}\} satisfy the relations e_{j}e_{k}+e_{k}e_{j} = 2δ_{jk} and e_{0}e_{1}e_{2} = −iI_{2}. Π_{+}(Σg) is a line bundle over S^{2} which generates K(S^{2}). We compute: Σ^{2}g(x_{0}, x_{1}, x_{2}, x_{3}) = x_{0}e_{0} + x_{1}e_{1} + x_{2}e_{2} − ix_{3}I_{2}. If we introduce z_{1} = x_{0} + ix_{1} and z_{2} = x_{2} + ix_{3} then Σ^{2}g(z_{1}, z_{2}) = ( {z̄2 z1 \atop z̄1 −z2} ) . Consequently Σ^{2}g generates π_{3}(U(∞)). We suspend once to construct: Σ^{3}g(x_{0}, x_{1}, x_{2}, x_{3}, x_{4}) = x_{0}e_{0} ⊗ e_{0} + x_{1}e_{0} ⊗ e_{1} + x_{2}e_{0} ⊗ e_{2} + x_{3}e_{1} ⊗ I + x_{4}e_{2} ⊗ I. The bundle Π_{+}(Σ^{3}g) is a 2-plane bundle over S^{4} which generates K(S^{4}). We express Σ^{3}g = x_{0}e_{4 0} + ⋯ + x_{4}e_{4 4} then these matrices satisfy the com- mutation relations e_{4 j}e_{4 j} + e_{4 k}e_{4 j} = 2δ_{jk} and e_{4 0}e_{4 1}e_{4 2}e_{4 3}e_{4 4} = −I.

We proceed inductively to define matrices e_{2k j} for 0 ≤ j ≤ 2k so that e_{i}e_{j} + e_{j}e_{i} = 2δ_{ij} and e_{2k 0} … e_{2k 2k} = (−i)^{k}I. These are matrics of shape 2^{k} × 2^{k} such that Σ^{2k−1}(g)(x) = ∑ x_{j}e_{2k j}. In Lemma 2.1.5 we computed that: ∫_{S 2k} ch_{k}(Π_{+}Σ^{2k−1}g) = i^{k}2^{−k} Tr(e_{2k 0} … e_{2k 2k}) = 1. These bundles generate K(S^{2k}) and Σ^{2k}(g) generates π_{2k+1} (U(∞)). We summarize these calculations as follows: Lemma 3.8.9. Let {e_{0},…, e_{2k}} be a collection of self-adjoint matrices of shape 2^{k} ×2^{k} such that e_{i}e_{j} +e_{j}e_{i} = 2δ_{ij} and such that e_{0} … e_{2k} = (−i)^{k}I. We define e(x) = x_{0}e_{0}+ ⋯+x_{2k}e_{2k} for x ∈ S^{2k}. Let Π_{+}(e) be the bundle of +1 eigenvectors of e over S^{2k}. Then Π_{+}(e) generates K(S^{2k}) ≃ Z. Σe(y) = e(x) − ix_{2k+1} generates π_{2k+1} (U(∞)) ≃ Z. ∫_{S 2k} ch_{k}(V): K(S^{2k}) → Z is an isomorphism and ∫_{S 2k} ch_{k}(Π_{+}e) = 1. Proof: We note that ∫_{S 2k} ch_{k}(V) is the index of the spin complex with coefficients in V since all the Pontrjagin forms of T (S^{2k}) vanish except for p_{0}. Thus this integral, called the topological charge, is always an integer. We have checked that the integral is 1 on a generator and hence the map is surjective. Since K(S^{2k}) = Z, it must be bijective. It was extremely convenient to have the bundle with clutching function Σ^{2k−2}g so concretely given so that we could apply Lemma 2.1.5 to compute the topological charge. This will also be important in the next chapter.

3.9. The Atiyah-Singer Index Theorem. In this section, we shall discuss the Atiyah-Singer theorem for a general elliptic complex by interpreting the index as a map in K-theory. Let M be smooth, compact, and without boundary. For the moment we make no assumptions regarding the parity of the dimension m. We do not as- sume M is orientable. Let P: C^{∞}(V_{1}) → C^{∞}(V_{2}) be an elliptic complex with leading symbol p(x, ξ): S(T^{∗}M) → HOM(V_{1}, V_{2}). We let Σ(T^{∗}M) be the fiberwise suspension of the unit sphere bundle S(T^{∗}M). We identify Σ(T^{∗}M) = S(T^{∗}M ⊕ 1). We generalize the construction of section 3.8 to define Σp: Σ(T^{∗}M) → END(V_{1} ⊕ V_{2}) by: Σp(x, ξ, θ) = ( (sin θ)IV1 (cos θ)p^{∗}(x, ξ) (cos θ)p(x, ξ) − (sin θ)IV2 ) . This is a self-adjoint invertible endomorphism. We let Π_{±}(Σp) be the sub- bundle of V_{1} ⊕ V_{2} over Σ(T^{∗}M) corresponding to the span of the eigenvec- tors of Σp with positive/negative eigenvalues. If we have given connections on V_{1} and V_{2}, we can project these connections to define natural connec- tions on Π_{±}(Σp). The clutching function of Π_{+}(Σp) is p in a sense we explain as follows: We form the disk bundles D_{±}(M) over M corresponding to the north- ern and southern hemispheres of the fiber spheres of Σ(T^{∗}M). Lemma 3.8.6 generalizes immediately to let us identify Π_{+}(Σp) with the bundle V_{+ 1} ∪ V_{− 2} over the disjoint union D_{+}(M) ∪ D_{−}(M) attached using the clutching function p over their common boundary S(T^{∗}M). If dim V_{1} = k, then Π_{+}(Σp) ∈ Vect_{k}(Σ(T^{∗}M)). Conversely, we suppose given a bundle V ∈ Vect_{k}(Σ(T^{∗}M)). Let N: M → Σ(T^{∗}M) and S: M → Σ(T^{∗}M) be the natural sections mapping M to the northern and southern poles of the fiber spheres; N(x) = (x, 0, 1) and S(x) = (x, 0, −1) in S(T^{∗}M ⊕ 1). N and S are the centers of the disk bundles D_{±}(M). We let N^{∗}(V) = V1 and S^{∗}(V) = V_{2} be the induced vector bundles over M. D_{±}(M) deforma- tion retracts to M × {N} and M × {S}. Thus V restricted to D_{±}(M) is cannonically isomorphic to the pull back of V_{1} and V_{2}. On the in- tersection S(T^{∗}M) = D_{+}(M) ∩ D_{−}(M) we have a clutching or glueing function relating the two decompositions of V. This gives rise to a map p: S(T^{∗}M) → HOM(V_{1}, V_{2}) which is non-singular. The same argument as that given in the proof of Lemma 3.8.5 shows that V is completely deter- mined by the isomorphism class of V_{1} and of V_{2} together with the homotopy class of the map p: S(T^{∗}M) → HOM(V_{1}, V_{2}). Given an order ν we can recover the leading symbol p by extending p from S(T^{∗}M) to T^{∗}M to be homogeneous of order ν. We use this to define an elliptic operator P_{ν}: C^{∞}(V_{1}) → C^{∞}(V_{2}) with leading symbol p_{ν}. If Q_{ν} is another operator with the same leading symbol, then we define

The Atiyah-Singer Theorem 225

P_{ν}(t) = tP_{ν} + (1 − t)Q_{ν}. This is a 1-parameter family of such operators with the same leading symbol. Consequently by Lemma 1.4.4 index(P_{ν}) = index(Q_{ν}). Similarly, if we replace p by a homotopic symbol, then the index is unchanged. Finally, suppose we give two orders of homgeneity ν_{1} > ν_{2}. We can choose a self-adjoint pseudo-differential operator R on C^{∞}(V_{2}) with leading symbol |ξ|^{ν_{1}−ν2} I_{V2}. Then we can let P_{ν1} = RP_{ν2} so index(P_{ν1} ) = index(R)+index(P_{ν2} ). Since R is self-adjoint, its index is zero so index(P_{ν1} ) = index(P_{ν2} ). This shows that the index only depends on the homotopy class of the clutching map over S(T^{∗}M) and is independent of the order of homogeneity and the extension and the particular operator chosen. Consequently, we can regard index: Vect(Σ(T^{∗}M)) → Z so that index(Π_{+}(Σp)) = index(P) if P is an elliptic operator. (We can always “roll up” an elliptic complex to give a 2-term elliptic complex in computing the index so it suffices to consider this case). It is clear from our definition that Σ(p ⊕ q) = Σ(p) ⊕ Σ(q) and therefore Π_{+}(Σ(p ⊕ q)) = Π_{+}(Σ(p)) ⊕ Π_{+}(Σ(q)). Since index(P ⊕ Q) = index(P) + index(Q) we conclude: index(V ⊕ W) = index(V) + index(W) for V, W ∈ Vect(Σ(T^{∗}M)). This permits us to extend index: K(Σ(T^{∗}M)) → Z to be Z-linear. We tensor with the complex numbers to extend index: K(Σ(T^{∗}M); C) → C, so that: Lemma 3.9.1. There is a natural map index: K(Σ(T^{∗}M); C) → C which is linear so that index(P) = index(Π_{+}(Σp)) if P: C^{∞}V_{1} → C^{∞}V_{2} is an elliptic complex over M with symbol p. There is a natural projection map π: Σ(T^{∗}M) → M. This gives a natural map π^{∗}: K(M; C) → K(Σ(T^{∗}M); C). If N denotes the north pole section, then πN = 1_{M} so N^{∗}π^{∗} = 1 and consequently π^{∗} is injective. This permits us to regard K(M; C) as a subspace of K(Σ(T^{∗}M); C). If V = π^{∗}V_{1}, then the clutching function defining V is just the identity map. Consequently, the corresponding elliptic operator P can be taken to be a self-adjoint operator on C^{∞}(V) which has index zero. This proves: Lemma 3.9.2. If V ∈ K(Σ(T^{∗}M); C) can be written as π^{∗}V_{1} for V_{1} ∈ K(M; C) then index(V) = 0. Thus index: K(Σ(T^{∗}M); C)/K(M; C) → C. These two lemmas show that all the information contained in an elliptic complex from the point of view of computing its index is contained in the corresponding description in K-theory. The Chern character gives an iso- morphism of K(X; C) to the even dimensional cohomology. We will exploit this isomorphism to give a formula for the index in terms of cohomology. In addition to the additive structure on K(X; C), there is also a ring structure. This ring structure also has its analogue with respect to elliptic

operators as we have discussed previously. The multiplicative nature of the four classical elliptic complexes played a fundamental role in determining the normalizing constants in the formula for their index. We give Σ(T^{∗}M) the simplectic orientation. If x = (x_{1},…, x_{m}) is a system of local coordinates on M, let ξ = (ξ_{1},…, ξ_{m}) be the fiber coordi- nates on T^{∗}M. Let u be the additional fiber coordinate on T^{∗}M ⊕ 1. We orient T^{∗}M ⊕ 1 using the form: ω_{2m+1} = dx_{1} ∧ dξ_{1} ∧ ⋯ ∧ dx_{m} ∧ dξ_{m} ∧ du. Let N be the outward normal on S(T^{∗}M⊕1) and define ω_{2m} on S(T^{∗}M⊕1) by:  ω_{2m+1} = N ∧ ω_{2m} = ω_{2m} ∧ N. This gives the orientation of Stokes theorem. Lemma 3.9.3. (a) Let P: C^{∞}(V_{1}) → C^{∞}(V_{2}) be an elliptic complex over M_{1} and let Q: C^{∞}(W_{1}) → C^{∞}(W_{2}) be an elliptic complex over M_{2}. We assume P and Q are partial differential operators of the same order and we let M = M_{1} × M_{2} and define over M R = (P ⊗ 1 + 1 ⊗ Q) ⊕ (P^{∗} ⊗ 1 − 1 ⊗ Q^{∗}): C^{∞}(V_{1} ⊗ W_{1} ⊕ V_{2} ⊗ W_{2}) → C^{∞}(V_{2} ⊗ W_{1} ⊕ V_{1} ⊗ W_{2}). Then R is elliptic and index(R) = index(P) · index(Q). (b) The four classical elliptic complexes discussed earlier can be decom- posed in this fashion over product manifolds. (c) Let p and q be arbitrary elliptic symbols over M_{1} and M_{2} and define r = ( {p ⊗ 1 −1 ⊗ q∗ \atop 1 ⊗ q p∗ ⊗ 1} )  over M = M_{1} × M_{2}. Let θ_{i} ∈ H^{even} (M_{i}; C) for i = 1, 2, then: ∫_{Σ(T^{∗}M)} θ_{1} ∧ θ_{2} ∧ ch(Π_{+}(Σr)) = ∫_{Σ(T^{∗}M_{1})} θ_{1} ∧ ch(Π_{+}(Σp)) · ∫_{Σ(T^{∗}M_{2})} θ_{2} ∧ ch(Π_{+}(Σq)). (d) Let q be an elliptic symbol over M_{1} and let p be a self-adjoint elliptic symbol over M_{2}. Over M = M_{1} ×M_{2} define the self-adjoint elliptic symbol r by:  r = ( {1 ⊗ p q∗ ⊗ 1 \atop q ⊗ 1 −1 ⊗ p} ) .

Let θ_{1} ∈ Λ^{e}(M_{1}) and θ_{2} ∈ Λ^{o}(M_{2}) be closed differential forms. Give S(T^{∗}M), Σ(T^{∗}M_{1}) and S(T^{∗}M_{2}) the orientations induced from the sim- plectic orientations. Then: ∫_{S(T^{∗}M)} θ_{1} ∧ θ_{2} ∧ ch(Π_{+}r) = ∫_{Σ(T^{∗}M_{1})} θ_{1} ∧ ch(Π_{+}(Σq)) · ∫_{S(T^{∗}M_{2})} θ_{1} ∧ ch(Π_{+}(p)). Remark: We will use (d) to discuss the eta invariant in Chapter 4; we include this integral at this point since the proof is similar to that of (c). Proof: We let (p, q, r) be the symbols of the operators involved. Then: r = ( {p ⊗ 1 −1 ⊗ q∗ \atop 1 ⊗ q p∗ ⊗ 1} )  r^{∗} = ( {p∗ ⊗ 1 1 ⊗ q∗ \atop −1 ⊗ q p ⊗ 1} ) so that:  r^{∗}r = ( {p^{∗}p ⊗ 1 + 1 ⊗ q^{∗}q  0 \atop 0  pp^{∗} ⊗ 1 + 1 ⊗ qq∗} ) rr∗ = ( {pp^{∗} ⊗ 1 + 1 ⊗ q^{∗}q  0 \atop 0  p^{∗}p ⊗ 1 + 1 ⊗ qq∗} ) . r^{∗}r and rr^{∗} are positive self-adjoint matrices if (ξ^{1}, ξ^{2}) ≠ (0, 0). This verifies the ellipticity. We note that if (P, Q) are pseudo-differential, R will still be formally elliptic, but the symbol will not in general be smooth at ξ^{1} = 0 or ξ^{2} = 0 and hence R will not be a pseudo-differential operator in that case. We compute: R^{∗}R = ( {P^{∗}P ⊗ 1 + 1 ⊗ Q^{∗}Q  0 \atop 0  P P^{∗} ⊗ 1 + 1 ⊗ QQ∗} ) RR∗ = ( {P P^{∗} ⊗ 1 + 1 ⊗ Q^{∗}Q  0 \atop 0  P^{∗}P ⊗ 1 + 1 ⊗ QQ∗} ) N(R^{∗}R) = N(P^{∗}P) ⊗ N(Q^{∗}Q) ⊕ N(P P^{∗}) ⊗ N(QQ^{∗}) N(RR^{∗}) = N(P P^{∗}) ⊗ N(Q^{∗}Q) ⊕ N(P^{∗}P) ⊗ N(QQ^{∗}) index(R) = \{dim N(P^{∗}P) − dim N(P P^{∗})\} × \{dim N(Q^{∗}Q) − dim N(QQ^{∗})\} = index(P) index(Q) which completes the proof of (a).

We verify (b) on the symbol level. First consider the de Rham complex and decompose: Λ(T^{∗}M_{1}) = Λ_{e 1} ⊕ Λ_{o 1},  Λ(T^{∗}M_{2}) = Λ_{e 2} ⊕ Λ_{o 2}, Λ(T^{∗}M) = (Λ_{e 1} ⊗ Λ_{e 2} ⊕ Λ_{o 1} ⊗ Λ_{o 2}) ⊕ (Λ_{o 1} ⊗ Λ_{e 2} ⊕ Λ_{e 1} ⊗ Λ_{o 2}). Under this decomposition: σ_{L}((d + δ)_{M})(ξ^{1}, ξ^{2}) = ( {c(ξ^{1}) ⊗ 1  −1 ⊗ c(ξ^{2}) \atop 1 ⊗ c(ξ^{2}) c(ξ^{1}) ⊗ 1} ) . c(·) denotes Clifford multiplication. This verifies the de Rham complex decomposes properly. The signature complex is more complicated. We decompose Λ^{±}(T^{∗}M_{1}) = Λ_{±,e 1} ⊕ Λ^{±,o} 1 to decompose the signature complex into two complexes: (d + δ): C^{∞}(Λ_{+,e 1}) → C^{∞}(Λ_{−,o 1}) (d + δ): C^{∞}(Λ_{+,o 1}) → C^{∞}(Λ_{−,e 1}). Under this decomposition, the signature complex of M decomposes into four complexes. If, for example, we consider the complex: (d + δ): C^{∞}(Λ_{+,e 1} ⊗ Λ_{+,e 2} ⊕ Λ_{−,o 1} ⊗ Λ_{−,o 2}) → C^{∞}(Λ_{−,o 1} ⊗ Λ_{+,e 2} ⊕ Λ_{+,e 1} ⊗ Λ_{−,o 2}) then the same argument as that given for the de Rham complex applies to show the symbol is: ( {c(ξ^{1}) ⊗ 1  −1 ⊗ c(ξ^{2}) \atop 1 ⊗ c(ξ^{2}) c(ξ^{1}) ⊗ 1} ) . If we consider the complex: (d + δ): C^{∞}(Λ_{+,o 1} ⊗ Λ_{+e 2} ⊕ Λ_{−,e 1} ⊗ Λ_{−,o 2}) → C^{∞}(Λ_{−,e 1} ⊗ Λ_{+,e 2} ⊕ Λ_{+,o 1} ⊗ Λ_{−,o 2}) then we conclude the symbol is ( {c(ξ^{1}) ⊗ 1  1 ⊗ c(ξ^{2}) \atop −1 ⊗ (ξ^{2}) c(ξ^{1}) ⊗ 1} )

which isn’t right. We can adjust the sign problem for ξ^{2} either by changing one of the identifications with Λ^{+}(M) or by changing the sign of Q (which won’t affect the index). The remaining two cases are similar. The spin and Dolbeault complexes are also similar. If we take coefficients in an auxilary bundle V, the symbols involved are unchanged and the same arguments hold. This proves (b). The proof of (c) is more complicated. We assume without loss of gener- ality that p and q are homogeneous of degree 1. Let (ξ^{1}, ξ^{2}, u) parametrize the fibers of T^{∗}M ⊕ 1. At ξ^{1} = (1/√2, 0,…, 0), ξ^{2} = (1/√2, 0,…, 0), u = 0 the orientation is given by: − dx_{1 1} ∧ dξ_{1 1} ∧ dx_{1 2} ∧ dξ_{1 2} ∧ ⋯ ∧ dx_{1 m1} ∧ dξ_{1 m1} ∧ dx_{2 1} ∧ dx_{2 2} ∧ dξ_{2 2} ∧ ⋯ ∧ dx_{2 m2} ∧ dξ_{2 m2} ∧ du. We have omitted dξ_{1 2} (and changed the sign) since it points outward at this point of the sphere to get the orientation on Σ(T^{∗}M). In matrix form we have on (V_{1} ⊗W_{1})⊕(V_{2} ⊗W_{2})⊕(V_{2} ⊗W_{1})⊕(V_{1} ⊗W_{2}) that: Σr = u ( {1 0 \atop 0 1} )  ( {p∗ ⊗ 1 1 ⊗ q∗ \atop −1 ⊗ q p ⊗ 1} ) ; ( {p ⊗ 1 −1 ⊗ q∗ \atop 1 ⊗ q p∗ ⊗ 1} )  −u ( {1 0 \atop 0 1} ) . This is not a very convenient form to work with. We define γ_{1} = ( {1 0 \atop 0 −1} ) and α = ( {0 p∗ \atop p 0} )  on V_{1} ⊕ V_{2} = V γ_{2} = ( {1 0 \atop 0 −1} ) and β = ( {0 q∗ \atop q 0} ) on W_{1} ⊕ W_{2} = W and compute that Σr = uγ_{1} ⊗ γ_{2} + α ⊗ I_{W} + γ_{1} ⊗ β. We replace p and q by homotopic symbols so that: p^{∗}p = |ξ^{1}|^{2}I_{V1}, pp^{∗} = |ξ^{1}|^{2}I_{V2}, q^{∗}q = |ξ^{2}|^{2}I_{W1}, qq^{∗} = |ξ^{2}|^{2}I_{W2}. Since \{γ_{1} ⊗ γ_{2}, α ⊗ I_{W}, γ_{1} ⊗ β\} all anti-commute and are self-adjoint, (Σr)^{2} = (u^{2} + |ξ^{1}|^{2} + |ξ^{2}|^{2})I  on V ⊗ W. We parametrize Σ(T^{∗}M) by S(T^{∗}M_{1}) × [0, _{π 2} ] × Σ(T^{∗}M_{2}) in the form: (ξ^{1} cos θ, ξ^{2} sin θ, u sin θ).

230 3.9. The Atiyah-Singer Theorem We compute the orientation by studying θ = _{π 4} , ξ^{1} = (1, 0,…, 0), ξ^{2} = (1, 0,…, 0), u = 0. Since dξ_{1 1} = − sin θ dθ, the orientation is given by: dx_{1 1} ∧ dθ ∧ dx_{1 2} ∧ dξ_{1 2} ∧ ⋯ ∧ dx_{1 m1} ∧ dξ_{1 m1} ∧ dx_{2 1} ∧ dx_{2 2} ∧ dξ_{2 2} ∧ ⋯ ∧ dx_{2 m2} ∧ dξ_{2 m2} ∧ du. If we identify S(T^{∗}M_{1}) × [0, _{π 2} ] with D_{+}(T^{∗}M_{1}) then this orientation is: ω^{1 2m1} ∧ ω^{2} 2m2 so the orientations are compatible. In this parametrization, we compute Σr(ξ^{1}, θ, ξ^{2}, u) = (sin θ)γ_{1} ⊗ (uγ_{2} + β(ξ^{2})) + (cos θ)α(ξ^{1}) ⊗ I_{W}. Since Σq = uγ_{2} + β(ξ^{2}) satisfies (Σq)^{2} = |ξ^{2}|^{2} + u_{2 2} on W, we may decompose W = Π_{+}(Σq) ⊕ Π_{−}(Σq). Then: Σr = {(sin θ)γ_{1} + (cos θ)α(ξ^{1})} ⊗ I = Σp(ξ^{1}, θ) ⊗ I on V ⊗ Π_{+}(Σq) Σr = {sin(−θ)γ_{1} + (cos θ)α(ξ^{1})} ⊗ I = Σp(ξ^{1}, −θ) ⊗ I on V ⊗ Π_{−}(Σq). Consequently: Π^{+}(Σr) = {Π^{+}(Σp)(ξ^{1}, θ) ⊗ Π^{+}(Σq)} ⊕ {Π^{+}(Σp)(ξ^{1}, −θ) ⊗ Π^{−}(Σq)} over (D_{+}M_{1}) × Σ(T^{∗}M_{2}). If we replace −θ by θ in the second factor, we may replace Π_{+}(Σp)(ξ^{1}, −θ)⊗ Π_{−}(Σq) by Π_{+}(Σp)(ξ^{1}, θ) ⊗ Π_{−}(Σq). Since we have changed the orienta- tion, we must change the sign. Therefore: ∫_{Σ(T^{∗}M)} θ_{1} ∧ θ_{2} ∧ ch(Π_{+}(Σr)) = ∫_{D_{+}M1} θ_{1} ∧ ch(Π_{+}(Σp)) · ∫_{Σ(T^{∗}M_{2})} θ_{2} ∧ ch(Π_{+}(Σq)) − ∫_{D_{−}M1} θ_{1} ∧ ch(Π_{+}(Σp)) · ∫_{Σ(T^{∗}M_{2})} θ_{2} ∧ ch(Π_{−}(Σq)). ch(Π_{+}(Σq)) + ch(Π_{−}(Σq)) = ch(V_{2}) does not involve the fiber coordinates of Σ(T^{∗}M_{2}) and thus ∫_{Σ(T^{∗}M_{2})} θ_{2} ∧ ch(V_{2}) = 0. We may therefore replace −ch(Π_{−}(Σq)) by ch(Π_{+}(Σq)) in evaluating the integral over D_{−}(M_{1}) × Σ(T^{∗}M_{2}) to complete the proof of (c).

We prove (d) in a similar fashion. We suppose without loss of generality that p and q are homogeneous of degree 1. We parameterize S(T^∗M) = S(T^∗M_1) × [0, π/2] × S(T^∗M_2) in the form (ξ^1 cos θ, ξ^2 sin θ). Then

r = ( sin θ 0 0 − sin θ )

⊗ p(ξ^2) + (

0 (cos θ)q^∗(ξ^1)

(cos θ)q(ξ^1)  0  )

.

Again we decompose V_2 = Π_+(p) ⊕ Π_−(p) so that

Σr = ((sin θ)γ_1 + (cos θ)α^1(ξ^1)) on V ⊗ Π_+(p)

Σr = (sin(−θ)γ_1 + (cos θ)α^1(ξ^1)) on V ⊗ Π_−(p)

where

γ_1 = ( 1 0 0 −1 ) α^1 = ( 0 q^∗ q 0 ) .

The remainder of the argument is exactly as before (with the appropriate change in notation from (c)) and is therefore omitted.

We now check a specific example to verify some normalizing constants:

Lemma 3.9.4. Let M = S^1 be the unit circle. Define P: C^∞(M) → C^∞(M) by:

P (e^{inθ}) = { ne^{i(n−1)θ} for n ≥ 0 ne^{inθ} for n ≤ 0

then P is an elliptic pseudo-differential operator with index(P) = 1. Furthermore: ∫

Σ(T^∗S^1)

ch(Π_+(Σp)) = −1.

Proof: Let P_0 = −i∂/∂θ and let P_1 = {−∂^2/∂θ^2}^{1/2}. P_0 is a differential operator while P_1 is a pseudo-differential operator by the results of section 1.10. It is immediate that:

σ_L(P_0) = ξ, P_0(e^{inθ}) = ne^{inθ} σ_L(P_1) = |ξ|, P_1(e^{inθ}) = |n|e^{inθ}.

We define:

Q_0 = 1 / 2 e^{−iθ}(P_0 + P_1)

σ_LQ_0 = { ξe^{−iθ} ξ ≥ 0 0 ξ ≤ 0

Q_0(e^{inθ}) = { ne^{i(n−1)θ} n ≥ 0 0 n ≤ 0

and  Q1 = 1 2(P_{0} − P_{1}) σ_{L}Q1 = \{ 0  ξ ≥ 0 ξ  ξ ≤ 0 Q_{1}(e^{inθ}) = \{ 0  n ≥ 0^{neinθ}  n ≤ 0. Consequently, P = Q_{0} + Q_{1} is a pseudo-differential operator and σ^{L}P = p = \{ ξe−iθ  ξ > 0 ξ  ξ < 0. It is clear P is surjective so N(P^{∗}) = 0. Since N(P) is the space of constant functions, N(P) is one dimensional so index(P) = 1. We compute: Σp(θ, ξ, t) = ( ξe−iθ −t^{t  ξe+iθ} ) if ξ ≥ 0 ( {t ξ \atop ξ −t} )  if ξ ≤ 0. Since Σp does not depend on θ for ξ ≤ 0, we may restrict to the region ξ ≥ 0 in computing the integral. (We must smooth out the symbol to be smooth where ξ = 0 but suppress these details to avoid undue technical complications). It is convenient to introduce the parameters: u = t,  v = ξ cos θ,  w = ξ sin θ  for u^{2} + v^{2} + w^{2} = 1 then this parametrizes the region of Σ(T^{∗}S^{1}) where ξ ≥ 0 in a 1-1 fashion except where ξ = 0. Since du ∧ dv ∧ dw = −ξ dθ ∧ dξ ∧ dt, S^{2} inherits the reversed orientation from its natural one. Let e^{0} = ( {1 0 \atop 0 −1} ) ,  e^{1} = ( {0 1 \atop 1 0} ) ,  e^{2} = ( {0 i \atop −i 0} ) so Σp(u, v, w) = ue_{0} + ve_{1} + we_{2}. Then by Lemma 2.1.5 we have − ∫_{S 2} ch(Π_{+}(Σp)) = (−1) · ( i2) · Tr(e_{0}e_{1}e_{2}) = −1 which completes the proof. We can now state the Atiyah-Singer index theorem:

The Atiyah-Singer Theorem 233

Theorem 3.9.5 (The index theorem). Let P: C^{∞}(V_{1}) → C^{∞}(V_{2}) be an elliptic pseudo-differential operator. Let Td_{r}(M) = Td(T M ⊗ C) be the Todd class of the complexification of the real tangent bundle. Then: index(P) = (−1)^{dim M} ∫_{Σ(T^{∗}M)} Td_{r}(M) ∧ ch(Π_{+}(Σp)). We remark that the additional factor of (−1)^{dim M} could have been avoided if we changed the orientation of Σ(T^{∗}M). We begin the proof by reducing to the case dim M = m even and M orientable. Suppose first that m is odd. We take Q: C^{∞}(S^{1}) → C^{∞}(S^{1}) to be the operator defined in Lemma 3.9.4 with index +1. We then form the operator R with index(R) = index(P) index(Q) = index(P) defined in Lemma 3.9.3. Although R is not a pseudo-differential operator, it can be uniformly approximated by pseudo-differential operators in the natural Fredholm topology (once the order of Q is adjusted suitably). (This process does not work when discussing the twisted eta invariant and will involve us in additional technical complications in the next chapter). Therefore: (−1)^{m+1} ∫_{Σ(T^{∗}(M×S^{1}))} Td_{r}(M × S^{1}) ∧ ch(Π_{+}(Σr)) = (−1)^{m} ∫_{Σ(T^{∗}M)} Td_{r}(M) ∧ ch(Π_{+}(Σp)) · (−1) ∫_{Σ(T^{∗}S^{1})} ch(Π_{+}(Σq)) = (−1)^{m} ∫_{Σ(T^{∗}M)} Td_{r}(M) ∧ ch(Π_{+}(Σp)). To show the last integral gives index(P) it suffices to show the top integral gives index(R) and therefore reduces the proof of Theorem 3.9.3 to the case dim M = m even. If M is not orientable, we let M′ be the orientable double cover of M. It is clear the formula on the right hand side of the equation multiplies by two. More careful attention to the methods of the heat equation for pseudo-differential operators gives a local formula for the index even in this case as the left hand side is also multiplied by two under this double cover. This reduces the proof of Theorem 3.9.3 to the case dim M = m even and M orientable. We fix an orientation on M henceforth. Lemma 3.9.6. Let P: C^{∞}(Λ^{+}) → C^{∞}(Λ^{−}) be the operator of the signa- ture complex. Let ω = ch_{m/2}(Π_{+}(Σp)) ∈ H^{m}(Σ(T^{∗}M); C). Then if ωM is the orientation class of M (a) ω_{M} ∧ ω gives the orientation of Σ(T^{∗}M).

234 3.9. The Atiyah-Singer Theorem (b) If S^{m} is a fiber sphere of Σ(T^{∗}M), then ∫_{S m} ω = 2^{m/2}. Proof: Let (x_{1},…, x_{m}) be an oriented local coordinate system on M so that the {dx_{j}} are orthonormal at x_{0} ∈ M. If ξ = (ξ,…, ξ_{m}) are the dual fiber coordinates for T^{∗}M), then: p(ξ) = _{∑ j} iξ_{j}(c(dx_{j})) = _{∑ j} iξ_{j}(ext(dx_{j}) − int(ξ_{j})) gives the symbol of (d + δ); c(·) denotes Clifford multiplication as defined previously. We let e_{j} = ic(dx_{j}); these are self-adjoint matrices such that e_{j}e_{k} + e_{k}e_{j} = 2δ_{jk}. The orientation class is defined by: e_{0} = i^{m/2}c(dx_{1}) … c(dx_{m}) = (−i)^{m/2}e_{1} … e_{m}. The bundles Λ^{±} are defined as the ±1 eigenspaces of e_{0}. Consequently, Σp(ξ, t) = te_{0} + ∑ ξ_{j}e_{j}. Therefore by Lemma 2.1.5 when S^{m} is given its natural orientation, ∫_{S m} ch_{m/2} Π_{+}(Σp) = i^{m/2}2^{−m/2} Tr(e_{0}e_{1} … e_{m}) = i^{m/2}2^{−m/2} Tr(e_{0}i^{m/2}e_{0}) = (−1)^{m/2}2^{−m/2} Tr(I) = (−1)^{m/2}2^{−m/2}2m = (−1)^{m/2}2^{m/2}. However, S^{m} is in fact given the orientation induced from the orientation on Σ(T^{∗}M) and on M. At the point (x, 0,…, 0, 1) in T^{∗}M ⊕R the natural orientations are: of X: dx_{1} ∧ ⋯ ∧ dx_{m}, of Σ(T^{∗}M): dx_{1} ∧ dξ_{1} ∧ ⋯ ∧ dx_{m} ∧ dξm = (−1)^{m/2}dx_{1} ∧ ⋯ ∧ dx_{m} ∧ dξ_{1} ∧ ⋯ ∧ dξm of S^{m}: (−1)^{m/2}dξ_{1} ∧ ⋯ ∧ dξ_{m}. Thus with the induced orientation, the integral becomes 2^{m/2} and the lemma is proved. Consequently, ω provides a cohomology extension and: Lemma 3.9.7. Let ρ: Σ(T^{∗}M) → M where M is orientable and even dimensional. Then (a) ρ^{∗}: H^{∗}(M; C) → H^{∗}(Σ(T^{∗}M); C) is injective.

The Atiyah-Singer Theorem 235

(b) If ω is as defined in Lemma 3.9.6, then we can express any α ∈ H^{∗}(Σ(T^{∗}M); C) uniquely as α = ρ^{∗}α_{1} + ρ^{∗}α_{2} ∧ ω for α_{i} ∈ H^{∗}(M; C). Since ρ^{∗} is injective, we shall drop it henceforth and regard H^{∗}(M; C) as being a subspace of H^{∗}(Σ(T^{∗}M); C). The Chern character gives an isomorphism K(X; C) ≃ H^{e}(X; C). When we interpret Lemma 3.9.7 in K-theory, we conclude that we can decom- pose K(Σ(T^{∗}M); C) = K(M; C) ⊕ K(M; C) ⊗ Π_{+}(Σp); ch(V) generates H^{e}(M; C) as V ranges over K(M; C). Therefore K(Σ(T^{∗}M); C)/K(M; C) is generated as an additive module by the twisted signature complex with coefficients in bundles over M. Π_{+}(Σp_{V}) = V ⊗Π_{+}(Σp) if p_{V} is the symbol of the signature complex with coefficients in V. In Lemma 3.9.2, we interpreted the index as a map in K-theory. Since it is linear, it suffices to compute on the generators given by the signature complex with coefficients in V. This proves: Lemma 3.9.8. Assume M is orientable and of even dimension m. Let PV be the operator of the signature complex with coefficients in V. The bun- dles {Π_{+}(Σp)_{V} }_{V ∈Vect (M)} generate K(Σ(T^{∗}M); C)/K(M; C) additively. It suffices to prove Theorem 3.9.5 in general by checking it on the special case of the operators P_{V}. We will integrate out the fiber coordinates to reduce the integral of The- orem 3.9.5 from Σ(T^{∗}M) to M. We proceed as follows. Let W be an oriented real vector bundle of fiber dimension k + 1 over M equipped with a Riemannian inner product. Let S(W) be the unit sphere bundle of W. Let ρ: S(W) → M be the natural projection map. We define a map I: C^{∞}(Λ(S(W))) → C^{∞}(Λ(M)) which is a C^{∞}(Λ(M)) module homomor- phism and which commutes with integration—i.e., if α ∈ C^{∞}(Λ(S(W))) and β ∈ C^{∞}(Λ(M)), we require the map α ↦ I (α) to be linear so that I (ρ^{∗}β ∧ α) = β ∧ I (α) and ∫_{S(W)} α = ∫_{M} I (α). We construct I as follows. Choose a local orthonormal frame for W to define fiber coordinates u = (u_{0},…, u_{k}) on W. This gives a local rep- resentation of S(W) = V × S^{k} over the coordinate patch U on M. If α ∈ C^{∞}(Λ(S(W))) has support over V, we can decompose α = ∑_{ν} β_{ν} ∧αν for β_{ν} ∈ C^{∞}(Λ(U)) and α_{ν} ∈ C^{∞}(Λ(S^{k})). We permit the α_{ν} to have coefficients which depend upon x ∈ U. This expression is, of course, not unique. Then I (α) is necessarily defined by: I (α)(x) = ∑ β_{ν} ∫_{S k} α_{ν}(x). It is clear this is independent of the particular way we have decomposed α. If we can show I is independent of the frame chosen, then this will define I in general using a partition of unity argument.

236 3.9. The Atiyah-Singer Theorem Let u_{′i} = a_{ij}(x)u_{j} be a change of fiber coordinates. Then we compute: du_{′i} = a_{ij}(x) du_{j} + da_{ij}(x)u_{j}. Clearly if a is a constant matrix, we are just reparamatrizing S^{k} so the integral is unchanged. We fix x_{0} and suppose a(x_{0}) = I. Then over x_{0}, du_{I} = dv_{I} +_{∑ |I|<|J|} c_{I,J} ∧ dv_{J} where c_{I,J} ∈ Λ^{|I|−|J|}(M). To integrate and get an answer different from 0 over S^{k}, we must have |I| = k so these error terms integrate to zero and I is invariantly defined. We specialize to the case W = T^{∗}M ⊕ 1. The orientation of M induces a natural orientation of T^{∗}M ⊕ 1 as a bundle in such a way as to agree with the orientation of T^{∗}M ⊕1 as a topological space. Let α = ch(Π_{+}(Σp)), so I (α) ∈ C^{∞}(Λ(M)). If we reverse the orientation of M, then we interchange the roles of Λ^{+} and of Λ^{−}. This has the effect of replacing the parameter u by −u which is equivalent to reversing the orientation of T^{∗}M ⊕ 1 as a topological space. Since both orientations have been reversed, the orien- tation of the fiber is unchanged so I (α) is invariantly defined independent of any local orientation of M. It is clear from the definition that I (α) is a polynomial in the jets of the metric and is invariant under changes of the metric by a constant factor. Therefore Theorems 2.5.6 and Lemma 2.5.3 imply I (α) is a real characteristic form. By Lemma 3.9.6, we can expand I (α) = 2^{m/2} + · · ·. We solve the equation: {I (ch(Π_{+}(Σp))) ∧ Todd(m)}_{m−4s} = 2^{(m−4s)/2} Ls recursively to define a real characteristic class we shall call Todd(m) for the moment. It is clear Todd(m) = 1 + · · ·. In this equation, L_{s} is the Hirzebruch genus. Lemma 3.9.9. Let Todd(m) be the real characteristic class defined above. Then if P is any elliptic pseudo-differential operator, index(P) = ∫_{Σ(T^{∗}M)} Todd(m) ∧ ch(Π_{+}(Σp)). Proof: By Lemma 3.9.8 it suffices to prove this identity if P is the oper- ator of the twisted signature complex. By Theorem 3.1.5, index(P_{signature V} ) = ∫_{M} _{∑ 2t+4s=m} ch_{t}(V)2^{t} ∧ Ls = ∫_{M} ch(V) ∧ I (ch(Π_{+}(Σp))) ∧ Todd(m) = ∫_{Σ(T^{∗}M)} ch(V) ∧ ch(Π_{+}(Σp)) ∧ Todd(m) = ∫_{Σ(T^{∗}M)} Todd(m) ∧ ch(Π^{+}(Σp^{V})).

The Atiyah-Singer Theorem 237

It is clear Todd(m) is uniquely determined by Lemma 3.9.9. Both the index of an elliptic operator and the formula of Lemma 3.9.9 are multi- plicative with respect to products by Lemma 3.9.3 so Todd(m) is a multi- plicative characteristic form. We may therefore drop the dependence upon the dimension m and simply refer to Todd. We complete the proof of the Atiyah-Singer index theorem by identifying this characteristic form with the real Todd polynomial of T (M). We work with the Dolbeault complex instead of with the signature com- plex since the representations involved are simpler. Let m be even, then (−1)^{m} = 1. Let M be a holomorphic manifold with the natural orienta- tion. We orient the fibers of T^{∗}M using the natural orientation which arises from the complex structure on the fibers. If ξ are the fiber coordinates, this gives the orientation: dx_{1} ∧ dy_{1} ∧ ⋯ ∧ dx_{n} ∧ dy_{n} ∧ dξ_{1} ∧ ⋯ ∧ dξ_{m} where m = 2n. This gives the total space T^{∗}M an orientation which is (−1)^{n} times the simplectic orientation. Let S^{m} denote a fiber sphere of Σ(T^{∗}M) with this orientation and let q be the symbol of the Dolbeault complex. Then: index( ¯∂) = ∫_{Σ(T^{∗}M)} Todd ∧ ch(Π_{+}Σq) = (−1)^{n} ∫_{M} Todd ∧I (ch(Π_{+}Σq)). We define the complex characteristic form S = (−1)^{m/2}I (ch(Π_{+}(Σq))) then the Riemann-Roch formula implies that: S ∧ Todd(m) = Todd(T_{c}M). It is convenient to extend the definition of the characteristic form S to arbitrary complex vector bundles V. Let W = Λ^{∗}(V) and let q(v): V → END(W) be defined by Lemma 3.5.2(a) to be the symbol of ¯∂ + δ′′ if V = T_{c}(M). We let ext: V → HOM(W, W) be exterior multiplication. This is complex linear and we let int be the dual of ext; int(λv) = λ̄ int(v) for λ ∈ C. ext is invariantly defined while int requires the choice of a fiber matric. We let q = ext(v) − int(v) (where we have deleted the factor of i/2 which appears in Lemma 3.5.2 for the sake of simplicity). We regard q as a section to the bundle HOM(V, HOM(W, W)). Fix a Riemannian connection on V and covariantly differentiate q to compute ∇q ∈ C^{∞}(T^{∗}M ⊗HOM(V, HOM(W, W))). Since the connection is Rieman- nian, ∇q = 0; this is not true in general for non-Riemannian connections. If V is trivial with flat connection, the bundles Π_{±}(Σq) have curvature π_{±}dπ_{±}dπ_{±} as computed in Lemma 2.1.5. If V is not flat, the curvature of 238 3.9. The Atiyah-Singer Theorem V enters into this expression. The connection and fiber metric on V define a natural metric on T^{∗}V. We use the splitting defined by the connection to decompose T^{∗}V into horizontal and vertical components. These compo- nents are orthogonal with respect to the natural metric on V. Over V, q becomes a section to the bundle HOM(V, V). We let ∇^{V} denote covariant differentiation over V, then ∇^{V} q has only vertical components in T^{∗}V and has no horizontal components. The calculation performed in Lemma 2.1.5 shows that in this more gen- eral setting that the curvatures of ∇_{±} are given by: Ω_{±} = π_{+}(∇^{V} π_{+} ∧ ∇^{V} π_{+} + ρ^{∗}Ω_{W}). If we choose a frame for V and W which is covariant constant at a point x_{0}, then ∇^{V} π_{+} = dπ_{+} has only vertical components while ρ^{∗}Ω_{W} has only horizontal components. If Ω_{V} is the curvature of V, then Ω_{W} = Λ(Ω_{V}). Instead of computing S on the form level, we work with the corresponding invariant polynomial. Lemma 3.9.10. Let A be an n × n complex skew-adjoint matrix. Let B = Λ(A) acting on Λ(C^{n}) = C^{2n}. Define: S (A) = _{∑ ν}(−1)^{n} ( i 2π )^{ν} 1 ν! ∫_{S 2n} Tr\{(π_{+}dπ_{+}dπ_{+} + B)^{ν}\}. If x_{j} = iλ_{j}/2π are the normalized eigenvectors of A, then S (A) = _{∏ j} e^{xj} − 1 xj . Proof: If V = V_{1} ⊕ V_{2} and if A = A_{1} ⊕ A_{2}, then the Dolbeault com- plex decomposes as a tensor product by Lemma 3.9.3. The calculations of Lemma 3.9.3 using the decomposition of the bundles Π_{±} shows S (A) is a multiplicative characteristic class. To compute the generating function, it suffices to consider the case n = 1. If n = 1, A = λ so that B = ( {0 0 \atop 0 λ} ), if we decompose W = Λ^{0,0}⊕Λ^{0,1} = 1 ⊕ V. If x + iy give the usual coordinates on V = C, then: q(x, y) = x( {0 1 \atop 1 0} ) + y( {0 −i \atop i 0} ) by Lemma 3.5.2 which gives the symbol of the Dolbeault complex. There- fore: q(x, y, u) = x( {0 1 \atop 1 0} ) + y( {0 −i \atop i 0} ) + u( {1 0 \atop 0 −1} ) = xe^{0} + ye^{1} + ue^{2}.

We compute: e_{0}e_{1}e_{2} = i( {1 0 \atop 0 1} ) and π_{+} dπ_{+} dπ_{+} = i2π_{+} dvol . Since n = 1, (−1)^{n} = −1 and: S (A) = _{∑ j>0} − i2 ( i 2π )^{j} 1 j! ∫ Tr((π_{+} dvol +π_{+}B)^{j}) = _{∑ j>0} − i2 ( i 2π )^{j} 1 (j − 1)! ∫ Tr(π_{+}B)^{j−1} dvol . We calculate that: π_{+}B = 1 2( {1 + u x − iy \atop x + iy 1 − u} )( {0 0 \atop 0 λ} ) = 1 2( {0  ∗ \atop 0 (1 − u)λ} ) (π^{+}B)j−1 = 2^{1−j} λ^{j−1}(1 − u)^{j−1}( {0 ∗ \atop 0 1} ) where “∗” indicates a term we are not interested in. We use this identity and re-index the sum to express: S (A) = _{∑ j≥0} 1 4π ( iλ 2π )^{j} 1 j!2^{−j} ∫_{S 2}(1 − u)^{j} dvol . We introduce the integrating factor of e^{−r2} to compute: ∫_{R3} u^{2k}e^{−r2} dx dy du = π ∫_{R} u^{2k}e^{−u2} du = ∫_{∞ 0} r^{2k+2} e^{−r2} dr · ∫_{S 2} u^{2k} dvol = (2k + 1)/2 ∫_{∞ 0} r^{2k}e^{−r2} dr · ∫_{S 2} u^{2k} dvol so that:  ∫_{S 2} u^{2k} dvol = 4π/(2k + 1). The terms of odd order integrate to zero so: ∫_{S 2}(1 − u)^{j} dvol = 4π ∑ ( j 2k) · 1 2k + 1 = 4π ∫_{1 0} ∑ ( j 2k)t^{2k} dt = 2π ∫_{1 0}(1 + t)^{j} + (1 − t)^{j} dt = 2π (1 + t)^{j+1} − (1 − t)j+1 j + 1  ||||^{1} 0 = 4π · 2j j + 1.

240 3.9. The Atiyah-Singer Theorem We substitute this to conclude: S (A) = _{∑ j≥0} ( iλ 2π )^{j} 1 (j + 1)!. If we introduce x = iλ/2π then S (x) = ^{∑ j≥0} xj (j + 1)! = e^{x} − 1 x which gives the generating function for S. This completes the proof of the lemma. We can now compute Todd. The generating function of Todd(T_{c}) is x/(1 − e^{−x}) so that Todd = S^{−1} · Todd(T_{c}) will have generating function: x 1 − e−x · x e^{x} − 1 = x 1 − e−x · −x 1 − ex which is, of course, the generating function for the real Todd class. This completes the proof. We have gone into some detail to illustrate that it is not particularly difficult to evaluate the integrals which arise in applying the index theorem. If we had dealt with the signature complex instead of the Dolbeault complex, the integrals to be evaluated would have been over S^{4} instead of S^{2} but the computation would have been similar.