On the body of ample angles of asymptotically log Fano varieties

TL;DR

This paper studies the geometric properties of asymptotically log Fano varieties, showing that their bodies of ample angles are rational polytopes and reducing classification problems in dimension two.

Contribution

It establishes the rationality of the body of ample angles in any dimension and simplifies the classification of asymptotically log Fano pairs in dimension two.

Findings

Bodies of ample angles are always rational polytopes.

Classification in dimension two reduces to generality conditions on blow-ups.

Proves rationality of these convex bodies in all dimensions.

Abstract

In dimension two, we reduce the classification problem for asymptotically log Fano pairs to the problem of determining generality conditions on certain blow-ups. In any dimension, we prove the rationality of the body of ample angles of an asymptotically log Fano pair, i.e., these convex bodies are always rational polytopes.