Functionals for the Study of LCK Metrics on Compact Complex Manifolds

TL;DR

This paper introduces a new functional approach to investigate the existence of locally conformally Kähler metrics on compact complex manifolds, with distinct methods depending on the complex dimension.

Contribution

It proposes a novel functional framework tailored to the complex dimension, advancing the understanding of LCK metrics on compact complex manifolds.

Findings

Functional varies with complex dimension

Provides criteria for existence of LCK metrics

Differentiates methods for dimension 2 and higher

Abstract

We propose an approach to the existence problem for locally conformally K\"ahler metrics on compact complex manifolds by introducing and studying a functional that is different according to whether the complex dimension of the manifold is $2$ or higher.