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.

Twisted polytope sheaves and coherent–constructible correspondence for toric varieties

Abstract: Given a smooth projective toric variety $$X_\Sigma $$ of complex dimension n, Fang–Liu–Treumann–Zaslow (Invent Math 186(1):79–114, 2011) showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves $$Coh(X_\Sigma )$$ into the dg derived category of constructible sheaves on a torus $$Sh(T^n, \Lambda _\Sigma )$$ . Recently, Kuwagaki (The nonequivariant coherent-constructible correspondence for toric stacks, 2016. arXiv:1610.03214 ) proved that the quasi-embedding is a quasi-equivalence, and generalized the result to toric stacks. Here we give a different proof in the smooth projective case, using non-characteristic deformation of sheaves to find twisted polytope sheaves that co-represent the stalk functors.

Qubits from Adinkra Graph Theory via Colored Toric Geometry

Abstract: We develop a new approach to deal with qubit information systems using toric geometry and its relation to Adinkra graph theory. More precisely, we link three different subjects namely toric geometry, Adinkras and quantum information theory. This one to one correspondence may be explored to attack qubit system problems using geometry considered as a powerful tool to understand modern physics including string theory. Concretely, we examine in some details the cases of one, two, and three qubits, and we find that they are associated with \bf CP^1, \bf CP^1\times CP^1 and \bf CP^1\times CP^1\times CP^1 toric varieties respectively. Using a geometric procedure referred to as colored toric geometry, we show that the qubit physics can be converted into a scenario handling toric data of such manifolds by help of Adinkra graph theory. Operations on toric information can produce universal quantum gates.

Extremal metrics on toric surfaces, I

Abstract: The paper develops a continiuty method for solutions of the Abreu equation, which include extremal metrics on toric surfaces. Results are obtained, assuming a hypothesis (the "M-condition") on the solutions.

The algebraic de Rham theorem for toric varieties

Abstract: On an arbitrary toric variety, we introduce the logarithmic double complex, which is essentially the same as the algebraic de Rham complex in the nonsingular case, but which behaves much better in the singular case. Over the field of complex numbers, we prove the toric analog of the algebraic de Rham theorem which Grothendieck formulated and proved for general nonsingular algebraic varieties re-interpreting an earlier work of Hodge-Atiyah. Namely, for a finite simplicial fan which need not be complete, the complex cohomology groups of the corresponding toric variety as an analytic space coincide with the hypercohomology groups of the single complex associated to the logarithmic double complex. They can then be described combinatorially as Ishida's cohomology groups for the fan. We also prove vanishing theorems for Ishida's cohomology groups. As a consequence, we deduce directly that the complex cohomology groups vanish in odd degrees for toric varieties which correspond to finite simplicial fans with full-dimensional convex support. In the particular case of complete simplicial fans, we thus have a direct proof for an earlier result of Danilov and the author.