Abstract. By analyzing the local and
infinitesimal behavior of degenerating polarized variations of Hodge
structure the notion of ifinitesimal variation of Hodge structure
(IVHS) at infinity is introduced. It is shown that all such structures
can be integrated to polarized variations of Hodge structure and that,
conversely, all are limits of infinitesimal variations of Hodge
structure at finite points. As an illustration of the rich information
encoded in this new structure, some instances of the maximal
dimension problem for this type of infinitesimal variation are presented
and contrasted with the "classical" case of IVHS at finite points.
Abstract. The Hard Lefschetz Theorem (HLT) and
the Hodge-Riemann bilinear relations (HRR) hold in various contexts:
they impose restrictions on the cohomology algebra of a smooth compact
Kähler manifold; they restrict the local monodromy of a polarized
variation of Hodge structure; they impose conditions on the
f-vectors of convex polytopes. While the statements of these
theorems depend on the choice of a Kähler class, or its analog,
there is usually a cone of possible choices. It is then natural
to ask whether the HLT and HRR remain true in a mixed context. In
this note we present a unified approach to proving the mixed HLT and
HRR, generalizing the known results, and proving it in new cases such
as the intersection cohomology of non-rational polytopes.
Abstract. 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 prove a Gale dual characterization of dual
defect toric varieties and deduce from it the classsification of such
varieties in codimension less than or equal to four. This
classification motivates a decomposition theorem which yields a
sufficient condition for a toric variety to be dual defect. For
codimension less than or equal to four, this condition is also
necessary and we expect this to be the case in general.
Abstract. We show that the problem of
counting the
total
number of affine solutions (with and without multiplicities)
of a system of n binomials in n variables is
#P-hard. We use commutative algebra tools to reduce
the study of these solutions to a combinatorial problem
on a graph associated to the exponents occurring in the given
binomials.
Abstract. We present examples which 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.
Abstract. These are the lecture notes for a
mini-course
on Hypergeometric Functions at the 2006 ElENA (El Encuentro Nacional de
Algebra)
meeting in Cordoba, Argentina. We study multivariate hypergeometric
functions
in the sense of Gel'fand, Kapranov, and Zelevinsky (GKZ systems).
These functions generalize the classical hypergeometric functions of
Gauss, Horn, Appell, and Lauricella. Throughout we emphasize the
algebraic methods of Saito, Sturmfels, and Takayama to construct
hypergeometric series and the connection with deformation techniques in
commutative algebra. We end with a brief discussion of the
classification problem for rational hypergeometric functions.
E. Cattani, A.
Dickenstein , Introduction
to Residues and Resultants. Chapter 1 in:
A.Dickenstein,
I.Z.Emiris (Eds.): Solving Polynomial Equations: Foundations,
Algorithms, and Applications. Algorithms and Computation in Mathematics
14, Springer-Verlag, 2005.
Abstract. 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.
Abstract. 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 complex projective space P^6 are those varieties
associated with an essential Cayley configuration. In this case, we
study a suitable hyperplane arrangement and show that all rational
A-hypergeometric functions may be described in terms of toric residues.
Abstract. We exhibit a direct correspondence
between the potential defining the $H^{1,1}$ 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.
Abstract. 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 Kaehler 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.
Abstract. A binomial residue is a rational
function defined by a hypergeometric integral whose kernel is singular
along binomial divisors. Binomial residues provide an integral
representation for rational solutions of A-hypergeometric systems of
Lawrence type. The space of binomial residues of a given degree, modulo
those which are polynomial in some variable, has dimension equal to the
Euler characteristic of the matroid associated with A.
Abstract.
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.
Abstract.
We make a detailed analysis of the A-hypergeometric system (or GKZ
system) associated with a monomial curve and integral, hence resonant,
exponents. We characterize the Laurent polynomial solutions and show
that these are the only rational solutions. 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 associated with A. We determine the
holonomic rank r for all integral exponents and show that it is
constantly equal to the degree d of X if and only if X is
arithmetically Cohen-Macaulay.
Abstract.
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.
Abstract.
We study the total sum of Grothendieck residues of a Laurent polynomial
relative to a family $f_1,\dots,f_n$ of sparse Laurent polynomials in
$n$-variables with a finite number 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.
Abstract.
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.
Abstract.
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.
E. Cattani, P. Deligne and A. Kaplan, On the
Locus of Hodge Classes, Journal of the American
Mathematical
Society,8,
483-506, 1995. Published version available from JSTOR.
Abstract. Let $f: X \rightarrow S$ be a family
of non singular projective varieties parametrized by a complex
algebraic variety $S$. Fix $s \in S$, an integer $p$, and a class $h
\in {\rm H}^{2p}(X_s,\Z)$ of Hodge type $(p,p)$. We show that the
locus, on $S$, where $h$ remains of type $(p,p)$ is algebraic. This
result, which in the geometric case would follow from the rational
Hodge conjecture, is obtained in the setting of variations of Hodge
structures.