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 grassmanian of semi-infinite subspaces in H* (X, ℂ) [[ħ±1]]. Periods associated with such semi-infinite Hodge structures enter into mirror symmetry formulas in dimensions greater then three.
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Serguei Barannikov (2001) studied this question.
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