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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, it was shown that toric geometry can be used to translate a brane configuration to geometry and that the skeletons of toric space are identified with the brane configurations.
Abstract: We show that toric geometry can be used rather effectively to translate a brane configuration to geometry. Roughly speaking the skeletons of toric space are identified with the brane configurations. The cases where the local geometry involves hypersurfaces in toric varieties (such as P^2 blown up at more than 3 points) presents a challenge for the brane picture. We also find a simple physical explanation of Batyrev's construction of mirror pairs of Calabi-Yau manifolds using T-duality.

17 citations

Journal ArticleDOI
TL;DR: In this article, the integral homology, topological K-theory, and Hodge structure on cohomology of Calabi-Yau threefold hypersurfaces and semi-ample complete intersections in toric varieties associated with maximal projective triangulations of reflexive polytopes are computed.
Abstract: We compute the integral homology (including torsion), the topological K‐theory, and the Hodge structure on cohomology of Calabi‐Yau threefold hypersurfaces and semiample complete intersections in toric varieties associated with maximal projective triangulations of reflexive polytopes. The methods are purely topological. 14J32; 32Q25 One of the most fruitful sources of Calabi‐Yau threefolds is hypersurfaces, or more generally complete intersections, in toric varieties. This is especially true since there is a proposal for the mirror of any such Calabi‐Yau threefold. Usually the toric varieties associated to convex lattice polytopes are singular, causing the Calabi‐Yau threefolds in them also to be singular, so that to get smooth Calabi‐Yau threefolds we must resolve the ambient singularities and take the preimage in the resolution of the singular Calabi‐Yau threefold. This can be done torically by the combinatorial device of taking a triangulation of the boundary of the convex lattice polytope defining the toric variety where the vertices of the triangulation are exactly the lattice points contained in the boundary of the polytope. In general, there will be many such triangulations of a given lattice polytope, leading to different ambient resolutions producing different families of Calabi‐Yau threefolds associated with the original toric variety. In spite of the existence of many different such resolutions of a given singular object, there are lattice-theoretic formulas for the Hodge numbers of these resolutions expressed in terms of the lattice polytope and its polar. Thus, the Hodge numbers of all the different resolutions coming from different triangulations are the same. The proofs of these combinatorial formulas rely on the Griffiths‐Dwork method of computing Hodge numbers using residues of meromorphic differentials on the complement of the Calabi‐Yau threefolds in the toric ambient space (Batyrev and Borisov [1; 2]). The same methods were used in (Mavlyutov [12]) to describe the Hodge components in complex cohomology explicitly. In this paper we study the resolution process from a more topological point of view. In the topological study of these objects, one treats both complete intersections and

17 citations

Journal ArticleDOI
TL;DR: In this paper, the authors show that divisors contributing to the superpotential are always "exceptional" for the Calabi-Yau 4-fold X. In fact, this is a consequence of the (log-minimal model algorithm in dimension 4.

17 citations

Posted Content
TL;DR: In this article, the authors classify terminal Fano threefolds of Picard number one that come with an effective action of a two-torus and generalize the correspondence between toric Fano varieties and lattice polytopes, which leads, for example, to simple characterizations of terminality and canonicity.
Abstract: We classify the terminal Fano threefolds of Picard number one that come with an effective action of a two-torus. Our approach applies also to higher dimensions and generalizes the correspondence between toric Fano varieties and lattice polytopes: to any Fano variety with a complete intersection Cox ring we associate its "anti canonical complex", which is a certain polyhedral complex living in the lattice of one parameter groups of an ambient toric variety. For resolutions constructed via the tropical variety, the lattice points inside the anticanonical complex control the discrepancies. This leads, for example, to simple characterizations of terminality and canonicity.

17 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that all n-dimensional toric singularities with log-discrepancy greater than (or equal to) ϵ$ form finitely many "series".
Abstract: In 1988 S. Mori, D. Morrison, and I. Morrison gave a computer-based conjectural classification of four-dimensional cyclic quotient singularities of prime index. It was partially proven in 1990 by G. Sankaran. In 1991 Jim Lawrence basically proved this and much more, been unaware of algebro-geometric meaning of what he did. In this short note I just bring this all together to prove that all n-dimensional toric singularities with log-discrepancy greater than (or equal to) $\epsilon$ form finitely many "series".

17 citations


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