Algebraic approximations of compact K\"ahler manifolds of algebraic codimension 1

TL;DR

This paper proves that compact Kähler manifolds with algebraic dimension one less than their complex dimension can be deformed into projective manifolds, advancing understanding of their geometric structure.

Contribution

It establishes that such Kähler manifolds can be approximated arbitrarily closely by projective manifolds through small deformations.

Findings

Kähler manifolds of algebraic dimension $ ext{dim} X - 1$ can be deformed into projective manifolds

Provides a method for algebraic approximation of these manifolds

Enhances understanding of the relationship between Kähler and projective geometries

Abstract

For every compact K\"ahler manifold $X$ of algebraic dimension $a(X) = \dim X - 1$, we prove that $X$ has arbitrarily small deformations to some projective manifolds.