Positivity of direct image sheaves a geometric point of view

Abstract

These are extended notes of talks I gave at the workshop "Rencontre positivité'' in Rennes. The aim of these talks was to illustrate the interaction between the geometry of a fibration and the positivity of direct image sheaves. We give numerous examples and an introduction to the Kollár-Viehweg method for proving direct image theorems. As an application, we study the geometry of projective manifolds with nef anticanonical bundle.