Holomorphic Connections

on Filtered Bundles over Curves

Indranil Biswas and Viktoria Heu

Received: July 1, 2013

Communicated by Edward Frenkel

Abstract. Let X be a compact connected Riemann surface and E_{P} a holomorphic principal P –bundle over X, where P is a parabolic subgroup of a complex reductive affine algebraic group G. If the Levi bundle associated to E_{P} admits a holomorphic connection, and the reduction E_{P} ⊂ E_{P} ×^{P} G is rigid, we prove that E_{P} admits a holomorphic connection. As an immediate consequence, we obtain a sufficient condition for a filtered holomorphic vector bundle over X to admit a filtration preserving holomorphic connection. Moreover, we state a weaker sufficient condition in the special case of a filtration of length two.

2010 Mathematics Subject Classification: 14H60, 14F05, 53C07 Keywords and Phrases: Holomorphic connection, filtration, Atiyah bundle, parabolic subgroup

1. Introduction

Let X be a compact connected Riemann surface. A holomorphic vector bundle E over 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 context of principal bundles over X with a complex reductive affine algebraic group as the structure group [AB1]. Note that since there are no nonzero (2 , 0)–forms on X, holomorphic connections on a holomorphic bundle on X are the same as flat connections compatible with the holomorphic structure of the bundle. Our aim here is to consider flat connections on vector bundles compatible with a given filtration of the bundle. Let

(1.1)  0 = E_{0} ⊂ E_{1} ⊂ ⋯ ⊂ E_{ℓ−1} ⊂ E_{ℓ} = E

be a filtration of a holomorphic vector bundle E on X. If E admits a flat connection

D : E ⟶ E ⊗ Ω_{X}^{1}

preserving the filtration, meaning D(E_{i}) ⊂ E_{i} ⊗ Ω_{X}^{1} for every i, then this connection induces a flat connection D_{i} on each successive quotient E_{i}/E_{i−1} with i ∈ [1 , ℓ]. The question is the following: which supplementary condition is needed in order to ensure the existence of a filtration preserving holomorphic connection D? Suppose for example that E is semi-stable of degree zero such that each successive quotient in (1.1) admits a flat connection. Then it follows immediately that each subbundle E_{i}, i ∈ [1 , ℓ], is also semi-stable of degree zero. According to Corollary 3.10 in [Si, p. 40], the filtered vector bundle E then admits a filtration preserving holomorphic connection D. In this paper, we show that the rigidity of the filtration (1.1) is another sufficient supplementary condition for the existence of a filtration preserving holomorphic connection on E. We note that a related example is quoted in [Bi] (see [Bi, p. 119, Example 3.6]). More generally, we consider holomorphic connections on principal bundles with a parabolic group as the structure group. Let P be a parabolic subgroup of a complex reductive affine algebraic group G, and let E_{P} be a holomorphic principal P –bundle over X. Let L(P) := P/R_{u}(P) be the Levi quotient of P, where R_{u}(P) is the unipotent radical of P. Assume that the associated holomorphic principal L(P)–bundle E_{P} /R_{u}(P) admits a holomorphic connection. We are interested in the question of finding sufficient conditions for the existence of a holomorphic connection on E_{P}. Let E_{P} ×^{P} G be the holomorphic principal G–bundle obtained by extending the structure group E_{P} using the inclusion of P in G. We shall prove that the rigidity of the reduction of structure group E_{P} ⊂ E_{P} ×^{P} G ensures the existence of a holomorphic connection on E_{P} (see Theorem 2.1).

2. Connections on principal bundles with parabolic structure group

Let G be a connected reductive affine algebraic group defined over ℂ. Let P ⊂ G be a parabolic subgroup, i.e., P is a Zariski closed connected algebraic subgroup of G such that the quotient variety G/P is complete. The unipotent radical of P will be denoted by R_{u}(P). The quotient L(P) := P/R_{u}(P), which is a connected reductive complex affine algebraic group, is called the Levi quotient of P. The Lie algebra of G (respectively, P) will be denoted by 𝔤 (respectively, 𝔭). Let X be a compact connected Riemann surface. Let

(2.1)  f : E_{P} ⟶ X

be a holomorphic principal P –bundle. The quotient

(2.2)  E_{L(P)} := E_{P} /R_{u}(P)

is a holomorphic principal L(P)–bundle on X. We note that E_{L(P)} is identified with the principal L(P)-bundle obtained by extending the structure group of E_{P} using the quotient map P ⟶ L(P).

Let

E_{G} := E_{P} ×^{P} G ⟶ X

be the holomorphic principal G–bundle obtained by extending the structure group of E_{P} using the inclusion of P in G. Let

ad(E_{G}) := E_{G} ×^{G} 𝔤 and ad(E_{P}) := E_{P} ×^{P} 𝔭

be the adjoint vector bundles for E_{G} and E_{P} respectively. The reduction of structure group E_{P} ⊂ E_{G} is called rigid if

H^{0}(X, ad(E_{G})/ad(E_{P})) = 0 .

Let us give a brief geometric interpretation of this property. Recall that the space of infinitesimal deformations of the principal bundle E_{G} (respectively, E_{P}) can be identified with H^{1}(X, ad(E_{G})) (respectively, H^{1}(X, ad(E_{P}))) [SU]. We have a short exact sequence of vector bundles

0 ⟶ ad(E_{P}) ⟶ ad(E_{G}) ⟶ ad(E_{G})/ad(E_{P}) ⟶ 0 .

The rigidity of the reduction of structure group E_{P} ⊂ E_{G} thus translates as

H^{1}(X, ad(E_{P})) ↪ H^{1}(X, ad(E_{G})),

i.e. the infinitesimal deformations of E_{P} are uniquely determined by the infinitesimal deformations of E_{G} that they induce. In other words, if we fix the principal bundle E_{G}, then the parabolic subbundle E_{P} cannot be deformed.

Theorem 2.1. Assume that the holomorphic principal L(P)–bundle E_{L(P)} in (2.2) admits a holomorphic connection, and the reduction of structure group E_{P} ⊂ E_{G} is rigid. Then the holomorphic principal P –bundle E_{P} admits a holomorphic connection.

Proof. Let At(E_{P}) := (f_{∗}T E_{P})^{P} ⊂ f_{∗}T E_{P} be the Atiyah bundle for E_{P}, where f is the projection in (2.1) [At]. It fits in a short exact sequence of holomorphic vector bundles on X

(2.3)  0 ⟶ ad(E_{P}) ⟶ At(E_{P}) ⟶^{p_{0}} T X ⟶ 0 ,

where p_{0} is given by the differential df : T E_{P} ⟶ f^{∗}T X of f. We recall that a holomorphic connection on E_{P} is a holomorphic splitting of (2.3) [At]. Let R_{n}(𝔭) be the Lie algebra of the unipotent radical R_{u}(P). We note that R_{n}(𝔭) is the nilpotent radical of the Lie algebra 𝔭. Let

(2.4)  V_{0} := E_{P} ×^{P} R_{n}(𝔭) ⟶ X

be the holomorphic vector bundle associated to the principal P –bundle E_{P} for the P –module R_{n}(𝔭). Let f̂ : E_{L(P)} ⟶ X be the projection induced by f. Let

At(E_{L(P)}) := (f̂_{∗}T E_{L(P)})^{L(P)} ⊂ f̂_{∗}T E_{L(P)} be the Atiyah bundle for E_{L(P)}. We have a commutative diagram

(2.5)  [figure: 0; 0; V_{0}; V_{0}; 0; ad(E_{P}); At(E_{P}); p_{0}; q; T X; 0; 0; ad(E_{L(P)}); At(E_{L(P)}); p_{1}; T X; 0; 0; 0]

where V_{0} is defined in (2.4). By assumption, E_{L(P)} admits a holomorphic connection. Hence there is a holomorphic homomorphism

(2.6)  β : T X ⟶ At(E_{L(P)})

such that p_{1} ∘ β = Id_{T X}, where p_{1} is the projection in (2.5). Therefore, we have a short exact sequence of holomorphic vector bundles

(2.7)  0 ⟶ V_{0} ⟶ V := q^{−1}(β(T X)) ⟶^{p_{0}} T X ⟶ 0 ,

where q is the projection in (2.5). The short exact sequence in (2.3) splits holomorphically if the the short exact sequence in (2.7) splits holomorphically. The obstruction for splitting of (2.7) is a cohomology class

(2.8)  ψ ∈ H^{1}(X, V_{0} ⊗ (T X)^{∗}) = H^{0}(X, V_{0}^{∗})^{∗}

(by Serre duality). Since the group G is reductive, its Lie algebra 𝔤 has a G–invariant symmetric non-degenerate bilinear form. For example, let B be the direct sum of the Killing form on [𝔤 , 𝔤] and a symmetric non-degenerate bilinear form on the center of 𝔤. Note that 𝔭 ⊂ R_{n}(𝔭)^{⊥} (the annihilator of R_{n}(𝔭)^{⊥}) and actually

𝔭 = R_{n}(𝔭)^{⊥}

since they have the same dimension. We thus have

R_{n}(𝔭)^{∗} = 𝔤/R_{n}(𝔭)^{⊥} = 𝔤/𝔭 .

As the above isomorphism between R_{n}(𝔭)^{∗} and 𝔤/𝔭 is P –equivariant, it follows that

V_{0}^{∗} = E_{P} ×^{P} R_{n}(𝔭)^{∗} = ad(E_{G})/ad(E_{P}) .

Now the given condition that E_{P} ⊂ E_{G} is rigid implies that that

H^{0}(X, V_{0}^{∗}) = 0 .

Therefore, ψ in (2.8) vanishes. Consequently, the short exact sequence in (2.7) splits, implying that the short exact sequence in (2.3) splits. □

Some criteria for the existence of a holomorphic connection on E_{L(P)} can be found in [AB1] and [AB2]. Theorem 2.1 has the following immediate corollary:

Corollary 2.2. Let

0 = E_{0} ⊂ E_{1} ⊂ ⋯ ⊂ E_{ℓ−1} ⊂ E_{ℓ} = E

be a filtration of holomorphic vector bundles on X, and let End(E_{•}) ⊂ End(E) be the subbundle defined by the sheaf of filtration preserving endomorphisms. Assume that each successive quotient E_{i}/E_{i−1}, with i ∈ [1 , ℓ], admits a holomorphic connection, and

(2.9)  H^{0}(X, End(E)/End(E_{•})) = 0 .

Then E admits a holomorphic connection D such that D preserves each subbundle E_{i} with i ∈ [1 , ℓ].

Note that (2.9) is not a necessary condition for the existence of a filtration preserving connection D, as one can see by the example of trivial bundles filtered by trivial subbundles. In the next section, we state a weaker sufficient condition when the of length ℓ of the filtration is two.

3. Holomorphic connections on extensions

Let E and F be holomorphic vector bundles on X admitting holomorphic connections. A holomorphic connection on E and a holomorphic connection on F together define a holomorphic connection on the vector bundle Hom(E , F) = E^{∗} ⊗ F.

Proposition 3.1. Assume that E and F admit holomorphic connections D_{E} and D_{F} respectively, such that every holomorphic section of Hom(E , F) is flat with respect to the connection on Hom(E , F) given by D_{E} and D_{F}. Then for any holomorphic extension

0 ⟶ E ⟶ W ⟶ F ⟶ 0 ,

the holomorphic vector bundle W admits a holomorphic connection that preserves the subbundle E.

Proof. Let r_{1} and r_{2} be the ranks of E and F respectively. Take the group

G = GL(r_{1} + r_{2}, ℂ) ;

let P ⊂ G be the parabolic subgroup that preserves the subspace ℂ^{r_{1}} ⊂ ℂ^{r_{1}+r_{2}} given by the first r_{1} vectors of the standard basis. We note that L(P) = GL(r_{1}) × GL(r_{2}). Take an extension W as in the proposition. Then the pair (W , E) defines a holomorphic principal P –bundle E_{P} over X and E⊕F defines the associated L(P)−bundle E_{L(P)}. The holomorphic connection D_{E} ⊕ D_{F} on E ⊕ F gives a section β as in (2.6).

After we fix the above set-up, the vector bundle V_{0} in (2.4) is E ⊗F^{∗}. Consider

ψ ∈ H^{1}(X, E ⊗ F^{∗} ⊗ K_{X}) = H^{0}(X, E^{∗} ⊗ F)^{∗} = H^{0}(X, Hom(E , F))^{∗}

in (2.8). Given any T ∈ H^{0}(X, Hom(E , F)), we will explicitly describe the evaluation ψ(T) ∈ ℂ. Fix a C^{∞} splitting

η : F ⟶ W

of the short exact sequence in the proposition. We will identity F with η(F) ⊂ W. Let \overline{∂}_{E} (respectively, \overline{∂}_{F}) be the Dolbeault operator defining the holomorphic structure of E (respectively, F). Using the C^{∞} isomorphism

(3.1)  W ⟶ E ⊕ F

given by η, the Dolbeault operator of W is

(\begin{matrix}\overline{∂}_{E} & A \\ 0 & \overline{∂}_{F} \end{matrix} ) ,

where A is a smooth section of Hom(F , E) ⊗ Ω_{X}^{0,1}. Let D_{F,E} be the holomorphic connection on Hom(F , E) given by D_{E} and D_{F}. We have

D_{F,E}(A) ∈ C^{∞}(X; Hom(F , E) ⊗ Ω_{X}^{1,1}) .

Take any T ∈ H^{0}(X, Hom(E , F)). We will show that

(3.2)  ψ(T) = ∫_{X} trace(D_{F,E}(A) ∘ T) ∈ ℂ .

To prove this, consider the holomorphic connection D_{E} ⊕ D_{F} on E ⊕ F. Using the C^{∞} isomorphism in (3.1), this connection produces a C^{∞} connection ∇^{W} on W. We should clarify that ∇^{W} is holomorphic if and only if the isomorphism in (3.1) is holomorphic. Let

K(∇^{W}) ∈ C^{∞}(X; End(W) ⊗ Ω_{X}^{1,1})

be the curvature of the connection ∇^{W}. Since D_{E} ⊕ D_{F} is a flat connection on E ⊕ F, and the inclusion of E in W is holomorphic, it follows that K(∇^{W}) lies in the subspace

C^{∞}(X; E ⊗ F^{∗} ⊗ Ω_{X}^{1,1}) ⊂ C^{∞}(X; End(W) ⊗ Ω_{X}^{1,1})

constructed using the inclusion of the vector bundle Hom(F , E) in End(W). From the definition of the cohomology class ψ ∈ H^{1}(X, E⊗F^{∗}⊗K_{X}) it follows that the Dolbeault cohomology class in H^{1}(X, E ⊗ F^{∗} ⊗ K_{X}) represented by the form K(∇^{W}) ∈ C^{∞}(X; E ⊗ F^{∗} ⊗ Ω_{X}^{1,1}) coincides with ψ. On the other hand, the form

D_{F,E}(A) ∈ C^{∞}(X; E ⊗ F^{∗} ⊗ Ω_{X}^{1,1})

coincides with the curvature K(∇^{W}). Therefore, the equality in (3.2) follows. We note that ∫_{X} trace(D_{F,E}(A) ∘ T) is independent of the choice of the homomorphism η. Indeed, for a different choice of η, the section A is replaced by A + \overline{∂}_{E⊗F^{∗}}(A′), where A′ is a smooth section of Hom(F , E), and \overline{∂}_{F,E} is the Dolbeault operator defining the holomorphic structure of Hom(F , E). Now

∫_{X} trace(D_{F,E}(\overline{∂}_{F,E}(A′)) ∘ T) = ∫_{X} trace(\overline{∂}_{F,E}(D_{F,E}(A′)) ∘ T)

since the connection D_{F,E} is flat and compatible with the holomorphic structure, and we also have

∫_{X} trace(\overline{∂}_{F,E}(D_{F,E}(A′)) ∘ T) = ∫_{X} \overline{∂}(trace(D_{F,E}(A′) ∘ T)) = 0

because the section T is holomorphic. Therefore, ∫_{X} trace(D_{F,E}(A) ∘ T) is independent of the choice of η. We also note that trace(D_{F,E}(A) ∘ T) = trace(T ∘ D_{E,F} (A)). Let D_{E,E} be the holomorphic connection on End(E) induced by D_{E}. Let D_{E,F} be the holomorphic connection on Hom(E , F) induced by D_{E} and D_{F}. Note that

D_{E,F} (T) = 0

by the condition given in the proposition. Therefore, we have

D_{F,E}(A) ∘ T = D_{F,E}(A) ∘ T + A ∘ D_{E,F} (T) = D_{E,E}(A ∘ T) .

On the other hand,

∫_{X} trace(D_{E,E}(A ∘ T)) = ∫_{X} ∂(trace(A ∘ T)) = 0 .

Combining these, from (3.2) it follows that ψ = 0. The principal P –bundle E_{P} thus admits a holomorphic connection. In other words, the holomorphic vector bundle W admits a holomorphic connection that preserves the subbundle E. □

Corollary 3.2. Let E be a holomorphic vector bundle on X of degree zero such that

H^{0}(X, End(E)) = ℂ · Id_{E}.

Then given any short exact sequence of holomorphic vector bundles

0 ⟶ E ⟶ W ⟶ E ⟶ 0 ,

the holomorphic vector bundle W admits a holomorphic connection that preserves the subbundle E.

Proof. The holomorphic vector bundle E is indecomposable because

H^{0}(X, End(E)) = ℂ · Id_{E} .

Therefore, the given condition that degree(E) = 0 implies that E admits a holomorphic connection [We], [At, p. 203, Theorem 10]. For any holomorphic connection on E, the corresponding connection on End(E) has the property that the section Id_{E} is flat with respect to it. Hence Proposition 3.1 completes the proof. □

Acknowledgements

We are very grateful to Claude Sabbah for going through the paper carefully and making helpful suggestions. The first-named author acknowledges the support of the J. C. Bose Fellowship.

References

[At] M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207.

[AB1] H. Azad and I. Biswas, On holomorphic principal bundles over a compact Riemann surface admitting a flat connection, Math. Ann. 322 (2002), 333–346.

[AB2] H. Azad and I. Biswas, On holomorphic principal bundles over a compact Riemann surface admitting a flat connection, II, Bull. London Math. Soc. 35 (2003), 440–444.

[Bi] I. Biswas, A criterion for flatness of sections of adjoint bundle of a holomorphic principal bundle over a Riemann surface, Documenta Math. 18 (2013), 111–120.

[SU] Y. Shimizu and K. Ueno, Advances in moduli theory, Iwanami Series in Modern Mathematics, American Mathematical Society, Providence, RI, 2002.

[Si] C. T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95.

[We] A. Weil, Generalisation des fonctions abeliennes, Jour. Math. Pure Appl. 17 (1938), 47–87.

Indranil Biswas

School of Mathematics

Tata Institute of Fundamental Research Homi Bhabha Road Bombay 400005 India indranil@math.tifr.res.in Viktoria Heu IRMA UMR 7501 7 rue René-Descartes 67084 Strasbourg Cedex France heu@math.unistra.fr