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.

Test configurations and Okounkov bodies

Abstract: We associate to a test configuration of an ample line bundle a filtration of the section ring of the line bundle. Using the recent work of Boucksom-Chen we get a concave function on the Okounkov body whose law with respect to Lebesgue measure determines the asymptotic distribution of the weights of the test configuration. We show that this is a generalization of a well-known result in toric geometry. As an application, we prove that the pushforward of the Lebesgue measure on the Okounkov body is equal to a Duistermaat-Heckman measure of a certain deformation of the manifold. Via the Duisteraat-Heckman formula, we get as a corollary that in the special case of an effective $\mathbb{C^{\times}}$-action on the manifold lifting to the line bundle, the pushforward of the Lebesgue measure on the Okounkov body is piecewise polynomial.

On moduli spaces of quiver representations associated with dimer models (Higher Dimensional Algebraic Varieties and Vector Bundles)

Abstract: We give a sufficient condition for the moduli space of quiver representations associated with a dimer model to be smooth for a general stability parameter. We also show that the moduli space in this case is a crepant resolution of the toric variety determined by the Newton polygon of the characteristic polynomial.

K-Theory of Algebraic Tori and Toric Varieties

Abstract: It is proved that under certain conditions the group Kn(X) of a smooth projective variety X over a field F is a natural direct summand of Kn(A) for some separable F -algebra. As an application we study the K-groups of toric models and toric varieties. A presentation in terms of generators and relations of the group K0(T ) for an algebraic torus T is given. In [14] the second author has computed K-groups of homogeneous projective varieties. It was proved that for a variety X in this class defined over a field F there exists a separable F -algebra A and a natural isomorphism Kn(X) ' Kn(A). In fact, this isomorphism was obtained by applying the “K-theory functor” to an isomorphism X ' A is a certain motivic category C “containing” varieties and algebras. In the present paper we find certain sufficient conditions under which for a variety X over a field F there exists a separable algebra A and morphisms u : X → A and v : A→ X in C such that the composition X u → A v → X is the identity, i.e X is a “direct summand” of A in C. As an application we consider K-groups of toric models and toric varieties. The paper is organized as follows. In the first section we define the motivic category C, the main technical tool. We prove that there are natural functors i from the category of smooth projective F -varieties and j from the category of separable F -algebras to the motivic category C and also the functor from Support from Alexander von Humboldt-Stiftung is gratefully acknowledged The second author thanks the Soros Fund for support

Optimal test-configurations for toric varieties

Abstract: On a K-unstable toric variety we show the existence of an optimal destabilising convex function. We show that if this is piecewise linear then it gives rise to a decomposition into semistable pieces analogous to the Harder-Narasimhan filtration of an unstable vector bundle. We also show that if the Calabi flow exists for all time on a toric variety then it minimizes the Calabi functional. In this case the infimum of the Calabi functional is given by the supremum of the normalized Futaki invariants over all destabilising test-configurations, as predicted by a conjecture of Donaldson.