On the local monodromy of a variation of Hodge structure

Abstract

Associated to a variation of polarized Hodge structure there is a period mapping ty : S —• Y\P, where S is the parameter space and T\P denotes the corresponding modular variety of polarized Hodge structures (the primary example to keep in mind is that arising from a family of smooth projective varieties parametrized by S) [3], [4]. The local study of the singularities of \p ([5]) reduces to the case when S = (A*) x A, a product of punctured disks and disks. Given a lifting \jj: U x A —> D (U — upper half-plane) of \p to the universal covering of S there are monodromy transformations yl, . . . , yt G T such that