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: It is proved that in some cases the general fiber, which the authors christen a brick manifold, is a toric variety, and a nice description of the toric varieties of the associahedron is given.
Abstract: Bott-Samelson varieties are a twisted product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational to the image; however in this paper we study a fiber of this map when it is not birational. We prove that in some cases the general fiber, which we christen a brick manifold, is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into one in terms of the "subword complexes" of Knutson and Miller. Pilaud and Stump realized certain subword complexes as the dual of the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg, Lange and Thomas. These stories connect in a nice way: we show that the moment polytope of the brick manifold is the brick polytope. In particular, we give a nice description of the toric variety of the associahedron. We give each brick manifold a stratification dual to the subword complex. In addition, we relate brick manifolds to Brion's resolutions of Richardon varieties.
23 citations
TL;DR: In this article, an explicit stabilisation of all closed string moduli (including dilaton, complex structure and Kaehler moduli) in fluxed type IIB Calabi-Yau compactifications with chiral matter is studied.
Abstract: We address the open question of performing an explicit stabilisation of all closed string moduli (including dilaton, complex structure and Kaehler moduli) in fluxed type IIB Calabi-Yau compactifications with chiral matter. Using toric geometry we construct Calabi-Yau manifolds with del Pezzo singularities. D-branes located at such singularities can support the Standard Model gauge group and matter content. In order to control complex structure moduli stabilisation we consider Calabi-Yau manifolds which exhibit a discrete symmetry that reduces the effective number of complex structure moduli. We calculate the corresponding periods in the symplectic basis of invariant three-cycles and find explicit flux vacua for concrete examples. We compute the values of the flux superpotential and the string coupling at these vacua. Starting from these explicit complex structure solutions, we obtain AdS and dS minima where the Kaehler moduli are stabilised by a mixture of D-terms, non-perturbative and perturbative alpha'-corrections as in the LARGE Volume Scenario. In the considered example the visible sector lives at a dP_6 singularity which can be higgsed to the phenomenologically interesting class of models at the dP_3 singularity.
23 citations
TL;DR: Denef and Loeser as discussed by the authors proved that the Poincar\'{e} series associated with the image of a singular point in some suitable localization of the Grothendieck ring of algebraic varieties over a toric surface singularity is a rational function.
Abstract: Let $H$ denote the set of formal arcs going through a singular point of an algebraic variety $V$ defined over an algebraically closed field $k$ of characteristic zero. In the late sixties, J. Nash has observed that for any nonnegative integer $s$, the set $j^s(H)$ of $s$-jets of arcs in $H$ is a constructible subset of some affine space. Recently (1999), J. Denef and F. Loeser have proved that the Poincar\'{e} series associated with the image of $j^s(H)$ in some suitable localization of the Grothendieck ring of algebraic varieties over $k$ is a rational function. We compute this function for normal toric surface singularities.
23 citations
TL;DR: In this article, Song et al. developed a global Poincare residue formula to study period integrals of families of complex manifolds for any compact complex manifold $X$ equipped with a linear system of generically smooth CY hypersurfaces.
Abstract: We develop a global Poincare residue formula to study period integrals of families of complex manifolds For any compact complex manifold $X$ equipped with a linear system $V^*$ of generically smooth CY hypersurfaces, the formula expresses period integrals in terms of a canonical global meromorphic top form on $X$ Two important ingredients of our construction are the notion of a CY principal bundle, and a classification of such rank one bundles We also generalize our construction to CY and general type complete intersections When $X$ is an algebraic manifold having a sufficiently large automorphism group $G$ and $V^*$ is a linear representation of $G$, we construct a holonomic D-module that governs the period integrals The construction is based in part on the theory of tautological systems we have developed in the paper \cite{LSY1}, joint with R Song The approach allows us to explicitly describe a Picard-Fuchs type system for complete intersection varieties of general types, as well as CY, in any Fano variety, and in a homogeneous space in particular In addition, the approach provides a new perspective of old examples such as CY complete intersections in a toric variety or partial flag variety
22 citations
TL;DR: In this paper, the cohomological Brauer group of a normal toric variety with singular locus having codimension less than or equal to 2 everywhere is computed for a toric set.
Abstract: We compute the cohomological Brauer group of a normal toric variety whose singular locus has codimension less than or equal to 2 everywhere
22 citations