Bergman bundles and applications to the geometry of compact complex manifolds

We introduce the concept of Bergman bundle attached to a hermitian manifold $X$, assuming the manifold $X$ to be compact—although the results are local for a large part. The Bergman bundle is some sort of infinite dimensional very ample Hilbert bundle whose fibers are isomorphic to the standard $L^2$ Hardy space on the complex unit ball; however the bundle is locally trivial only in the real analytic category, and its complex structure is strongly twisted. We compute the Chern curvature of the Bergman bundle, and show that it is strictly positive. As a potential application, we investigate a long standing and still unsolved conjecture of Siu on the invariance of plurigenera in the general situation of polarized families of compact Kähler manifolds.

Keywords

Bergman metric, Hardy space, Stein manifold, Grauert tubular neighborhood, Hermitian metric, Hilbert bundle, very ample vector bundle, compact Kähler manifold, invariance of plurigenera

2010 Mathematics Subject Classification

32F32, 32J25

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The author is supported by the Advanced ERC grant ALKAGE, no 670846 from September 2015, attributed by the European Research Council.

Received 5 March 2020

Accepted 11 April 2021

Published 10 February 2022