Canonical extensions of manifolds with nef tangent bundle

Abstract

To any compact Kähler manifold $(X,\omega )$ one may associate a bundle of affine spaces $Z_X\to X$ 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.

13 pages. Exposition severely improved, main results unchanged. To appear in Arkiv för Matematik