Eduardo Cattani

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-

Residues in toric varieties

We study residues on a complete toric variety X, which are defined in terms of the homogeneous coordinate ring of X.We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When X is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees, we also show how to represent X as a quotient (Y\{0})/C* such that the toric residue becomes the local residue at 0 in Y.

Computing multidimensional residues

Given n polynomials in n variables with a finite number of complex roots, for any of their roots there is a local residue operator assigning a complex number to any polynomial. This is an algebraic, but generally not rational, function of the coefficients. On the other hand, the global residue, which is defined as the sum of the local residues over all roots, has invariance properties which guarantee its rational dependence on the coefficients [9], [27]. In this paper we present symbolic algorithms for evaluating that rational function.

The A-hypergeometric System Associated with a Monomial Curve

Introduction. Inthis paper we make a detailed analysis of the -hypergeometric system (or GKZ system) associated with a monomial curve and integral, hence resonant, exponents. We describe all rational solutions and show in Theorem 1.10 that they are, in fact, Laurent polynomials. We also show that for any exponent there are at most two linearly independent Laurent solutions and that the upper bound is reached if and only if the curve is not arithmetically Cohen-Macaulay. We then construct, for all integral parameters, a basis of local solutions in terms of the roots of the generic univariate polynomial (0.5) associated with . We also determine in Theorem 3.7 the holonomic rank r(α)for all α ∈ Z 2 and show that d ≤ r(α)≤ d +1, where d is the degree of the curve. Moreover, the value d +1 is attained only for those exponents α for which there are two linearly independent rational solutions, and, therefore, r(α)= d for all α if and only if the curve is arithmetically Cohen-Macaulay. Inorder to place these results intheir appropriate con text, we recall the defin ition

Introduction to residues and resultants

This chapter is an expanded version of the lecture notes prepared by the second-named author for her introductory course at the CIMPA Graduate School on Systems of Polynomial Equations held in Buenos Aires, Argentina, in July 2003. We present an elementary introduction to residues and resultants and outline some of their multivariate generalizations. Throughout we emphasize the application of these ideas to polynomial system solving.

Counting solutions to binomial complete intersections

We study the problem of counting the total number of affine solutions of a system of n binomials in n variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is finite. We give a combinatorial formula for computing the total number of affine solutions (with or without multiplicity) from which we deduce that this counting problem is #P-complete. We discuss special cases in which this formula may be computed in polynomial time; in particular, this is true for generic exponent vectors.

Complete intersections in toric ideals

We present examples that show that in dimension higher than one or codimension higher than two, there exist toric ideals I A such that no binomial ideal contained in I A and of the same dimension is a complete intersection. This result has important implications in sparse elimination theory and in the study of the Horn system of partial differential equations.

Planar Configurations of Lattice Vectors and GKZ-Rational Toric Fourfolds in \Bbb P 6

We introduce a notion of balanced configurations of vectors. This is motivated by the study of rational A-hypergeometric functions in the sense of Gelfand, Kapranov and Zelevinsky. We classify balanced configurations of seven plane vectors up to GL(2,ℝ)-equivalence and deduce that the only gkz-rational toric four-folds in ℙ6 are those varieties associated with an essential Cayley configuration. We show that in this case, all rational A-hypergeometric functions may be described in terms of toric residues. This follows from studying a suitable hyperplane arrangement.

Frobenius Modules and Hodge Asymptotics

AbstractWe exhibit a direct correspondence between the potential defining the H1,1 small quantum module structure on the cohomology of a Calabi-Yau manifold and the asymptotic data of the A-model variation of Hodge structure. This is done in the abstract context of polarized variations of Hodge structure and Frobenius modules.