Bergman bundles and applications to the geometry of compact complex manifolds

Jean-Pierre Demailly (1)

1 IF - Institut Fourier

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 L² 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.

Mots clés: Compact Kähler manifold; very ample vector bundle; Bergman metric; Hardy space; Stein manifold; Grauert tubular neighborhood; Hermitian metric; Hilbert bundle; invariance of plurigenera

Domaines: Variables complexes [math.CV]

Jean-Pierre Demailly. Bergman bundles and applications to the geometry of compact complex manifolds. 2020. ⟨hal-02501656v2⟩