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.

Which 4-manifolds are toric varieties?

Abstract: Let 2; be a complete d-dimensional fan, i.e. a complex of closed convex polyhedral cones in R d with apex 0, generated by primitive lattice points Vl . . . . . v, ~ Z d, such that ~ ) ~ a = R d. Denote the q-skeleton of 2; by S ~ = { a ~ X l d i m a = q}. The polyhedral cells obtained by intersecting each cone a ~ X, a 4: {0}, with the unit sphere S d l c R d form a spherical complex C. Let S(C) be the barycentric subdivision of C and for a e Z, a 4: {0}, let # be the union of all simplices of S(C) whose vertices are barycenters of cells of C which contain a n S d-1 For a = {0} we set # = B d, the unit ball of R d, and call 2~ = { # l a s S } the dual complex of 2;. Let T d denote the d-dimensional torus Rd/Z d. Each k-dimensional cone a EZ spans a k-dimensional subspace of R d which, since it has a rational basis, maps under the projection R a ~ T d to a k-dimensional subtorus T,~ ~ T d. The toric variety X z is defined to be the. quotient Bax Td/ where (x, t ) ~ (x', t ') if and only if x = x ' e i n t # and t and t' belong to the same orbit of the natural action of T~ on T d. Note that the torus T d over a point x e int # is collapsed by the relation ~ to the quotient 7~, = Td/T,, which is a torus of dimension d dim a = dim ~. We denote by p the obvious projection of X~ onto B d and will identify p 1 (int ~) with int ~ x 7~. The proofs of the following properties can be found e.g. in I-2].

Branes at generalized conifolds and toric geometry.

Abstract: We use toric geometry to investigate the recently proposed relation between a set of D3 branes at a generalized conifold singularity and type IIA configurations of D4 branes stretched between a number of relatively rotated NS5 branes. In particular we investigate how various resolutions of the singularity corresponds to moving the NS branes and how Seiberg's duality is realized when two relatively rotated NS-branes are interchanged.

Fonction zêta des hauteurs des variétés toriques non déployées

Abstract: The author investigates the anticanonical height zeta function of a (not necessarily split) toric variety defined over a global field of positive characteristic, drawing inspiration from the method used by Batyrev and Tschinkel to deal with the analogous problem over a number field. The author includes a detailed account of their method.