Showing papers on "Toric variety published in 1995"

Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces

Abstract: Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.

The functor of a smooth toric variety

Abstract: (1) Y 7→ {line bundle quotients of O Y } . This is easy to prove since a surjection O Y → L gives n + 1 sections of L which don’t vanish simultaneously and hence determine a map Y → Pnk . The goal of this paper is to generalize this representation to the case of an arbitrary smooth toric variety. We will work with schemes over an algebraically closed field k of characteristic zero, and we will fix a smooth n-dimensional toric variety X determined by the fan ∆ in NR = R. As usual, M denotes the dual lattice of N and ∆(1) denotes the 1-dimensional cones of ∆. We will use ∑

The topology of the space of rational curves on a toric variety

Abstract: Let $X$ be a compact toric variety. Let $Hol$ denote the space of based holomorphic maps from $CP^1$ to $X$ which lie in a fixed homotopy class. Let $Map$ denote the corresponding space of continuous maps. We show that $Hol$ has the same homotopy groups as $Map$ up to some (computable) dimension. The proof uses a description of $Hol$ as a space of configurations of labelled points, where the labels lie in a partial monoid determined by the fan of $X$.

GKZ-Generalized Hypergeometric Systems in Mirror Symmetry of Calabi-Yau Hypersurfaces

Abstract: We present a detailed study of the generalized hypergeometric system introduced by Gel'fand, Kapranov and Zelevinski (GKZ-hypergeometric system) in the context of toric geometry. GKZ systems arise naturally in the moduli theory of Calabi-Yau toric varieties, and play an important role in applications of the mirror symmetry. We find that the Grobner basis for the so-called toric ideal determines a finite set of differential operators for the local solutions of the GKZ system. At the special point called the large radius limit, we find a close relationship between the principal parts of the operators in the GKZ system and the intersection ring of a toric variety. As applications, we analyze general three dimensional hypersurfaces of Fermat and non-Fermat types with Hodge numbers up to $h^{1,1}=3$. We also find and analyze several non Landau-Ginzburg models which are related to singular models.

Equivariant sheaves on toric varieties

Abstract: © Foundation Compositio Mathematica, 1995, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Mirror Symmetry, Mirror Map and Applications to

Abstract: Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to es- tablish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.

Manin's conjecture for toric varieties

Abstract: We prove an asymptotic formula conjectured by Manin for the number of $K$-rational points of bounded height with respect to the anticanonical line bundle for arbitrary smooth projective toric varieties over a number field $K$.

Topological invariants of a fan associated to a toric variety

Abstract: Associated to a toric variety X of dimension r over a field k is a fan Δ on R1. The fan Δ is a finite set of cones which are in one-to-one correspondence with the orbits of the torus action on X. The fan Δ inherits the Zariski topology from X. In this article some cohomological invariants of X are studied in terms of whether or not they depend only on Δ and not k. Secondly some numerical invariants of X are studied in terms of whether or not they are topological invariants of the fan Δ. That is, whether or not they depend only on the finite topological space defined on Δ. The invariants with which we are mostly concerned are the class group of Weil divisors, the Picard group, the Brauer group and the dimensions of the torsion free part of the etale cohomology groups with coefficients in the sheaf of units. The notion of an open neighborhood of a fan is introduced and examples are given for which the above invariants are sufficiently fine to give nontrivial stratifications of an open neighborhood of a fan ...

On the homogeneous ideal of a projective nonsingular toric variety

Abstract: Reason for withdrawal: There is a serious mistake in the calculation of the divisor of the rational section used in the proof of Prop 221, and with the correct value the argument does not work

Some homogeneous coordinate rings that are Koszul algebras

Abstract: Using reduction to positive characteristic and the method of Frobenius splitting of diagonals, due to Mehta and Ramanathan, it is shown that homogeneous coordinate rings for either proper and smooth toric varieties or Schubert varieties are Koszul algebras.

Periodic Hamiltonian flows on four dimensional manifolds

Abstract: We classify the periodic Hamiltonian flows on compact four dimensional symplectic manifolds up to isomorphism of Hamiltonian S^1 spaces. Additionally, we show that all these spaces are Kaehler, that every such space is obtained from a simple model by a sequence of symplectic blowups, and that if the fixed points are isolated then the space is a toric variety.

On the Connectedness of the Moduli Space of Calabi--Yau Manifolds

Abstract: We show that the moduli space of all Calabi-Yau manifolds that can be realized as hypersurfaces described by a transverse polynomial in a four dimensional weighted projective space, is connected. This is achieved by exploiting techniques of toric geometry and the construction of Batyrev that relate Calabi-Yau manifolds to reflexive polyhedra. Taken together with the previously known fact that the moduli space of all CICY's is connected, and is moreover connected to the moduli space of the present class of Calabi-Yau manifolds (since the quintic threefold P_4[5] is both CICY and a hypersurface in a weighted P_4, this strongly suggests that the moduli space of all simply connected Calabi-Yau manifolds is connected. It is of interest that singular Calabi-Yau manifolds corresponding to the points in which the moduli spaces meet are often, for the present class, more singular than the conifolds that connect the moduli spaces of CICY's.

Varieties of CM-type

Abstract: We introduce the notion of a variety (or more generally a motive) of CM-type which generalises the well known notion of abelian variety of CM-type. Just as in that particular case it will turn out that the cohomology of the variety is determined by purely combinatorial data; the type of the variety. As applications we will show that the \l-adic representations are given by algebraic Hecke characters whose algebraic parts are determined by the type and give a method for computing the discriminant of the Neron-Severi group of super-singular Fermat surfaces.

Mirror Symmetry on Arbitrary Dimensional Calabi-Yau Manifold with a Few Moduli

Abstract: We calculate the B-model on the mirror pair of X2N−2(2, 2, …, 2, 1, 1), which is an (N−2)-dimensional Calabi-Yau manifold and has two marginal operators, i.e. h1,1(X2N−2(2, 2, …, 2, 1, 1))=2. In Ref. 1 we have discussed about mirror symmetry on XN(1, 1, …, 1) and its mirror pair. However, XN(1, 1, …, 1) had only one moduli. In this letter, we extend its methods to the case with a few moduli using toric geometry.

On a class of K\"ahler manifolds whose geodesic flows are integrable

Abstract: We study $n$-dimensional Kahler manifolds whose geodesic flows possess $n$ first integrals in involution that are fibrewise hermitian forms and simultaneously normalizable. Under some mild assumption, one can associate with such a manifold an $n$-dimensional commutative Lie algebra of infinitesimal automorphisms. This, combined with the given $n$ first integrals, makes the geodesic flow integrable. If the manifold is compact, then it becomes a toric variety.

Flips in low codimension : classification and quantitative theory

Abstract: A flip is a birational map of 3-folds X- ---> X+ which is an isomorphism away from curves C- c X- and C+ c X+ and does not extend across these curves. Flips are the primary object of study of this thesis. I discuss their formal definition and history in Chapter 1. Flips are well known in toric geometry. In Chapter 2, I calculate how the numbers K3 and χ(nK) differ between X- and X+ for toric flips. These numbers are also related in a primary way by Riemann-Roch theorems but I keep that quiet until Chapter 5. In Chapter 3, I describe a technique, which I learned from Miles Reid, for constructing a flip as C* quotients of a local variety 0 E A, taken in different ways. The codimension of my title refers to the minimal embedding dimension of 0 E A. The case of codimension 0 turns out to be exactly the case of toric geometry as studied in Chapter 2. The main result of Chapter 3 classifies the cases when A c C5 is a singular hypersurface, that is, when A defines a flip in codimension 1. Chapters 4 and 5 concern themselves with computing new examples of flips in higher codimension and studying changes in general flips. I indicate one benefit of knowing how these changes work. The main results of Chapters 2 and 3 have been circulated informally as [2] and [3] respectively.

Algebraic cycles and realification of complex projective varieties

Abstract: We investigate some properties of underlying real algebraic structure on complex projective varieties.

Dual Polyhedra, Mirror Symmetry and Landau-Ginzburg Orbifolds

Abstract: New geometrical features of the Landau-Ginzburg orbifolds are presented, for models with a typical type of superpotential We show the one-to-one correspondence between some of the $(a,c)$ states with $U(1)$ charges $(-1,1)$ and the integral points on the dual polyhedra, which are useful tools for the construction of mirror manifolds Relying on toric geometry, these states are shown to correspond to the $(1,1)$ forms coming from blowing-up processes In terms of the above identification, it can be checked that the monomial-divisor mirror map for Landau-Ginzburg orbifolds, proposed by the author, is equivalent to that mirror map for Calabi-Yau manifolds obtained by the mathematicians

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.

Frobenius morphisms over Z/p^2 and Bott vanishing

Abstract: Let $X$ be a smooth projective algebraic variety over $Z/p$, which has a flat lift to a scheme $X'$ over $Z/p^2$. If the absolute Frobenius morphism $F$ on $X$ lifts to a morphism on $X'$, then an old trick by Mazur shows that push-down of the de Rham complex under $F$ decomposes. We show that the quasi-isomorphism in question is split. This is then applied to toric varieties (where a glueing argument gives lifting of Frobenius to $Z/p^2$) and we derive natural characteristic $p$ proofs of Bott vanishing and degeneration of the Danilov spectral sequence. For flag varieties we obtain generalizations of a result of Paranjape and Srinivas about non-lifting of Frobenius to the Witt vectors.

Topological invariants of a fan associated to a toric variety

Abstract: Associated to a toric variety $X$ of dimension $r$ over a field $k$ is a fan $\Delta$ on $\Bbb R^r$. The fan $\Delta$ is a finite set of cones which are in one-to-one correspondence with the orbits of the torus action on $X$. The fan $\Delta$ inherits the Zariski topology from $X$. In this article some cohomological invariants of $X$ are studied in terms of whether or not they depend only on $\Delta$ and not $k$. Secondly some numerical invariants of $X$ are studied in terms of whether or not they are topological invariants of the fan $\Delta$. That is, whether or not they depend only on the finite topological space defined on $\Delta$. The invariants with which we are mostly concerned are the class group of Weil divisors, the Picard group, the Brauer group and the dimensions of the torsion free part of the \'etale cohomology groups with coefficients in the sheaf of units. The notion of an open neighborhood of a fan is introduced and examples are given for which the above invariants are sufficiently fine to give nontrivial stratifications of an open neighborhood of a fan all of whose maximal cones are nonsimplicial.