Canonical extensions of manifolds with nef tangent bundle

TL;DR

This paper proves that for compact Kähler manifolds with nef tangent bundle, their canonical extension is Stein, providing partial answers to a question about the relationship between these properties.

Contribution

It establishes that the canonical extension of a Kähler manifold with nef tangent bundle is Stein, advancing understanding of the geometric structure of such manifolds.

Findings

Canonical extension is Stein for manifolds with nef tangent bundle

Partially answers Greb-Wong's question on equivalence of properties

Provides additional results for surfaces in the converse direction

Abstract

To any compact K\"ahler manifold $(X, \omega)$ one may associate a bundle of affine spaces $Z_X\rightarrow X$ called a \emph{canonical extension} of $X$. In this paper we prove that if the tangent bundle of $X$ is nef, then the total space $Z_X$ is a Stein manifold. This partially answers a question raised by Greb-Wong of whether these two properties are actually equivalent. We also complement some known results for surfaces in the converse direction.