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Splitting jet sequences. (English) Zbl 1059.32004

Here the author proves the following result which is an infinitesimal converse of a result of I. Biswas [J. Math. Pures Appl. 78, 1–26 (1999; Zbl 0921.53007)]: Let \(E\) be a holomorphic vector bundle on the compact Kähler manifold \(X\). Assume that the \(k\)-th jet sequence of \(E\) splits for some \(k>1\). Then \(X\) admits a holomorphic normal projective connection and the jet bundle \(J_{k-1}(E)\) admits a holomorphic connection. If \(\det(E)\) is not nef and either \(X\) is projective or it contains a rational curve, then \(X \cong P_n\) and \(E(1-k)\) is trivial. If \(\det(E)^\ast \equiv 0\), then \(X\) is covered by a torus and \(E\) admits a holomorphic affine connection. If \(\det(E)^\ast\) is ample, then \(X\) is covered by the unit ball.

MSC:

32Q15 Kähler manifolds
32L05 Holomorphic bundles and generalizations
58H10 Cohomology of classifying spaces for pseudogroup structures (Spencer, Gelfand-Fuks, etc.)
58A20 Jets in global analysis

Citations:

Zbl 0921.53007
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