Locally Constant Fibrations and Positivity of Curvature

Abstract

Up to finite étale 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.

20 pages, Comments welcome! v3: Slight changes in accordance with suggestions made by the referee, main results unchanged. To appear in Bull. London Math. Soc