Curvatures of direct image sheaves of vector bundles and applications I

TL;DR

This paper derives curvature formulas for direct image sheaves of vector bundles over Kähler fibrations, showing negativity properties under certain conditions and exploring implications for moduli spaces of flat vector bundles.

Contribution

It provides new, simplified curvature formulas for direct image sheaves and applies them to establish negativity and automorphism results in vector bundle moduli spaces.

Findings

Direct image sheaves are Nakano-negative when the family is infinitesimally trivial and the bundle is Nakano-negative.

Curvature formulas connect the vanishing of Chern curvature to automorphisms of the pair (X, E).

Applications include insights into the geometry of moduli spaces of projectively flat vector bundles.

Abstract

Let $p:\sXS$ be a proper K\"ahler fibration and $\sE\sX$ a Hermitian holomorphic vector bundle. As motivated by the work of Berndtsson(\cite{Berndtsson09a}), by using basic Hodge theory, we derive several general curvature formulas for the direct image $p_*(K_{\sX/S}\ts \sE)$ for general Hermitian holomorphic vector bundle $\sE$ in a simple way. A straightforward application is that, if the family $\sXS$ is infinitesimally trivial and Hermitian vector bundle $\sE$ is Nakano-negative along the base $S$, then the direct image $p_*(K_{\sX/S}\ts \sE)$ is Nakano-negative. We also use these curvature formulas to study the moduli space of projectively flat vector bundles with positive first Chern classes and obtain that, if the Chern curvature of direct image $p_*(K_{X}\ts E)$--of a positive projectively flat family $(E,h(t))_{t\in \mathbb D}X$--vanishes, then the curvature forms of this family are connected by holomorphic automorphisms of the pair $(X,E)$.