Showing papers on "Toric variety published in 2004"

The orbifold Chow ring of toric Deligne-Mumford stacks

Abstract: The orbifold Chow ring of a Deligne-Mumford stack, defined by Abramovich, Graber and Vistoli [2], is the algebraic version of the orbifold cohomology ring in troduced by W. Chen and Ruan [7], [8]. By design, this ring incorporates numerical invariants, such as the orbifold Euler characteristic and the orbifold Hodge num bers, of the underlying variety. The product structure is induced by the degree zero part of the quantum product; in particular, it involves Gromov-Witten invariants. Inspired by string theory and results in Batyrev [3] and Yasuda [28], one expects that, in nice situations, the orbifold Chow ring coincides with the Chow ring of a resolution of singularities. Fantechi and G?ttsche [14] and Uribe [25] verify this conjecture when the orbifold is Symn(5) where 5 is a smooth projective surface with Ks = 0 and the resolution is Hilbn(?>). The initial motivation for this project was to compare the orbifold Chow ring of a simplicial toric variety with the Chow ring of a cr?pant resolution. To achieve this goal, we first develop the theory of toric Deligne-Mumford stacks. Modeled on simplicial toric varieties, a toric Deligne-Mumford stack corresponds to a combinatorial object called a stacky fan. As a first approximation, this object is a simplicial fan with a distinguished lattice point on each ray in the fan. More precisely, a stacky fan S is a triple consisting of a finitely generated abelian group N, a simplicial fan E in Q z N with n rays, and a map ?: Zn ?> N where the image of the standard basis in Zn generates the rays in E. A rational simplicial fan E produces a canonical stacky fan S := (N, E, ?) where N is the distinguished lattice and ? is the map defined by the minimal lattice points on the rays. Hence, there is a natural toric Deligne-Mumford stack associated to every simplicial toric variety. A stacky fan ? encodes a group action on a quasi-affine variety and the toric Deligne-Mumford stack #(?) is the quotient. If E corresponds to a smooth toric variety X?E) and S is the canonical stacky fan associated to E, then we simply have #(!?) = X?Z). We show that many of the basic concepts, such as open and closed toric substacks, line bundles, and maps between toric Deligne Mumford stacks, correspond to combinatorial notions. We expect that many more results about toric varieties lift to the realm of stacks and hope that toric Deligne Mumford stacks will serve as a useful testing ground for general theories.

Toric degenerations of toric varieties and tropical curves

Abstract: We show that the counting of rational curves on a complete toric variety that are in general position to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraic-geometric and relies on degeneration techniques and log deformation theory. This generalizes results of Mikhalkin obtained by different methods in the surface case to arbitrary dimensions.

Topological strings and their physical applications

Abstract: We give an introductory review of topological strings and their application to various aspects of superstrings and supersymmetric gauge theories. This review includes developing the necessary mathematical background for topological strings, such as the notions of Calabi-Yau manifold and toric geometry, as well as physical methods developed for solving them, such as mirror symmetry, large N dualities, the topological vertex and quantum foam. In addition, we discuss applications of topological strings to N=1,2 supersymmetric gauge theories in 4 dimensions as well as to BPS black hole entropy in 4 and 5 dimensions. (These are notes from lectures given by the second author at the 2004 Simons Workshop in Mathematics and Physics.)

Multigraded Castelnuovo-Mumford regularity

Abstract: We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the standard graded case, our definition of multigraded regularity involves the vanishing of graded components of local cohomology. We establish the key properties of regularity: its connection with the minimal generators of a module and its behavior in exact sequences. For an ideal sheaf on a simplicial toric variety X , we prove that its multigraded regularity bounds the equations that cut out the associated subvariety. We also provide a criterion for testing if an ample line bundle on X gives a projectively normal embedding.

Chow rings of toric varieties defined by atomic lattices

Abstract: We study a graded algebra \(D=D(\mathcal{L},\mathcal{G})\) over ℤ defined by a finite lattice ℒ and a subset \(\mathcal{G}\) in ℒ, a so-called building set. This algebra is a generalization of the cohomology algebras of hyperplane arrangement compactifications found in work of De Concini and Procesi [2]. Our main result is a representation of D, for an arbitrary atomic lattice ℒ, as the Chow ring of a smooth toric variety that we construct from ℒ and \(\mathcal{G}\). We describe this variety both by its fan and geometrically by a series of blowups and orbit removal. Also we find a Grobner basis of the relation ideal of D and a monomial basis of D.

Hamiltonian 2-forms in Kähler geometry, II Global Classification

Abstract: We present a classification of compact Kahler manifolds admitting a hamiltonian 2-form (which were classified locally in part I of this work). This involves two components of independent interest. The first is the notion of a rigid hamiltonian torus action. This natural condition, for torus actions on a Kahler manifold, was introduced locally in part I, but such actions turn out to be remarkably well behaved globally, leading to a fairly explicit classification: up to a blow-up, compact Kahler manifolds with a rigid hamiltonian torus action are bundles of toric Kahler manifolds. The second idea is a special case of toric geometry, which we call orthotoric. We prove that orthotoric Kahler manifolds are diffeomorphic to complex projective space, but we extend our analysis to orthotoric orbifolds, where the geometry is much richer. We thus obtain new examples of Kahler–Einstein 4-orbifolds. Combining these two themes, we prove that compact Kahler manifolds with hamiltonian 2-forms are covered by blow-downs of projective bundles over Kahler products, and we describe explicitly how the Kahler metrics with a hamiltonian 2-form are parameterized. We explain how this provides a context for constructing new examples of extremal Kahler metrics—in particular a subclass of such metrics which we call weakly Bochner-flat. We also provide a self-contained treatment of the theory of compact toric Kahler manifolds, since we need it and find the existing literature incomplete.

Hard Lefschetz theorem for nonrational polytopes

Abstract: The Hard Lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope. One can construct the intersection cohomology combinatorially from the polytope, hence it is well defined even for nonrational polytopes when there is no variety associated to it. We prove the Hard Lefschetz theorem for the intersection cohomology of a general polytope.

Hamiltonian 2-forms in Kahler geometry, II Global Classification

Abstract: We present a classification of compact Kaehler manifolds admitting a hamiltonian 2-form (which were classified locally in part I of this work). This involves two components of independent interest. The first is the notion of a rigid hamiltonian torus action. This natural condition, for torus actions on a Kaehler manifold, was introduced locally in part I, but such actions turn out to be remarkably well behaved globally, leading to a fairly explicit classification: up to a blow-up, compact Kaehler manifolds with a rigid hamiltonian torus action are bundles of toric Kaehler manifolds. The second idea is a special case of toric geometry, which we call orthotoric. We prove that orthotoric Kaehler manifolds are diffeomorphic to complex projective space, but we extend our analysis to orthotoric orbifolds, where the geometry is much richer. We thus obtain new examples of Kaehler--Einstein 4-orbifolds. Combining these two themes, we prove that compact Kaehler manifolds with hamiltonian 2-forms are covered by blow-downs of projective bundles over Kaehler products, and we describe explicitly how the Kaehler metrics with a hamiltonian 2-form are parameterized. We explain how this provides a context for constructing new examples of extremal Kaehler metrics - in particular a subclass of such metrics which we call weakly Bochner-flat. We also provide a self-contained treatment of the theory of compact toric Kaehler manifolds, since we need it and find the existing literature incomplete.

Phylogenetic Algebraic Geometry

Abstract: Phylogenetic algebraic geometry is concerned with certain complex projective algebraic varieties derived from finite trees Real positive points on these varieties represent probabilistic models of evolution For small trees, we recover classical geometric objects, such as toric and determinantal varieties and their secant varieties, but larger trees lead to new and largely unexplored territory This paper gives a self-contained introduction to this subject and offers numerous open problems for algebraic geometers

Tilting sheaves on toric varieties

Abstract: In [19], A. King states the following conjecture: Any smooth complete toric variety has a tilting bundle whose summands are line bundles. The goal of this paper is to prove King’s conjecture for the following types of smooth complete toric varieties: (i) Any d-dimensional smooth complete toric variety with splitting fan. (ii) Any d-dimensional smooth complete toric variety with Picard number ≤2. (iii) The blow up of any smooth complete minimal toric surface at T-invariants points.

Toric Geometry, Sasaki-Einstein Manifolds and a New Infinite Class of AdS/CFT Duals

Abstract: Recently an infinite family of explicit Sasaki-Einstein metrics Y^{p,q} on S^2 x S^3 has been discovered, where p and q are two coprime positive integers, with q

Homotopy groups of K-contact toric manifolds

Abstract: Contact toric manifolds of Reeb type are a subclass of contact toric manifolds which have the property that they are classified by the images of the associated moment maps. We compute their first and second homotopy group terms of the images of the moment map. We also explain why they are K-contact.

Graded rings and equivariant sheaves on toric varieties

Abstract: In this note we derive a formalism for describing equivariant sheaves over toric varieties. This formalism is a generalization of a correspondence due to Klyachko, which states that equivariant vector bundles on toric varieties are equivalent to certain sets of filtrations of vector spaces. We systematically construct the theory from the point of view of graded ring theory and this way we clarify earlier constructions of Kaneyama and Klyachko. We also connect the formalism to the theory of fine-graded modules over Cox' homogeneous coordinate ring of a toric variety. As an application we construct minimal resolutions of equivariant vector bundles of rank two on toric surfaces. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Multiplier ideals and modules on toric varieties

Abstract: A formula computing the multiplier ideal of a monomial ideal on an arbitrary affine toric variety is given. Variants for the multiplier module and test ideals are also treated.

Introduction to the toric mori theory

Abstract: The main purpose of this paper is to give a simple and non-combinatorial proof of the toric Mori theory. Here, the toric Mori theory means the (log) Minimal Model Program (MMP, for short) for toric varieties. We minimize the arguments on fans and their decompositions. We recommend this paper to the following people: (A) those who are uncomfortable with manipulating fans and their decompositions, (B) those who are familiar with toric geometry but not with the MMP. People in the category (A) will be relieved from tedious combina- torial arguments in several problems. Those in the category (B) will discover the potential of the toric Mori theory. As applications, we treat the Zariski decomposition on toric varieties and the partial resolutions of non-degenerate hypersurface singularities. By these applications, the reader will learn to use the toric Mori theory.

Properties of the residual circle action on a hypertoric variety

Abstract: We consider an orbifold X obtained by a Kahler reduction of C n , and we define its hyperkahler analogue M as a hyperkahler reduction of T*C n ≅ H n by the same group. In the case where the group is abelian and X is a toric variety, M is a toric hyperkahler orbifold, as defined in Bielawski and Dancer, 2000, and further studied by Konno and by Hausel and Sturmfels. The variety M carries a natural action of S 1 , induced by the scalar action of S 1 on the fibers of T*C n . In this paper we study this action, computing its fixed points and its equivariant cohomology. As an application, we use the associated Z 2 action on the real locus of M to compute a deformation of the Orlik-Solomon algebra of a smooth, real hyperplane arrangement H, depending nontrivially on the affine structure of the arrangement. This deformation is given by the Z 2 -equivariant cohomology of the complement of the complexification of H, where Z 2 acts by complex conjugation.

Toric Codes over Finite Fields

Abstract: In this note, a class of error-correcting codes is associated to a toric variety defined over a finite field * q, analogous to the class of AG codes associated to a curve. For small q, many of these codes have parameters beating the Gilbert-Varshamov bound. In fact, using toric codes, we construct a (n,k,d)=(49,11,28) code over * 8, which is better than any other known code listed in Brouwer’s tables for that n, k and q. We give upper and lower bounds on the minimum distance. We conclude with a discussion of some decoding methods. Many examples are given throughout.

Cycles representing the Todd class of a toric variety

Abstract: In this paper, we describe a way to construct cycles which represent the Todd class of a toric variety. Given a lattice with an inner product we assign a rational number m(s) to each rational polyhedral cone s in the lattice, such that for any toric variety X with fan S, the Todd class of X is the sum over all cones s in S of m(s)[V(s)]. This constitutes an improved answer to an old question of Danilov. In a similar way, beginning with the choice of a complete flag in the lattice, we obtain the cycle Todd classes constructed by Morelli. Our construction is based on an intersection product on cycles of a simplicial toric variety developed by the second-named author. Important properties of the construction are established by showing a connection to the canonical representation of the Todd class of a simplicial toric variety as a product of torus-invariant divisors developed by the first-named author.

Stable reductive varieties I: Affine varieties

Abstract: The motivation of this work is to construct an analog of compactified moduli of abelian varieties and toric pairs in the case of non-commutative algebraic group G. We introduce a class of "stable reductive varieties" which contain connected reductive groups and their equivariant compactifications, and is closed under flat reduced degenerations. We classify them all, describe their degenerations, and establish a connection between these varieties and "reductive semigroups" which we also define. Finally, we construct a Hilbert scheme of embedded G-varieties by applying and generalizing a construction of Haiman and Sturmfels. The second version adds some cosmetic changes.

Lower Bounds for Real Solutions to Sparse Polynomial Systems

Abstract: We show how to construct sparse polynomial systems that have non-trivial lower bounds on their numbers of real solutions. These are unmixed systems associated to certain polytopes. For the order polytope of a poset P this lower bound is the sign-imbalance of P and it holds if all maximal chains of P have length of the same parity. This theory also gives lower bounds in the real Schubert calculus through sagbi degeneration of the Grassmannian to a toric variety, and thus recovers a result of Eremenko and Gabrielov.

Lattice polarized toric K3 surfaces

Abstract: When studying mirror symmetry in the context of K3 surfaces, the hyperkaehler structure of K3 makes the notion of exchanging Kaehler and complex moduli ambiguous. On the other hand, the metric is not renormalized due to the higher amount of supersymmetry of the underlying superconformal field theory. Thus one can define a natural mapping from the classical K3 moduli space to the moduli space of conformal field theories. Apart from the generalization of mirror constructions for Calabi-Yau threefolds, there is a formulation of mirror symmetry in terms of orthogonal lattices and global moduli space arguments. In many cases both approaches agree perfectly - with a long outstanding exception: Batyrev's mirror construction for K3 hypersurfaces in toric varieties does not fit into the lattice picture whenever the Picard group of the K3 surface is not generated by the pullbacks of the equivariant divisors of the ambient toric variety. In this case, not even the ranks of the corresponding Picard lattices add up as expected. In this paper the connection is clarified by refining the lattice picture. We show (by explicit calculation with a computer) mirror symmetry for all families of toric K3 hypersurfaces corresponding to dual reflexive polyhedra, including the formerly problematic cases.

Bunches of Cones in the Divisor Class Group : a new Combinatorial Language for Toric Varieties

Abstract: As an alternative to the description of a toric variety by a fan in the lattice of one-parameter subgroups, we present a new language in terms of what we call bunches—these are certain collections of cones in the vector space of rational divisor classes. The correspondence between these bunches and fans is based on the classical Gale duality. The new combinatorial language allows a much more natural description of geometric phenomena around divisors of toric varieties than the usual method by fans does. For example, the numerically effective cone and the ample cone of a toric variety can be read off immediately from its bunch. Moreover, the language of bunches appears to be useful for classification problems.

Distribution laws for integrable eigenfunctions

Abstract: We determine the asymptotics of the joint eigenfunctions of the torus action on a toric Kahler variety. Such varieties are models of completely integrable systems in complex geometry. We first determine the pointwise asymptotics of the eigenfunctions, which show that they behave like Gaussians centered at the corresponding classical torus. We then show that there is a universal Gaussian scaling limit of the distribution function near its center. We also determine the limit distribution for the tails of the eigenfunctions on large length scales. These are not universal but depend on the global geometry of the toric variety and in particular on the details of the exponential decay of the eigenfunctions away from the classically allowed set.

Lattice polygons and Green’s theorem

Abstract: Associated to an n-dimensional integral convex polytope P is a toric variety X and divisor D, such that the integral points of P represent H 0 (O X (D)) We study the free resolution of the homogeneous coordinate ring ○+ m ∈ Z H 0 (mD) as a module over Sym(H 0 (O X (D))), It turns out that a simple application of Green's theorem yields good bounds for the linear syzygies of a projective toric surface In particular, for a planar polytope P = H 0 (O X (D)), D satisfies Green's condition Np if ∂P contains at least p + 3 lattice points