2022/11/18 by Dan Popovici, Popovici, Dan · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2211.10267
openalex publication_date 2022/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a class of Hermitian metrics, that we call pluriclosed star split, generalising both the astheno-Kähler metrics of Jost and Yau and the (n-2)-Gauduchon metrics of Fu-Wang-Wu on complex manifolds. They have links with Gauduchon and balanced metrics through the properties of a smooth function associated with any Hermitian metric. After pointing out several examples, we generalise the property to pairs of Hermitian metrics and to triples consisting of a holomorphic map between two complex manifolds and two Hermitian metrics, one on each of these manifolds. Applications include an attack on the Fino-Vezzoni conjecture predicting that any compact complex manifold admitting both SKT and balanced metrics must be Kähler, that we answer affirmatively under extra assumptions. We also introduce and study a Laplace-like differential operator of order two acting on the smooth (1, 1)-forms of a Hermitian manifold. We prove it to be elliptic and we point out its links with the pluriclosed star split metrics and pairs defined in the first part of the paper.