Deformations of maps with fixed target and curvature of direct image bundles

TL;DR

This paper explores the relationship between deformations of maps with fixed targets and the curvature properties of associated direct image bundles, deriving formulas and establishing a seminegativity result.

Contribution

It introduces new curvature formulas linking map deformations to direct image bundle curvature and proves a seminegativity property for related vector bundles.

Findings

Derived curvature formulas for deformations of maps with fixed targets.

Established a seminegativity result for a vector bundle of relative forms.

Connected deformation theory with curvature properties of direct image bundles.

Abstract

Consider a smooth proper holomorphic fibration of complex manifolds. It is known that the semipositivity of the curvature of the direct image of the relative canonical bundle can be read in terms of the cup product with the Kodaira-Spencer class of the fibers. Motivated by this result, in this paper we study the relation between deformations of maps with fixed target and the curvature of certain direct image bundles. As a result, we find some curvature formulas and use them to prove a seminegativity result for a vector bundle of relative forms naturally related to the deformation data.