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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: Using mirror pairs (M3, W3) in type II superstring compactifications on Calabi-Yau threefolds, this paper showed the duality between M-theory on S1 × M3/Z2 with G2 holonomy and F-theories on elliptically fibred Calabi Yau fourfolds with SU(4) holonomy.
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.

24 citations

Posted Content
TL;DR: In this paper, the authors characterize combinatorial objects corresponding to toric and spherical embeddings with group action, and construct an example of a smooth toric variety under a 3-dimensional nonsplit torus over $k$ whose fan is Galois-stable but which admits no $k-form.
Abstract: We are interested in two classes of varieties with group action, namely toric varieties and spherical embeddings. They are classified by combinatorial objects, called fans in the toric setting, and colored fans in the spherical setting. We characterize those combinatorial objects corresponding to varieties defined over an arbitrary field $k$. Then we provide some situations where toric varieties over $k$ are classified by Galois-stable fans, and spherical embeddings over $k$ by Galois-stable colored fans. Moreover, we construct an example of a smooth toric variety under a 3-dimensional nonsplit torus over $k$ whose fan is Galois-stable but which admits no $k$-form. In the spherical setting, we offer an example of a spherical homogeneous space $X_0$ over $\mr$ of rank 2 under the action of SU(2,1) and a smooth embedding of $X_0$ whose fan is Galois-stable but which admits no $\mr$-form.

24 citations

Posted Content
TL;DR: In this article, it was shown that there is a fully faithful embedding of the perfect derived category of a proper toric variety into the derived categories of constructible sheaves on a compact torus.
Abstract: We prove the following result of Bondal's: that there is a fully faithful embedding $\kappa$ of the perfect derived category of a proper toric variety into the derived category of constructible sheaves on a compact torus. We compare this result to a torus-equivariant version considered in joint work with Fang, Liu, and Zaslow. There we showed that in the torus-equivariant version the image of the embedding is cut out by microlocal conditions. To establish a similar characterization of the image of $\kappa$ is an open problem.

24 citations

Posted Content
TL;DR: In this paper, it was shown that the moduli spaces of spatial polygons degenerate to polarized toric varieties with the moment polytopes defined by the lengths of their diagonals.
Abstract: We show that a weight variety, which is a quotient of a flag variety by the maximal torus, admits a flat degeneration to a toric variety. In particular, we show that the moduli spaces of spatial polygons degenerate to polarized toric varieties with the moment polytopes defined by the lengths of their diagonals. We extend these results to more general Flaschka-Millson hamiltonians on the quotients of products of projective spaces. We also study boundary toric divisors and certain real loci.

24 citations

Journal ArticleDOI
TL;DR: In this article, the de Rham cohomology classes represented by the Kahler form and the complex symplectic form were studied and the variation of its complex structure according to these parameters.
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.

24 citations


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