Locally Constant Fibrations and Positivity of Curvature

TL;DR

This paper characterizes smooth complex projective varieties with nef anti-canonical bundles as fiber bundles over K-trivial varieties with locally constant transition functions, and explores the positivity of curvature on tangent bundles.

Contribution

It establishes a precise structural description of varieties with nef anti-canonical bundles and relates this to locally constant fibrations, also analyzing curvature properties of tangent bundles.

Findings

Varieties with nef anti-canonical bundle are fibered over K-trivial varieties after finite étale cover.

Projective fiber bundles with locally constant transition functions over K-trivial varieties have nef anti-canonical bundles.

Results connect the structure of tangent bundles with positive curvature metrics.

Abstract

Up to finite \'etale cover, any smooth complex projective variety $X$ with nef anti-canonical bundle is a holomorphic fibre bundle over a $K$-trivial variety with locally constant transition functions. We show that this result is optimal by proving that any projective fibre bundle with locally constant transition functions over a $K$-trivial variety has a nef anti-canonical bundle. Moreover, we complement some results on the structure theory of varieties whose tangent bundle admits a singular hermitean metric of positive curvature.