On Barycentric transformations of Fano polytopes

TL;DR

This paper introduces barycentric transformations for Fano polytopes to classify their proximity to being Kähler-Einstein, verifying the classification in low-dimensional cases and establishing a precise criterion for certain dimensions.

Contribution

It defines a new barycentric transformation-based classification for Fano polytopes and links this classification to their Kähler-Einstein property in specific dimensions.

Findings

Fano polytopes can be assigned a type indicating their closeness to Kähler-Einstein condition.

In dimensions 1, 3, and 5, being Kähler-Einstein is equivalent to being of type B_infinity.

The classification aligns with known properties in low-dimensional cases.

Abstract

We introduce the notion of barycentric transformation of Fano polytopes, from which we can assign a certain type to each Fano polytope. The type can be viewed as a measure of the extent to which the given Fano polytope is close to be K\"ahler-Einstein. In particular, we expect that every K\"ahler-Einstein or symmetric Fano polytope is of type $B_\infty$. We verify this expectation for some low dimensional cases. We emphasize that for a Fano polytope $X$ of dimension $1$, $3$ or $5$, $X$ is K\"ahler-Einstein if and only if it is of type $B_\infty$.