II. Curvature of direct images and deformations of maps with fixed target  (16:10 - 17:10)

Abstract: Motivated by the interplay between curvature of direct images and deformations of complex manifolds discussed in the first talk, it is natural to ask whether a similar setting can be explored for other types of deformation theories. 

 Based on recent work, this talk explores the relationship between the so-called deformations of maps with fixed target and the curvature of appropriate direct image bundles. We discuss curvature formulas for these bundles and use them to prove a seminegativity result for a vector bundle of relative forms naturally related to the deformation data. This negativity result can be seen as a natural extension, in the case of deformations of maps with fixed target, of a classical result of Zuo, asserting the seminegativity of the kernels of the Kodaira-Spencer maps on Hodge bundles.