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Exotic deformations of Calabi-Yau manifolds. (Déformations exotiques des variétés de Calabi-Yau.) (English. French summary) Zbl 1293.32016

The authors study Quantum Inner State (QIS) manifolds, which are defined as quadruples \((M,J, \omega, \Omega)\), where \((M,\omega)\) is a compact \(2n\)-dimensional symplectic manifold, \(J\) is an almost complex structure on \(M\) tamed by \(\omega\), and \(\Omega\) is a smooth nowhere vanishing \((n,0)\)-form (with respect to \(J\)) such that \(\overline \partial_J \Omega =0\). If \(J\) is integrable and compatible with \(\omega\), then a QIS manifold is a Calabi-Yau manifold (with holomorphically trivial canonical bundle). A classical result of G. Tian [Adv. Ser. Math. Phys. 1, 629–646 (1987; Zbl 0696.53040)] and A. N. Todorov [Commun. Math. Phys. 126, No. 2, 325–346 (1989; Zbl 0688.53030)] shows that in the Calabi-Yau case the deformation space is unobstructed. The main result of this paper is a generalization of this statement to QIS manifolds. The authors prove that the deformation space is still unobstructed. In the last section several examples of homogeneous non-Kähler QIS manifolds are discussed.
Reviewer: Anna Fino (Torino)

MSC:

32G05 Deformations of complex structures
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
17B30 Solvable, nilpotent (super)algebras
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References:

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