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.
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Andreas Höring, Positivity of direct image sheaves a geometric point of view. Enseign. Math. 56 (2010), no. 1, pp. 87–142