The Lee--Gauduchon cone on complex manifolds

TL;DR

This paper studies the Lee-Gauduchon cone on compact complex manifolds, proving it is a bimeromorphic invariant and computing it for various non-Kähler examples, thus advancing understanding of Hermitian metrics.

Contribution

It establishes the Lee-Gauduchon cone as a bimeromorphic invariant and provides explicit computations for specific non-Kähler manifolds.

Findings

Lee-Gauduchon cone is a bimeromorphic invariant

Computed Lee-Gauduchon cone for several non-Kähler manifolds

Provides new insights into Hermitian metrics on complex manifolds

Abstract

Let $M$ be a compact complex $n$-manifold. A Gauduchon metric is a Hermitian metric whose fundamental 2-form $\omega$ satisfies the equation $dd^c(\omega^{n-1})=0$. Paul Gauduchon has proven that any Hermitian metric is conformally equivalent to a Gauduchon metric, which is unique (up to a constant multiplier) in its conformal class. Then $d^c(\omega^{n-1})$ is a closed $(2n-1)$-form; the set of cohomology classes of all such forms, called the Lee-Gauduchon cone, is a convex cone, superficially similar to the Kahler cone. We prove that the Lee-Gauduchon cone is a bimeromorphic invariant, and compute it for several classes of non-Kahler manifolds.