Showing papers on "Toric variety published in 1993"

Introduction to Toric Varieties.

Abstract: Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories.The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are quoted without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.

The Homogeneous coordinate ring of a toric variety, revised version

Abstract: This submission consists of two papers: 1) an erratum that corrects an error in the proof of Proposition 4.3 in my paper "The Homogeneous Coordinate Ring of a Toric Variety", and 2) the original (unchanged) version of the paper, published in 1995. The original paper introduced the homogeneous coordinate ring of a toric variety (now called the total coordinate ring or Cox ring) and gave a quotient construction. The paper also studied sheaves on a toric variety, and in Section 4 described its automorphism group. The error in the proof of Proposition 4.3 resulted from the faulty assumption that a certain set of graded endomorphisms forms a ring; rather, it is a monoid under composition. The erratum notes this error and gives a correct proof of the proposition.

Towards the Mirror Symmetry for Calabi-Yau Complete intersections in Gorenstein Toric Fano Varieties

Abstract: We propose a combinatorical duality for lattice polyhedra which conjecturally gives rise to the pairs of mirror symmetric families of Calabi-Yau complete intersections in toric Fano varieties with Gorenstein singularities. Our construction is a generalization of the polar duality proposed by Batyrev for the case of hypersurfaces.

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 $X$.

Singular toric fano varieties

Abstract: The authors prove that the number of types of toric Fano varieties with certain constraints on the singularities is finite.

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.

Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varieties

Abstract: We formulate general conjectures about the relationship between the A-model connection on the cohomology of a $d$-dimensional Calabi-Yau complete intersection $V$ of $r$ hypersurfaces $V_1, \ldots, V_r$ in a toric variety ${\bf P}_{\Sigma}$ and the system of differential operators annihilating the special hypergeometric function $\Phi_0$ depending on the fan $\Sigma$. In this context, the Mirror Symmetry phenomenon can be interpreted as the following twofold characterization of the series $\Phi_0$. First, $\Phi_0$ is defined by intersection numbers of rational curves in ${\bf P}_{\Sigma}$ with the hypersurfaces $V_i$ and their toric degenerations. Second, $\Phi_0$ is the power expansion near a boundary point of the moduli space of the monodromy invariant period of the holomorphic differential $d$-form on an another Calabi-Yau $d$-fold $V'$ which is called Mirror of $V$. Using the generalized hypergeometric series, we propose a general construction for Mirrors $V'$ of $V$ and canonical $q$-coordinates on the moduli spaces of Calabi-Yau manifolds.

On the Brauer group of toric varieties

Abstract: We compute the cohomological Brauer group of a normal toric variety whose singular locus has codimension less than or equal to 2 everywhere

The algebraic de Rham theorem for toric varieties

Abstract: On an arbitrary toric variety, we introduce the logarithmic double complex, which is essentially the same as the algebraic de Rham complex in the nonsingular case, but which behaves much better in the singular case. Over the field of complex numbers, we prove the toric analog of the algebraic de Rham theorem which Grothendieck formulated and proved for general nonsingular algebraic varieties re-interpreting an earlier work of Hodge-Atiyah. Namely, for a finite simplicial fan which need not be complete, the complex cohomology groups of the corresponding toric variety as an analytic space coincide with the hypercohomology groups of the single complex associated to the logarithmic double complex. They can then be described combinatorially as Ishida's cohomology groups for the fan. We also prove vanishing theorems for Ishida's cohomology groups. As a consequence, we deduce directly that the complex cohomology groups vanish in odd degrees for toric varieties which correspond to finite simplicial fans with full-dimensional convex support. In the particular case of complete simplicial fans, we thus have a direct proof for an earlier result of Danilov and the author.

Picard Groups of compact toric Varieties and combinatorial Classes of Fans

Abstract: We consider the question what can be said about the rank of the Picard group Pic Xσ of a compact toric variety Xσ if we know only the combinatorial type of the associated fan σ. We establish upper and lower bounds for the rank of Pic Xσ and give conditions for Pic Xσ to be determined by the combinatorial type of σ. Furthermore, we show that for simple fans Pic Xσ is necessary isomorphic to {0} or Z and give an example for a compact toric variety having a trivial Picard group. Moreover in the projective case we study the relation between addition of T-invariant Cartier divisors on Xσ, taking tensor product of elements of Pic Xσ and piecewise linear functions on σ with Minkowski-addition of polytopes, where the latter operation is extended to a group operation. Finally, we explain the relation to strong cohomology in the projective case.

On the Hodge Structure of Projective Hypersurfaces in Toric Varieties

Abstract: This paper generalizes classical results of Griffiths, Dolgachev and Steenbrink on the cohomology of hypersurfaces in weighted projective spaces. Given a $d$-dimensional projective simplicial toric variety $P$ and an ample hypersurface $X$ defined by an polynomial $f$ in the homogeneous coordinate ring $S$ of $P$ (as defined in an earlier paper of the first author), we show that the graded pieces of the Hodge filtration on $H^d(P - X)$ are naturally isomorphic to certain graded pieces of $S/J(f)$, where $J(f)$ is the Jacobian ideal of $f$. We then discuss how this relates to the primitive cohomology of $X$. Also, if $T$ is the torus contained in $X$, then the intersection of $X$ and $T$ is an affine hypersurface in $T$, and we show how recent results of the second author can be stated using various ideals in the ring $S$. To prove our results, we must give a careful description (in terms of $S$) of $d$-forms and $(d-1)$-forms on the toric variety $P$. For completeness, we also provide a proof of the Bott-Steenbrink-Danilov vanishing theorem for simplicial toric varieties. Other topics considered in the paper include quasi-smooth hypersurfaces and $V$-submanifolds, the structure of the complement of $U$ when $P$ is represented as the quotient of an open subset $U$ of affine space, a generalization of the Euler exact sequence on projective space, and the relation between graded pieces of $R/J(f)$ and the moduli of ample hypersurfaces in $P$.

Lefschetz Fixed Point Theorem and Lattice Points in Convex Polytopes

Abstract: A simple convex lattice polytope $\Box$ defines a torus-equivariant line bundle $\LB$ over a toric variety $\XB.$ Atiyah and Bott's Lefschetz fixed-point theorem is applied to the torus action on the $d''$-complex of $\LB$ and information is obtained about the lattice points of $\Box$. In particular an explicit formula is derived, computing the number of lattice points and the volume of $\Box$ in terms of geometric data at its extreme points. We show this to be equivalent the results of Brion \cite{brion} and give an elementary convex geometric interpretation by performing Laurent expansions similar to those of Ishida \cite{ishida}.

The Functor of a Smooth Toric Variety

Abstract: A map Y -> P^n is determined by a line bundle quotient of (O_Y)^{n+1}. In this paper, we generalize this description to the case of maps from Y to an arbitrary smooth toric variety. The data needed to determine such a map consists of a collection of line bundles on Y together with a section of each line bundle. Further, the line bundles must satisfy certain compatbility conditions, and the sections must be nondegenerate in an appropriate sense. In the case of maps from P^m to a smooth toric variety, we get an especially simple description that generalizes the usual way of specifying maps between projective spaces in terms of homogeneous polynomials (of the same degree) that don't vanish simultaneously.

Algebraic Geometry and Convexity

Abstract: Publisher Summary This chapter reviews the basic issues of developments that have inter-related the combinatorial theory of convex polytopes in many ways to algebraic geometry. Combinatorial methods have been usually used as a tool to solve algebraic-geometric problems. There have been only a few examples of the opposite relationship. One is Stanley's proof of the necessity of McMullen's conjecture on ƒ-vectors. Thereafter, in 1988 a book by Oda, Convex Bodies and Algebraic Geometry, has appeared that provides a compact survey of the subject. In the work of Oda and Ishida, an extensive study has been made about classification of regular compact toric varieties. Starting point is usually a coarse classification based on the cell complex structure—combinatorial equivalence classes—of the fan or the polytope which spans the fan in the strongly polytopal case.

Mirror Symmetry for Hypersurfaces in Weighted Projective Space and Topological Couplings

Abstract: By means of toric geometry we study hypersurfaces in weighted projective space of dimension four. In particular we compute for a given manifold its intrinsic topological coupling. We find that the result agrees with the calculation of the corresponding coupling on the mirror model in the large complex structure limit.

Real, complex, and quaternionic toric spaces

Abstract: In this thesis we give topological generalizations of complex toric varieties to the real numbers and quaternions. The resulting spaces, called, respectively, real toric spaces and quaternionic toric spaces, are characterized by a convex polytope P together with some algebraic data in the form of a characteristic function A on the faces of P . In all cases we discuss conditions for nonsingularity and compute the cohomology ring for these nonsingular examples in terms of P and A. The real part of a complex toric variety is the motivating example in the real case, and for these real varieties we give conditions on P for the existence of topological embeddings into real projective space. These conditions are shown to be weaker than those for projective embeddings of complex toric varieties. In contrast to the real and complex cases, quaternionic toric spaces can be topologically nonsingular but fail to be smooth. Examples which arise easily are the Thom spaces of Milnor's exotic 7-spheres (given as 3-sphere bundles over the 4-sphere). Focusing on dimension 2 (8 real dimensions), we study a certain class of smooth examples. These 8-manifolds, being 3-connected, are classified by their intersection forms and first Pontrjagin classes. Formulas are given for these invariants, as well as other characteristic numbers, in terms of the characteristic function A. Thesis Supervisor: Dr. Robert MacPherson Title: Professor of Mathematics

On the Betti numbers of the real part of a three-dimensional torus embedding

Abstract: and α1, . . . , αk are some primitive vectors in M (N). The dimension of σ is by definition the dimension of the linear space spanned by α1, . . . , αk in MR (NR). In this article we consider only convex, rational, polyhedral cones in NR which do not include any line. If σ ⊂ NR is a cone then the set σ̂ = {y ∈ MR : 〈x, y〉 ≥ 0} is also a cone. We call it the dual cone. The face of the cone σ is the set