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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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Journal ArticleDOI
TL;DR: In this article, it was shown that the coordinate ring of the quantum Grassmannian is an algebra with straightening law, which is normal, Cohen-Macaulay, and Koszul, and the ideal of quantum Plucker relations has a quadratic Grobner basis.

38 citations

Journal ArticleDOI
TL;DR: In this article, an analog of compactified moduli of abelian varieties and toric pairs in the case of non-commutative algebraic groups G is presented.
Abstract: The motivation of this work is to construct an analog of compactified moduli of abelian varieties and toric pairs in the case of non-commutative algebraic group G. We introduce a class of "stable reductive varieties" which contain connected reductive groups and their equivariant compactifications, and is closed under flat reduced degenerations. We classify them all, describe their degenerations, and establish a connection between these varieties and "reductive semigroups" which we also define. Finally, we construct a Hilbert scheme of embedded G-varieties by applying and generalizing a construction of Haiman and Sturmfels. The second version adds some cosmetic changes.

37 citations

Posted Content
TL;DR: In this paper, the authors constructed a free action of the group R m n on the complement of a simple polytope P n with m codimension-one faces.
Abstract: An n-dimensional polytope P n is called simple if exactly n codimension-one faces meet at each vertex. The lattice of faces of a sim- ple polytope P n with m codimension-one faces defines an arrangement of even-dimensional planes in R 2m . We construct a free action of the group R m n on the complement of this arrangement. The corresponding quotient is a smooth manifold ZP invested with a canonical action of the compact torus T m with the orbit space P n . For each smooth projective toric variety M 2n defined by a simple polytope P n with the given lattice of faces there exists a subgroup T m nT m acting freely on ZP such that ZP /T m n = M 2n . We calculate the cohomology ring of ZP and show that it is isomorphic to the cohomology ring of the face ring of P n regarded as a module over the polynomial ring. In this way the cohomology of ZP acquires a bigraded al- gebra structure, and the additional grading allows to catch the combinatorial invariants of the polytope. At the same time this gives an example of explicit calculation of the cohomology ring for the complement of an arrangement of planes, which is of independent interest.

37 citations

Journal ArticleDOI
TL;DR: In this paper, a detailed study of the case of a toric variety of the geodesic rays ϕt defined by Phong and Sturm corresponding to test configurations T in the sense of Donaldson is presented, and the connection between Bergman metrics, Bergman kernels and the theory of large deviations is made.

37 citations

BookDOI
15 Aug 2012
TL;DR: Equivariant cohomology as mentioned in this paper encodes information about how the topology of a space interacts with a group action, and has found many applications in enumerative geometry, Gromov-Witten theory, and the study of toric varieties and homogeneous spaces.
Abstract: Introduced by Borel in the late 1950’s, equivariant cohomology encodes information about how the topology of a space interacts with a group action. Quite some time passed before algebraic geometers picked up on these ideas, but in the last twenty years, equivariant techniques have found many applications in enumerative geometry, Gromov-Witten theory, and the study of toric varieties and homogeneous spaces. In fact, many classical algebro-geometric notions, going back to the degeneracy locus formulas of Giambelli, are naturally statements about certain equivariant cohomology classes. These lectures survey some of the main features of equivariant cohomology at an introductory level. The first part is an overview, including basic definitions and examples. In the second lecture, I discuss one of the most useful aspects of the theory: the possibility of localizing at fixed points without losing information. The third lecture focuses on Grassmannians, and describes some recent “positivity” results about their equivariant cohomology rings.

37 citations


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