Approximation results for compact Vaisman manifolds

TL;DR

This paper extends approximation theorems to Vaisman manifolds, a class of non-Kähler complex manifolds, by studying how their metrics can be approximated via immersions into Hopf manifolds.

Contribution

It generalizes Tian's approximation theorem from projective to Vaisman manifolds, a significant class of non-Kähler manifolds.

Findings

Vaisman metrics can be approximated by metrics induced by immersions into Hopf manifolds.

The approximation extends Tian's theorem to a broader class of complex manifolds.

Provides new tools for studying non-Kähler geometry.

Abstract

We extend the Tian approximation theorem for projective manifolds to a class of complex non-K\"ahler manifolds, the so-called Vaisman manifolds. More precisely, we study the problem of approximating compact regular, respectively quasi-regular, Vaisman metrics by metrics induced by immersions, respectively embeddings, into Hopf manifolds.