I. Curvature of direct images and deformations  (14:30 - 15:45) 

Abstract: Consider a proper holomorphic fibration of Kähler manifolds $f\colon X\to B$ and denote by $\omega_{X/B}:=\omega_X\otimes f^*\omega_B^\vee$ the relative dualizing sheaf.

 It is classically known that the vector bundles associated to the direct image $f_*\omega_{X/B}$ can be naturally endowed with a Hermitian metric with semipositive curvature. Furthermore, this semipositivity can be read in terms of the cup product with the Kodaira-Spencer class of the fibers, giving an interesting interplay between curvature and deformations of complex varieties.

 More generally, there is a vast literature on the curvature properties of higher direct images of sheaves of twisted relative forms (and their subbundles), in which the deformation data associated to the fibration play a key role.

 In this first talk I will give an introduction and overview of the topic, with particular emphasis on this relationship between curvature and deformations.