Aroldo Kaplan

ON THE LOCUS OF HODGE CLASSES

Let S be a complex algebraic variety and {Xs}s∈S a family of non singular projective varieties parametrized by S: the Xs are the fibers of f : X → S, with X projective and smooth over S. Fix s ∈ S, an integer p, and a class h ∈ H(Xs,Z) of Hodge type (p, p). Let U be an open, simply connected neighborhood of s. The H(Xt,Z), t ∈ S, form a local system on S, necessarily trivial on U , so that for t ∈ U they can all be identified with H(Xs,Z). The Hodge filtration Ft of H (Xt,C), t ∈ U , can be viewed as a variable filtration on the fixed complex vector space H(Xs,C). It varies holomorphically with t. It follows that the locus T ⊂ U where h remains of type (p, p), i.e., in F, is a complex analytic subspace of U .

Polarized Mixed Hodge Structures and the Local Monodromy of a Variation of Hodge Structure.

A variation of polarized Hodge structures on the complement of a divisor with normal crossings gives rise, locally (on the divisor) to a commuting set {Ni} of nilpotent endomorphisms of the vector space underlying the variation: the logarithms of the unipotent parts of the various Picard-Lefschetz transformations. These reflect properties of the singularities of the period map associated to the variation (cf. [5, 7, 9] and Sect. 3). For example, a variation depending on a single parameter defines asymptotically a mixed Hodge structure whose weight filtration is the monodromy weight filtration of the corresponding endomorphism N ([9, 11]). This paper is mainly concerned with the properties of the set {N/} arising in the several parameters case. In particular, Theorem 3.3 asserts that all the elements in the open polyhedral cone C spanned over R by the N,.s, determine the same monodromy weight filtration. It also describes the relationship between this common filtration and those associated to the faces of the cone. This statement was conjectured by Deligne based on his analogous result ([4], 1.9.2) for the Q-cone associated to those variations which arise from families of polarized, non-singular algebraic varieties. In w 1 we discuss some general properties of commuting sets of nilpotent endomorphisms and their associated filtrations. Section2 focuses on the case when each element in such set defines in the sense of (2.4) a polarized mixed Hodge structure with a fixed Hodge filtration F (as is the case for the monodromy cone C of a variation). These conditions imply (cf. (2.16)) a generic uniqueness for the monodromy weight filtration. We also consider those pola-

On the local monodromy of a variation of Hodge structure

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