Holomorphic connections on holomorphic bundles on Riemann surfaces

We investigate aspects of holomorphic connections on holomorphic principal bundles over a Riemann surface.

The assumption Theorem 1.1 that G is reductive is rather essential; see Sect. 4. In the final section, we consider logarithmic connections on a compact Riemann surface. The following theorem was proved in [4]: When D = 0, Theorem 1.2 coincides with the above-mentioned criterion of Atiyah and Weil for the existence of a holomorphic connection on a holomorphic vector bundle over M.

Holomorphic connection on principal bundles
Let X be a compact connected complex manifold. Let G be a complex Lie group. A holomorphic principal G-bundle over X consists of • a complex manifold E G , • a surjective holomorphic map f : E G −→ X , and • a holomorphic right action ϕ : E G × G −→ E G of G on E G , such that the following two conditions hold: (1) f • ϕ = f • p 1 , where p 1 : E G × G −→ E G is the natural projection, and (2) the map to the fiber product is a biholomorphism.
Note that the first condition implies that (z, ϕ(z, g)) ∈ E G × X E G .
For notational convenience, ϕ(z, g), where z ∈ E G and g ∈ G, will be denoted by zg. The holomorphic tangent (respectively, cotangent) bundle of a complex manifold Y will be denoted by T Y (respectively, 1 Y ). The Lie algebra of G will be denoted by g. Let f : E G −→ X be a holomorphic principal G-bundle over X . Consider the holomorphic right action of G on the holomorphic tangent bundle T E G induced by the right action of G on E. The quotient is a complex manifold; in fact, it is a holomorphic vector bundle over E G /G = X . This holomorphic vector bundle At(E G ) over X is called the Atiyah bundle for E G [2]. The differential of the projection f is evidently G-equivariant for the natural action of G on f * T X that lifts the action of G on E G . The action of G on E G produces a holomorphic homomorphism from the trivial holomorphic bundle E G × g −→ E G with fiber g which is in fact an isomorphism. From this it follows that we have a short exact sequence of holomorphic vector bundles on E G all the homomorphisms in (2.3) are G-equivariant. The quotient kernel(d f )/G is called the adjoint vector bundle, and it is denoted by ad(E G ). This holomorphic vector bundle ad(E G ) over X is identified with the holomorphic vector bundle E G (g) associated to E G for the adjoint action of G on the Lie algebra g; this identification is produced by the isomorphism in (2.2). Taking quotient of the holomorphic vector bundles in (2.3), by the actions of G, the following short exact sequence of holomorphic vector bundles on X is obtained: where d f is the descent of the homomorphism d f (see [2]). The short exact sequence in (2.4) is known as the Atiyah exact sequence for E G .
A holomorphic connection on E G is a holomorphic homomorphism of vector bundles such that where d f is the projection in (2.4) (see [2]). Therefore, E G admits a holomorphic connection if and only if the Atiyah exact sequence in (2.4) splits holomorphically. Giving a holomorphic connection on E G is equivalent to giving a g-valued holomorphic 1-form on E G such that the following two conditions hold: • the homomorphism is G-equivariant for the adjoint action of G on g and the action of G on T E G given by the right action of G on E G , and • the restriction of ω to any fiber of f coincides with the Maurer-Cartan form, equivalently, the restriction of ω to any fiber of f coincides with the inverse of the isomorphism in (2.2).
For a form ω as in (2.5) giving a holomorphic connection on E G , let kernel(ω) ⊂ T E G be the kernel of the homomorphism T E G −→ E G × g given by ω. The action of G on T E G preserves this holomorphic subbundle kernel(ω). Consequently, is a holomorphic subbundle. There is a unique holomorphic homomorphism such that Therefore, D defines a holomorphic connection on E G . Conversely, given a holomorphic homomorphism Then there is a unique holomorphic 1-form ω ∈ H 0 (E G , 1 E G ⊗ C g) such that • the kernel of the homomorphism T E G −→ E G × g given by ω coincides with the subbundle V ⊂ T E G in (2.6), and • the restriction of ω to any fiber of f coincides with the Maurer-Cartan form.
The above homomorphism ω : T E G −→ g is G-equivariant, and hence we conclude that ω defines a holomorphic connection on E G .
For a complex manifold Z , the Lie bracket operation of locally defined holomorphic vector fields on Z makes the coherent analytic sheaf T Z a sheaf of Lie algebras. Note that the Lie bracket operation on locally defined holomorphic vector fields on Z is not O Z -linear.
The Lie bracket of two G-invariant vector fields on E G | U , where U ⊂ X is an open subset, is again G-invariant. Consequently, the Lie algebra structure on the coherent analytic sheaf T E G makes the coherent analytic sheaf At(E G ) a sheaf of Lie algebras. Take a holomorphic connection on E G . The curvature of D is the obstruction for the homomorphism D to be compatible with the Lie algebra structures on the sheaves At(E G ) and T X. More precisely, given holomorphic vector fields s, t defined on an open subset U ⊂ X , consider

Holomorphic connections on bundles over a Riemann surface
In this section we will consider holomorphic principal G-bundles on compact Riemann surfaces and holomorphic connections on them.
Let M be a compact connected Riemann surface. As before, G is a complex Lie group. Given a holomorphic principal G-bundle E G on M, it is natural to ask whether E G admits a holomorphic connection. A well-known theorem of Atiyah and Weil answers this question when G = GL(r, C) [2,17]. This result of [2,17] will be recalled below.
Using the standard representation of GL(r, C), holomorphic principal GL(r, C)-bundles on M are identified with holomorphic vector bundles on M of rank r .
A holomorphic vector bundle W on a compact connected complex manifold X is called decomposable if where W 1 and W 2 are holomorphic vector bundles of positive rank. A holomorphic vector bundle W on X is called indecomposable if it is not decomposable. Every holomorphic vector bundle on X is holomorphically isomorphic to a direct sum of indecomposable holomorphic vector bundles.
The following very important theorem was proved by Atiyah.
The above-mentioned theorem of [2,17] says that a holomorphic vector bundle E over a compact connected Riemann surface M admits a holomorphic connection if and only if each direct summand of E is of degree zero. Let be a decomposition of E into a direct sum of indecomposable vector bundles (as in Theorem 3.1). Then E admits a holomorphic connection if and only if We shall now describe a generalization of the above theorem of [2] and [17] to the context of principal bundles.
Let G be a connected reductive affine algebraic group defined over C. A Zariski closed connected subgroup P ⊂ G is called a parabolic subgroup if G/P is a projective variety [8, 11.2], [13]. The unipotent radical of a parabolic subgroup P ⊂ G will be denoted by R u (P). The quotient group P/R u (P) is called the Levi quotient of P. A Levi factor of P is a Zariski closed connected subgroup L ⊂ P such that the composition is an isomorphism [13, p. 184]. We note that P admits Levi factors, and any two Levi factors of P are conjugate by an element of R u (P) [13,§ 30.2,p. 185,Theorem]. A Levi factor L ⊂ P satisfies the following condition: L contains a maximal torus of P, and moreover L is a maximal reductive subgroup of P (see [8,13]).
Given a holomorphic principal G-bundle E G on M and a complex Lie subgroup H ⊂ G, a holomorphic reduction of E G to H is given by a holomorphic section of the holomorphic fiber bundle E G /H over M. Let If E H is a holomorphic principal H -bundle on M, and χ is a holomorphic character of H , then the associated holomorphic line bundle The following theorem is proved in [3] (see [3,Theorem 4.1]).

is a holomorphic reduction of structure group to H , and (3) λ is a holomorphic character of H , the associated line bundle E H
Note that if we set G = GL(n, C) in Theorem 3.2, then it coincides with the mentioned criterion of Atiyah and Weil for the existence of a holomorphic connection on a holomorphic vector bundle over M.
A sketch of the proof of Theorem 3.2 will be given below. Let E G be a holomorphic principal G-bundle over M equipped with a holomorphic connection ∇. Take any triple (H, E H , λ) as in Theorem 3.2. We will first show that a holomorphic connection ∇ on E G produces a holomorphic connection on the principal H -bundle E H .
Let g and h denote the Lie algebras of G and H respectively. The group H has adjoint actions on both h and g. To construct the connection on E H , fix a splitting of the injective homomorphism of H -modules be the chosen splitting. Since a holomorphic connection on E G is a given by a holomorphic splitting of the Atiyah exact sequence for E G , a holomorphic connection ∇ on E G produces a g-valued holomorphic 1-form ω on E G satisfying the following two conditions: • ω is G-equivariant (G acts on g by inner automorphism), and • the restriction of ω to any fiber of E G is the Maurer-Cartan form on the fiber.
Using the chosen splitting homomorphism the connection form ω on E G defines a h-valued holomorphic one-form ω on E G . The restriction of ω to the complex submanifold E H ⊂ E G satisfies the two conditions needed for a holomorphic h-valued 1-form on E H to define a holomorphic connection on E H .
Therefore, E H admits a holomorphic connection. A holomorphic connection on E H induces a holomorphic connection on the associated line bundle E H (λ). Any line bundle admitting a holomorphic connection must be of degree zero [2]. Therefore, if E G admits a holomorphic connection then we know that the degree of E H (λ) is zero.
To prove the converse, let E G be a holomorphic principal G-bundle over M such that degree(E H (λ)) = 0 for all triples (H, E H , λ) of the above type. We need to show that the Atiyah exact sequence for E G (see (2.4)) splits holomorphically.
As the first step for it, in [3] the following is proved: it is enough to prove that the Atiyah exact sequence for E G splits holomorphically under the assumption that E G does not admit any holomorphic reduction of structure group to any proper Levi subgroup of G. Therefore, we assume that E G does not admit any holomorphic reduction of structure group to any proper Levi subgroup of G.
Let K M denote the holomorphic cotangent bundle of M. The obstruction for splitting of the Atiyah exact sequence for E G is an element We note that Indeed, the Lie algebra g admits a G-invariant symmetric nondegenerate bilinear form. Such a form on g produces an isomorphism of ad(E G ) with ad(E G ) * . Consequently, by Serre duality, Any homomorphic section φ of the adjoint bundle ad(E G ) has a Jordan decomposition where f s is pointwise semisimple and f n is pointwise nilpotent. From the assumption that E G does not admit any holomorphic reduction of structure group to any proper Levi subgroup of G it can be deduced that the semisimple section f s is given by some element of the center of g. Indeed, the conjugacy class of f s (x) is independent of x ∈ M; this follows from the fact that there is no nonconstant holomorphic map from M to the affine variety that parametrizes the conjugacy classes of semisimple elements in g. Fix an element z 0 ∈ g in the conjugacy class determined by f s (x). Then the centralizer of z 0 in G is a Levi subgroup L(z 0 ) of some parabolic subgroup of G. Note that for any y ∈ E G and z ∈ g, the pair (y, z ) represents an element of ad(E G ). It is straightforward to check that is a holomorphic reduction of structure group of E G to the subgroup L(z 0 ) of G. Now from the given assumption that E G does not admit any holomorphic reduction of structure group to any proper Levi subgroup of G we conclude that L(z 0 ) = G. Consequently, the semisimple section f s is given by some element of the center of g.
Since the semisimple section f s is given by some element of the center of g, from the given condition on E G it can be deduced that The nilpotent section f n of ad(E G ) gives a holomorphic reduction of structure group of E G to a proper parabolic subgroup P of G. This reduction E P has the property that f n lies in the image where ad(E P ) is the adjoint bundle of E P , and is the nilpotent radical bundle of ad(E P ). On the other hand, the cohomology class τ (E G ) ∈ H 1 (M, K M ⊗ ad(E G )) in (3.2) lies in the image of the homomorphism induced by the inclusion map ad(E P ) → ad(E G ). next we observe the following. Let p be the Lie algebra of P, and let R(p) ⊂ p be the nilpotent radical of p. Then for any G-invariant symmetric bilinear form B on g, we have This, and the above observations that , which implies that τ (E G ) = 0. Therefore, the Atiyah exact sequence for E G splits holomorphically, implying that the holomorphic principal G-bundle E G admits a holomorphic connection.

Non-reductive algebraic groups
Now let G be a unipotent affine algebraic group defined over C. It is know that any holomorphic principal Gbundle over a compact connected Riemann surface M admits a holomorphic connection [7, p. 4015, Corollary 3.8].
Let G be a connected affine algebraic group defined over C. The unipotent radical of G will be denoted by R u (G). Let

L(G) := G/R u (G)
be the reductive quotient. Let E G be a holomorphic principal G-bundle over a compact connected Riemann surface M. Assume that the holomorphic principal L(G)-bundle E G /R u (G) admits a holomorphic connection. The following is a natural question to ask:

Does the holomorphic principal G-bundle E G admit a holomorphic connection?
We will show in Sect. 4 that the above question has a negative answer.

A construction
This section is an exposition of [6]. Let M be a compact connected Riemann surface of genus g, with g ≥ 2. Denote by K M the holomorphic cotangent bundle of M. The linear equivalence class of K M can be expressed as where P ∈ M is a single point and D is an effective divisor of degree 2g − 3 such that P is disjoint from the support of D. We note that this can be deduced from the fact that the linear system K M is globally generated and Indeed, let be the map corresponding to K M . Take a hyperplane H ⊂ P(H 0 (M, K M )) such that there is a point z ∈ H f (M) satisfying the following two conditions: (1) H and f (M) intersects transversally at z, and (2) f is unramified over z.
Fix P and D as in (4.1). Let us now split the divisor D into two parts, namely where D Q and D R are effective divisors with By Serre duality, In particular, we can choose a nonzero element (which is actually unique up to multiplication by a scalar) the cohomology class θ in (4.5) produces a short exact sequence of holomorphic vector bundles on M. This exact sequence does not split holomorphically because θ = 0. From (4.3), it follows that degree(V ) = −1. Consider the holomorphic vector bundle on M of rank three and degree zero. We have (4.6)). Let s P (respectively, s Q ) be the holomorphic section of O M (P) (respectively, O M (D Q )) given by the constant function 1 on M. So s P (respectively, s Q ) vanishes over P (respectively, the support of D Q ) of order one, and is nonzero everywhere else. Now consider the holomorphic section defined by x −→ (s P (x) , s Q (x)). Note that σ 1 does not vanish anywhere because P is disjoint from the support of D Q according to (4.1) and (4.2). Let σ be the composition (see (4.8)). Since σ is nowhere vanishing, we get a short exact sequence of holomorphic vector bundles on M By construction, O M (P) is a direct summand of E. Since degree(O M (P)) = 0, from the criterion of Atiyah-Weil we conclude that E does not admit a holomorphic connection.
We will now prove that both F and Q in (4.11) admit a holomorphic connection. Of course, F := O M admits the trivial holomorphic connection. So we need to show that Q in (4.11) admits a holomorphic connection. The following lemma would be needed for that.

Lemma 4.1 The holomorphic vector bundle Q in (4.11) is a nontrivial extension of the line bundle O M (−D R ) by O M (P + D Q ).
Proof On the one hand, we have On the other hand, (see (4.9)). It follows that (4.12) (4.10)) produces an inclusion of the quotient (4.11)). Therefore, from (4.12), we have as a subbundle. Using (4.6), (4.7) we have Note that its degree is zero (4.3). Therefore, from (4.13), Consequently, from (4.13), we get a short exact sequence of vector bundles To complete the proof of the lemma, we need to show that the short exact sequence in (4.14) does not split. Let be the extension class for the exact sequence in (4.14). We will now compute ω. From (4.6) and (4.7) we have the short exact sequence be the cohomology class for this exact sequence. Evidently, θ coincides with where θ is the class in (4.5). Next, consider the homomorphism γ defined by the composition (see (4.12)), where the homomorphism is the inclusion of the second factor. Clearly, this composition γ coincides with the natural inclusion of the coherent sheaf O M (D Q ) in O M (P + D Q )). Therefore, the cohomology classes ω and θ (constructed in (4.15) and (4.5)) satisfy the equation is the homomorphism induced by the natural inclusion map of coherent analytic sheaves. Consider the short exact sequence of coherent sheaves where O M (P + D Q + D R ) P is the torsion sheaf supported at P with its stalk being the fiber of the line bundle be the long exact sequence of cohomologies associated with it. We have and, by Riemann-Roch and (4.4), These imply that α 1 in (4.17) is surjective. Therefore, α 2 in (4.17) is the zero homomorphism. This implies that ρ in (4.17) is injective.

Proposition 4.2
The holomorphic vector bundle Q in (4.14) admits a holomorphic connection.
Proof Assume that Q does not admit any holomorphic connection. Since degree(Q) = 0, and Q does not admit any holomorphic connection, the criterion of Atiyah-Weil says that Q holomorphically decomposes as where the inclusion is constructed in (4.13). Since and L are line subbundles on Q, this implies that the two subbundles O M (P + D Q ) and L coincide. Hence, Lemma 4.1). Therefore, the decomposition Q = L ⊕ M in (4.18) produces a splitting of the short exact sequence in (4.14). But we know from Lemma 4.1 that the short exact sequence in (4.14) does not split. In view of the above contradiction, we conclude that Q admits a holomorphic connection.
As we have seen, E does not admit a holomorphic connection. On the other hand, consider the short exact sequence in (4.11). The trivial holomorphic line bundle F = O M admits the trivial holomorphic connection. The quotient bundle Q is flat by Proposition 4.2.
Therefore, the question in Sect. 3.1 has a negative answer.

Logarithmic connections on a Riemann surface
The multiplicative group C\{0} will be denoted by G m . A torus is a product of copies of G m . Any two maximal tori in a complex algebraic group are conjugate [8, p. 158, Proposition 11.23(ii)]. By a homomorphism between algebraic groups or by a character we will always mean a holomorphic homomorphism or a holomorphic character.
Let M be a compact connected Riemann surface. Fix a finite subset The reduced effective divisor x 1 + · · · + x n will also be denoted by The exact sequence in (5.1) will be called the logarithmic Atiyah exact sequence for E G . A logarithmic connection on E G singular over D is a holomorphic homomorphism , where σ is the homomorphism in (5.1). Note that giving such a homomorphism θ is equivalent to giving a homomorphism such that • i 0 = Id ad(E H ) , where i 0 is the homomorphism in (5.1). So, a holomorphic principal G-bundle admits a logarithmic connection singular over D if and only if the logarithmic Atiyah exact sequence in (5.1) splits holomorphically.

Residue of a logarithmic connection
Given a vector bundle W on M, the fiber of W over any point x ∈ M will be denoted by W x . For any O M -linear homomorphism f : W −→ V of holomorphic vector bundles, its restriction W x −→ V x will be denoted by f (x).
From (2.4) and (5.1), we have the commutative diagram of homomorphisms Note that ι(x) = 0 if x ∈ D; therefore, in that case d f (x) • j (x) = 0. Consequently, for every x ∈ D there is a homomorphism where i 0 is the homomorphism in (5.3). Therefore, from (5.1), we have note that the composition of homomorphisms is an isomorphism.
For any x ∈ D, the fiber T M(−D) x is identified with C using the Poincaré adjunction formula [12, p. 146]. Indeed, for any holomorphic coordinate z around x with z(x) = 0, the image of z ∂ ∂z in T M(−D) x is independent of the choice of the coordinate function z; the above-mentioned identification between T M(−D) x and C sends this independent image to 1 ∈ C. Therefore, from (5.5), we have for all x ∈ D.
where R x is the homomorphism in ( for all x ∈ D. This proves the following: Lemma 5.1 With the above notation, if E G admits a logarithmic connection θ singular over D with residue w x ∈ ad(E G ) x at each x ∈ D, then E S admits a logarithmic connection θ = A • θ singular over D with residue α(w x ) at each x ∈ D.
Now we set G = GL(r, C). Take a holomorphic vector bundle E on M. The group of all holomorphic automorphisms of E will be denoted by Aut(E). Let be an endomorphism such that φ(x) • T = T • φ(x) for all φ ∈ Aut(E).

Lemma 5.2 Let V ⊂ E be a holomorphic subbundle which is a holomorphic direct summand. Then T
Proof Take a holomorphic subbundle W ⊂ E such that E = V W . For nonzero complex numbers a, b, we have the automorphism A(a, b) ∈ Aut(E) which acts on V (respectively, W ) as multiplication by a (respectively, b). Since T commutes with A(a, b)(x) for all a, b ∈ C\{0}, it follows that T (V x ) ⊂ V x .
The following theorem is proved in [4]: