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Toric variety

About: Toric variety is a research topic. Over the lifetime, 2630 publications have been published within this topic receiving 65604 citations. The topic is also known as: torus embedding.


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TL;DR: In this article, the twisted sectors of a compact simplicial toric variety were identified and the same was done for a generic non-egenerate Calabi-Yau hypersurface of an n-dimensional simplicial Fano Toric variety.
Abstract: We identify the twisted sectors of a compact simplicial toric variety. We do the same for a generic nondegenerate Calabi-Yau hypersurface of an n-dimensional simplicial Fano toric variety and then explicitly compute h 1,1 orb and h n-2,1 orb for the hypersurface. We give applications to the orbifold string theory conjecture and orbifold mirror symmetry.

16 citations

Posted Content
TL;DR: In this article, the authors present a unified approach to proving the mixed Hodge-Riemann bilinear relations (HRR) and prove it in new cases such as intersection cohomology of non-rational polytopes.
Abstract: Statements analogous to the Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in a variety of contexts: they impose restrictions on the cohomology algebra of a smooth compact K\"ahler manifold or on the intersection cohomology of a projective toric variety; they restrict the local monodromy of a polarized variation of Hodge structure; they impose conditions on the possible $f$-vectors of convex polytopes. While the statements of these theorems depend on the choice of a K\"ahler class, or its analog, there is usually a cone of possible K\"ahler classes. It is then natural to ask whether the HLT and HRR remain true in a mixed context. In this note we present a unified approach to proving the mixed HLT and HRR, generalizing the previously known results, and proving it in new cases such as the intersection cohomology of non-rational polytopes.

16 citations

Journal ArticleDOI
TL;DR: Danilov et al. as mentioned in this paper showed that the Todd class of simplicial toric varieties has a canonical expression in terms of products of torus-invariant divisors.
Abstract: The purpose of this paper is to show that the Todd class of a simplicial toric variety has a canonical expression in terms of products of torus-invariant divisors. The coefficients in this expression, which are generalizations of the classical Dedekind sum, are shown to satisfy a reciprocity relation which characterizes them uniquely. We achieve these results by giving an explicit formula for the push-forward of a product of cycles under a proper birational map of simplicial toric varieties. Since the introduction of toric varieties in the 1970s, finding formulas for their Todd class has been an interesting and important problem. This is partly due to a well-known application of the Riemann-Roch theorem which allows a formula for the Todd class of a toric variety to be translated directly into a formula for the number of lattice points in a lattice polytope (cf. [Dan]). An example of this application is contained in [Pom], where a formula for the Todd class of a toric variety in terms of Dedekind sums is used to obtain new lattice point formulas. Danilov [Dan] posed the problem of finding a formula for the Todd class of a toric variety in terms of the orbit closures under the torus action. Specifically, he asked if it is possible, given a lattice, to assign a rational number to each cone in the lattice such that given any fan in the lattice, the Todd class of the corresponding toric variety equals the sum of the orbit closures with coefficients given by these assigned rational numbers. Morelli [Mor] showed that such an assignment is indeed possible in a natural way if the coefficients, instead of being rational numbers, are allowed to take values in the field of rational functions on a Grassmannian of linear subspaces of the lattice. However, if it is required that the coefficients be rational numbers invariant under lattice automorphisms, such an assignment is clearly impossible. For example, the nonsingular cone a in 22 generated by (1, 0) and (0, 1) when subdivided by the ray through (1, 1) yields two cones a, and o2 which are both lattice equivalent to a. By additivity, a consequence of the fact that the Todd class pushes forward, we deduce that the coefficient assigned to a must equal 0, which is absurd. In this paper, we show that there is a canonical expression for the Todd class of a simplicial toric variety in terms of products of the torus invariant divisors. Furthermore, this expression is invariant under lattice automorphisms. That is, the coefficient of each product depends only on the set of rays with multiplicities

16 citations

Posted Content
TL;DR: In this article, a volume for Cartier divisors on normal complex algebraic varieties of dimension greater than one with a distinguished point is studied. But this is not a generalization of the volume of isolated singularities.
Abstract: In this paper we study a notion of volume for Cartier divisors on arbitrary blow-ups of normal complex algebraic varieties of dimension greater than one, with a distinguished point. We apply this to study a volume for normal isolated singularities, generalizing work on surfaces done by Wahl. We also compare this volume of isolated singularities to a different generalization due to Boucksom, de Fernex and Favre.

16 citations

Posted Content
TL;DR: In this article, the authors presented a complete classification of pseudo-symmetric simplicial reflexive polytopes together with some applications, which generalizes a result of Ewald on pseudo symmetric nonsingular toric Fano varieties.
Abstract: A reflexive polytope, respectively its associated Gorenstein toric Fano variety, is called pseudo-symmetric, if the polytope has a centrally symmetric pair of facets. Here we present a complete classification of pseudo-symmetric simplicial reflexive polytopes together with some applications. This generalizes a result of Ewald on pseudo-symmetric nonsingular toric Fano varieties.

16 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202346
2022112
2021107
2020107
2019100
201894