Aeppli Cohomology Classes Associated with Gauduchon Metrics on Compact Complex Manifolds

Abstract

We propose the study of a Monge-Ampère-type equation in bidegree $(n-1,n-1)$ rather than $(1,1)$ on a compact complex manifold $X$ of dimension $n$ for which we prove uniqueness of the solution subject to positivity and normalisation restrictions. Existence will hopefully be dealt with in future work. The aim is to construct a special Gauduchon metric uniquely associated with any Aeppli cohomology class of bidegree $(n-1,n-1)$ lying in the Gauduchon cone of $X$ that we hereby introduce as a subset of the real Aeppli cohomology group of type $(n-1,n-1)$ and whose first properties we study. Two directions for applications of this new equation are envisaged: to moduli spaces of Calabi-Yau $\partial \bar{\partial }$-manifolds and to a further study of the deformation properties of the Gauduchon cone beyond those given in this paper.

37 pages, to appear in Bulletin de la Société Mathématique de France (accepted in July 2014), acknowledgments added

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