Exotic Deformations of Calabi-Yau Manifolds

de Bartolomeis, Paolo; Tomassini, Adriano

Annales de l'Institut Fourier, Tome 63 (2013) no. 2, pp. 391-415.

On considère la classe des variétés QIS (Quantum Inner State variétés), à savoir la classe des variétés symplectiques, compactes et de dimension $2n$, munies d’une structure presque complexe $J$ modérée par $k$ et d’une section $ϵ$ du fibré ${\Lambda }_J^{n,0}\left(M\right)$, qui ne s’annule nulle part, normalisée et satisfaisant la condition ${\overline{\partial }}_Jϵ=0$.

Le but du papier est d’étudier l’espace $𝔐$ des modules des déformations QIS d’une variété de Calabi-Yau. À ce propos, on calcule l’espace tangent de $𝔐$ et on montre que $𝔐$ n’a pas d’obstructions. Plusieurs exemples de variétés QIS sont aussi exhibés.

We introduce Quantum Inner State manifolds (QIS manifolds) as (compact) $2n$-dimensional symplectic manifolds $(M,\kappa )$ endowed with a $\kappa$-tamed almost complex structure $J$ and with a nowhere vanishing and normalized section $ϵ$ of the bundle ${\Lambda }_J^{n,0}\left(M\right)$ satisfying the condition ${\overline{\partial }}_Jϵ=0$.

We study the moduli space $𝔐$ of QIS deformations of a given Calabi-Yau manifold, computing its tangent space and showing that $𝔐$ is non obstructed. Finally, we present several examples of QIS manifolds.

Classification : 32G05, 53C15, 17B30
Keywords: tamed symplectic structure, Calabi-Yau manifold, quantum inner state structure, deformation, moduli space
Mot clés : variétés de Calabi-Yau

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