# A Criterion for Flatness of Sections of Adjoint Bundle of a Holomorphic Principal Bundle over a Riemann Surface

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6 116 Indranil Biswas 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 E G E Z X E H1 X X E Hl. 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 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 (EG ) 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

8 118 Indranil Biswas 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 vectorbundle 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 Let (3.5) E = β H 0 (X, End(E)). be the generalized eigen-bundle decomposition of E for β. Therefore, l i=1 E i β Ei = λ i Id Ei +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 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.

9 A Criterion for Flatness of Sections of Adjoint Bundle 119 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. References [AAB] B. Anchouche, H. Azad and I. Biswas, Harder Narasimhan reduction for principal bundles over a compact Kähler manifold, Math. Ann. 323 (2002), [At] M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), [AB] H. Azad and I. Biswas, On holomorphic principal bundles over a compact Riemann surface admitting a flat connection, Math. Ann. 322 (2002), [BG] I. Biswas and T. L. Gómez, Connections and Higgs fields on a principal bundle, Ann. Global Anal. Geom. 33 (2008), [DM] F. Digne and J. Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, [Hu] [Si] J. E. Humphreys, Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, 43, American Mathematical Society, Providence, RI, C. T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), [We] A. Weil, Generalisation des fonctions abeliennes, Jour. Math. Pure Appl. 17 (1938),

10 120 Indranil Biswas Indranil Biswas School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Bombay India

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