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: In this paper, the authors construct global F-theory GUT models on del Pezzo surfaces in compact Calabi-Yau fourfolds realized as complete intersections of two hypersurface constraints, which allow for the phenomenologically relevant Yukawa couplings and GUT breaking to the MSSM via hypercharge flux while preventing dimension-4 proton decay.
271 citations
TL;DR: The operational Chow cohomology classes of a complete toric variety are identified with certain functions, called Minkowski weights, on the corresponding fan as discussed by the authors, and the natural product of these functions makes the Minkowowski weights into a commutative ring; the product is computed by a displacement in the lattice, which corresponds to a deformation in the toric manifold.
252 citations
TL;DR: In this article, a 1-parameter family of extremal Kahler metrics of non-constant scalar curvature on convex polytopes is recast using Guillemin's approach.
Abstract: A (symplectic) toric variety X, of real dimension 2n, is completely determined by its moment polytope Δ ⊂ ℝn Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on X, using only data on Δ In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction In particular, a nice combinatorial formula for the scalar curvature R is given, and the Euler–Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is proven to be R being an affine function on Δ ⊂ ℝn A construction, due to Calabi, of a 1-parameter family of extremal Kahler metrics of non-constant scalar curvature on is recast very simply and explicitly using Guillemin's approach Finally, we present a curious combinatorial identity for convex polytopes Δ ⊂ ℝn that follows from the well-known relation between the total integral of the scalar curvature of a Kahler metric and the wedge product of the first Chern class of the underlying complex manifold with a suitable power of the Kahler class
248 citations
TL;DR: The basic theory of toric dynamical systems is developed in the context of computational algebraic geometry and it is shown that the associated moduli space is a toric variety, which has a unique point within each invariant polyhedron.
242 citations
TL;DR: In this paper, it was shown that rational curves on a complete toric variety which are in general position relative to the toric prime divisors coincide with the counting of certain tropical curves.
Abstract: We show that the counting of rational curves on a complete toric variety which are in general position relative to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraic-geometric and relies on degeneration techniques and log deformation theory
240 citations