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arXiv · 1310.6326

Hermitian metrics, (n-1, n-1) forms and Monge-Amp\`ere equations

Valentino Tosatti·Ben Weinkove
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Abstract

We show existence of unique smooth solutions to the Monge-Ampere equation for (n-1)-plurisubharmonic functions on Hermitian manifolds, generalizing previous work of the authors. As a consequence we obtain Calabi-Yau theorems for Gauduchon and strongly Gauduchon metrics on a class of non-Kahler manifolds: those satisfying the Jost-Yau condition known as Astheno-Kahler. Gauduchon conjectured in 1984 that a Calabi-Yau theorem for Gauduchon metrics holds on all compact complex manifolds. We discuss another Monge-Ampere equation, recently introduced by Popovici, and show that the full Gauduchon conjecture can be reduced to a second order estimate of Hou-Ma-Wu type.

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Valentino Tosatti, Ben Weinkove. 2013-10-23. Hermitian metrics, (n-1, n-1) forms and Monge-Amp\`ere equations. https://doi.org/10.1515/crelle-2017-0017

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