# Dror Varolin: Direct images of holomorphic vector bundles

Title: Direct images of holomorphic vector bundles.

Abstract: For a holomorphic submersion f:X -->Y of complex manifolds the (zeroth) direct image of a holomorphic vector bundle E --> X is the presheaf on Y that assigns to an open set U in Y the sections H^0(f^{-1}(U), O(E)). If X is Stein, for example, these groups are huge, so it is natural to cut them down somewhat by introducing L^2 norms. An L^2 norm can be defined if X has a Hermitian Riemannian metric g and E --> X has a Hermitian vector bundle metric h. By letting U shrink down to a point, one obtains over each point y of Y a Hilbert space of holomorphic sections of the restriction of E to f^{-1}(y). Under stronger conditions on f, X and E, this family of vector spaces might be locally trivial, but in general this is not the case, even if f is proper (in which case these vector spaces are finite dimensional).

Nevertheless these fibrations, which have a natural Hermitian fiber metric, namely the L^2 metric, share some properties with vector bundles. For example, one can talk about their Chern connection and curvature.

Berndtsson showed that when E is a line bundle, g is a Kahler metric, and the direct image is a vector bundle, then the curvature of the direct image bundle is positive in the sense of Nakano as soon as R(h) + Ricci(g) is positive, where R(h) is the curvature of the metric h for the line bundle E-->X, and Ricci(g) = - dd^c log det (g).

In this talk, I will discuss an analogue of Berndtsson's Theorem in the cases in which (i) E has higher rank and (ii) the direct image is not necessarily a vector bundle. I will also discuss some of the applications of the theorem.

Seminar bo v predavalnici 3.06 na Jadranski 21. Vljudno vabljeni!

Vodji seminarja

Franc Forstnerič in Barbara Drinovec Drnovšek