Stein complements in compact K\"ahler manifolds

TL;DR

This paper investigates conditions under which the complement of a hypersurface in a compact Kähler manifold is Stein, with applications to the projectivization of canonical extensions of tangent bundles.

Contribution

It provides new criteria for Steinness of complements in Kähler manifolds and applies these to the case of projectivized tangent bundle extensions.

Findings

Identifies conditions for Stein complements in Kähler manifolds

Applies results to projectivizations of tangent bundle extensions

Advances understanding of complex geometric structures

Abstract

Given a projective or compact K\"ahler manifold X and a (smooth) hypersurface Y, we study conditions under which $X \setminus Y$ could be Stein. We apply this in particular to the case when X is the projectivization of the so-called canonical extension of the tangent bundle $T_M$ of a projective manifold M with Y being the projectivization of $T_M$ itself.