The Lee--Gauduchon cone 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.