Showing papers on "Toric variety published in 1997"

Hamiltonian torus actions on symplectic orbifolds and toric varieties

Abstract: In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus actions are classified by convex simple rational polytopes with a positive integer attached to each facet and that all such orbifolds are algebraic toric varieties.

Stringy Hodge numbers of varieties with Gorenstein canonical singularities

Abstract: We introduce the notion of stringy E-function for an arbitrary normal irreducible algebraic variety X with at worst log-terminal singularities. We prove some basic properties of stringy E-functions and compute them explicitly for arbitrary Q-Gorenstein toric varieties. Using stringy E-functions, we propose a general method to define stringy Hodge numbers for projective algebraic varieties with at worst Gorenstein canonical singularities. This allows us to formulate the topological mirror duality test for arbitrary Calabi-Yau varieties with canonical singularities. In Appendix we explain non-Archimedian integrals over spaces of arcs. We need these integrals for the proof of the main technical statement used in the definition of stringy Hodge numbers.

Birational Calabi–Yau n -folds have equal Betti numbers

Abstract: Let X and Y be two smooth projective n-dimensional algebraic varieties X and Y over C with trivial canonical line bundles. We use methods of p-adic analysis on algebraic varieties over local number fields to prove that if X and Y are birational, they have the same Betti numbers.

Lattice points in simple polytopes

Abstract: in terms of fP(h) q(x)dx where the polytope P(h) is obtained from P by independent parallel motions of all facets. This extends to simple lattice polytopes the EulerMaclaurin summation formula of Khovanskii and Pukhlikov [8] (valid for lattice polytopes such that the primitive vectors on edges through each vertex of P form a basis of the lattice). As a corollary, we recover results of Pommersheim [9] and Kantor-Khovanskii [6] on the coefficients of the Ehrhart polynomial of P. Our proof is elementary. In a subsequent article, we will show how to adapt it to compute the equivariant Todd class of any complete toric variety with quotient singularities. The Euler-Maclaurin summation formula for simple lattice polytopes has been obtained independently by Ginzburg-Guillemin-Karshon [4]. They used the dictionary between convex polytopes and projective toric varieties with an ample divisor class, in combination with the Riemann-Roch-Kawasaki formula ([1], [7]) for complex manifolds with quotient singularities. A counting formula for lattice points in lattice simplices has been announced by Cappell and Shaneson [2], as a consequence of their computation of the Todd class of toric varieties with quotient singularities.

The versal deformation of an isolated toric Gorenstein singularity

Abstract: Given a lattice polytope Q ⊆ ℝ n , we define an affine scheme that reflects the possibilities of splitting Q into a Minkowski sum. Denoting by Y the toric Gorenstein singularity induced by Q, we construct a flat family over with Y as special fiber. In case Y has an isolated singularity, this family is versal.

Basic properties of convex polytopes

Abstract: Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their importance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry (toric varieties) to linear and combinatorial optimization. In this chapter we try to give a short introduction, provide a sketch of “what polytopes look like” and “how they behave,” with many explicit examples, and briefly state some main results (where further details are in the subsequent chapters of this Handbook). We concentrate on two main topics:

An equivariant Riemann-Roch theorem for complete, simplicial toric varieties.

Abstract: Introduction. The theory of toric varieties establishes a now classical connection between algebraic geometry and convex polytopes. In particular, äs observed by Danilov in the seventies, finding a closed formula for the Todd class of complete toric varieties would have important consequences for enumeration of lattice points in convex lattice polytopes. Since then, a number of such formulas have been proposed; see [M], [Pl], [P2] The Todd class of complete simplicial toric varieties is computed in [G-G-K], using the Riemann-Roch formula of T. Kawasaki [Ka].

Residues and Resultants

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 re- sultants for both the toric residue and the global residue in the torus. A byproduct is a determinantal formula for resultants based on Jaco- bians.

Residues in toric varieties

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.

Residues and Resultants

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.

A mirror theorem for toric complete intersections

Abstract: We prove a generalized mirror conjecture for non-negative complete intersections in symplectic toric manifolds. Namely, we express solutions of the PDE system describing quantum cohomology of such a manifold in terms of suitable hypergeometric functions. Revision 03.03.97: we correct an error in Introduction.

Toric Degenerations of Fano Varieties and Constructing Mirror Manifolds

Abstract: For an arbitrary smooth n-dimensional Fano variety $X$ we introduce the notion of a small toric degeneration. Using small toric degenerations of Fano n-folds $X$, we propose a general method for constructing mirrors of Calabi-Yau complete intersections in $X$. Our mirror construction is based on a generalized monomial-divisor mirror correspondence which can be used for computing Gromov-Witten invariants of rational curves via specializations of GKZ-hypergeometric series.

Two Classes of Riemannian Manifolds Whose Geodesic Flows Are Integrable

Abstract: Part 1. Liouville Manifolds: Introduction Preliminary remarks and notations Local structure of proper Liouville manifolds Global structure of proper Liouville manifolds Proper Liouville manifolds of rank one Appendix. Simply connected manifolds of constant curvature Part 2. Kahler-Liouville manifolds: Introduction Preliminary remarks and notations Local calculus on $M^1$ Summing up the local data Structure of $M-M^1$ Torus action and the invariant hypersurfaces Properties as a toric variety Bundle structure associated with a subset of $\mathcal A$ The case where $ No. \mathcal A=1$ Existence theorem.

Divisors on elliptic Calabi-Yau 4-folds and the superpotential in F-theory, I

Abstract: Each smooth elliptic Calabi-Yau 4-fold determines both a three-dimensional physical theory (a compactification of ``M-theory'') and a four-dimensional physical theory (using the ``F-theory'' construction). A key issue in both theories is the calculation of the ``superpotential''of the theory. We propose a systematic approach to identify these divisors, and derive some criteria to determine whether a given divisor indeed contributes. We then apply our techniques in explicit examples, in particular, when the base B of the elliptic fibration is a toric variety or a Fano 3-fold. When B is Fano, we show how divisors contributing to the superpotential are always "exceptional" (in some sense) for the Calabi-Yau 4-fold X. This naturally leads to certain transitions of X, that is birational transformations to a singular model (where the image of D no longer contributes) as well as certain smoothings of the singular model. If a smoothing exists, then the Hodge numbers change. We speculate that divisors contributing to the superpotential are always "exceptional" (in some sense) for X, also in M-theory. In fact we show that this is a consequence of the (log)-minimal model algorithm in dimension 4, which is still conjectural in its generality, but it has been worked out in various cases, among which toric varieties.

Minimal discrepancies of toric singularities

Abstract: The main purpose of this paper is to prove that minimal discrepancies ofn-dimensional toric singularities can accumulate only from above and only to minimal discrepancies of toric singularities of dimension less thann. I also prove that some lower-dimensional minimal discrepancies do appear as such limit.

K-Theory of Algebraic Tori and Toric Varieties

Abstract: It is proved that under certain conditions the group Kn(X) of a smooth projective variety X over a field F is a natural direct summand of Kn(A) for some separable F -algebra. As an application we study the K-groups of toric models and toric varieties. A presentation in terms of generators and relations of the group K0(T ) for an algebraic torus T is given. In [14] the second author has computed K-groups of homogeneous projective varieties. It was proved that for a variety X in this class defined over a field F there exists a separable F -algebra A and a natural isomorphism Kn(X) ' Kn(A). In fact, this isomorphism was obtained by applying the “K-theory functor” to an isomorphism X ' A is a certain motivic category C “containing” varieties and algebras. In the present paper we find certain sufficient conditions under which for a variety X over a field F there exists a separable algebra A and morphisms u : X → A and v : A→ X in C such that the composition X u → A v → X is the identity, i.e X is a “direct summand” of A in C. As an application we consider K-groups of toric models and toric varieties. The paper is organized as follows. In the first section we define the motivic category C, the main technical tool. We prove that there are natural functors i from the category of smooth projective F -varieties and j from the category of separable F -algebras to the motivic category C and also the functor from Support from Alexander von Humboldt-Stiftung is gratefully acknowledged The second author thanks the Soros Fund for support

The variety of complete pairs of zero-dimensional subschemes of an algebraic surface

Abstract: We investigate the variety of zero-dimensional subschemes (that is, systems of points) and of given lengths and on a smooth projective algebraic surface . The variety is realized as the blowing-up of the direct product of Hilbert schemes of points along the incidence graph. It is proved that is naturally isomorphic to the variety of biflags , where . We also study the problem of the smoothness of . It is proved that is smooth for and an arbitrary using the Kodaira-Spenser rank map in the theory of determinantal varieties and also in the case when by means of a direct geometric consideration.

Classification of G-varieties of complexity?1

Abstract: We consider the problem of finding a combinatorial description of the algebraic varieties in a given birational class that admit an action of a reductive group G. This is a direct generalization of the theory of toric varieties. A general approach to this problem is described, and the solution is given for varieties in which the orbits in general position of a Borel subgroup G have codimension 1 (varieties of complexity 1).

Toric laminations, sparse generalized characteristic polynomials, and a refinement of Hilbert's tenth problem

Abstract: This paper reexamines univariate reduction from a toric geometric point of view We begin by constructing a binomial variant of the u-resultant and then retailor the generalized characteristic polynomial to sparse polynomial systems We thus obtain a fast new algorithm for univariate reduction and a better understanding of the underlying projections As a corollary, we show that a refinement of Hilbert’s Tenth Problem is decidable in single-exponential time We also obtain interesting new algebraic identities for the sparse resultant and certain multisymmetric functions

Values of zeta functions at negative integers, Dedekind sums and toric geometry

Abstract: This is an expanded version. We study relations among special values of zeta functions, invariants of toric varieties, and generalized Dedekind sums. In particular, we use invariants arising in the Todd class of a toric variety to give a new explicit formula for the values of the zeta function of a real quadratic field at nonpositive integers. We also express these invariants in terms of the generalized Dedekind sums studied previously by several authors. The paper includes conceptual proofs of the above mentioned relations and explicit computations of the various zeta values and Dedekind sums involved.

Branes and Toric Geometry

Abstract: We show that toric geometry can be used rather effectively to translate a brane configuration to geometry. Roughly speaking the skeletons of toric space are identified with the brane configurations. The cases where the local geometry involves hypersurfaces in toric varieties (such as P^2 blown up at more than 3 points) presents a challenge for the brane picture. We also find a simple physical explanation of Batyrev's construction of mirror pairs of Calabi-Yau manifolds using T-duality.