Deformation of K\"ahler manifolds

TL;DR

This paper demonstrates that certain compact K"ahler manifolds can be approximated by projective manifolds when their canonical bundle satisfies specific positivity or negativity conditions, extending previous deformation results.

Contribution

It establishes new deformation approximation results for K"ahler manifolds under semipositivity or seminegativity conditions on the canonical bundle.

Findings

K"ahler manifolds with specific canonical bundle conditions can be approximated by projective manifolds

Extension of Voisin's negative result to positive cases under additional conditions

Provides new insights into the deformation theory of K"ahler manifolds

Abstract

It has been shown by Claire Voisin in 2003 that one cannot always deform a compact K\"ahler manifold into a projective algebraic manifold, thereby answering negatively a question raised by Kodaira. In this article, we prove that under an additional semipositivity or seminegativity condition on the canonical bundle, the answer becomes positive, namely such a compact K\"ahler manifold can be approximated by deformations of projective manifolds.