Small toric resolutions of toric varieties of string polytopes with small indices

TL;DR

This paper investigates the combinatorics of string polytopes for $SL_{n+1}(b C)$, providing conditions for small toric resolutions of associated toric varieties, and explores implications for symplectic topology and Fano properties.

Contribution

It offers explicit constructions of small toric resolutions for string polytopes with small indices and demonstrates their integrality and Gorenstein Fano properties, extending to symplectic topology applications.

Findings

Explicit small toric resolutions using Bott manifolds for small index cases

String polytopes with small indices are integral for any dominant weight

Toric varieties of string polytopes are Gorenstein Fano when indices are small

Abstract

Let $G$ be a semisimple algebraic group over $\mathbb{C}$. For a reduced word $\bf i$ of the longest element in the Weyl group of $G$ and a dominant integral weight $\lambda$, one can construct the string polytope $\Delta_{\bf i}(\lambda)$, whose lattice points encode the character of the irreducible representation $V_{\lambda}$. The string polytope $\Delta_{\bf i}(\lambda)$ is singular in general and combinatorics of string polytopes heavily depends on the choice of $\mathbf i$. In this paper, we study combinatorics of string polytopes when $G = SL_{n+1}(\mathbb{C})$, and present a sufficient condition on $\mathbf i$ such that the toric variety $X_{\Delta_{\mathbf i}(\lambda)}$ of the string polytope $\Delta_{\mathbf i}(\lambda)$ has a small toric resolution. Indeed, when $\mathbf i$ has small indices and $\lambda$ is regular, we explicitly construct a small toric resolution of the toric variety $X_{\Delta_{\bf i}(\lambda)}$ using a Bott manifold. Our main theorem implies that a toric variety of any string polytope admits a small toric resolution when $n < 4$. As a byproduct, we show that if $\mathbf i$ has small indices then $\Delta_{\mathbf i}(\lambda)$ is integral for any dominant integral weight $\lambda$, which in particular implies that the anticanonical limit toric variety $X_{\Delta_{\bf i}(\lambda_P)}$ of a partial flag variety $G/P$ is Gorenstein Fano. Furthermore, we apply our result to symplectic topology of the full flag manifold $G/B$ and obtain a formula of the disk potential of the Lagrangian torus fibration on $G/B$ obtained from a flat toric degeneration of $G/B$ to the toric variety $X_{\Delta_{\bf i}(\lambda)}$.