Balanced metrics and Gauduchon cone of locally conformally Kahler manifolds

TL;DR

This paper investigates the relationships between balanced, SKT, and LCK Hermitian structures on complex manifolds, proposing conjectures about their implications for the existence of Kähler metrics and verifying these for known LCK classes.

Contribution

It introduces conjectures linking the existence of two Hermitian structures to Kähler metrics and verifies these conjectures for known classes of LCK manifolds.

Findings

Partial results supporting the conjecture that two Hermitian structures imply Kähler metrics.

Verification of the conjecture for all known classes of LCK manifolds.

Proposal that a certain (1,1)-form is Bott--Chern homologous to a positive current.

Abstract

A complex Hermitian $n$-manifold $(M,I, \omega)$ is called locally conformally Kahler (LCK) if $d\omega=\theta\wedge\omega$, where $\theta$ is a closed 1-form, balanced if $\omega^{n-1}$ is closed, and SKT if $dId\omega=0$. We conjecture that any compact complex manifold admitting two of these three types of Hermitian forms (balanced, SKT, LCK) also admits a Kahler metric, and prove partial results towards this conjecture. We conjecture that the (1,1)-form $-d(I\theta)$ is Bott--Chern homologous to a positive (1,1)-current. This conjecture implies that $(M,I)$ does not admit a balanced Hermitian metric. We verify this conjecture for all known classes of LCK manifolds.