Generically nef vector bundles and geometric applications

TL;DR

This paper proves that the cotangent bundle of a general type manifold is generically ample and explores the implications for manifolds with nef and ample tangent bundles, connecting to classical vector field theorems.

Contribution

It establishes the generically ample property of cotangent bundles for manifolds of general type and analyzes cases with intermediate Kodaira dimension.

Findings

Cotangent bundle of a general type manifold is generically ample.

Manifolds with generically nef and ample tangent bundles are characterized.

Connections to classical theorems on vector fields are explored.

Abstract

The cotangent bundle of a non-uniruled projective manifold is generically nef, due to a theorem of Miyaoka. We show that the cotangent bundle is actually generically ample, if the manifold is of general type and study in detail the case of intermediate Kodaira dimension. Moreover, manifolds with generically nef and ample tangent bundles are investigated as well as connections to classical theorems on vector fields on projective manifolds.