An integral invariant from the view point of locally conformally K\"ahler geometry

TL;DR

This paper investigates an integral invariant related to locally conformally K"ahler (LCK) geometry, which obstructs the existence of certain volume forms and K"ahler-Einstein metrics, and explores its properties and vanishing conditions.

Contribution

It introduces a new integral invariant for coverings of complex manifolds with automorphic volume forms and shows its equivalence to a known invariant in LCK geometry.

Findings

Invariant obstructs K"ahler-Einstein metric existence.

Invariant vanishes for compact Vaisman manifolds.

Invariant defined for coverings with automorphic volume forms.

Abstract

In this paper we study an integral invariant which obstructs the existence on a compact complex manifold of a volume form with the determinant of its Ricci form proportional to itself, in particular obstructs the existence of a K\"ahler-Einstein metric, and has been studied since 1980's. We study this invariant from the view point of locally conformally K\"ahler geometry. We first see that we can define an integral invariant for coverings of compact complex manifolds with automorphic volume forms. This situation typically occurs for locally conformally K\"ahler manifolds. Secondly, we see that this invariant coincides with the former one. We also show that the invariant vanishes for any compact Vaisman manifolds.