Documenta Math. 111 A Criterion for Flatness of Sections of Adjoint Bundle of a Holomorphic Principal Bundle over a Riemann Surface

Let EG be a holomorphic principal G-bundle over a com- pact connected Riemann surface, where G is a connected reductive affine algebraic group defined overC, such that EG admits a holo- morphic connection. Take any β ∈ H 0 (X, ad(EG)), where ad(EG) is the adjoint vector bundle for EG, such that the conjugacy class β(x) ∈ g/G, x ∈ X, is independent of x. We give a sufficient condi- tion for the existence of a holomorphic connection on EG such that β is flat with respect to the induced connection on ad(EG).


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:

Indranil Biswas
Let E be a holomorphic vector bundle of rank r on X, and β ∈ H 0 (X, End(E)).Let be the generalized eigen-bundle decomposition for β.So β| Ei = λ i • Id Ei + 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 ri−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 ri−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).

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 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 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 the above projection At(E G ) −→ T X, where T X is the holomorphic tangent bundle of X, is defined by the differential df : 3) [At].
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 2 X = 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.

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 Hbundle 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.
) each H i is simple of adjoint type (the center is trivial), and (2) the natural homomorphism is surjective, and the kernel of ϕ is a finite group.Let Documenta Mathematica 18 (2013) 111-120 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 Hi ) be the adjoint vector bundles for E Z and E Hi respectively.Since the homomorphism ϕ in (3.1) induces an isomorphism of Lie algebras, we have ad(E Hi )) .
Let β Z (respectively, β i ) be the holomorphic section of ad(E Z ) (respectively, ad(E Hi )) 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 Hi .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 Hi ).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 Hi 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 b x of ad(E Hi ) x such that β i (x) ∈ b x [Hu,p. 62,Theorem].Let b ⊂ ad(E Hi ) be the Borel subalgebra bundle such that for every point x the fiber ( b) x is b x .Fix a Borel subgroup B ⊂ H i .Using b, we will construct a holomorphic reduction of structure group of E Hi 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 Hi ) is the quotient of E Hi × h i where two points (z 1 , v 1 ) and (z 2 , v 2 ) of E Hi × 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 Hi ) x be the complex submanifold consisting of all z ∈ (E Hi ) x such that for all v ∈ b, the image of ( 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 Hi ) 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 The corresponding filtration of vector bundles associated to E B is a filtration of ad(E Hi ) 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 Hi ) such that each successive quotient is the trivial line bundle.This immediately implies that the vector bundle ad(E Hi ) is semistable.Hence the holomorphic principal H i -bundle E Hi is semistable [AAB,p. 698,Lemma 3].Since H i is simple, and E Hi is semistable, there is a natural holomorphic connection on E Hi [BG,p. 20, Theorem 1.1] (set the Higgs field in [BG,Theorem 1.1] to be zero).Let D Hi denote this connection.The vector bundle ad(E Hi ) 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 Hi ).This connection D ad coincides with the one induced by D Hi (see the construction of the connection in [BG]).Any holomorphic section of ad(E Hi ) is flat with respect to D ad .To see this, let φ : O X −→ ad(E Hi ) be the homomorphism given by a nonzero holomorphic section of ad(E Hi ).Since image(φ) is a semistable subbundle of ad(E Hi ) 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 Hi ) induced by D Hi 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 Hi , [1 , ℓ], then we get a holomorphic connection on E G ; simply pullback the connection form using the map The connection on E G given by the connections on E Z and E Hi , [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 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 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 x ∈ X, is independent of x, where φ is defined in (2.2).This implies that Documenta Mathematica 18 (2013) 111-120 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 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 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 β 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 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 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 Documenta Mathematica 18 (2013) 111-120 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 a holomorphic vector bundle of rank r on X.Take any be the generalized eigen-bundle decomposition of E for β.Therefore, 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 ri−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 ri−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.
Documenta Mathematica 18 (2013) 111-120 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 not admit a holomorphic connection because it is of nonzero degree.

Indranil Biswas
Hi be the complex submanifold such that E B (E Hi ) 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 Hi to the subgroup B. Consider the adjoint action of B on b 1 z , v) in ad(E Hi ) x lies in b 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 Hi ) x , with the action of B on E B,x being transitive.Let E B ⊂ E