Positivity of direct image sheaves. A geometric point of view (Q993657)

Suppose that \(\varphi : X \to Y\) is a fibration (a morphism with connected fibers) between two complex projective manifolds. Over the last 30 years, it has been incredibly important to understand the positivity properties of \(\varphi_* \omega_{X/Y}^{\otimes n}\) or more generally of \(\varphi_*( \omega_{X/Y} \otimes L)\) where \(L\) is a line bundle which is also in some sense positive. The goal of collection of lecture notes is to explain, with many examples, the connections between these positivity properties and the geometry of \(\varphi\). The history and a description of some of the many and varied applications of these ideas are laid out in the introduction. In the second section, a series of examples is given showing what can (and cannot) happen. In the third section, some of the ideas of the proofs of these positivity theorems are explained (including variants for higher direct images). Finally, in section 4, these ideas are applied to the study of projective manifolds with nef anti-canonical line bundle. An appendix on reflexive sheaves, singularities and duality is also included.