Holomorphic Connections on Filtered Bundles over Curves

Let X be a compact connected Riemann surface and EP 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 EP admits a holomorphic connection, and the reduction EP ⊂ EP × P G is rigid, we prove that EP 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


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 be a filtration of a holomorphic vector bundle E on X.If E admits a flat connection preserving the filtration, meaning D(E i ) ⊂ E i ⊗ Ω 1 X 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).

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 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 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 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 The rigidity of the reduction of structure group E P ⊂ E G thus translates as 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 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) be the holomorphic vector bundle associated to the principal P -bundle E P for the P -module R n (p).Let f : E L(P ) −→ X be the projection induced by f .Let be the Atiyah bundle for E L(P ) .We have a commutative diagram By assumption, E L(P ) admits a holomorphic connection.Hence there is a holomorphic homomorphism (2.6) 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 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 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 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 .
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: 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 Then E admits a holomorphic connection D such that D preserves each subbundle 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.

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 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 r1 ⊂ C r1+r2 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 in (2.8).Given any T ∈ H 0 (X, Hom(E , F )), we will explicitly describe the evaluation ψ(T of the short exact sequence in the proposition.We will identity F with η(F ) ⊂ W .Let ∂ E (respectively, ∂ F ) be the Dolbeault operator defining the holomorphic structure of E (respectively, F ). Using the given by η, the Dolbeault operator of W is where A is a smooth section of Hom(F , E) ⊗ Ω 0,1 X .Let D F,E 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 1,1 X ) .Take any T ∈ H 0 (X, Hom(E , F )).We will show that 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 X ) 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 X ) coincides with ψ.On the other hand, the form X ) 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 Documenta Mathematica 18 (2013) 1473-1480 A + ∂ E⊗F * (A ′ ), where A ′ is a smooth section of Hom(F , E), and ∂ F,E is the Dolbeault operator defining the holomorphic structure of Hom(F , E).
since the connection D F,E is flat and compatible with the holomorphic structure, and we also have 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 the holomorphic vector bundle W admits a holomorphic connection that preserves the subbundle E.
Proof.The holomorphic vector bundle E is indecomposable because 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.
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 haveD 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 .