Test Configurations, Large Deviations and Geodesic Rays on Toric Varieties

Lectures on Geodesics in the Space of Kaehler Metrics, Lecture 3: Hrma, Hcma and Toric Kaehler Manifolds

It is very hard to find computable examples of geodesics in the space of Kaehler metrics. It seems that they come in two types: (i) geodesics of toric Kaehler metrics, the topic of Lecture 2; (iii) geodesics coming from special test configurations and Hele-Shaw flows. Besides Donaldson’s construction (reviewed in §1) there are not many examples of geodesic rays. Almost the only other explicit e...

Notes on Toric Varieties from Mori Theoretic Viewpoint

The main purpose of this notes is to supplement the paper [Re], which treated Minimal Model Program (also called Mori’s Program) on toric varieties. We calculate lengths of negative extremal rays of toric varieties. As an application, we obtain a generalization of Fujita’s conjecture for singular toric varieties. We also prove that every toric variety has a small projective toric Q-factorializa...

On the Regularity of Geodesic Rays Associated to Test Configurations

Geodesic rays of class C1,1 are constructed for any test configuration of a positive line bundle L → X, using resolution of singularities. The construction reduces to finding a subsolution of the corresponding Monge-Ampère equation. Geometrically, this is accomplished by the use a positive line bundle on the resolution which is trivial outside of the exceptional divisor.

Test Configurations and Geodesic Rays

Restriction of A-discriminants and dual defect toric varieties

We study the A-discriminant of toric varieties. We reduce its computation to the case of irreducible configurations and describe its behavior under specialization of some of the variables to zero. We give characterizations of dual defect toric varieties in terms of their Gale dual and classify dual defect toric varieties of codimension less than or equal to four.