Classification of Holomorphic Vector Bundles

on Noncommutative Two-Tori

A. Polishchuk^{1}

Received: August 27, 2003 Revised: May 18, 2004

Communicated by Joachim Cuntz

Abstract. We prove that every holomorphic vector bundle on a noncommutative two-torus T can be obtained by successive extensions from standard holomorphic bundles considered in [2]. This implies that the category of holomorphic bundles on T is equivalent to the heart of a certain t-structure on the derived category of coherent sheaves on an elliptic curve.

1. Introduction

In this paper we continue the study of holomorphic bundles on noncommutative two-tori that was begun in [2]. Recall that for every θ ∈ ℝ \ ℚ and τ ∈ ℂ \ ℝ we considered in [2] holomorphic vector bundles on a noncommutative complex torus T = T_{θ,τ}. By definition, the algebra A_{θ} of smooth functions on T consists of series ∑_{(m,n)∈ℤ^{2}} a_{m,n}U_{1}^{m}U_{2}^{n} where the coefficients a_{m,n} ∈ ℂ decrease rapidly at infinity and the multiplication is defined using the rule

U_{1}U_{2} = exp(2πiθ)U_{2}U_{1}.

We consider the derivation δ = δ_{τ} : A_{θ} → A_{θ} defined by

δ(∑ a_{m,n}U_{1}^{m}U_{2}^{n}) = 2πi ∑_{m,n}(mτ + n)a_{m,n}U_{1}^{m}U_{2}^{n}

as an analogue of the \overline{∂}-operator. A holomorphic bundle over T is a pair (E, \overline{∇}) consisting of a finitely generated projective right A_{θ}-module E and an operator \overline{∇} : E → E satisfying the Leibnitz identity

\overline{∇}(ea) = \overline{∇}(e)a + eδ(a),

where e ∈ E, a ∈ A_{θ}. There is an obvious definition of a holomorphic map between holomorphic bundles, so we can define the category C(T) of holomorphic bundles on T.

1This work was partially supported by NSF grant DMS-0302215

For every pair of relatively prime integers (c, d) such that cθ + d > 0 and a complex number z we define a standard holomorphic bundle (E_{d,c}(θ), \overline{∇}_{z}) as follows. If c ≠ 0 then

E_{d,c}(θ) = S(ℝ × ℤ/cℤ) = S(ℝ)^{|c|},

where S(ℝ) is the Schwartz space of functions on ℝ, with the A_{θ}-action defined by

fU_{1}(x, α) = f(x − \frac{1}{μ}, α − 1), fU_{2}(x, α) = exp(2πi(x − \frac{αd}{c}))f(x, α),

where x ∈ ℝ, α ∈ ℤ/cℤ, μ = \frac{c}{cθ+d}. The operator \overline{∇}_{z} on this space is given by

\overline{∇}_{z}(f) = \frac{∂f}{∂x} + 2πi(τμx + z)f.  (1.1)

For c = 0 and d = 1 we set E_{1,0}(θ) = A_{θ} with the natural right A_{θ}-action and the operator \overline{∇}_{z} is given by

\overline{∇}_{z}(a) = δ(a) + 2πiza.

We define degree, rank and slope of a bundle E = E_{d,c}(θ) by setting deg(E) = c, rk(E) = cθ + d and μ(E) = deg(E)/ rk(E). Note that rk(E) > 0 and μ = μ(E) in the formulae above. According to the theorem of Rieffel (see [5]) every finitely generated projective right A_{θ}-module is isomorphic to E = E_{d,c}(θ)^{⊕n} for some (c, d) as above and n ≥ 0. Moreover, the degree and rank defined above extend to additive functions on the category of finitely generated projective A_{θ}-modules. The category of holomorphic bundles C = C(T) has a natural structure of a ℂ-linear exact category. In particular, for every pair of holomorphic bundles E_{1} and E_{2} we can form the vector space Ext_{C}^{1}(E_{1}, E_{2}) parametrizing extensions of E_{1} by E_{2}. Sometimes we will also use the notation Ext_{C}^{0} := Hom_{C}. Let C′ ⊂ C be the minimal full subcategory of C containing all standard holomorphic bundles and closed under extensions. Our main result is the following theorem.

Theorem 1.1. One has C′ = C.

Combining this theorem with the study of the category C′ in [2] we obtain the following result.

Corollary 1.2. The category C is abelian. It is equivalent to the heart C^{θ} of the t-structure on the derived category of coherent sheaves on the elliptic curve ℂ/ℤ + ℤτ, associated with θ (see section 3 of [2] or section 3.1 below).

Remark. Recall that we always assume θ to be irrational. For rational θ the category C will not be abelian.

Corollary 1.3. For every indecomposable holomorphic bundle E on T there exists a standard holomorphic bundle \overline{E} and a filtration 0 = E_{0} ⊂ E_{1} ⊂ … ⊂ E_{n} = E by holomorphic subbundles such that all quotients E_{i}/E_{i−1} are isomorphic to \overline{E}.

Holomorphic Vector Bundles … 165

The proof of Theorem 1.1 consists of two steps. First, we develop the cohomology theory for holomorphic bundles on T and prove the analogues of the standard theorems for them (such as finiteness, Riemann-Roch and Serre duality). Then we combine these results with the techniques of [4] where the category C′ was described in terms of coherent modules over a certain algebra. Acknowledgments. Parts of this paper were written during the author’s visits to Max-Planck-Institut für Mathematik in Bonn and the Institut des Hautes Études Scientifiques. I’d like to thank these institutions for hospitality and support.

2. Cohomology of holomorphic bundles on noncommutative two-tori

2.1. Cohomology and Ext-spaces. Let (E, \overline{∇}) be a holomorphic bundle on T = T_{θ,τ}. Then the cohomology of E is defined by

H^{i}(E) = H^{i}(E, \overline{∇}) = H^{i}(E →^{\overline{∇}} E)

where i = 0 or i = 1. Thus, H^{0}(E) = ker(\overline{∇}), H^{1}(E) = coker(\overline{∇}). These spaces are closely related to Ext^{i}-spaces in the category of holomorphic bundles (where i = 0 or i = 1). To explain this connection we have to use Morita equivalences between noncommutative tori. Recall that for every standard bundle E_{0} = E_{d,c}(θ) the algebra of endomorphisms End_{A_{θ}}(E_{0}) can be identified with the algebra A_{θ′} for some θ′ ∈ ℝ. In fact, θ′ = \frac{aθ+b}{cθ+d}, where a and b are

chosen in such a way that (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ). Furthermore, if E_{0} is equipped

with a standard holomorphic structure \overline{∇} then the formula ϕ ↦ [\overline{∇}, ϕ] defines a derivation of End_{A_{θ}}(E_{0}) ≃ A_{θ′}, hence the corresponding torus T_{θ′} is equipped with a complex structure. In fact, this derivation on A_{θ′} is equal to δ_{τ} / rk(E_{0}), where τ is the same parameter that was used to define the complex structure on T_{θ} (see Proposition 2.1 of [2]). Now one can define the Morita equivalence

C(T_{θ′,τ}) → C(T_{θ,τ}) : E ↦ E ⊗_{A_{θ′}} E_{0},

where the tensor product is equipped with the complex structure

\overline{∇}(e ⊗ e_{0}) = \frac{1}{rk(E_{0})}\overline{∇}_{E}(e) ⊗ e_{0} + e ⊗ \overline{∇}_{E_{0}}(e_{0})

(see Propositions 2.1 and 3.2 of [2]). This functor sends standard holomorphic bundles on T_{θ′,τ} to standard holomorphic bundles on T_{θ,τ}. The inverse functor is

C(T_{θ,τ}) → C(T_{θ′,τ}) : E ↦ Hom_{A_{θ}}(E_{0}, E),  (2.1)

where the latter space has a natural right action of A_{θ′} ≃ End_{A_{θ}}(E_{0}). Now we can formulate the connection between the cohomology and Ext-groups. For every holomorphic bundle E and a standard holomorphic bundle E_{0} on T = T_{θ,τ} one has a natural isomorphism

Ext_{C}^{i}(E_{0}, E) ≃ H^{i}(Hom_{A_{θ}}(E_{0}, E)),  (2.2)

where Hom_{A_{θ}}(E_{0}, E) is viewed as a holomorphic bundle on T_{θ′} (the proof is similar to Proposition 2.4 of [2]). Note that for an arbitrary pair of holomorphic bundles E_{1} and E_{2} one can still define an operator \overline{∇} on Hom_{A_{θ}}(E_{1}, E_{2}) such that the analogue of isomorphism (2.2) holds. However, we have a natural interpretation of Hom_{A_{θ}}(E_{1}, E_{2}) as a holomorphic bundle on some noncommutative two-torus only in the case when one of the bundles E_{1} or E_{2} is standard (see (2.3) below).

2.2. Duality and metrics. One can define the category of left holomorphic bundles on T by replacing right A_{θ}-modules with left ones and changing the Leibnitz identity appropriately. The definition of cohomology for these bundles remains the same. There is a natural duality functor E ↦ E^{∨} that associates to a (right) holomorphic bundle E the left holomorphic bundle Hom_{A_{θ}}(E, A_{θ}). More generally, for every standard holomorphic bundle E_{0} we can consider Hom_{A_{θ}}(E, E_{0}) as a left module over End_{A_{θ}}(E_{0}) ≃ A_{θ′} equipped with an induced holomorphic structure. Then the natural isomorphism

Ext_{C}^{i}(E, E_{0}) ≃ H^{i}(Hom_{A_{θ}}(E, E_{0}))  (2.3)

allows to view Ext_{C}^{i}(E, E_{0}) as cohomology of a holomorphic bundle on T_{θ′}. Using duality, the functor (2.1) for a standard holomorphic bundle E_{0} can be rewritten as the usual Morita functor due to the isomorphism

Hom_{A_{θ}}(E_{0}, E) ≃ E ⊗_{A_{θ}} E_{0}^{∨}.  (2.4)

For a standard holomorphic bundle E_{0} on T_{θ,τ} the dual bundle E_{0}^{∨} can also be considered as a right holomorphic bundle on T_{θ′,τ}, where End_{A_{θ}}(E_{0}) ≃ A_{θ′}. In fact, it is again a standard holomorphic bundle (see Corollary 2.3 of [2]). More

precisely, for E_{0} = E_{d,c}(θ) we have θ′ = \frac{aθ+b}{cθ+d}, where (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ). The

left action of A_{θ′} on E_{d,c}(θ) = S(ℝ × ℤ/cℤ) (where c ≠ 0) is defined by the formulae

U_{1}f(x, α) = f(x − \frac{1}{c}, α − a), U_{2}f(x, α) = exp(2πi(\frac{x}{cθ + d} − \frac{α}{c}))f(x, α),

where x ∈ ℝ, α ∈ ℤ/cℤ. We can identify E_{d,c}(θ)^{∨} considered as a right A_{θ′}- module with E_{a,−c}(θ′) using the natural pairing

t : E_{a,−c}(θ′) ⊗ E_{d,c}(θ) → A_{θ}

constructed as follows (see Proposition 1.2 of [2]). First, we define the map

b : E_{a,−c}(θ′) ⊗ E_{d,c}(θ) → ℂ

by the formula

b(f_{1}, f_{2}) = ∑_{α∈ℤ/cℤ} ∫_{x∈ℝ} f_{1}(\frac{x}{cθ + d}, α)f_{2}(x, −aα)dx.

Then t is given by

t(f_{1}, f_{2}) = ∑_{(m,n)∈ℤ^{2}} U_{1}^{m}U_{2}^{n}b(U_{2}^{−n}U_{1}^{−m}f_{1} ⊗ f_{2}).

The corresponding isomorphism

E_{d,c}(θ)^{∨} ≃ E_{a,−c}(θ′)

is compatible with the A_{θ} − A_{θ′}-bimodule structures and with holomorphic structures (see Corollary 2.3 of [2]). Note that b = tr ∘t, where tr : A_{θ} → ℂ is the trace functional sending ∑ a_{m,n}U_{1}^{m}U_{2}^{n} to a_{0,0}. On the other hand, we can define a ℂ-antilinear isomorphism

σ : E_{d,c}(θ) → E_{a,−c}(θ′)

by the formula

σ(f)(x, α) = \overline{f((cθ + d)x, −aα)}.

This isomorphism satisfies

σ(bea) = a^{∗}σ(e)b^{∗}  (2.5)

for e ∈ E_{d,c}(θ), a ∈ A_{θ}, b ∈ A_{θ′}, where ∗ : A_{θ} → A_{θ} is the ℂ-antilinear antiinvolution sending U_{i} to U_{i}^{−1}. In view of the identification of E_{a,−c}(θ′) with the dual bundle to E_{d,c}(θ) the isomorphism σ should be considered as an analogue of the Hermitian metric on E_{d,c}(θ). The corresponding analogue of the scalar product on global sections is simply the Hermitian form on E_{d,c}(θ) given by the formula

⟨f_{1}, f_{2}⟩ = b(σ(f_{2}), f_{1}) = ∑_{α∈ℤ/cℤ} ∫_{x∈ℝ} f_{1}(x, α)\overline{f_{2}(x, α)}dx,  (2.6)

where f_{1}, f_{2} ∈ E_{d,c}(θ). We can also define the corresponding L^{2}-norm: ||f||_{0}^{2} = ⟨f, f⟩. The above Hermitian form is related to the structure of A_{θ′} − A_{θ}- bimodule on E_{d,c}(θ) in the following way:

⟨f_{1}, af_{2}b⟩ = ⟨a^{∗}f_{1}b^{∗}, f_{2}⟩,

where a ∈ A_{θ′}, b ∈ A_{θ} (this is a consequence of (2.5) and of Lemma 1.1 of [2]). In the case of the trivial bundle E_{1,0}(θ) = A_{θ} we can easily modify the above definitions. First of all, θ′ = θ and the dual bundle is still A_{θ}. The role of σ is played by ∗ : A_{θ} → A_{θ} and the Hermitian form on A_{θ} is given by ⟨a, b⟩ = tr(ab^{∗}). The corresponding L^{2}-norm is

|| ∑ a_{m,n}U_{1}^{m}U_{2}^{n}||_{0}^{2} = ∑ |a_{m,n}|^{2}.

Note that the operator \overline{∇}_{z} on E_{d,c}(θ) admits an adjoint operator \overline{∇}_{z}^{∗} with respect to the Hermitian metrics introduced above. Namely, for c ≠ 0 it is given by

\overline{∇}_{z}^{∗}(f) = −\frac{∂f}{∂x} − 2πi(\overline{τ}μx + \overline{z})f,

while for c = 0 we have \overline{∇}_{z}^{∗} = −δ\overline{τ} − 2πiz id on A_{θ}. In either case we have

\overline{∇}_{z}\overline{∇}_{z}^{∗} − \overline{∇}_{z}^{∗}\overline{∇}_{z} = λ · id  (2.7)

for some constant λ ∈ ℝ. It follows that for an arbitrary holomorphic structure \overline{∇} on E = E_{d,c}(θ)^{⊕n} there exists an adjoint operator \overline{∇}^{∗} : E → E with respect to the above Hermitian metric. Indeed, we can write \overline{∇} = \overline{∇}_{0}+ϕ, where \overline{∇}_{0} is the standard holomorphic structure and set \overline{∇}^{∗} = \overline{∇}_{0}^{∗} + ϕ^{∗}.

2.3. Sobolev spaces. The idea to consider Sobolev spaces for bundles on noncommutative tori is due to M. Spera (see [7], [8]). Let (E, \overline{∇}) be a standard holomorphic bundle on T_{θ,τ}. For s ∈ ℤ, s ≥ 0 we define the s-th Sobolev norm on E by setting

||e||_{s}^{2} = ∑_{i=0}^{s} ||\overline{∇}^{i}e||_{0}^{2},

where ||e||_{0} is the L^{2}-norm on E. We define W_{s}(E) to be the completion of E with respect to this norm. Note that there is a natural embedding W_{s+1}(E) ⊂ W_{s}(E). We can define analogous spaces for E^{⊕n} in an obvious way. All the definitions above make sense also for rational θ. Moreover, for θ ∈ ℤ the space E can be identified with the space of smooth section of a holomorphic vector bundle V on an elliptic curve ℂ/ℤ + ℤτ in such a way that \overline{∇} corresponds to the \overline{∂}-operator. Furthermore, the L^{2}-norm above corresponds to the L^{2}-norm with respect to a Hermitian metric on V that has constant curvature. This implies that in this case the Sobolev spaces W_{s}(E) coincide with the corresponding Sobolev spaces constructed for the holomorphic bundle V. Indeed, using the equation (2.7) it is easy to see that the norm ||e||_{s} is equivalent to the norm given by

(||e||′_{s})^{2} = ∑_{i=0}^{s}⟨e, (\overline{∇}^{∗}\overline{∇})^{i}e⟩

which is equivalent to the standard Sobolev norm. An important observation is that the operator \overline{∇}_{z} defined by (1.1) depends only on τμ, where μ is the slope of the bundle, so it is the same for the bundle E_{d,c}(θ) on T_{θ,τ} and the bundle E_{d,c}(sign(c)N) on the commutative torus T_{sign(c)N,τ′}, where N is a large enough integer so that |c|N +d > 0, τ′ = (|c|N +d)/(cθ+d). Therefore, the sequences of spaces (W_{s}(E)) in these two cases are the same. Hence, the following standard results about Sobolev spaces in the commutative case extend immediately to our situation (the first two are analogues of Rellich’s lemma and Sobolev’s lemma). In all these results E is a direct sum of a finite number of copies of a standard holomorphic bundle.

Lemma 2.1. The embedding W_{s}(E) ⊂ W_{s−1}(E) is a compact operator.

Lemma 2.2. One has E = ∩_{s≥0}W_{s}(E).

Lemma 2.3. The operator \overline{∇} extends to a bounded operator W_{s}(E) → W_{s−1}(E)

The following result is the only noncommutative contribution to the techniques of Sobolev spaces, however, it is quite easy.

Lemma 2.4. For every ϕ ∈ End_{A_{θ}}(E) the operator ϕ : E → E extends to a bounded operator W_{s}(E) → W_{s}(E).

Proof. It suffices to prove that for every s ≥ 0 one has

||ϕe||_{s} ≤ C · ||e||_{s}

for some constant C > 0. By our assumption E ≃ E_{0}^{⊕N} for some standard bundle E_{0}. Identifying End_{A_{θ}}(E_{0}) with A_{θ′} for some θ′ ∈ ℝ we can write ϕ = ∑ a_{m,n}U_{1}^{m}U_{2}^{n} where U_{1} and U_{2} are unitary generators of A_{θ′}, a_{m,n} are complex N × N matrices. Since U_{1} and U_{2} act on E by unitary operators, it follows that

||ϕe||_{0} ≤ C(ϕ) · ||e||_{0}

for e ∈ E, where C(ϕ) = ∑_{m,n} ||a_{m,n}|| (here ||a|| denote the norm of a matrix a). Applying the Leibnitz rule repeatedly we derive similarly that

∑_{i=0}^{s} ||\overline{∇}^{i}(ϕe)||_{0}^{2} ≤ ∑_{i=0}^{s} c_{i} · ||\overline{∇}^{i}e||_{0}^{2}

for some constants c_{i} > 0 which implies the result. □

It is convenient to extend the definition of the chain … ⊂ W_{1}(E) ⊂ W_{0}(S) to the chain of embedded spaces

… ⊂ W_{1}(E) ⊂ W_{0}(S) ⊂ W_{−1}(E) ⊂ …

by setting W_{−s}(E) = \overline{W_{s}(E)}^{∗} (the space of ℂ-antilinear functionals) and using the natural Hermitian form of W_{0}(E). It is easy to see that the results of this section hold for all integer values of s.

Lemma 2.5. Let \overline{∇} : E → E be a (not necessarily standard) holomorphic structure on E. Then the operators \overline{∇} and \overline{∇}^{∗} can be extended to bounded operators W_{s}(E) → W_{s−1}(E) for every s ∈ ℤ.

Proof. Let \overline{∇}_{0} be a standard holomorphic structure on E. Then \overline{∇} = \overline{∇}_{0} +ϕ for some ϕ ∈ End_{A_{θ}}(E). By Lemma 2.3 (resp., Lemma 2.4) there exist a continuous extension \overline{∇}_{0} : W_{s}(E) → W_{s−1}(E) (resp., ϕ : W_{s}(E) → W_{s}(E)). Hence, \overline{∇} extends to a family of continuous operators \overline{∇}(s) : W_{s}(E) → W_{s−1}(E) for s ∈ ℤ. The extensions of \overline{∇}^{∗} are given by the adjoint operators \overline{∇}(−s + 1)^{∗} : W_{s}(E) → W_{s−1}(E). □

2.4. Applications to cohomology. We begin our study of cohomology with standard holomorphic bundles.

Proposition 2.6. Let (E, \overline{∇}) be a direct sum of several copies of a standard holomorphic bundle on T_{θ,τ}.

(i) The cohomology spaces H^{0}(E) and H^{1}(E) are finite-dimensional and for Im(τ) < 0 one has

χ(E) = dim H^{0}(E) − dim H^{1}(E) = deg(E).

(ii) There exists an operator Q : E → E such that

id −Q\overline{∇} = π_{ker \overline{∇}},
id −\overline{∇}Q = π_{\overline{∇}(E)^{⊥}},

where \overline{∇}(E)^{⊥} ⊂ E is the orthogonal complement to \overline{∇}(E) ⊂ E, for a finitedimensional subspace V ⊂ E we denote by π_{V} : E → V the orthogonal projection. (iii) If Im(τ) < 0 and deg(E) > 0 then H^{1}(E) = 0. (iv) The operator Q : E → E extends to a bounded operator W_{s}(E) → W_{s+1}(E).

(v) For every e ∈ W_{0}(E) one has

||Qe||_{0} ≤ \frac{1}{2\sqrt{π| Im(τ)μ(E)|}}||e||_{0}.

Proof. In the commutative case the assertions (i)-(iii) are well known. For example, the operator Q is given by \overline{∂}^{∗}G, where G is the Green operator for the \overline{∂}-Laplacian. The condition Im(τ) < 0 corresponds to the way we define the operator δ_{τ} on A_{θ} (see Proposition 3.1 of [2]). As before we can deduce (i)- (iii) in general from the commutative case. One can also prove these assertions directly in the noncommutative case (see Proposition 2.5 of [2] for the proofs of (i) and (iii)). The assertion (iv) follows immediately from the identity

\overline{∇}^{n}Q = \overline{∇}^{n−1}(id −π_{\overline{∇}(E)^{⊥}}).

To prove (v) we can assume that E = E_{d,c}(θ), where c ≠ 0 and \overline{∇} = \overline{∇}_{z} for some z ∈ ℂ. Then the space W_{0}(E) is the orthogonal sum of |c| copies of L^{2}(ℝ). Moreover, the operator \overline{∇} respects this decomposition and restricts to the operator

f ↦ f′ + (ax + z)f

on each copy, where a = 2πiτμ(E). Hence, the operator Q also respects this decomposition and it suffices to consider its restriction to one copy of L^{2}(ℝ). Since Re(a) ≠ 0, by making the unitary transformation of the form f̃(x) = exp(itx)f(x + t′) for some t, t′ ∈ ℝ we can reduce ourselves to the case z = 0. Furthermore, the transformation of the form f̃ = exp(i Im(a)x^{2}/2)f gives a unitary equivalence with the operator \overline{∇} : f ↦ f′ + λxf where λ = Re(a). Consider the following complete orthogonal system of functions in L^{2}(ℝ):

f_{n}(x) = H_{n}(\sqrt{|λ|}x) exp(−|λ|\frac{x^{2}}{2} ), n = 0, 1, 2, …,

where H_{n}(x) = (−1)^{n} exp(x^{2})\frac{d^{n}}{dx^{n}} (exp(−x^{2})) are Hermite polynomials ((f_{n}) is an eigenbasis of the operator f ↦ −f′′ + λ^{2}x^{2}). Note that

||f_{n}||_{0}^{2} = \frac{1}{\sqrt{λ}} ∫_{ℝ} H_{n}(x)^{2} exp(−x^{2})dx = \frac{2^{n} · n! · \sqrt{π}}{\sqrt{λ}} .

Assume first that λ > 0. Then using the formula H′_{n}(x) = 2nH_{n−1}(x) we obtain

\overline{∇}(f_{n}) = 2n\sqrt{λ}f_{n−1}

for n > 0 and \overline{∇}(f_{0}) = 0. Therefore, in this case

Q(f_{n}) = \frac{1}{(2n + 2)\sqrt{λ}}f_{n+1}

for all n ≥ 0. Hence,

\frac{||Q(f_{n})||_{0}}{||f_{n}||_{0}} = \frac{||f_{n+1}||_{0}}{(2n + 2)\sqrt{λ}||f_{n}||_{0}} = \frac{1}{\sqrt{(2n + 2)λ}} ≤ \frac{1}{\sqrt{2λ}}

which implies (v) in this case. Now assume that λ < 0. Then using the formula H′_{n}(x) = 2xH_{n}(x) − H_{n+1}(x) we find

\overline{∇}(f_{n}) = −\sqrt{|λ|}f_{n+1}.

Hence,

Q(f_{n}) = −\frac{1}{\sqrt{|λ|}}f_{n−1}

for n > 0 and Q(f_{0}) = 0. It follows that

\frac{||Q(f_{n})||_{0}}{||f_{n}||_{0}} = \frac{||f_{n−1}||_{0}}{\sqrt{|λ|}||f_{n}||_{0}} = \frac{1}{\sqrt{2n|λ|}} ≤ \frac{1}{\sqrt{2|λ|}}

for n ≥ 1, which again implies our statement. □

Now we are ready to prove results about cohomology of arbitrary holomorphic bundles. We will use the following well known lemma.

Lemma 2.7. Let L : W → W′ and L′ : W′ → W be bounded operators between Banach spaces such that L′L = id +C, LL′ = id +C′ for some compact operators C : W → W and C′ : W′ → W′. Then the operator L is Fredholm.

Theorem 2.8. (i) For every holomorphic bundle (E, \overline{∇}) on T_{θ,τ} the spaces H^{0}(E) and H^{1}(E) are finite-dimensional. (ii) If Im τ < 0 then

χ(E) = dim H^{0}(E) − dim H^{1}(E) = deg(E).

(iii) Let us equip E with a metric by identifying it with the direct sum of several copies of a standard bundle. Then one has the following orthogonal decompositions

E = ker(\overline{∇}) ⊕ \overline{∇}^{∗}(E),
E = ker(\overline{∇}^{∗}) ⊕ \overline{∇}(E),

where \overline{∇}^{∗} : E → E is the adjoint operator to \overline{∇}.

Proof. Let us write the holomorphic structure on E in the form

\overline{∇} = \overline{∇}_{0} + ϕ,

where (E, \overline{∇}_{0}) is holomorphically isomorphic to the direct sum of several copies of a standard holomorphic bundle, ϕ ∈ End_{A_{θ}}(E). By Lemma 2.5, \overline{∇} has a bounded extension to an operator \overline{∇} : W_{s}(E) → W_{s−1}(E) for every s ∈ ℤ. Consider the operator Q : E → E constructed in Proposition 2.6 for the holomorphic structure \overline{∇}_{0}. Then Q extends to a bounded operator W_{s}(E) → W_{s+1}(E) for every s ∈ ℤ. We have

Q\overline{∇} = Q\overline{∇}_{0} + Qϕ = id −π_{0} + Qϕ,

where π_{0} is the orthogonal projection to the finite-dimensional space ker(\overline{∇}_{0}) ⊂ E. Clearly, π_{0} defines a bounded operator W_{0}(E) → ker(\overline{∇}_{0}). Hence, the operator C = Q\overline{∇}−id : W_{s}(E) → W_{s}(E) factors as a composition of some bounded operator W_{s}(E) → W_{s+1}(E) with the embedding W_{s+1}(E) → W_{s}(E). By Lemma 2.1 this implies that C is a compact operator. Similarly, the operator C′ = \overline{∇}Q − id : W_{s}(E) → W_{s}(E) is compact. Applying Lemma 2.7 we deduce that \overline{∇} : W_{s}(E) → W_{s−1}(E) is a Fredholm operator. This immediately implies that H^{0}(E) is finite-dimensional. Moreover, we claim that

ker(\overline{∇} : W_{s}(E) → W_{s−1}(E)) = H^{0}(E) ⊂ E

for any s ∈ ℤ. Indeed, it suffices to check that if \overline{∇}(e) = 0 for e ∈ W_{s}(E) then e ∈ E. Let us prove by induction in t ≥ s that e ∈ W_{t}(E). Assume that this is true for some t. Then

e = Q\overline{∇}(e) − C(e) = −C(e) ∈ W_{t+1}(E).

Since ∩_{t}W_{t}(E) = E by Lemma 2.2 we conclude that e ∈ E. Let Q^{∗} : W_{s}(E) → W_{s+1}(E) be the adjoint operator to Q : W_{−s−1}(E) → W_{−s}(E), where s ∈ ℤ. Then the operators Q^{∗}\overline{∇}^{∗} −id = (C′)^{∗} and \overline{∇}^{∗}Q^{∗} −id = C^{∗} are compact. Thus, the same argument as before shows that for every s ∈ ℤ the operator \overline{∇}^{∗} : W_{s}(E) → W_{s−1}(E) is Fredholm and one has

ker(\overline{∇}^{∗} : W_{s}(E) → W_{s−1}(E)) ⊂ E.

Next we claim that

E = ker(\overline{∇}^{∗}) ⊕ \overline{∇}(E).

Since the orthogonal complement to ker(\overline{∇}^{∗}) in W_{0}(E) coincides with the image of \overline{∇} : W_{1}(E) → W_{0}(E), it suffices to prove that E ∩\overline{∇}(W_{1}(E)) ⊂ \overline{∇}(E). But if e = \overline{∇}(e_{1}) for some e ∈ E, e_{1} ∈ W_{1}(E), then we can easily prove by induction in s ≥ 1 that e_{1} ∈ W_{s}(E). Indeed, assuming that e_{1} ∈ W_{s}(E) we have

e_{1} = Q\overline{∇}(e_{1}) − C(e_{1}) = Q(e) − C(e_{1}) ∈ W_{s+1}(E).

A similar argument using the operator Q^{∗} shows that

E = ker(\overline{∇}) ⊕ \overline{∇}^{∗}(E).

Thus, we checked that H^{0}(E) and H^{1}(E) are finite-dimensional and that χ(E) coincides with the index of the Fredholm operator \overline{∇} = \overline{∇}_{0} + ϕ : W_{1}(E) → W_{0}(E). Note that

\overline{∇}_{t} := \overline{∇}_{0} + tϕ : W_{1}(E) → W_{0}(E)

is a continuous family of Fredholm operators depending on t ∈ [0, 1]. It follows that the index of \overline{∇} = \overline{∇}_{1} is equal to the index of \overline{∇}_{0} computed in Proposition 2.6. □

Corollary 2.9. If one of the holomorphic bundles E_{1} and E_{2} is standard then the spaces Hom_{C}(E_{1}, E_{2}) and Ext_{C}^{1}(E_{1}, E_{2}) are finite-dimensional and

χ(E_{1}, E_{2}) := dim Hom_{C}(E_{1}, E_{2}) − dim Ext_{C}^{1}(E_{1}, E_{2})
= rk(E_{1}) deg(E_{2}) − rk(E_{2}) deg(E_{1}).

The following vanishing result will play a crucial role in the proof of Theorem 1.1.

Theorem 2.10. Assume that Im(τ) < 0. For every holomorphic bundle E on T = T_{θ,τ} there exists a constant C = C(E) ∈ ℝ such that for every standard holomorphic bundle E_{0} on T with μ(E_{0}) < C one has Ext_{C}^{1}(E_{0}, E) = 0.

Proof. Let us choose a (non-holomorphic) isomorphism E ≃ E_{1}^{⊕N}, where E_{1} is a standard holomorphic bundle on T_{θ,τ}. Then we can write the holomorphic structure on E as

\overline{∇}_{E} = \overline{∇}_{0} + ϕ,

where \overline{∇}_{0} comes from the standard holomorphic structure on E_{1}, ϕ ∈ End_{A_{θ}}(E). Then for every standard holomorphic bundle E_{0} we can consider the holomorphic bundle E′ = Hom_{A_{θ}}(E_{0}, E) on T_{θ′,τ}, where End_{A_{θ}}(E_{0}) = A_{θ′}. Note that Ext_{C}^{1}(E_{0}, E) ≃ H^{1}(E′), so we want to prove that the latter group vanishes for μ(E_{0}) << 0. Recall that the holomorphic structure \overline{∇}′ on E′ is given by

\overline{∇}′(f)(e_{0}) = rk(E_{0}) · [\overline{∇}(f(e_{0})) − f(\overline{∇}_{E_{0}}(e_{0}))],

where e_{0} ∈ E_{0}, f ∈ E′ (see section 2.2 of [2]). The isomorphism E ≃ E_{1}^{⊕N} induces an isomorphism E′ ≃ (E′_{1})^{⊕N}, where E′_{1} is the standard bundle Hom_{A_{θ}}(E_{0}, E_{1}) on T_{θ′,τ}. Therefore, we have

\overline{∇}′ = \overline{∇}′_{0} + rk(E_{0})ϕ,

where \overline{∇}′_{0} corresponds to the standard holomorphic structure on (E′_{1})^{⊕N} and ϕ is now considered as an A_{θ′}-linear endomorphism of E′. Note that by Proposition 2.6(iii) we have H^{1}(E′, \overline{∇}′_{0}) = 0 as long as μ(E′) > 0. It is easy to compute that

rk(E′) = rk(E)/ rk(E_{0}), deg(E′) = rk(E) rk(E_{0})(μ(E) − μ(E_{0})),

hence

μ(E′) = rk(E_{0})^{2}(μ(E) − μ(E_{0})).

Therefore, μ(E′) > 0 provided that μ(E_{0}) < μ(E). In this case the operator Q on E′ constructed in Proposition 2.6 for the standard holomorphic structure \overline{∇}′_{0} satisfies \overline{∇}′_{0}Q = id. Let C_{0} be a constant such that

||Qe′||_{0} ≤ \frac{C_{0}}{\sqrt{μ(E′)}}||e′||_{0}

for e′ ∈ W_{0}(E′) (see Proposition 2.6(v)). Also, let us write ϕ ∈ End_{A_{θ}}(E) ≃ End_{A_{θ}}(E_{1}^{⊕N}) in the form ϕ = ∑ a_{m,n}U_{1}^{m}U_{2}^{n}, where U_{1} and U_{2} are unitary generators of End_{A_{θ}}(E_{1}) and a_{m,n} are N × N complex matrices. Then we set C(ϕ) = ∑_{m,n} ||a_{m,n}|| (the sum of the matrix norms of all coefficients). Now we choose the constant C < μ(E) in such a way that

\frac{C_{0}C(ϕ)}{\sqrt{μ(E) − C}} < 1.

Then for μ(E_{0}) < C we will have

rk(E_{0}) · ||ϕQe′||_{0} ≤ \frac{rk(E_{0})C_{0}C(ϕ)}{\sqrt{μ(E′)}}||e′||_{0} < \frac{C_{0}C(ϕ)}{\sqrt{μ(E) − C}} ||e′||_{0} < r · ||e′||_{0}

for some 0 < r < 1. It follows from the above estimate that the operator id + rk(E_{0})ϕQ : W_{0}(E′) → W_{0}(E′) is invertible. Therefore, we can define the operator

Q̃ = Q(id + rk(E_{0})ϕQ)^{−1} : W_{0}(E′) → W_{1}(E′)

that satisfies

(\overline{∇}′_{0} + rk(E_{0})ϕ)Q̃ = id .

Hence, the operator \overline{∇}′ = \overline{∇}′_{0} + rk(E_{0})ϕ : W_{1}(E′) → W_{0}(E′) is surjective. But H^{1}(E′) can be identified with the cokernel of this operator (see the proof of Theorem 2.8), so H^{1}(E′) = 0. □

2.5. Serre duality. For every holomorphic bundle E we have a natural pairing

E ⊗_{A_{θ}} E^{∨} → A_{θ} : e ⊗ f ↦ f(e).

It is compatible with the \overline{∂}-operators, so it induces a pairing

H^{1−i}(E) ⊗ H^{i}(E^{∨}) → H^{1}(A_{θ}) ≃ ℂ  (2.8)

for i = 0, 1.

Theorem 2.11. The pairing (2.8) is perfect.

Proof. Since we can switch E and E^{∨}, it suffices to consider the case i = 0. We choose an isomorphism of E with a direct sum of several copies of a standard holomorphic bundle E_{0}, so that we can talk about standard metrics and Sobolev spaces. Note that the isomorphism H^{1}(A_{θ}) is induced by the trace functional tr : A_{θ} → ℂ : ∑ a_{m,n}U_{1}^{m}U_{2}^{n} ↦ a_{0,0}. Hence, the pairing (2.8) is induced by the pairing

b : E ⊗ E^{∨} → ℂ : e ⊗ f ↦ tr(f(e)) that satisfies the identity

rk(E_{0})b(\overline{∇}_{E}(e), e^{∨}) + b(e, \overline{∇}_{E^{∨}}(e^{∨})) = 0,  (2.9)

where e ∈ E, e^{∨} ∈ E^{∨} (see Proposition 2.2 of [2]). According to Theorem 2.8(iii) we have orthogonal decompositions

E^{∨} = ker(\overline{∇}_{E^{∨}}) ⊕ \overline{∇}_{E^{∨}}^{∗}(E^{∨}),
E = ker(\overline{∇}_{E}^{∗}) ⊕ \overline{∇}_{E}(E).

Therefore, it suffices to check that b induces a perfect pairing between ker(\overline{∇}_{E}^{∗}) and ker(\overline{∇}_{E^{∨}}). Let σ : E → E^{∨} be the ℂ-antilinear isomorphism defined in section 2.2. We claim that σ maps ker(\overline{∇}_{E}^{∗}) isomorphically onto ker(\overline{∇}_{E^{∨}}). Since b(e_{1}, σ(e_{2})) = ⟨e_{1}, e_{2}⟩ for e_{1}, e_{2} ∈ E, the theorem would immediately follow this. To prove the claim it is enough to check that \overline{∇}_{E^{∨}} = − rk(E_{0})σ\overline{∇}_{E}^{∗}σ^{−1}. To this end let us rewrite (2.9) as follows:

rk(E_{0})⟨\overline{∇}_{E}(e), σ^{−1}e^{∨}⟩ = −⟨e, σ^{−1}\overline{∇}_{E^{∨}}(e^{∨})⟩.

Since the left-hand side is equal to rk(E_{0})⟨e, \overline{∇}_{E}^{∗}σ^{−1}e^{∨}⟩ we conclude that rk(E_{0})\overline{∇}_{E}^{∗}σ^{−1} = −σ^{−1}\overline{∇}_{E^{∨}} as required. □

Recall that we denote by C′ ⊂ C the full subcategory consisting of all successive extensions of standard holomorphic bundles. Theorem 3.8 of [2] implies that the derived category of C′ is equivalent to the derived category of coherent sheaves on an elliptic curve. Therefore, the standard Serre duality gives a functorial isomorphism

Ext_{C′}^{1}(E_{1}, E_{2}) ≃ Hom_{C′}(E_{2}, E_{1})^{∗}

for E_{1}, E_{2} ∈ C′. Note that we can replace here Ext_{C′}^{i} with Ext_{C}^{i}. Now using the above theorem we can extend this isomorphism to the case when only one of the objects E_{1}, E_{2} belongs to C′.

Corollary 2.12. For every holomorphic bundles E and E_{0} such that E_{0} ∈ C′ the natural pairings

Ext_{C}^{i}(E_{0}, E) ⊗ Ext_{C}^{1−i}(E, E_{0}) → Ext_{C}^{1}(E_{0}, E_{0}) → ℂ

for i = 0, 1 are perfect. Here the functional Ext_{C}^{1}(E_{0}, E_{0}) → ℂ is induced by the Serre duality on C′. Therefore, we have functorial isomorphisms

Ext_{C}^{1−i}(E, E_{0})˜→ Ext_{C}^{i}(E_{0}, E)^{∗}  (2.10)

for E ∈ C, E_{0} ∈ C′.

Proof. If E_{0} is standard the assertion follows from Theorem 2.11. It remains to observe that if for fixed E ∈ C the map (2.10) is an isomorphism for some E_{0}, E′_{0} ∈ C′ then it is also an isomorphism for any extension of E_{0} by E′_{0}. □

3. Ampleness

3.1. Ample sequences of standard holomorphic bundles. Let us start by recalling some basic notions concerning ample sequences in abelian categories and associated ℤ-algebras. The reader can consult [3] for more details.

Definition.(see [9],[3]): Let (E_{n})_{n∈ℤ} be a sequence of objects in a ℂ-linear abelian category A such that Hom(E_{n}, E) is finite-dimensional for every E ∈ A and every n ∈ ℤ. Then (E_{n}) is called ample if the following two conditions hold:

(i) for every surjection E → E′ in A the induced map Hom(E_{n}, E) → Hom(E_{n}, E′) is surjective for all n << 0; (ii) for every object E ∈ A and every N ∈ ℤ there exists a surjection ⊕_{i=1}^{s}E_{n_{i}} → E where n_{i} < N for all i.

To a sequence (E_{n})_{n∈ℤ} one can associate a so called ℤ-algebra A = ⊕_{i≤j}A_{ij}, where A_{ii} = ℂ, A_{ij} = Hom(E_{i}, E_{j}) for i < j, the multiplications A_{jk} ⊗ A_{ij} → A_{ik} are induced by the composition in C. One can define for ℤ-algebras all the standard notions associated with graded algebras (see [3]). In particular, we can talk about right A-modules: these have form M = ⊕_{i∈ℤ}M_{i} and the right A-action is given by the maps M_{j} ⊗ A_{ij} → M_{i}. The analogues of free A-modules are direct sums of the modules P_{n}, n ∈ ℤ, defined by (P_{n})_{i} = A_{ni}. We say that an A-module M is finitely generated if there exists a surjection ⊕_{i=1}^{s}P_{n_{i}} → M. A finitely generated A-module M is called coherent if for every morphism f : P → M, where P is a finitely generated free module, the module ker(f) is finitely generated. Finally, a ℤ-algebra A is called coherent if all the modules P_{n} are coherent and in addition all one-dimensional A-modules are coherent. The main theorem of [3] asserts that if (E_{n}) is ample then the ℤ-algebra A is coherent and the natural functor E ↦ ⊕_{i<0} Hom(E_{i}, E) gives an equivalence of categories

A ≃ cohproj A  (3.1)

where cohproj A is the quotient of the category of coherent right A-modules by the subcategory of finite-dimensional modules. We are going to apply this theorem to the category C′ generated by standard holomorphic bundles on T = T_{θ,τ}. Recall that in [2] we identified this category with a certain abelian subcategory C^{θ} of the derived category D^{b}(X) of coherent sheaves on the elliptic curve X = ℂ/ℤ + ℤτ. To define C^{θ} one has to consider two full subcategories in the category Coh(X) of coherent sheaves on X: Coh_{<θ} (resp., Coh_{>θ}) is the minimal subcategory of Coh(X) closed under extensions and containing all stable bundles of slope < θ (resp., all stable bundles of slope > θ and all torsion sheaves). Then by the definition

C^{θ} = {K ∈ D^{b}(X) : H^{>0}(K) = 0, H^{0}(K) ∈ Coh_{>θ},
H^{−1}(K) ∈ Coh_{<θ}, H^{<−1}(K) = 0}.

Thus, C^{θ} contains Coh_{>θ} and Coh_{<θ}[1] and these two subcategories generate C^{θ} in an appropriate sense. The fact that C^{θ} is abelian follows from the torsion theory (see [1]). Note that the vectors (deg(K), rk(K)) ∈ ℤ^{2} for K ∈ C^{θ} are characterized by the inequality

deg(K) − θ rk(K) > 0.

In [2] we showed that a version of Fourier-Mukai transform gives an equivalence S : C′˜→C^{θ} (this S differs from the transform studied in section 3.3 of [2] by the shift K ↦ K[1]). Standard holomorphic bundles correspond under S to stable objects of C^{θ}: the latter are structure sheaves of points and objects of the form V [n] where V is a stable bundle, n ∈ {0, 1}. Moreover, one has

deg S(E_{d,c}(θ)) = d, rk S(E_{d,c}(θ)) = −c.

It follows that

rk(S(E)) = − deg(E), μ(S(E)) = θ − μ(E)^{−1}.

The following criterion of ampleness in C′ is essentially contained in the proof of Theorem 3.5 of [4], where we showed the existence of ample sequences in C′.

Theorem 3.1. Let (E_{n}) be a sequence of standard holomorphic bundles on T such that μ(E_{n}) → −∞ as n → −∞ and rk(E_{n}) > c for all n << 0 for some constant c > 0. Then (E_{n}) is an ample sequence in C′. Moreover, for every E ∈ C′ the natural morphism Hom(E_{n}, E) ⊗_{ℂ} E_{n} → E in C is surjective for n << 0.

Proof. Let F_{n} = S(E_{n}) be the corresponding sequence of stable objects of C^{θ}. Then rk(F_{n}) = − deg(E_{n}) → +∞ and μ(F_{n}) = θ − μ(E_{n})^{−1} → θ as n → −∞. Moreover, we have

μ(F_{n}) − θ = \frac{rk(E_{n})}{rk(F_{n})} > \frac{c}{rk(F_{n})}.

Therefore, the same proof as in Theorem 3.5 of [4] (where we considered only the special case c = 1) shows that the sequence (F_{n}) is ample in C^{θ} and that for every F ∈ C^{θ} the morphism Hom(F_{n}, F) ⊗_{ℂ} F_{n} → F is surjective for n << 0. Hence, the same assertions hold for the sequence (E_{n}) in C′. □

Theorem 3.2. Let (E_{n}) be a sequence as in Theorem 3.1 and let A = ⊕_{i≤j} Hom_{C}(E_{i}, E_{j}) be the corresponding ℤ-algebra. Then for every holomorphic bundle E on T the A-module M(E) = ⊕_{i<0} Hom_{C}(E_{i}, E) is coherent. Also, for every sufficiently small i_{0} the canonical morphism of A-modules

Hom_{C}(E_{i_{0}}, E) ⊗ P_{i_{0}} → M(E)

has finite-dimensional cokernel.

First, we need a criterion for finite generation of modules M(E) with a weaker assumption on (E_{n}).

Lemma 3.3. Let E be a holomorphic bundle on T and let C = C(E) be the corresponding constant from Theorem 2.10. Let (E_{i}) be a sequence of standard holomorphic bundles such that μ(E_{i}) → μ as i → −∞, where μ ∈ ℝ ∪ {−∞}. Assume that for some ϵ > 0 one has rk(E_{i})^{2}(μ(E_{i}) − μ) > 1 + ϵ for all i << 0 (this condition is vacuous if μ = −∞). Then

(i) for every sufficiently small i_{0} ∈ ℤ there exists i_{1} = i_{1}(i_{0}) such that for all i < i_{1} there exists a standard holomorphic bundle F_{i} fitting into the following short exact sequence in C′:

0 → E_{i} →^{can} Hom_{C}(E_{i}, E_{i_{0}})^{∗} ⊗ E_{i_{0}} → F_{i} → 0,  (3.2)

where can is the canonical morphism. (ii) Assume in addition that μ < C and if μ = −∞ then for some c > 0 one has rk(E_{i}) > c for all i << 0. Then for all sufficiently small i_{0} there exists i_{1} such that for i < i_{1} one has Ext^{1}(F_{i}, E) = 0, where F_{i} is defined by (3.2). Under the same assumptions the A-module M(E) is finitely generated.

Proof. (i) Let us denote r_{i} = rk(E_{i}), μ_{i} = μ(E_{i}). If μ is finite then for every sufficiently small i_{0} one has r_{i_{0}}^{2}(μ_{i_{0}} − μ) > 1. Therefore, we can find i_{1} < i_{0} such that for i < i_{1} one has r_{i_{0}}^{2}(μ_{i_{0}} − μ_{i}) > 1. If μ = −∞ then we can take any i_{0} and then still find i_{1} < i_{0} such that for i < i_{1} the above inequality holds. We are going to construct F_{i} in this situation. Using the equivalence S : C′˜→C^{θ} ⊂ D^{b}(X) we can first define F̃_{i} ∈ D^{b}(X) from the exact triangle

S(E_{i}) → Hom_{C}(E_{i}, E_{i_{0}})^{∗} ⊗ S(E_{i_{0}}) → F̃_{i} → S(E_{i})[1].

In other words, F̃_{i} is the image of E_{i} under the equivalence R_{E_{i_{0}}} : D^{b}(X) → D^{b}(X) given by the right twist with respect to E_{i_{0}} (see [4], sec.2.3, or [6]; our functor differs from that of [6] by a shift). It follows that Hom_{D^{b}(E)}(F̃_{i}, F̃_{i}) ≃ Hom_{C}(E_{i}, E_{i}) ≃ ℂ, so F̃_{i} is a stable object and either F̃_{i} ∈ C^{θ} or F̃_{i} ∈ C^{θ}[1]. To prove that F̃_{i} ∈ C^{θ} it suffices to check that deg(F̃_{i}) − θ rk(F̃_{i}) > 0. But

deg(F̃_{i}) − θ rk(F̃_{i}) = χ(E_{i}, E_{i_{0}})r_{i_{0}} − r_{i} = ((μ_{i_{0}} − μ_{i})r_{i_{0}}^{2} − 1)r_{i} > 0

by our choice of i. Hence, we have F̃_{i} ∈ C^{θ} and we can set F_{i} = S^{−1}(F̃_{i}). (ii) If μ is finite then we can choose i_{0} such that r_{i_{0}}^{2}(μ_{i_{0}} − μ) > 1 + ϵ and μ_{i_{0}} + ϵ^{−1}(μ_{i_{0}} − μ) < C. If μ = −∞ then we choose i_{0} such that μ_{i_{0}} + 2r_{i_{0}}^{−2} < C (here we use the assumption that r_{i} > c for i << 0). In either case for sufficiently small i we have short exact sequence (3.2). Applying to it the functor Hom(?, E) we get the long exact sequence

0 → Hom_{C}(F_{i}, E) → Hom_{C}(E_{i_{0}}, E) ⊗ Hom_{C}(E_{i}, E_{i_{0}})
→ Hom_{C}(E_{i}, E) → Ext_{C}^{1}(F_{i}, E).

Thus, vanishing of Ext_{C}^{1}(F_{i}, E) for all i << 0 would imply that the A-module M(E) is finitely generated. By the definition of the constant C this vanishing would follow from the inequality μ(F_{i}) < C. In the case μ ≠ −∞ we have for i << 0

μ(F_{i}) = μ_{i_{0}} + \frac{μ_{i_{0}} − μ_{i}}{r_{i_{0}}^{2}(μ_{i_{0}} − μ_{i}) − 1} < μ_{i_{0}} + ϵ^{−1}(μ_{i_{0}} − μ_{i}),

so the required inequality follows for i << 0 from our choice of i_{0}. In the case μ = −∞ we can finish the proof similarly using the inequality

μ(F_{i}) < μ_{i_{0}} + \frac{2}{r_{i_{0}}^{2}}

that holds for all i << 0. □

Proof of Theorem 3.2. By Lemma 3.3 for any sufficiently small i_{0} there exists i_{1} (depending on i_{0}) such that we have short exact sequence (3.2) and the induced sequence of A-modules

0 → ⊕_{i<i_{1}} Hom_{C}(F_{i}, E) → Hom_{C}(E_{i_{0}}, E) ⊗ (P_{i_{0}})_{<i_{1}} → M(E)_{<i_{1}} → 0

is exact, where for every A-module M = ⊕M_{i} we set M_{<n} = ⊕_{i<n}M_{i}. This immediately implies the last assertion of the theorem. Note that the structure of the A-module on ⊕ Hom_{C}(F_{i}, E) is defined using the natural isomorphisms Hom_{C}(F_{i}, F_{j}) ≃ Hom_{C}(E_{i}, E_{j}) coming from the equality F_{i} = R_{E_{i_{0}}} (E_{i}), where R_{E_{i_{0}}} is the right twist with respect to E_{i_{0}}. It suffices to prove that the module M′(E) := ⊕_{i<i_{1}} Hom_{C}(F_{i}, E) is finitely generated. Indeed, this would imply that the module M(E)_{<i_{1}} is finitely presented and hence coherent (since A is coherent), therefore, the module M(E) is also coherent. To check that M′(E) is finitely generated we will use the criterion of Lemma 3.3 for the sequence (F_{i}). We have

μ(F_{i}) → μ = μ_{i_{0}} + 1/r_{i_{0}}^{2}

as i → −∞. Also,

rk(F_{i})^{2}(μ(F_{i}) − μ) = \frac{((μ_{i_{0}} − μ_{i})r_{i_{0}}^{2} − 1)r_{i}^{2}}{r_{i_{0}}^{2}} > \frac{((μ_{i_{0}} − μ_{i})r_{i_{0}}^{2} − 1)c^{2}}{r_{i_{0}}^{2}} → +∞

as i → −∞. Hence, the conditions of Lemma 3.3 will be satisfied once we show that μ = μ_{i_{0}} + 1/r_{i_{0}}^{2} can be made smaller than any given constant by an appropriate choice of i_{0}. But this is of course true since μ_{i_{0}} +1/r_{i_{0}}^{2} < μ_{i_{0}} +1/c^{2} and μ_{i_{0}} → −∞ as i_{0} → −∞. □

3.2. Proof of Theorem 1.1. Let us pick a sequence (E_{n})_{n∈ℤ} of stable holomorphic bundles satisfying conditions of Theorem 3.1 (it is easy to see that such a sequence exists, see the proof of Theorem 3.5 in [4]). Let E be a holomorphic bundle on T. Then by Theorem 3.2 the module M = ⊕_{i} Hom_{C}(E_{i}, E) is coherent, hence, we can consider the object E′ ∈ C′ corresponding to this module via the equivalence (3.1). By the definition this means that there is an isomorphism of A-modules

M(E′)_{<i_{0}} ≃ M(E)_{<i_{0}}  (3.3)

for some i_{0}. By Theorem 3.2 assuming that i_{0} is small enough we can ensure that for all i < i_{0} the canonical morphism M(E′)_{i} ⊗ P_{i} → M(E′) has finitedimensional cokernel. We claim that there exists a morphism f : E′ → E in C that induces the same isomorphism of A-modules M(E′)_{<i_{1}} ≃ M(E)_{<i_{1}} for some i_{1} < i_{0} as the isomorphism (3.3). Indeed, by Theorem 3.1 we can find a resolution for E′ in C′ of the form

… → V_{1} ⊗ E_{n_{1}} → V_{0} ⊗ E_{n_{0}} → E′ → 0  (3.4)

where V_{0} = Hom(E_{n_{0}}, E′) and n_{j} < i_{0} for all j ≥ 0. Using this resolution we can compute Hom_{C}(E′, E):

Hom_{C}(E′, E) ≃ ker(V_{0}^{∗} ⊗ Hom_{C}(E_{n_{0}}, E) → V_{1}^{∗} ⊗ Hom_{C}(E_{n_{1}}, E)).

Using isomorphism (3.3) we can identify this space with

ker(V_{0}^{∗} ⊗ Hom_{C}(E_{n_{0}}, E′) → V_{1}^{∗} ⊗ Hom_{C}(E_{n_{1}}, E′)) ≃ Hom_{C}(E′, E′).

Thus, we obtain an isomorphism Hom_{C}(E′, E) ≃ Hom_{C}(E′, E′). We define f ∈ Hom_{C}(E′, E) to be the element corresponding to the identity in Hom_{C}(E′, E′). Let us check that f induces the same isomorphism as (3.3) on some truncations of the modules M(E′) and M(E). The definition of f implies that the composition of the induced morphism f_{∗} : M(E′) → M(E) with the natural morphism V_{0} ⊗ P_{n_{0}} = M(V_{0} ⊗ E_{n_{0}}) → M(E′) coincides with the morphism V_{0} ⊗ P_{n_{0}} → M(E) induced by the isomorphism V_{0} = Hom(E_{n_{0}}, E′) ≃ Hom(E_{n_{0}}, E) induced by (3.3). Therefore, our claim follows from the fact that the above morphism V_{0} ⊗ P_{n_{0}} → M(E′) induces a surjective morphism on appropriate truncations. Thus, we can assume from the beginning that the isomorphism (3.3) is induced by a morphism f : E′ → E. Next, we are going to construct a morphism g : E′ → E such that g ∘ f = id_{E′}. To do this we note that by Serre duality Hom_{C}(E, E′) ≃ Ext_{C}^{1}(E′, E)^{∗} (see Corollary 2.12). Let us make n_{0} smaller if needed so that Ext_{C}^{1}(E_{n_{0}}, E) = Ext_{C}^{1}(E_{n_{0}}, E′) = 0. Then the space Ext_{C}^{1}(E′, E) can be computed using resolution (3.4):

Ext_{C}^{1}(E′, E) ≃ H^{1}[V_{0}^{∗} ⊗ Hom_{C}(E_{n_{0}}, E)
→ V_{1}^{∗} ⊗ Hom_{C}(E_{n_{1}}, E) → V_{2}^{∗} ⊗ Hom_{C}(E_{n_{2}}, E)]. (3.5)

Indeed, let us define K_{1} ∈ C′ from the short exact sequence

0 → K_{1} → V_{0} ⊗ E_{n_{0}} → E′ → 0,

so that we have the following resolution for K_{1}:

… → V_{2} ⊗ E_{n_{2}} → V_{1} ⊗ E_{n_{1}} → K_{1} → 0.

Then the isomorphism (3.5) can be derived from the induced exact sequences

Hom_{C}(V_{0}⊗, E_{n_{0}}, E) → Hom_{C}(K_{1}, E) → Ext_{C}^{1}(E′, E) → Ext_{C}^{1}(V_{0}⊗E_{n_{0}}, E) = 0,

0 → Hom_{C}(K_{1}, E) → Hom_{C}(V_{1} ⊗ E_{n_{1}}, E) → Hom_{C}(V_{2} ⊗ E_{n_{2}}, E).

Using the fact that isomorphism (3.5) is functorial in E such that Ext_{C}^{1}(E_{n_{0}}, E) = 0 we derive that the morphism Ext_{C}^{1}(E′, E′) → Ext_{C}^{1}(E′, E) induced by f is an isomorphism. But there is a natural functional ϕ′ ∈ Ext_{C}^{1}(E′, E′)^{∗} given by Serre duality. Let ϕ ∈ Ext_{C}^{1}(E′, E)^{∗} be the corresponding functional. The isomorphism Ext_{C}^{1}(E′, E)^{∗}˜→ Hom_{C}(E, E′) maps ϕ to some element g ∈ Hom_{C}(E, E′). By functoriality of the Serre duality the following diagram is commutative:

Hom_{C}(E, E′) \overline{\begin{matrix}α \\  \end{matrix}}Ext_{C}^{1}(E′, E)^{∗}

[figure: Hom_{C}(E′, E′); f^{∗}; α′; Ext_{C}^{1}(E′, E′)^{∗}; f^{∗}]

where the vertical arrows are induced by f. Since ϕ′ = α′(id_{E′}), f^{∗}(ϕ) = ϕ′ and α(g) = ϕ we deduce that f^{∗}(g) = id_{E′}, i.e. g∘f = id_{E′}. Therefore, we have E ≃ E′ ⊕ E′′ for some holomorphic bundle E′′ such that Hom_{C}(E_{i}, E′′) = 0 for i < i_{0}. But Theorem 2.10 implies that Ext_{C}^{1}(E_{i}, E′′) = 0 for all sufficiently negative i. Together with Corollary 2.9 this implies that E′′ = 0. □

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Alexander Polishchuk

Department of Mathematics University of Oregon Eugene, OR 97403 apolishuoregon.edu 182

Documenta Mathematica 9 (2004)