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Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. Math. 82 (1965), 540-564. 13. C. Simpson, Constructing variation of Hodge structure using Yang-Mills theorey and applications to uniformization, J. Amer. Math. Soc. 1 (1988), 867-918. 14. -T. Siu, A Hartogs type extension theorem for coherent analytic sheaves, Ann. of Math. 93 (1971), 166-188. 15. -T. Siu, Lectures on Hermitian-Einstein metrics for stable bundles and KahlerEinstein metrics, (Birkhauser, Basel-Boston, 1987).
It is easy to see that Hk
W')®Xj. Then u; and
G and M x G —> M respectively. The Maurer-Cartan equation, dw = [w,w], means that the curvature of w vanishes. It follows that the curvature of ip also vanishes since r) is a Lie-Hopf algebra homomorphism. Therefore we have an unique horizontal lift M —> M x G through the point (o, e) € M x G, which induces, via the projection M x G —» G, an unique map A: M —» G with A(o) = e. Condition "horizontal" is nothing but A* = TT'T] on H(G); this also implies that A is holomorphic since so is tp.
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