Gauduchon metric

In mathematics , Gauduchon metrics are a concept of complex geometry that generalizes the concept of Calabi-Yau metrics from Kähler manifolds to Hermitian manifolds .

definition

Is a complex manifold and so is a Hermitian metric on Gauduchon metric when ${\displaystyle M}$ ${\displaystyle n=\dim _{\mathbb {C} }M}$ ${\displaystyle h}$ ${\displaystyle M}$

${\displaystyle \partial {\overline {\partial }}(h^{n-1})=0}$

applies.

Gauduchon conjecture

The Gauduchon conjecture , proven by Gábor Székelyhidi , Valentino Tosatti and Ben Weinkove in 2015 , says that on a compact , complex manifold for every -form representing the first Chern class there is a Gauduchon metric whose Ricci-form is. Equivalent: Each volume form on there is a Gauduchon metric with . ${\displaystyle M}$ ${\displaystyle c_{1}(M)}$ ${\displaystyle (1,1)}$ ${\displaystyle \Psi }$ ${\displaystyle \Psi }$ ${\displaystyle \sigma }$ ${\displaystyle M}$ ${\displaystyle \omega }$ ${\displaystyle \omega ^{n}=\sigma }$

background

For Kähler metrics, the Gauduchon conjecture corresponds to the Calabi conjecture proven by Yau . The corresponding problem for Hermitian metrics without additional conditions could easily be solved by a conformally equivalent metric. For every Hermitian metric is conformally equivalent to a Gauduchon metric that is unique except for scaling, which motivated the Gauduchon conjecture. ${\displaystyle \dim _{\mathbb {C} }(M)>1}$

literature

• Gauduchon : La 1-forme de torsion d'une variété hermitienne compacte , Mathematische Annalen 267, 495-518 (1994).
• Székelyhidi, Tosatti, Weinkove: Gauduchon metrics with prescribed volume form , Acta Mathematica 219, 181-211 (2017).