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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 studied the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it, and derived an inhomogeneous Picard-Fuchs equation for the Abel-Jacobi map.
Abstract: We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a Calabi-Yau hypersurface in toric variety, and deriving a general formula for the $d$-exact form on one side of the equation. We also derive a double residue formula, giving a purely algebraic way to compute the inhomogeneous Picard-Fuchs equations for the Abel-Jacobi map, which has played an important role in the recent study of D-branes (by Morrison and Walcher). Using the variation formalism, we prove that the relative periods of toric B-branes on Calabi-Yau hypersurface satisfy the enhanced GKZ-hypergeometric system proposed in physics literature (by Alim, Hecht, Mayr, and Mertens), and discuss the relations between several works in the recent study of open string mirror symmetry. We also give the general solutions to the enhanced hypergeometric system.
43 citations
TL;DR: In this article, the Abreu equation on convex labeled quadrilaterals was resolved and the existence of extremal toric orbi-surfaces with constant scalar curvature was shown.
Abstract: We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kahler-Einstein and toric Sasaki-Einstein metrics constructed in [6,22,14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including Kahler-Einstein ones, and show that for a toric orbi-surface with 4 fixed points of the torus action, the vanishing of the Futaki invariant is a necessary and sufficient condition for the existence of Kahler metric with constant scalar curvature. Our results also provide explicit examples of relative K-unstable toric orbi-surfaces that do not admit extremal metrics.
43 citations
TL;DR: In this paper, the concepts of toric geometry that are necessary to understand applications in the context of string and F-theory dualities have been introduced, based on the definition of a toric variety in terms of homogeneous coordinates.
Abstract: This note is supposed to be an introduction to those concepts of toric geometry that are necessary to understand applications in the context of string and F-theory dualities The presentation is based on the definition of a toric variety in terms of homogeneous coordinates, stressing the analogy with weighted projective spaces We try to give both intuitive pictures and precise rules that should enable the reader to work with the concepts presented here
43 citations
TL;DR: Chan et al. as mentioned in this paper proved that the inverse of a mirror map for a toric Calabi-Yau manifold of the form K Y, where Y is a compact toric Fano manifold, can be expressed in terms of generating functions of genus 0 open Gromov-Witten invariants defined by Fukaya-Oh-Ohta-Ono.
43 citations