Gorenstein Fano Generic Torus Orbit closures in $G/P$

TL;DR

This paper characterizes when the closure of a generic torus orbit in a flag variety is a Gorenstein Fano toric variety, extending previous classifications and linking to reflexive polytopes.

Contribution

It provides a combinatorial criterion for Gorenstein Fano property of generic orbit closures in flag varieties, extending smooth cases and connecting to reflexive polytopes.

Findings

Characterization of Gorenstein Fano closures in combinatorial terms

Extension of smooth Fano classification to Gorenstein Fano cases

Identification of reflexive root polytopes in this context

Abstract

Given a reductive group $G$ and a parabolic subgroup $P\subset G$, with maximaltorus $T$, we consider (following Dabrowski's work) the closure $X$ of a generic $T$-orbit in $G/P$, and determine in combinatorial termswhen the toric variety $X$ is $\mathbb{Q}$-Gorenstein Fano, extending in this way the classification of smooth Fano generic closures given by Voskresenski\u{\i} and Klyachko. As an application, we apply the well known correspondence between Gorenstein Fano toric varieties and reflexive polytopes in order to exhibit which reflexive polytopes correspond to generic closures -- this list includes the reflexive root polytopes.