Aeppli Cohomology Classes Associated with Gauduchon Metrics on Compact Complex Manifolds

By Dan Popovici. 2013

108 citations recorded by the source corpus.

We propose the study of a Monge-Amp\`ere-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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bc17d0a4ac52f31e41350cfea532cad1dd15755e

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Citations in the corpus

$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations in complex and almost complex geometry (2014, background): “ill applies, because by the bounds (1.2) we conclude that k∇ukCβ(M) 6 C, for a uniform constant C, and hence χhas a uniform Cβbound. The equation (1.6) with this choice of χwas introduced by Popovici [48], motivated by some questions in algebraic geometry, and was also studied by the ﬁrst and third-”
The Monge–Ampère equation for non-integrable almost complex structures (2016, unknown): “1) is replaced by a Monge-Ampère type equation for (n − 1)-plurisubharmonic functions [26, 17, 18, 38, 50, 51].”
Gauduchon metrics with prescribed volume form (2015, methodology): “d 1.2. In [40], the second and third-named authors observed that to solve Theorem 1.3 it is enough to solve a certain partial diﬀerential equation, which was also independently introduced by Popovici [30]. This equation is a variant of one introduced by Fu-Wang-Wu [10] and related to Harvey-Lawson’s”