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Quantum periods. I: Semi-infinite variations of Hodge structures. (English) Zbl 1074.14510
Summary: We introduce a generalization of variations of Hodge structures living over moduli spaces of noncommutative deformations of complex manifolds. Hodge structure associated with a point of such moduli space is an element of Sato type Grassmannian of semi-infinite subspaces in $$H^{*} (X,\mathbb{C})[[h^{\pm 1}]]$$. Periods associated with such semi-infinite Hodge structures enter into mirror symmetry formulas in dimensions greater then three.

##### MSC:
 14D07 Variation of Hodge structures (algebro-geometric aspects) 14A22 Noncommutative algebraic geometry 32G20 Period matrices, variation of Hodge structure; degenerations 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 32Q25 Calabi-Yau theory (complex-analytic aspects)
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