The Lee–Gauduchon Cone on Complex Manifolds

Let M be a compact complex n-manifold. A Gauduchon metric is a Hermitian metric whose fundamental 2-form ω $$\omega $$ satisfies the equation d d c ( ω n −1 ) = 0 $$dd^c(\omega ^{n-1})=0$$ . 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 ( ω n −1 ) $$d^c(\omega ^{n-1})$$ is a closed ( 2 n −1 ) $$(2n-1)$$ -form; the set of cohomology classes of all such forms, called the Lee-Gauduchon cone, is a convex cone, superficially similar to the Kähler cone. We prove that the Lee-Gauduchon cone is a bimeromorphic invariant, and compute it for several classes of non-Kähler manifolds.