Vaisman manifolds with vanishing first Chern class

TL;DR

This paper classifies compact Vaisman manifolds with zero first Chern class, revealing their geometric structures, stability properties, and automorphism behaviors under deformations, with implications for Calabi-Yau and Beauville-Bogomolov type decompositions.

Contribution

It provides a detailed classification of Vaisman manifolds with vanishing first Chern class, including their metric properties, deformation stability, and automorphism group behavior.

Findings

Vaisman manifolds with non-positive Bott-Chern class admit canonical metrics.

Such manifolds are quasi-regular and stable under deformations.

Calabi-Yau Vaisman manifolds satisfy a Beauville-Bogomolov type decomposition.

Abstract

Compact Vaisman manifolds with vanishing first Chern class split into three categories, depending on the sign of the Bott-Chern class. We show that Vaisman manifolds with non-positive Bott-Chern class admit canonical metrics, are quasi-regular and are stable under deformations. We also show that Calabi-Yau Vaisman manifolds satisfy a version of the Beauville-Bogomolov decomposition and have torsion canonical bundle. Finally, we prove a general result concerning the behaviour of the automorphism group of a complex manifold under deformations.