Let S S be a nonsingular complex algebraic variety and V \mathcal {V} a polarized variation of Hodge structure of weight 2 p 2p with polarization form Q Q . Given an integer K K , let S ( K ) {S^{(K)}} be the space of pairs ( s , u ) (s,u) with s ∈ S s \in S , u ∈ V s u \in {\mathcal {V}_s} integral of type ( p , p ) (p,p) , and Q ( u , u ) ≤ K Q(u,u) \leq K . We show in Theorem 1.1 that S ( K ) {S^{(K)}} is an algebraic variety, finite over S S . When V \mathcal {V} is the local system H 2 p ( X s , Z ) {H^{2p}}({X_s},\mathbb {Z}) /torsion associated with a family of nonsingular projective varieties parametrized by S S , the result implies that the locus where a given integral class of type ( p , p ) (p,p) remains of type ( p , p ) (p,p) is algebraic.
Resultants, Jacobians and residues are basic invariants of multivariate polynomial systems. We examine their interrelations in the context of toric geometry. The global residue in the torus, studied by Khovanskii, is the sum over local Grothendieck residues at the zeros of n Laurent polynomials in n variables. Cox introduced the related notion of the toric residue relative to n + 1 divisors on an n-dimensional toric variety. We establish denominator formulas in terms of sparse resultants for both the toric residue and the global residue in the torus. A byproduct is a determinantal formula for resultants based on Jacobians.
We consider a polarized variation of Hodge structure (V, Vz, S, F) of weight k, over a complex manifold X 1-16]. Here V denotes a locally constant sheaf of finite dimensional complex vector spaces, V z a sheaf of lattices in V, and F a decreasing filtration of Ox | by locally free sheaves of Ox-modules F p. By assumption, the filtration F induces Hodge structure of weight k on the stalks of V, and satisfies the Riemann bilinear relations, as welt as the transversality relation, relative to the fiat bilinear form S. Typically variations of Hodge structure arise from the cohomology of the fibres in a family of smooth projective varieties.Henceforth the base X will be assumed to lie as a Zariski open subset in a compact K/ihler manifold X, such that X-X is a divisor with normal crossings. In other words, every x~X lies in some neighborhood U, with (1.1) U_~A", Uc~X~--(A*)mXA n-m(A=unit disc in C, A*=A-{0}), for some m=m(x) between 0 and n=dimX.Any smooth quasiprojective variety X has a projective completion )? of this sort. We equip X with a complete K/ihler metric g, whose restriction to any neighborhood U as in (1.1) is asymptotic to a product of metrics on the one dimensional factors: Euclidean on the n-m factors A, and of constant negative curvature, complete at the punctures, on the m factors A*; such a metric g always exists I-9]. Whether a V-valued form on X is square integrable over compact subsets of X', relative to the metric g on X and the Hodge metrics I-6, 18] on the stalks of V, depends only on the asymptotic behavior of g, not on the particular choice of g, nor on the choice of polarization. The complex of sheaves A~2)(V ) on X, defined by the assignment * Partially supported by NSF Grant DMS-8501949 ** Partially supported by NSF Grant DMS-8317436 218 E. Cattani et al.(1.2) U~--,complex of V-valued forms ~o on U n X, with measurable coefficients and measurable exterior derivative, such that 09 and d~o have finite L 2 norm on K n X, for every compact K c U, is therefore canonically attached to the completion X'. Because of the special nature of the metric g -on neighborhoods U as in (I.I) it splits into a product of metrics on the one dimensional factors, up to quasi-isometry -the sheaves A~z)(V ) are fine, so the complex of global sections computes the hypercohomo- The proof of Theorem (1.5) is a local problem -it suffices to identify the L 2 cohomology and the intersection cohomology of any neighborhood U~-A n as in (1.1). Both cohomologies have weight filtrations, which are induced by the local monodromy. The nature of the metric g and of the Hodge metrics on V enables us to compare the L 2 cohomology groups of A" to those of A "-{0}, via a Mayer-Vietoris spectral sequence: the L 2 cohomology of A n agrees with that of A n-{0} below a certain critical weight, and vanishes above it; the contribution of the critical weight itself is isomorphic to the tensor product of a L 2 and intersection cohomologies 219 certain infinite dimensional space with the L 2 cohomology of A"-{0}, also at the critical leve...
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