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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 proved an explicit formula for the sum of the Grothendieck residues of the form ω at all roots of the system of equations, where ω is any rational nform which is regular on (C \\ 0) outside the hypersurface P1... Pn = 0.
Abstract: We consider a system of n algebraic equations P1 = · · · = Pn = 0 in the space (C/0). It is assumed that the Newton polytopes of the equations are in a sufficiently general position with respect to one another. Let ω be any rational nform which is regular on (C \\ 0) outside the hypersurface P1 . . . Pn = 0. Formerly we have announced an explicit formula for the sum of the Grothendieck residues of the form ω at all roots of the system of equations. In the present paper this formula is proved.
33 citations
TL;DR: The Mirror Symmetry phenomenon can be interpreted as the following twofold characterization of the generalized hypergeometric series as mentioned in this paper : 1) the power expansion near a boundary point of the moduli space of the monodromy invariant period of the holomorphic differential differential $d$-form on an another Calabi-Yau manifold, which is called Mirror of $V'$ which is defined by intersection numbers of rational curves in ${\bf P}_{\Sigma}$ with the hypersurfaces and their toric degenerations.
Abstract: We formulate general conjectures about the relationship between the A-model connection on the cohomology of a $d$-dimensional Calabi-Yau complete intersection $V$ of $r$ hypersurfaces $V_1, \ldots, V_r$ in a toric variety ${\bf P}_{\Sigma}$ and the system of differential operators annihilating the special hypergeometric function $\Phi_0$ depending on the fan $\Sigma$. In this context, the Mirror Symmetry phenomenon can be interpreted as the following twofold characterization of the series $\Phi_0$. First, $\Phi_0$ is defined by intersection numbers of rational curves in ${\bf P}_{\Sigma}$ with the hypersurfaces $V_i$ and their toric degenerations. Second, $\Phi_0$ is the power expansion near a boundary point of the moduli space of the monodromy invariant period of the holomorphic differential $d$-form on an another Calabi-Yau $d$-fold $V'$ which is called Mirror of $V$. Using the generalized hypergeometric series, we propose a general construction for Mirrors $V'$ of $V$ and canonical $q$-coordinates on the moduli spaces of Calabi-Yau manifolds.
33 citations
TL;DR: The obstruction space T2 and the cup product T1 × T1 → T2 for toric singularities were computed in this article, where the obstruction space was shown to be the same as the cup space T1.
33 citations
TL;DR: In this paper, the authors classify the torus orbit closures in an arbitrary algebraic homogeneous space G/P that are toric varieties, i.e., the closure of a torus in an algebraic space is toric.
33 citations
TL;DR: In this paper, a mirror construction for monomial degenerations of Calabi-Yau varieties in toric Fano varieties was proposed using tropical geometry, and the construction reproduces the mirror constructions by Batyrev and Borisov for complete intersections.
Abstract: Using tropical geometry we propose a mirror construction for monomial degenerations of Calabi-Yau varieties in toric Fano varieties. The construction reproduces the mirror constructions by Batyrev for Calabi-Yau hypersurfaces and by Batyrev and Borisov for Calabi-Yau complete intersections. We apply the construction to Pfaffian examples and recover the mirror given by Rodland for the degree 14 Calabi-Yau threefold in PP^6 defined by the Pfaffians of a general linear 7x7 skew-symmetric matrix.
We provide the necessary background knowledge entering into the tropical mirror construction such as toric geometry, Groebner bases, tropical geometry, Hilbert schemes and deformations. The tropical approach yields an algorithm which we illustrate in a series of explicit examples.
33 citations