no. 4
Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds
[Classes de cohomologie d'Aeppli associées à des métriques de Gauduchon sur les variétés complexes compactes]
Bulletin de la Société Mathématique de France, Tome 143 (2015) no. 4, pp. 763-800

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 ellipticity and uniqueness of the solution subject to positivity and normalization 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 ∂∂¯-manifolds and to a further study of the deformation properties of the Gauduchon cone beyond those given in this paper.

Nous proposons l'étude d'une équation de type Monge-Ampère en bidegré (n-1,n-1) plutôt que (1,1) sur une variété complexe compacte X de dimension n pour laquelle nous démontrons l'ellipticité et l'unicité des solutions soumises à des contraintes de positivité et de normalisation. Nous espérons que la question de l'existence pourra être traitée dans un travail ultérieur. Le but est de construire une métrique de Gauduchon spéciale associée de manière unique à une classe de cohomologie d'Aeppli quelconque de bidegré (n-1,n-1) appartenant au cône de Gauduchon de X que nous introduisons comme un sous-ensemble du groupe de cohomologie d'Aeppli réel de type (n-1,n-1) et dont nous étudions les premières propriétés. Des applications de cette nouvelle équation sont envisagées dans deux directions : aux espaces de modules de ∂∂¯-variétés de Calabi-Yau et à une étude des propriétés de déformations du cône de Gauduchon au-delà de celles décrites dans ce travail.

Publié le :
DOI : 10.24033/bsmf.2704
Classification : 53C55, 32Q25, 32G05, 14C30
Keywords: Positivity in bidegree $(n-1, n-1)$, Gauduchon and sG cones, duality between the Bott-Chern and the Aeppli cohomologies, equation of the Monge-Ampére type in bidegree $(n-1, n-1)$.
Mots-clés : Positivité en bidegré $(n-1, n-1)$, cônes de Gauduchon et cône fG, dualité entre la cohomologie de Bott-Chern et la cohomologie d'Aeppli, équation du type Monge-Ampère en bidegré $(n-1, n-1)$. 
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     author = {Popovici, Dan},
     title = {Aeppli cohomology classes associated with {Gauduchon} metrics on compact complex manifolds},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {763--800},
     year = {2015},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {143},
     number = {4},
     doi = {10.24033/bsmf.2704},
     mrnumber = {3450501},
     zbl = {1347.53060},
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     url = {https://www.numdam.org/articles/10.24033/bsmf.2704/}
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Popovici, Dan. Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds. Bulletin de la Société Mathématique de France, Tome 143 (2015) no. 4, pp. 763-800. doi: 10.24033/bsmf.2704

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