  We introduce a generalization of variations of Hodge structures living over
moduli spaces of non-commutative 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)[[h^{-1},h]]. Periods
associated with such semi-infinite Hodge structures serve in order to extend
mirror symmetry relations in dimensions greater then three.
