Showing papers on "Toric variety published in 2003"

Theory of multi-fans

Abstract: We introduce the notion of a multi-fan. It is a generalization of that of a fan in the theory of toric variety in algebraic geometry. Roughly speaking a toric variety is an algebraic variety with an action of algebraic torus of the same dimension as that of the variety, and a fan is a combinatorial object associated with the toric variety. Algebro-geometric properties of the toric variety can be described in terms of the associated fan. We develop a combinatorial theory of multi-fans and define ``topological invariants'' of a multi-fan. A smooth manifold with an action of a compact torus of half the dimension of the manifold and with some orientation data is called a torus manifold. We associate a multi-fan with a torus manifold, and apply the combinatorial theory to describe topological invariants of the torus manifold. A similar theory is also given for torus orbifolds. As a related subject a generalization of the Ehrhart polynomial concerning the number of lattice points in a convex polytope is discussed.

Valuations, Deformations, and Toric Geometry

Abstract: A study of the relation between a noetherian local domain with a given valuation and its associated graded ring with respect to the valuation, which in some cases is an esentially toric variety, possibly of infinite embedding dimension, but of finite Krull dimension. After extension of the valuation to a suitable completion (whose existence is not established in the paper) the relation becomes much more precise, and suggests a possible way to establish local uniformization for an excellent equicharacteristic local domain with an algebraically closed residue field, by deforming a partial toric embedded resolution of the essentially toric spectrum of the associated graded ring.

Higher algebraic K-theory for actions of diagonalizable groups

Abstract: We study the K-theory of actions of diagonalizable group schemes on noetherian regular separated algebraic spaces: our main result shows how to reconstruct the K-theory ring of such an action from the K-theory rings of the loci where the stabilizers have constant dimension. We apply this to the calculation of the equivariant K-theory of toric varieties, and give conditions under which the Merkurjev spectral sequence degenerates, so that the equivariant K-theory ring determines the ordinary K-theory ring. We also prove a very refined localization theorem for actions of this type.

On the cohomology of torus manifolds

Abstract: A torus manifold is an even-dimensional manifold acted on by a half-dimensional torus with non-empty fixed point set and some additional orientation data. It may be considered as a far-reaching generalisation of toric manifolds from algebraic geometry. The orbit space of a torus manifold has a rich combinatorial structure, e.g., it is a manifold with corners provided that the action is locally standard. Here we investigate relationships between the cohomological properties of torus manifolds and the combinatorics of their orbit quotients. We show that the cohomology ring of a torus manifold is generated by two-dimensional classes if and only if the quotient is a homology polytope. In this case we retrieve the familiar picture from toric geometry: the equivariant cohomology is the face ring of the nerve simplicial complex and the ordinary cohomology is obtained by factoring out certain linear forms. In a more general situation, we show that the odd-degree cohomology of a torus manifold vanishes if and only if the orbit space is face-acyclic. Although the cohomology is no longer generated in degree two under these circumstances, the equivariant cohomology is still isomorphic to the face ring of an appropriate simplicial poset.

Notes on toric varieties from Mori theoretic viewpoint

Abstract: The main purpose of this notes is to supplement the paper by Reid: De- composition of toric morphisms, which treated Minimal Model Program (also called Mori's Program) on toric varieties. We compute lengths of negative extremal rays of toric varieties. As an application, a generalization of Fujita's c onjecture for singular toric varieties is obtained. We also prove that every toric variety has a small projective toric Q-factorialization. 0. Introduction. The main purpose of this notes is to supplement the paper by Reid (Re): Decomposition of toric morphisms, which treated Minimal Model Program (also called Mori's Program) on toric varieties. We compute lengths of negative extremal rays of toric varieties. This is an answer to (Ma, Remark-Question 10-3-6) for toric varieties, which is an easy exercise once we understand (Re). As a corollary, we obtain a strong version of Fujita's conjecture for singular toric varieties. Related topics are (Ft), (Ka), (La) and (Mu, Section 4). We will work, throughout this paper, over an algebraically closed field k of arbitrary characteristic.

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(L,G) defined by a finite lattice L and a subset 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. 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 G. We describe this variety both by its fan and geometrically by a series of blowups and orbit removal. Also we find a Groebner basis of the relation ideal of D and a monomial basis of D over Z.

On the integral cohomology of smooth toric varieties

Abstract: Let $X_\Sigma$ be a smooth, not necessarily compact toric variety. We show that a certain complex, defined in terms of the fan $\Sigma$, computes the integral cohomology of $X_\Sigma$, including the module structure over the homology of the torus. In some cases we can also give the product. As a corollary we obtain that the cycle map from Chow groups to integral Borel-Moore homology is split injective for smooth toric varieties. Another result is that the differential algebra of singular cochains on the Borel construction of $X_\Sigma$ is formal.

Calabi-Yau fourfolds with flux and supersymmetry breaking

Abstract: In Calabi-Yau fourfold compactifications of M-theory with flux, we investigate the possibility of partial supersymmetry breaking in the three-dimensional effective theory. To this end, we place the effective theory in the framework of general N = 2 gauged supergravities, in the special case where only translational symmetries are gauged. This allows us to extract supersymmetry-breaking conditions, and interpret them as conditions on the 4-form flux and Calabi-Yau geometry. For N = 2 unbroken supersymmetry in three dimensions we recover previously known results, and we find a new condition for breaking supersymmetry from N = 2 to N = 1, i.e. from four to two supercharges. An example of a Calabi-Yau hypersurface in a toric variety that satisfies this condition is provided.

Intersection Cohomology on Nonrational Polytopes

Abstract: We consider a fan as a ringed space (with finitely many points). We develop the corresponding sheaf theory and functors, such as direct image Rπ* (π is a subdivision of a fan), Verdier duality, etc. The distinguished sheaf \(\mathcal{L}_\Phi\), called the minimal sheaf plays the role of an equivariant intersection cohomology complex on the corresponding toric variety (which exists if Φ is rational). Using \(\mathcal{L}_\Phi\) we define the intersection cohomology space IH(Φ). It is conjectured that a strictly convex piecewise linear function on Φ acts as a Lefschetz operator on IH(Φ). We show that this conjecture implies Stanley's conjecture on the unimodality of the generalized h-vector of a convex polytope.

The total coordinate ring of a normal projective variety

Abstract: The total coordinate ring TC(X) of a normal variety is a generalization of the ring introduced and studied by Cox in connection with a toric variety. Consider a normal projective variety X with divisor class group Cl(X), and let us assume that it is a finitely generated free abelian group. We define the total coordinate ring of X to be TC(X) = oplus_{D} H^0 (X, O_X (D)), where the sum as above is taken over all Weil divisors of X contained in a fixed complete system of representatives of Cl(X). We prove that for any normal projective variety X, TC(X) is a UFD, this is a corollary of a more general theorem that is proved in the paper. (Berchtold and Haussen proved the unique factorization for a smooth variety independently.) We also prove that for X, the blow up of P^2 along a finite number of collinear points, TC(X) is Noetherian. We also give an example that TC(X) is not Noetherian but oplus_n H^0 (X, O(nD)) is Noetherian for any Weil divisor D.

Uniform bounds on multigraded regularity

Abstract: We give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety X with a given multigraded Hilbert polynomial. To establish this bound, we introduce a new combinatorial tool, called a Stanley filtration, for studying monomial ideals in the homogeneous coordinate ring of X. As a special case, we obtain a new proof of Gotzmann's regularity theorem. We also discuss applications of this bound to the construction of multigraded Hilbert schemes.

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 combinatorial 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.

Contractible classes in toric varieties

Abstract: Let X be a smooth, complete toric variety. Let A1(X) be the group of algebraic 1-cycles on X modulo numerical equivalence and N1(X) = A1(X)⊗ZQ . Consider inN1(X) the coneNE(X) generated by classes of curves on X . It is a well-known result due to M. Reid [13] that NE(X) is closed, polyhedral and generated by classes of invariant curves on X . The varietyX is projective if and only if NE(X) is strictly convex; in this case, a 1-dimensional face ofNE(X) is called an extremal ray. It is shown in [13] that every extremal ray admits a contraction to a projective toric variety. We think of A1(X) as a lattice in the Q -vector space N1(X). Suppose that X is projective. For every extremal ray R ⊂ NE(X), we choose the primitive class in R ∩ A1(X); we call this class an extremal class. The set E of extremal classes is a generating set for the cone NE(X), namely NE(X) = ∑ γ∈E Q≥0 γ. For many purposes it would be useful to have a linear decomposition with integral coefficients: for instance, what can we say about curves having minimal degree with respect to some ample line bundle onX? It is an open question whether extremal classes generate NE(X) ∩ A1(X) as a semigroup. In this paper we introduce a set C ⊇ E of classes in NE(X) ∩ A1(X) which is a set of generators of NE(X) ∩ A1(X) as a semigroup. Classes in C are geometrically characterized by “contractibility”:

Localizations on Moduli Spaces and Free Field Realizations of Feynman Rules

Abstract: We prove Iqbal's conjecture on the relationship between the free energy of closed string theory in local toric geometry and the Wess-Zumino-Witten model. This is achieved by first reformulating the calculations of the free energy by localization techniques in terms of suitable Feynman rule, then exploiting a realization of the Feynman rule by free bosons. We also use a formula of Hodge integrals conjectured by the author and proved jointly with Chiu-Chu Melissa Liu and Kefeng Liu.

Multiplier Ideals and Modules on Toric Varieties

Abstract: A simple 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.

Lifting of morphisms to quotient presentations

Abstract: In this article we investigate algebraic morphisms between toric varieties. Given presentations of toric varieties as quotients we are interested in the question when a morphism admits a lifting to these quotient presentations. We show that this can be completely answered in terms of invariant divisors. As an application we prove that two toric varieties, which are isomorphic as abstract algebraic varieties, are even isomorphic as toric varieties. This generalizes a well-known result of Demushkin on affine toric varieties.

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.

Toric Fano varieties and birational morphisms

Abstract: In this paper we study smooth toric Fano varieties using primitive relations and toric Mori theory. We show that for any irreducible invariant divisor D in a toric Fano variety X, we have $0\leq\rho_X-\rho_D\leq 3$, for the difference of the Picard numbers of X and D. Moreover, if $\rho_X-\rho_D>0$ (with some additional hypotheses if $\rho_X-\rho_D=1$), we give an explicit birational description of X. Using this result, we show that when dim X=5, we have $\rho_X\leq 9$. In the second part of the paper, we study equivariant birational morphisms f whose source is Fano. We give some general results, and in dimension 4 we show that f is always a composite of smooth equivariant blow-ups. Finally, we study under which hypotheses a non-projective toric variety can become Fano after a smooth equivariant blow-up.

Equivariant Todd Classes for Toric Varieties

Abstract: For a complete toric variety, we obtain an explicit formula for the localized equivariant Todd class in terms of the combinatorial data -- the fan. This is based on the equivariant Riemann-Roch theorem and the computation of the equivariant cohomology and equivariant homology of toric varieties.

Viro theorem and topology of real and complex combinatorial hypersurfaces

Abstract: We introduce a class of combinatorial hypersurfaces in the complex projective space. They are submanifolds of codimension 2 inℂP n and are topologically “glued” out of algebraic hypersurfaces in (ℂ*) n . Our construction can be viewed as a version of the Viro gluing theorem relating topology of algebraic hypersurfaces to the combinatorics of subdivisions of convex lattice polytopes. If a subdivision is convex, then according to the Viro theorem a combinatorial hypersurface is isotopic to an algebraic one. We study combinatorial hypersurfaces resulting from non-convex subdivisions of convex polytopes, show that they are almost complex varieties, and in the real case, they satisfy the same topological restrictions (congruences, inequalities etc.) as real algebraic hypersurfaces.

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 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.

F-theory duals of M-theory on G2 manifolds from mirror symmetry

Abstract: Using mirror pairs (M3, W3) in type II superstring compactifications on Calabi–Yau threefolds, we study, geometrically, F-theory duals of M-theory on seven manifolds with G2 holonomy. We first develop a way of obtaining Landau–Ginzburg (LG) Calabi–Yau threefolds W3, embedded in four complex-dimensional toric varieties, mirror to the sigma model on toric Calabi–Yau threefolds M3. This method gives directly the right dimension without introducing non-dynamical variables. Then, using toric geometry tools, we discuss the duality between M-theory on S1 × M3/Z2 with G2 holonomy and F-theory on elliptically fibred Calabi–Yau fourfolds with SU(4) holonomy, containing W3 mirror manifolds. Illustrative examples are presented.

Variation of toric hyperkähler manifolds

Abstract: A toric hyperKahler manifold is defined to be a smooth hyperKahler quotient of the quaternionic vector space ℍN by a subtorus of TN. It has two parameters corresponding to the de Rham cohomology classes represented by the Kahler form and the complex symplectic form respectively. We study the variation of its complex structure according to these parameters. After the detailed analysis of the stability condition depending on the first parameter, we show that toric hyperKahler manifolds with the same second parameter are related by a sequence of Mukai's elementary transformations. We also give a complete description of its Kahler cone and discuss when certain rational curves exist.

On the monomial birational maps of the projective space

Abstract: We describe the group structure of monomial Cremona transformations. It follows that every element of this group is a product of quadratic monomial transformations, and geometric descriptions in terms of fans.

Resolutions and Moduli for Equivariant Sheaves over Toric Varieties

Abstract: In this thesis the combinatorial framework of toric geometry is extended to equivariant sheaves over toric varieties. The central questions are how to extract combinatorial information from the so developed description and whether equivariant sheaves can, like toric varieties, be considered as purely combinatorial objects. The thesis consists of three main parts. In the first part, by systematically extending the framework of toric geometry, a formalism is developed for describing equivariant sheaves by certain configurations of vector spaces. In the second part, homological properties of a certain class of equivariant sheaves are investigated, namely that of reflexive equivariant sheaves. Several kinds of resolutions for these sheaves are constructed which depend only on the configuration of their associated vector spaces. Thus a partially positive answer to the question of combinatorial representability is given. As a particular result, a new way for computing minimal resolutions for Z^n - graded modules over polynomial rings is obtained. In the third part a complete classification of the simplest nontrivial sheaves, equivariant vector bundles of rank two over smooth toric surfaces, is given. A combinatorial characterization is given and parameter spaces (moduli spaces) are constructed which depend only on this characterization. In appendices a outlook on equivariant sheaves and the relation of Chern classes to their combinatorial classification is given, particularly focussing on the case of the projective plane. A classification of equivariant vector bundles of rank three over the projective plane is given.

The Denef-Loeser series for toric surface singularities

Abstract: Let $H$ denote the set of formal arcs going through a singular point of an algebraic variety $V$ defined over an algebraically closed field $k$ of characteristic zero. In the late sixties, J. Nash has observed that for any nonnegative integer $s$, the set $j^s(H)$ of $s$-jets of arcs in $H$ is a constructible subset of some affine space. Recently (1999), J. Denef and F. Loeser have proved that the Poincar\'{e} series associated with the image of $j^s(H)$ in some suitable localization of the Grothendieck ring of algebraic varieties over $k$ is a rational function. We compute this function for normal toric surface singularities.

Centrally symmetric generators in toric Fano varieties

Abstract: We give a structure theorem for n-dimensional smooth toric Fano varieties whose associated polytope has ``many'' pairs of centrally symmetric vertices.