Article Dans Une Revue Mathematische Zeitschrift Année : 2023

Pluriclosed Star Split Hermitian Metrics

Résumé

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.

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Dates et versions

hal-04930878 , version 1 (05-02-2025)

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Dan Popovici. Pluriclosed Star Split Hermitian Metrics. Mathematische Zeitschrift, 2023, 305 (1), pp.7. ⟨10.1007/s00209-023-03344-0⟩. ⟨hal-04930878⟩
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