Holomorphic Deformations of Balanced Calabi-Yau $\partial \overline{\partial }$-Manifolds

Popovici, Dan

Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 673-728.

Soit $X$ une variété complexe compacte lisse de dimension $n$, à fibré canonique trivial, qui satisfait le lemme du $\partial \overline{\partial }$ et possède une métrique hermitienne équilibrée (=semi-kählérienne) $\omega$. Nous introduisons le concept de déformations de $X$ co-polarisées par la classe équilibrée $\left[{\omega }^{n-1}\right]\in H^{n-1,n-1}(X,ℂ)\subset H^{2n-2}(X,ℂ)$ et montrons que la théorie des déformations équilibrées co-polarisées est une extension naturelle de la théorie classique des déformations kählériennes polarisées dans le contexte des variétés complexes compactes lisses de Calabi–Yau et dans celui des variétés holomorphes symplectiques. La notion de métrique de Weil–Petersson a encore un sens dans ce contexte strictement plus général, non nécéssairement kählérien, tandis que le théorème de Torelli local est encore valable.

Given a compact complex $n$-fold $X$ satisfying the $\partial \overline{\partial }$-lemma and supposed to have a trivial canonical bundle $K_X$ and to admit a balanced (=semi-Kähler) Hermitian metric $\omega$, we introduce the concept of deformations of $X$ that are co-polarised by the balanced class $\left[{\omega }^{n-1}\right]\in H^{n-1,n-1}(X,ℂ)\subset H^{2n-2}(X,ℂ)$ and show that the resulting theory of balanced co-polarised deformations is a natural extension of the classical theory of Kähler polarised deformations in the context of Calabi–Yau or holomorphic symplectic compact complex manifolds. The concept of Weil–Petersson metric still makes sense in this strictly more general, possibly non-Kähler context, while the Local Torelli Theorem still holds.

Classification : 32G05, 14C30, 14F43, 32Q25, 53C55
Keywords: co-polarisation by a balanced class, $\partial \bar{\partial }$-manifold, possibly non-Kähler Calabi–Yau manifold, deformations of complex structures, Weil–Petersson metric
Mot clés : co-polarisation par une classe équilibrée, $\partial \bar{\partial }$-variété, variété de Calabi–Yau non nécéssairement kählérienne, déformations de structures complexes, métrique de Weil–Petersson

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