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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: Toric codes are obtained by evaluating rational functions of a nonsingular toric variety at the algebraic torus, and they can be extended to generalized toric codes as mentioned in this paper, which consists of evaluating elements of an arbitrary polynomial algebra at the torus instead of a linear combination of monomials whose exponents are rational points of a convex polytope.
43 citations
TL;DR: For a toric Calabi-Yau (CY) orbifold, a non-toric Lagrangian torus fibration on $\mathcal{X}$ was constructed and studied in this article.
Abstract: For a toric Calabi-Yau (CY) orbifold $\mathcal{X}$ whose underlying toric variety is semi-projective, we construct and study a non-toric Lagrangian torus fibration on $\mathcal{X}$, which we call the Gross fibration. We apply the Strominger-Yau-Zaslow (SYZ) recipe to the Gross fibration of $\mathcal{X}$ to construct its mirror with the instanton corrections coming from genus 0 open orbifold Gromov-Witten (GW) invariants, which are virtual counts of holomorphic orbi-disks in $\mathcal{X}$ bounded by fibers of the Gross fibration.
We explicitly evaluate all these invariants by first proving an open/closed equality and then employing the toric mirror theorem for suitable toric (partial) compactifications of $\mathcal{X}$. Our calculations are then applied to
(1) prove a conjecture of Gross-Siebert on a relation between genus 0 open orbifold GW invariants and mirror maps of $\mathcal{X}$ -- this is called the open mirror theorem, which leads to an enumerative meaning of mirror maps, and
(2) demonstrate how open (orbifold) GW invariants for toric CY orbifolds change under toric crepant resolutions -- an open analogue of Ruan's crepant resolution conjecture.
43 citations
TL;DR: In this article, the authors studied algebraic properties of the small quantum homology algebra for the class of symplectic toric Fano manifolds and provided an easily verifiable sufficient condition for these properties independent of the symplectic form.
Abstract: We study certain algebraic properties of the small quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semisimplicity of this algebra, and the more general property of containing a field as a direct summand. Our main result provides an easily verifiable sufficient condition for these properties which is independent of the symplectic form. Moreover, we answer two questions of Entov and Polterovich negatively by providing examples of toric Fano manifolds with non-semisimple quantum homology, and others in which the Calabi quasi-morphism is not unique.
43 citations
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TL;DR: In this paper, the Picard lattice of certain K3 surfaces is computed using toric geometry, lattice theory, and elliptic surface techniques, and the results appear in a multipage table near the end of the paper.
Abstract: Using toric geometry, lattice theory, and elliptic surface techniques, we compute the Picard Lattice of certain K3 surfaces. In particular, we examine the generic member of each of M. Reid's list of 95 families of Gorenstein K3 surfaces which occur as hypersurfaces in weighted projective 3-spaces. The results appear in a multipage table near the end of the paper. As an application, we are able to determine whether the mirror family (in the sense of mirror symmetry for K3 surfaces) for each one is also on Reid's list.
42 citations
42 citations