Smooth Fano polytopes arising from finite directed graphs

TL;DR

This paper explores how finite directed graphs can generate smooth Fano polytopes, providing new examples of such polytopes with Kähler--Einstein metrics and characterizing those arising from symmetric graphs.

Contribution

It introduces a graph-theoretic approach to constructing smooth Fano polytopes and identifies classes of graphs that produce polytopes with special geometric properties.

Findings

Centrally symmetric smooth Fano polytopes can be derived from directed graphs.

New examples of smooth Fano polytopes with Kähler--Einstein metrics are constructed.

Characterization of graphs that yield smooth Fano polytopes.

Abstract

In this paper, we consider terminal reflexive polytopes arising from finite directed graphs and study the problem of deciding which directed graphs yield smooth Fano polytopes. We show that any centrally symmetric or pseudo-symmetric smooth Fano polytopes can be obtained from directed graphs. Moreover, by using directed graphs, we provide new examples of smooth Fano polytopes whose corresponding varieties admit K\"ahler--Einstein metrics.