Curvature of the total space of a Griffiths negative vector bundle and quasi-Fuchsian space

TL;DR

This paper investigates the curvature properties of holomorphic vector bundles, especially Griffiths and Nakano positivity, and applies these results to construct invariant K"ahler metrics on quasi-Fuchsian space, extending classical metrics.

Contribution

It provides new curvature estimates for Griffiths negative and Nakano positive bundles and constructs a mapping class group invariant K"ahler metric on quasi-Fuchsian space.

Findings

Curvature of total space of Griffiths negative bundles is non-positive along tautological directions.

Estimates for Nakano curvature operator on positive direct image bundles are established.

Constructed a K"ahler metric on quasi-Fuchsian space extending Weil-Petersson metric.

Abstract

For a holomorphic vector bundle $E$ over a Hermitian manifold $M$ there are two important notions of curvature positivity, the Griffiths positivity and Nakano positivity. We study the consequence of these positivities and the relevant estimates. If $E$ is Griffiths negative over K\"ahler manifold, then there is a K\"ahler metric on its total space $E$, and we calculate the curvature and prove the non-positivity of the curvature along the tautological direction. The Nakano positivity can be formulated as a positivity for the Nakano curvature operator and we give estimate the Nakano curvature operator associated with a Nakano positive direct image bundle. As applications we construct a mapping class group invariant K\"ahler metric on the quasi-Fuchsian space QF$(S)$, which extends the Weil-Petersson metric on the Teichm\"uller space $\mathcal{T}(S)\subset {\rm QF}(S)$, and we obtain estimates for the Nakano curvature operator for the dual Weil-Petersson metric on the holomorphic cotangent bundle of Teichm\"uller space.