A Criterion for Flatness of Sections

of Adjoint Bundle of a Holomorphic Principal Bundle

over a Riemann Surface

Indranil Biswas

Received: August 8, 2013

Communicated by Edward Frenkel

Abstract. Let E_{G} be a holomorphic principal G–bundle over a compact connected Riemann surface, where G is a connected reductive affine algebraic group defined over ℂ, such that E_{G} admits a holomorphic connection. Take any β ∈ H^{0}(X, ad(E_{G})), where ad(E_{G}) is the adjoint vector bundle for E_{G}, such that the conjugacy class β(x) ∈ 𝔤/G, x ∈ X, is independent of x. We give a sufficient condition for the existence of a holomorphic connection on E_{G} such that β is flat with respect to the induced connection on ad(E_{G}).

2010 Mathematics Subject Classification: 14H60, 14F05, 53C07 Keywords and Phrases: Holomorphic connection, adjoint bundle, flatness, canonical connection

1. Introduction

A holomorphic vector bundle E over a compact connected Riemann surface X admits a holomorphic connection if and only if every indecomposable component of E is of degree zero [We], [At]. This criterion generalizes to the holomorphic principal G–bundles over X, where G is a connected reductive affine algebraic group defined over ℂ [AB]. Let E_{G} be a holomorphic principal G–bundle over X, where X and G are as above. Let 𝔤 denote the Lie algebra of G. Let β be a holomorphic section of the adjoint vector bundle ad(E_{G}) = E_{G} ×^{G} 𝔤. Our aim here is to find a criterion for the existence of a holomorphic connection on E_{G} such that β is flat with respect to the induced connection on ad(E_{G}). A sufficient condition is obtained in Theorem 3.4. For G = GL(r, ℂ), Theorem 3.4 says the following:

Let E be a holomorphic vector bundle of rank r on X, and β ∈ H^{0}(X, End(E)). Let

E = ⨁_{i=1}^{ℓ} E_{i}

be the generalized eigen-bundle decomposition for β. So β|_{E_{i}} = λ_{i} · Id_{E_{i}} + N_{i}, where λ_{i} ∈ ℂ, and either N_{i} = 0 or N_{i} is nilpotent. If N_{i} ≠ 0, then assume that the section N^{r_{i}−1} is nowhere vanishing, where r_{i} is the rank of the vector bundle E_{i}. Also, assume that E admits a holomorphic connection. Then Theorem 3.4 says that E admits a holomorphic connection D such that β is flat with respect to the connection on End(E) induced by D. One may ask whether the above condition that N^{r_{i}−1} is nowhere vanishing whenever N_{i} ≠ 0 can be replaced by the weaker condition that the conjugacy class of β(x), x ∈ X, is independent of x. As example constructed by the referee shows that this cannot be done (see Example 3.6).

2. Flat sections of the adjoint bundle

Let X be a compact connected Riemann surface. Let G be a connected reductive affine algebraic group defined over ℂ. The Lie algebra of G will be denoted by 𝔤. The set of all conjugacy classes in 𝔤 will be denoted by 𝔤/G. Let

(2.1)  f : E_{G} ⟶ X

be a holomorphic principal G–bundle. Define the adjoint vector bundle

ad(E_{G}) := E_{G} ×^{G} 𝔤 .

In other words, ad(E_{G}) is the quotient of E_{G} × 𝔤 where any (z , v) ∈ E_{G} × 𝔤 is identified with (zg , Ad(g)(v)), g ∈ G; here Ad(g) is the automorphism of 𝔤 corresponding to the automorphism of G defined by g′ ⟼ g^{−1}g′g. Therefore, we have a set-theoretic map

(2.2)  ϕ : ad(E_{G}) ⟶ 𝔤/G

that sends any (z , v) ∈ E_{G} × 𝔤 to the conjugacy class of v. Let

At(E_{G}) := (f_{∗}T E_{G})^{G} ⊂ f_{∗}T E_{G}

be the Atiyah bundle for E_{G}, where f is the projection in (2.1), and T E_{G} is the holomorphic tangent bundle of E_{G} (the action of G on E_{G} produces an action of G on the direct image f_{∗}T E_{G}). The Atiyah bundle fits in a short exact sequence of vector bundles

(2.3)  0 ⟶ ad(E_{G}) ⟶ At(E_{G}) ⟶ T X ⟶ 0 ;

the above projection At(E_{G}) ⟶ T X, where T X is the holomorphic tangent bundle of X, is defined by the differential df : T E_{G} ⟶ f^{∗}T X of f. A holomorphic connection on E_{G} is a holomorphic splitting of the sequence in (2.3) [At].

A Criterion for Flatness of Sections of Adjoint Bundle 113

A holomorphic connection D on E_{G} induces a holomorphic connection on each holomorphic fiber bundle associated to E_{G}. In particular, the vector bundle ad(E_{G}) gets a holomorphic connection from D. A section β of ad(E_{G}) is said to be flat with respect to D if β is flat with respect to the connection on ad(E_{G}) induced by D.

Lemma 2.1. Take a holomorphic connection D on E_{G}, and let β ∈ H^{0}(X, ad(E_{G})) be flat with respect to D. Then the element ϕ ∘ β(x) ∈ 𝔤/G, where x ∈ X, is independent of x.

Proof. Any holomorphic connection on X is flat because Ω_{X}^{2} = 0. Using the flat connection D, we may holomorphically trivialize E_{G} on any connected simply connected open subset of X. With respect to such a trivialization, the section β is a constant one because it is flat with respect to D. This immediately implies that ϕ ∘ β(x) ∈ 𝔤/G is independent of x ∈ X. □

3. Holomorphic connections on Principal G-bundles

A nilpotent element v of the Lie algebra of a complex semisimple group H is called regular nilpotent if the dimension of the centralizer of v in H coincides with the rank of H [Hu, p. 53]. As before, G is a connected reductive affine algebraic group defined over ℂ. Take E_{G} as in (2.1).

Proposition 3.1. Take any β ∈ H^{0}(X, ad(E_{G})). Assume that • E_{G} admits a holomorphic connection,

• the element ϕ ∘ β(x) ∈ 𝔤/G, x ∈ X, is independent of x, where ϕ is defined in (2.2), and

• for every adjoint type simple quotient H of G, the section of the adjoint bundle ad(E_{H}) given by β, where E_{H} := E_{G} ×^{G} H is the principal H– bundle over X associated to E_{G} for the projection G ⟶ H, has the property that it is either zero or it is regular nilpotent at some point of X.

Then the principal G–bundle E_{G} admits a holomorphic connection for which β is flat.

Proof. Let Z := G/[G , G] be the abelian quotient. It is a product of copies of ℂ^{∗}. There are quotients H_{1} , ⋯, H_{ℓ} of G such that

(1) each H_{i} is simple of adjoint type (the center is trivial), and

(2) the natural homomorphism

(3.1)  φ : G ⟶ Z × ∏_{i=1}^{ℓ} H_{i}

is surjective, and the kernel of φ is a finite group. Let

E_{Z} := E_{G} ×^{G} Z and E_{H_{i}} := E_{G} ×^{G} H_{i} , i ∈ [1 , ℓ] ,

be the holomorphic principal Z–bundle and principal H_{i}–bundle associated to E_{G} for the quotient Z and H_{i} respectively. Let ad(E_{Z}) and ad(E_{H_{i}}) be the adjoint vector bundles for E_{Z} and E_{H_{i}} respectively. Since the homomorphism φ in (3.1) induces an isomorphism of Lie algebras, we have

(3.2)  ad(E_{G}) = ad(E_{Z}) ⊕ (⨁_{i=1}^{ℓ} ad(E_{H_{i}})) .

Let β_{Z} (respectively, β_{i}) be the holomorphic section of ad(E_{Z}) (respectively, ad(E_{H_{i}})) given by β using the decomposition in (3.2). Since the conjugacy class of β(x) is independent of x ∈ X (the second condition in the proposition), we conclude that the conjugacy class of β_{i}(x) is also independent of x ∈ X. A holomorphic connection on E_{G} induces a holomorphic connection on E_{Z}. Since E_{Z} admits a holomorphic connection, and Z is a product of copies of ℂ^{∗}, there is a unique holomorphic connection D^{Z} on E_{Z} whose monodromy lies inside the maximal compact subgroup of Z. The connection on ad(E_{Z}) induced by this connection D^{Z} has the property that any holomorphic section of ad(E_{Z}) is flat with respect to it. In particular, the section β_{Z} is flat with respect to this induced connection on ad(E_{Z}). Now take any i ∈ [1 , ℓ]. A holomorphic connection on E_{G} induces a holomorphic connection on E_{H_{i}}. If the section β_{i} is zero at some point, then β_{i} is identically zero because the conjugacy class of β_{i}(x) is independent of x. Hence, in that case β_{i} is flat with respect to any connection on ad(E_{H_{i}}). Therefore, assume that β_{i} is not zero at any point of X. By the assumption in the proposition, β_{i} is regularly nilpotent over some point of X. Since the conjugacy class of β_{i}(x), x ∈ X, is independent of x, we conclude that β_{i} is regular nilpotent over every point of X. We will now show that the holomorphic principal H_{i}–bundle E_{H_{i}} is semistable. For each point x ∈ X, from the fact that β_{i}(x) is regular nilpotent we conclude that there is a unique Borel subalgebra 𝔟̃_{x} of ad(E_{H_{i}})_{x} such that β_{i}(x) ∈ 𝔟̃_{x} [Hu, p. 62, Theorem]. Let

𝔟̃ ⊂ ad(E_{H_{i}})

be the Borel subalgebra bundle such that for every point x the fiber (𝔟̃)_{x} is 𝔟̃_{x}. Fix a Borel subgroup B ⊂ H_{i}. Using 𝔟̃, we will construct a holomorphic reduction of structure group of E_{H_{i}} to the subgroup B. Let 𝔟 be the Lie algebra of B. The Lie algebra of H_{i} will be denoted by 𝔥_{i}. We recall that ad(E_{H_{i}}) is the quotient of E_{H_{i}} × 𝔥_{i} where two points (z_{1} , v_{1}) and (z_{2} , v_{2}) of E_{H_{i}} × 𝔥_{i} are identified if there is an element h ∈ H_{i} such that z_{2} = z_{1}h and v_{2} = Ad(h)(v_{1}), where Ad(h) is the automorphism of 𝔥_{i} corresponding to the automorphism y ⟼ h^{−1}yh of H_{i}. For any point x ∈ X, let E_{B,x} ⊂ (E_{H_{i}})_{x} be the complex submanifold consisting of all z ∈ (E_{H_{i}})_{x} such that for all v ∈ 𝔟, the image of (z , v) in ad(E_{H_{i}})_{x} lies in 𝔟̃_{x}. Since any two Borel subalgebras of 𝔥_{i} are conjugate, it follows that E_{B,x} is nonempty. The normalizer of 𝔟 in H_{i} coincides with B. From this it follows that E_{B,x} is preserved by the action of B on (E_{H_{i}})_{x}, with the action of B on E_{B,x} being transitive. Let

E_{B} ⊂ E_{H_{i}}

be the complex submanifold such that E_{B} ⋂(E_{H_{i}})_{x} = E_{B,x} for every x ∈ X. From the above properties of E_{B,x} it follows immediately that E_{B} is a holomorphic reduction of structure group of the principal H_{i}–bundle E_{H_{i}} to the subgroup B. Consider the adjoint action of B on 𝔟_{1} := 𝔟/[𝔟 , 𝔟]. Let

E_{B}(𝔟_{1}) := E_{B} ×^{B} 𝔟_{1} ⟶ X

be the holomorphic vector bundle associated to E_{B} for the B–module 𝔟_{1}. Since β_{i} is everywhere regular nilpotent, it follows that the vector bundle E_{B}(𝔟_{1}) is trivial. Consequently, for any character χ of B which is a nonnegative integral combination of simple roots, the line bundle E_{B}(χ) ⟶ X associated to E_{B} for the character χ is trivial [AAB, p. 708, Theorem 5]. Therefore, for any character χ of B, the line bundle E_{B}(χ) associated to E_{B} for χ is trivial. Let d be the complex dimension of 𝔥_{i}. Consider the adjoint action on B on 𝔥_{i}. Note that ad(E_{H_{i}}) is identified with the vector bundle associated to the principal B–bundle E_{B} for this B–module 𝔥_{i}. Since B is solvable, there is a filtration of B–modules

0 = V_{0} ⊂ V_{1} ⊂ ⋯ ⊂ V_{d−1} ⊂ V_{d} = 𝔥_{i}

such that dim V_{j} = j for all j ∈ [1 , d]. The corresponding filtration of vector bundles associated to E_{B} is a filtration of ad(E_{H_{i}}) such that the successive quotients are the line bundles E_{B}(V_{j}/V_{j−1}), i ∈ [1 , d], associated to E_{B} for the B–modules V_{j}/V_{j−1}. We noted above that the line bundles associated to E_{B} for the characters of B are trivial. Therefore, we get a filtration of ad(E_{H_{i}}) such that each successive quotient is the trivial line bundle. This immediately implies that the vector bundle ad(E_{H_{i}}) is semistable. Hence the holomorphic principal H_{i}–bundle E_{H_{i}} is semistable [AAB, p. 698, Lemma 3]. Since H_{i} is simple, and E_{H_{i}} is semistable, there is a natural holomorphic connection on E_{H_{i}} [BG, p. 20, Theorem 1.1] (set the Higgs field in [BG, Theorem 1.1] to be zero). Let D^{H_{i}} denote this connection. The vector bundle ad(E_{H_{i}}) being semistable of degree zero has a natural holomorphic connection [Si, p. 36, Lemma 3.5]. See also [BG, p. 20, Theorem 1.1]. (In both [Si, Lemma 3.5] and [BG, Theorem 1.1] set the Higgs field to be zero.) Let D^{ad} denote this holomorphic connection on ad(E_{H_{i}}). This connection D^{ad} coincides with the one induced by D^{H_{i}} (see the construction of the connection in [BG]). Any holomorphic section of ad(E_{H_{i}}) is flat with respect to D^{ad}. To see this, let

ϕ : O_{X} ⟶ ad(E_{H_{i}})

be the homomorphism given by a nonzero holomorphic section of ad(E_{H_{i}}). Since image(ϕ) is a semistable subbundle of ad(E_{H_{i}}) of degree zero, the connection D^{ad} preserves image(ϕ), and, moreover, the restriction of D^{ad} to image(ϕ) coincides with the canonical connection of image(ϕ) [Si, p. 36, Lemma 3.5].

The canonical connection on the trivial holomorphic line bundle image(ϕ) is the trivial connection (the monodromy is trivial). In particular, the connection on ad(E_{H_{i}}) induced by D^{H_{i}} has the property that the section β_{i} is flat with respect to it. Since the homomorphism of Lie algebras corresponding to φ (in (3.1)) is an isomorphism, if we have holomorphic connections on E_{Z} and E_{H_{i}}, [1 , ℓ], then we get a holomorphic connection on E_{G}; simply pullback the connection form using the map

E_{G} ⟶ E_{Z} ×_{X} E_{H_{1}} ×_{X} ⋯ ×_{X} E_{H_{ℓ}} .

The connection on E_{G} given by the connections on E_{Z} and E_{H_{i}}, [1 , ℓ], constructed above satisfies the condition that β is flat with respect to it. This completes the proof of the proposition. □

Lemma 3.2. Take any semisimple section β_{s} ∈ H^{0}(X, ad(E_{G})) such that the element ϕ ∘ β_{s}(x) ∈ 𝔤/G, x ∈ X, is independent of x, where ϕ is defined in (2.2). Then β_{s} produces a holomorphic reduction of structure group of E_{G} to a Levi subgroup of a parabolic subgroup of G. The conjugacy class of the Levi subgroup is determined by ϕ ∘ β_{s}(x) ∈ 𝔤/G.

Proof. Fix an element

v_{0} ∈ 𝔤

such that the image of v_{0} in 𝔤/G coincides with ϕ ∘ β_{s}(x). Let 𝕃 ⊂ G be the centralizer of v_{0}. It is known that 𝕃 is a Levi subgroup of some parabolic subgroup of G [DM, p. 26, Proposition 1.22] (note that 𝕃 is the centralizer of the torus in G generated by v_{0}). In particular, 𝕃 is connected and reductive. For any point x ∈ X, let F_{x} ⊂ (E_{G})_{x} be the complex submanifold consisting of all points z such that the image of (z , v_{0}) in ad(E_{G})_{x} coincides with β_{s}(x). (Recall that ad(E_{G}) is a quotient of E_{G} × 𝔤.) Let

F_{𝕃} ⊂ E_{G}

be the complex submanifold such that F_{𝕃} ⋂(E_{G})_{x} = F_{x} for all x ∈ X. It is straightforward to check that F_{𝕃} is a holomorphic reduction of structure group of the principal G–bundle E_{G} to the subgroup 𝕃. □

Remark 3.3. If β_{s} ∈ H^{0}(X, ad(E_{G})) is such that β_{s}(x) is semisimple for every x ∈ X, then the conjugacy class of β_{s}(x) is in fact independent of x. But we do not need this here.

From the Jordan decomposition of a complex reductive Lie algebra we know that for any holomorphic section θ of ad(E_{G}), there is a naturally associated semisimple (respectively, nilpotent) section θ_{s} (respectively, θ_{n}) such that θ = θ_{s} + θ_{n}. Take any β ∈ H^{0}(X, ad(E_{G})). Let

β = β_{s} + β_{n}

be the Jordan decomposition. Assume that the element ϕ ∘ β(x) ∈ 𝔤/G, x ∈ X, is independent of x, where ϕ is defined in (2.2). This implies that ϕ∘β_{s}(x) ∈ 𝔤/G, x ∈ X, is also independent of x. Let (𝕃 , F_{𝕃}) be the principal bundle constructed in Lemma 3.2 from β_{s}. Let H be an adjoint type simple quotient of 𝕃. Let

E_{H} := F_{𝕃} ×^{𝕃} H ⟶ X

be the holomorphic principal H–bundle associated to F_{𝕃} for the projection 𝕃 ⟶ H. Since [β_{s} , β_{n}] = 0, from the construction of F_{𝕃} it follows that

β_{n} ∈ H^{0}(X, ad(F_{𝕃})) ⊂ H^{0}(X, ad(E_{G})) .

Therefore, using the natural projection ad(F_{𝕃}) ⟶ ad(E_{H}), given by the projection of the Lie algebra Lie(𝕃) ⟶ Lie(H), the above section β_{n} produces a holomorphic section of ad(E_{H}). Let

(3.3)  β̃_{n} ∈ H^{0}(X, ad(E_{H}))

be the section constructed from β_{n}.

Theorem 3.4. Take any β ∈ H^{0}(X, ad(E_{G})). Let β = β_{s} + β_{n} be the Jordan decomposition. Assume that

• E_{G} admits a holomorphic connection,

• the element ϕ ∘ β(x) ∈ 𝔤/G, x ∈ X, is independent of x, where ϕ is defined in (2.2), and

• for every adjoint type simple quotient H of 𝕃, the section β̃_{n} in (3.3) of ad(E_{H}) has the property that it is either zero or it is regular nilpotent at some point of X.

Then the principal G–bundle E_{G} admits a holomorphic connection for which β is flat.

Proof. Note that

β_{s} ∈ H^{0}(X, ad(F_{𝕃})) ⊂ H^{0}(X, ad(E_{G})) .

In fact, for each point x ∈ X, the element β_{s}(x) ∈ ad(F_{𝕃}))_{x} is in the center of ad(F_{𝕃}))_{x}. Consider the abelian quotient

Z_{𝕃} = 𝕃/[𝕃 , 𝕃] .

Let F_{Z_{𝕃}} be the holomorphic principal Z_{𝕃}–bundle over X obtained by extending the structure group of the principal 𝕃–bundle F_{𝕃} using the quotient map 𝕃 ⟶ Z_{𝕃}. The adjoint vector bundle ad(F_{Z_{𝕃}}) is a direct summand of ad(F_{𝕃}). In fact, for each x ∈ X, the subspace ad(F_{Z_{𝕃}})_{x} ⊂ ad(F_{𝕃})_{x} is the center of the Lie algebra ad(F_{𝕃})_{x}. A holomorphic connection on F_{𝕃} induces a holomorphic connection on E_{G}. We can now apply Proposition 3.1 to F_{𝕃} to complete the proof of the theorem. But for that we need to show that F_{𝕃} admits a holomorphic connection. Let 𝔩 be the Lie algebra of 𝕃. Consider the inclusion of 𝕃–modules 𝔩 ↪ 𝔤 given by the inclusion of 𝕃 in G. Since 𝕃 is reductive, there is a sub 𝕃–module S ⊂ 𝔤 such that the natural homomorphism

𝔩 ⊕ S ⟶ 𝔤

is an isomorphism (so S is a complement of 𝔩). Let

(3.4)  p : 𝔤 ⟶ 𝔩

be the projection given by the above decomposition of 𝔤. Let D be a holomorphic connection on E_{G}. So D is a holomorphic 1–form on the total space of E_{G} with values in the Lie algebra 𝔤. Let D′ be the restriction of this 1–form to the complex submanifold F_{𝕃} ⊂ E_{G}. Consider the 𝔩–valued 1–form p ∘ D′ on E_{𝕃}, where p is the projection in (3.4). This 𝔩–valued 1–form on F_{𝕃} defines a holomorphic connection of the principal 𝕃–bundle F_{𝕃}. Now Proposition 3.1 completes the proof of the theorem. □

We recall that a holomorphic vector bundle W on X has a holomorphic connection if and only if each indecomposable component of W is of degree zero [We], [At, p. 203, Theorem 10]. This criterion generalizes to holomorphic principal G–bundles on X (see [AB] for details). We now set G = GL(r, ℂ) in Theorem 3.4. Let E be a holomorphic vector bundle of rank r on X. Take any

β ∈ H^{0}(X, End(E)) .

Let

(3.5)  E = ⨁_{i=1}^{ℓ} E_{i}

be the generalized eigen-bundle decomposition of E for β. Therefore,

β|_{E_{i}} = λ_{i} · Id_{E_{i}} + N_{i} ,

where λ ∈ ℂ, and either N_{i} = 0 or N_{i} is nilpotent. Then Theorem 3.4 has the following corollary:

Corollary 3.5. For every N_{i} ≠ 0, assume that the section N^{r_{i}−1} of End(E_{i}) is nowhere vanishing, where r_{i} is the rank of the vector bundle E_{i} in (3.5). If the holomorphic vector bundle E admits a holomorphic connection, then it admits a holomorphic connection D such that the section β is flat with respect to the connection on End(E) induced by D.

Consider the condition on β in Corollary 3.5 which says that N^{r_{i}−1} is nowhere vanishing whenever N_{i} ≠ 0. This condition implies that the image of β(x) in M(r, ℂ)/GL(r, ℂ) is independent of x ∈ X (here GL(r, ℂ) acts on its Lie algebra M(r, ℂ) via conjugation). Therefore, one may ask whether the above mentioned condition in Corollary 3.5 can be replaced by the weaker condition that the conjugacy class of β(x) is independent of x ∈ X. Note that if this can be done, then the sufficient condition in Corollary 3.5 for the existence of a connection on E such that β is flat with respect to it actually becomes a necessary and sufficient condition. The following construction of the referee shows that the condition in Corollary 3.5 cannot be replaced by the weaker condition that the conjugacy class of β(x) is independent of x ∈ X.

Example 3.6 (Referee). Let X be of sufficiently high genus. Let L and M be holomorphic line bundles on X of degree 1 and degree −2 respectively. Then there exists an indecomposable holomorphic vector bundle E of rank three on X satisfying the following condition: it admits a filtration of holomorphic subbundles

L = E_{1} ⊂ E_{2} ⊂ E

such that E_{2}/L = M and E/E_{2} = L. We omit the detailed arguments given by the referee showing that such a vector bundle E exists. Let β denote the composition

E ⟶ E/E_{2} = L = E_{1} ↪ E .

Clearly, the conjugacy class of β(x) is independent of x ∈ X. The vector bundle E admits a holomorphic connection because it is indecomposable of degree zero. If D is a holomorphic connection on E such that β is flat with respect to the connection on End(E) induced by D, then the subsheaf image(β) ⊂ E is flat with respect to D. But image(β) = L does not admit a holomorphic connection because it is of nonzero degree.

Acknowledgements

The referee pointed out an error in an earlier version of the paper, and also made detailed comments that included Example 3.6. This helped the author to find the right formulation of Theorem 3.4. The author is immensely grateful to the referee for this.

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Indranil Biswas School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Bombay 400005 India indranil@math.tifr.res.in