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

L2 AND INTERSECTION COHOMOLOGIES FOR A POLARIZABLE VARIATION OF HODGE STRUCTURE
(1987) Cattani, E; KAPLAN, A; SCHMID, W
A global view of residues in the torus
(1997) Cattani, E; Dickenstein, A
We study the total sum of Grothendieck residues of a Laurent polynomial relative to a family f1,…,fn of sparse Laurent polynomials in n variables with a finite set of common zeroes in the torus T = (C*)n. Under appropriate assumptions we may embed T in a toric variety X in such a way that the total residue may be computed by a global object in X, the toric residue. This yields a description of some of its properties and new symbolic algorithms for its computation.
Planar configurations of lattice vectors and GKZ-rational toric fourfolds in P-6
(2004-01-01) Cattani, E; Dickenstein, A
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,)-equivalence and deduce that the only gkz-rational toric four-folds in 6 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.
Asymptotic Hodge theory and quantum products
(2000-01-01) Cattani, E; Fernandez, Javier
Assuming suitable convergence properties for the Gromov-Witten potential of a Calabi-Yau manifold $X$ one may construct a polarized variation of Hodge structure over the complexified K\"ahler cone of $X$. In this paper we show that, in the case of fourfolds, there is a correspondence between ``quantum potentials'' and polarized variations of Hodge structures that degenerate to a maximally unipotent boundary point. Under this correspondence, the WDVV equations are seen to be equivalent to the Griffiths' trasversality property of a variation of Hodge structure.
Rational hypergeometric functions
(2001-01-01) Cattani, E; Dickenstein, A; Sturmfels, B
Multivariate hypergeometric functions associated with toric varieties were introduced by Gel'fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors projectively dual to torus orbit closures. We show that most of these potential denominators never appear in rational hypergeometric functions. We conjecture that the denominator of any rational hypergeometric function is a product of resultants, that is, a product of special discriminants arising from Cayley configurations. This conjecture is proved for toric hypersurfaces and for toric varieties of dimension at most three. Toric residues are applied to show that every toric resultant appears in the denominator of some rational hypergeometric function.
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
Restriction of A-discriminants and dual defect toric varieties
(2007-01-01) Curran, R; Cattani, E
We study the A-discriminant of toric varieties. We reduce its computation to the case of irreducible configurations and describe its behavior under specialization of some of the variables to zero. We give characterizations of dual defect toric varieties in terms of their Gale dual and classify dual defect toric varieties of codimension less than or equal to four.
Complete intersections in toric ideals
(2007-01-01) Cattani, E; Curran, R; Dickenstein, A
We present examples that show that in dimension higher than one or codimension higher than two, there exist toric ideals such that no binomial ideal contained in 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.