Topic
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, a method to construct a partial embedded resolution of a nonnecessarily normal affine toric variety Z(Gamma) equivariantly embedded in a normal toric domain Z(rho) is presented.
25 citations
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TL;DR: In this article, it was shown that the stable tropical intersection along a (possibly positive-dimensional, possibly unbounded) connected component of Trop(X) of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, lifts to algebraic intersection points, with multiplicities.
Abstract: We prove that if X, X' are closed subschemes of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, then the stable tropical intersection along a (possibly positive-dimensional, possibly unbounded) connected component C of Trop(X) \cap Trop(X') lifts to algebraic intersection points, with multiplicities. This theorem requires potentially passing to a suitable toric variety X(\Delta) and its associated extended tropicalization N_R(\Delta); the algebraic intersection points lifting the stable tropical intersection will have tropicalization somewhere in the closure of C in N_R(\Delta). The proof involves a result on continuity of intersection numbers in the context of non-Archimedean analytic spaces.
24 citations
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TL;DR: In this paper, the authors carried out the SYZ program for local Calabi-Yau manifolds of type A by developing an equivariant SYZ theory for the toric Calabi Yau manifold of infinite-type.
24 citations
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01 Jan 1989TL;DR: In this article, a general theory of embeddings of homogeneous spaces has been developed, which works especially well in the so-called spherical case: G is reductive connected and a Borei subgroup of G has a dense orbit in G/H.
Abstract: When one studies complex algebraic homogeneous spaces it is natural to begin with the ones which are complete (i.e. compact) varieties. They are the “generalized flag manifolds”. Their occurence in many problems of representation theory, algebraic geometry, … make them an important class of algebraic varieties. In order to study a noncompact homogeneous space G/H, it is equally natural to compactify it, i.e. to embed it (in a G- equivariant way) as a dense open set of a complete G-variety. A general theory of embeddings of homogeneous spaces has been developed by Luna and Vust [LV]. It works especially well in the so-called spherical case: G is reductive connected and a Borei subgroup of G has a dense orbit in G/H. (This class includes complete homogeneous spaces as well as algebraic tori and symmetric spaces). A nice feature of a spherical homogeneous space is that any embedding of it (called a spherical variety) contains only finitely many G-orbits, and these are themselves spherical. So we can hope to describe these embeddings by combinatorial invariants, and to study their geometry. I intend to present here some results and questions on the geometry (see [LV], [BLV], [BP], [Lun] for a classification of embeddings).
24 citations
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TL;DR: In this paper, the authors provided a detailed description of the topology of Gelfand-Cetlin fibers and proved that any fiber over an interior point of a k-dimensional face of the Gelfandscetlin polytope is an isotropic submanifold and is diffeomorphic to some smooth manifold.
Abstract: A Gelfand-Cetlin system is a completely integrable system defined on a partial flag manifold whose image is a rational convex polytope called a Gelfand-Cetlin polytope. Motivated by the study of Nishinou-Nohara-Ueda on the Floer theory of Gelfand-Cetlin systems, we provide a detailed description of topology of Gelfand-Cetlin fibers. In particular, we prove that any fiber over an interior point of a k-dimensional face of the Gelfand-Cetlin polytope is an isotropic submanifold and is diffeomorphic to $(S^1)^k \times N$ for some smooth manifold $N$. We also prove that such $N$'s are exactly the vanishing cycles shrinking to points in the associated toric variety via the toric degeneration. We also devise an algorithm of reading off Lagrangian fibers from the combinatorics of the ladder diagram.
24 citations