Cattani, Eduardo

Job Title

Professor Emeritus, Department of Mathematics and Statistics, College of Natural Sciences

Last Name

Cattani

First Name

Eduardo

Discipline

Algebraic Geometry

Expertise

Toric Geometry Hypergeometric Functions Hodge Theory and Applications

Introduction

Professor Cattani's current research deals mainly with the study of rational hypergeometric functions in several variables and the Hodge theory of algebraic varieties.

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Now showing 1 - 10 of 26

Computing Multidimensional Residues
(1994) Cattani, E; Dickenstein, Alicia; Sturmfels, Bernd
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 the sum of the local residues over all roots, depends rationally on the coefficients. This paper deals with symbolic algorithms for evaluating that rational function. Under the assumption that the deformation to the initial forms is flat, for some choice of weights on the variables, we express the global residue as a single residue integral with respect to the initial forms. When the input equations are a Groebner basis, this leads to an efficient series expansion algorithm for global residues, and to a vanishing theorem with respect to the corresponding cone in the Groebner fan. The global residue of a polynomial equals the highest coefficient of its (Groebner basis) normal form, and, conversely, the entire normal form is expressed in terms of global residues. This yields a method for evaluating traces over zero-dimensional complete intersections. Applications include the counting of real roots, the computation of the degree of a polynomial map, and the evaluation of multivariate symmetric functions. All algorithms are illustrated for an explicit system in three variables.
SOME REMARKS ON L2 AND INTERSECTION COHOMOLOGIES
(1987) Cattani, E; KAPLAN, A; SCHMID, W
The A-hypergeometric system associated with a monomial curve
(1999) Cattani, E; D'Andrea, C; Dickenstein, A
DEGENERATING VARIATIONS OF HODGE STRUCTURE
(1989) Cattani, E; KAPLAN, A
Residues and Resultants
(1997) Cattani, E; Dickenstein, Alicia; Sturmfels, Bernd
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.
Residues in toric varieties
(1997) Cattani, E; Cox, D; Dickenstein, A
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.
ON THE LOCUS OF HODGE CLASSES
(1995) Cattani, E; DELIGNE, P; KAPLAN, A
Let S be a nonsingular complex algebraic variety and V a polarized variation of Hodge structure of weight 2p with polarization form Q. Given an integer K, let S(K) be the space of pairs (s, u) with s ∈ S, u ∈ Vs integral of type (p, p), and Q(u, u) ≤ K. We show in Theorem 1.1 that S(K) is an algebraic variety, finite over S. When V is the local system H2p (Xs, Z)/torsion associated with a family of nonsingular projective varieties parametrized by S, the result implies that the locus where a given integral class of type (p, p) remains of type (p, p) is algebraic.
ON THE LOCAL MONODROMY OF A VARIATION OF HODGE STRUCTURE
(1981) Cattani, E; KAPLAN, A
POLARIZED MIXED HODGE-STRUCTURES AND THE LOCAL MONODROMY OF A VARIATION OF HODGE STRUCTURE
(1982) Cattani, E; KAPLAN, A
Some of the work on this paper was done while the first author was on Sabbatical leave during 1979–80. He wishes to thank the Netherlands Organization for the Advancement of Pure Research (Z.W.O.), Leiden University, The Scuola Normale Superiore (Pisa) and the Institute des Hautes Études Scientifiques for their hospitality
ON THE L2-COHOMOLOGY AND THE INTERSECTION COHOMOLOGY OF A VARIATION OF HODGE-STRUCTURES
(1985) Cattani, E; KAPLAN, A