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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 paper, the authors develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra, and give the toric quiver varieties, in the sense of Nakajima.
Abstract: Extending work of Bielawski-Dancer (3) and Konno (12), we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawrence toric varieties, meaning GIT quotients of even-dimensional affine spaces by symplectic torus actions. A toric hyperkahler variety is a complete intersection in a Lawrence toric variety. Both varieties are non- compact, and they share the same cohomology ring, namely, the Stanley-Reisner ring of a matroid modulo a linear system of parameters. Familiar applications of toric geometry to combinatorics, including the Hard Lefschetz Theorem and the volume polynomials of Khovanskii-Pukhlikov (10), are extended to the hyperkahler setting. When the matroid is graphic, our construction gives the toric quiver varieties, in the sense of Nakajima (15).
96 citations
TL;DR: In this paper, a new method to find gravity duals to a large class of three-dimensional Chern-Simons-matter theories, using techniques from dimer models, was proposed.
Abstract: We propose a new method to find gravity duals to a large class of three-dimensional Chern-Simons-matter theories, using techniques from dimer models. The gravity dual is given by M-theory on AdS4 × Y7, where Y7 is an arbitrary seven-dimensional toric Sasaki-Einstein manifold. The cone of Y7 is a toric Calabi-Yau 4-fold, which coincides with a branch of the vacuum moduli space of Chern-Simons-matter theories.
96 citations
TL;DR: In this article, it was shown that the equivariant Chow cohomology ring of a toric vari- ety is naturally isomorphic to the ring of integral piecewise polynomial functions on the associated fan.
Abstract: We show that the equivariant Chow cohomology ring of a toric vari- ety is naturally isomorphic to the ring of integral piecewise polynomial functions on the associated fan. This gives a large class of singular spaces for which lo- calization holds in equivariant Chow cohomology with integer coe!cients. We also compute the equivariant Chow cohomology of toric prevarieties and general complex hypertoric varieties in terms of piecewise polynomial functions. If X = X(!) is a smooth, complete complex toric variety then the follow- ing rings are canonically isomorphic: the equivariant singular cohomology ring H ! T (X), the equivariant Chow cohomology ring A ! (X), the Stanley-Riesner ring SR(!), and the ring of integral piecewise polynomial functions PP ! (!). If X is simplicial but not smooth then H ! (X) may have torsion and the natural map from SR(!) takes monomial generators to piecewise linear functions with ra- tional, but not necessarily integral, coe"cients. In such cases, these rings are not isomorphic, but they become isomorphic after tensoring with Q. When X is not simplicial, there are still natural maps between these rings, for instance from A ! (X)Q to H ! (X)Q and from H ! T (X) to PP ! (!), but these maps are far
95 citations
TL;DR: In this article, the authors give an explicit construction of a quasitoric representative for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection.
Abstract: Our aim is to bring the theory of analogous polytopes to bear on the study of quasitoric manifolds, in the context of stably complex manifolds with compatible torus action By way of application, we give an explicit construction of a quasitoric representative for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection We suggest a systematic description for omnioriented quasitoric manifolds in terms of combinatorial data, and explain the relationship with non-singular projective toric varieties (otherwise known as toric manifolds) By expressing the first and third authors' approach to the representability of cobordism classes in these terms, we simplify and correct two of their original proofs concerning quotient polytopes; the first relates to framed embeddings in the positive cone, and the second involves modifying the operation of connected sum to take account of orientations Analogous polytopes provide an informative setting for several of the details
94 citations
TL;DR: The notion of genericity for lattice ideals was introduced in this paper, which includes ideals defining toric varieties, where the generators are generic with respect to their exponents, not their coefficients.
Abstract: Let S = k[x1, . . . , xn] be a polynomial ring over a field k and I a homogeneous ideal in S. A basic problem in commutative algebra is to construct the minimal free resolution FI of S/I over S. The resolution is nicely structured and simple when I is a complete intersection: in this case FI is the Koszul complex. Complete intersections are ideals whose generators have sufficiently general coefficients, so they might be regarded as generic among all ideals. Yet there is another, entirely different, notion of genericity: ideals whose generators are generic with respect to their exponents – not their coefficients. This point of view was developed for monomial ideals in [BPS]. In the present work we introduce a notion of genericity for lattice ideals, which include ideals defining toric varieties. If L is any sublattice of Z, then its associated lattice ideal in S is IL := 〈xa − x : a,b ∈ N and a− b ∈ L 〉, where monomials are denoted x = x1 1 · · ·xan n for a = (a1, . . . , an). We call a lattice ideal IL generic if it is generated by binomials with full support, i.e.,
94 citations