Compact K\"ahler manifolds with no projective specialization

TL;DR

This paper constructs a compact K"ahler manifold that cannot be deformed into a projective manifold within any proper flat family, providing new insights into the limitations of algebraic approximation.

Contribution

It demonstrates the existence of a compact K"ahler manifold with no projective specialization, strengthening previous counterexamples to the Kodaira algebraic approximation problem.

Findings

Existence of a compact K"ahler manifold without projective specialization

Topological version of the non-specialization result

Advancement in understanding algebraic approximation limitations

Abstract

We show the existence of a compact K\"ahler manifold which does not fit in a proper flat family over an irreducible base with one projective (possibly singular) fiber. We also give a topological version of this statement. This strengthens our earlier counterexamples to the Kodaira algebraic approximation problem.