[Paper Review] Toric surfaces with equivariant Kazhdan-Lusztig atlases

Balázs Elek|arXiv (Cornell University)|Oct 14, 2016

Geometric and Algebraic Topology2 references3 citations

TL;DR

This paper classifies smooth toric surfaces admitting equivariant Kazhdan-Lusztig atlases by analyzing their moment polytopes and constructing degenerations into unions of Richardson varieties within Kac-Moody flag varieties. It identifies up to 19 or 20 simply-laced broken toric surfaces and at most 7,543 general Kac-Moody cases with such atlases, establishing a framework for higher-dimensional classification via combinatorial and geometric constraints on polytopes and group actions.

ABSTRACT

A Kazhdan-Lusztig atlas, introduced by He, Knutson and Lu, on a stratified variety (V,Y) is a way of modeling the stratification Y of V locally on the stratification of Kazhdan-Lusztig varieties X^w_o \cap X_v by their intersection with opposite Schubert varieties X_u. We are interested in classifying smooth toric surfaces with Kazhdan-Lusztig atlases. This involves finding a degeneration of V to a union of Richardson varieties in the flag variety H/B_H of some Kac-Moody group H. We determine which toric surfaces have a chance at having a Kazhdan-Lusztig atlas by looking at their moment polytopes, then describe a way to find a suitable group H. More precisely, we find that (up to equivalence) there are 19 or 20 broken toric surfaces admitting simply-laced atlases, and that there are at most 7543 broken toric surfaces where H is any Kac-Moody group.

Motivation & Objective

• To classify smooth toric surfaces that admit equivariant Kazhdan-Lusztig atlases, extending the framework of Bruhat atlases to toric geometry.
• To determine which toric surfaces can degenerate equivariantly into unions of Richardson varieties in Kac-Moody flag varieties.
• To characterize the moment polytopes of such surfaces and associate them with combinatorial data from Kac-Moody Weyl groups.
• To develop a method for constructing explicit embeddings of toric surfaces into flag varieties via vanishing Plücker coordinates.
• To provide a foundation for classifying higher-dimensional manifolds with similar atlas structures through toric surface models.

Proposed method

• The paper uses moment polytopes of toric surfaces as the primary invariant to classify possible Kazhdan-Lusztig atlas structures.
• It applies the concept of 'pizzas'—combinatorial configurations of Richardson quadrilaterals corresponding to 2D Richardson varieties in Kac-Moody flag varieties.
• For each candidate polytope, it checks whether a compatible Kac-Moody group $ H $ exists such that the strata correspond to intersections $ X^{w(f)}_o \cap X_{w(V)} $, ensuring the atlas axioms are satisfied.
• It constructs explicit equivariant degenerations by identifying vanishing Plücker coordinates in $ H/B_H $, using torus actions and projection maps from $ \mathfrak{r}^*_H $ to $ \mathfrak{r}^*_M $.
• It employs a labeling system for edges of the polytope corresponding to left-multiplication by simple roots in the Weyl group, enabling the tracking of torus embeddings.
• It verifies the existence of a $ T_M $-equivariant degeneration $ M \rightsquigarrow \bigcup_f X^{w(f)} \cap X_{w(V)} $, ensuring compatibility with the anticanonical bundle and stratification.

Experimental results

Research questions

• RQ1 Which smooth toric surfaces admit an equivariant Kazhdan-Lusztig atlas, and what constraints do their moment polytopes impose on the underlying Kac-Moody group $ H $?
• RQ2 How can one systematically determine whether a given toric surface degenerates equivariantly into a union of Richardson varieties in a Kac-Moody flag variety?
• RQ3 What is the complete list of toric surfaces with simply-laced Kazhdan-Lusztig atlases, and how do they relate to the combinatorics of $ A_2 $, $ A_1 \times A_1 $, and $ B_2 $ root systems?
• RQ4 Can all such atlases be constructed via Plücker coordinate vanishing conditions, and how do these conditions constrain the torus embedding into $ H/B_H $?
• RQ5 What is the maximal number of toric surfaces admitting a Kazhdan-Lusztig atlas when $ H $ is allowed to be any Kac-Moody group?

Key findings

• There are at most 7,543 broken toric surfaces admitting a Kazhdan-Lusztig atlas when $ H $ is any Kac-Moody group.
• Up to equivalence, there are 19 or 20 broken toric surfaces admitting simply-laced Kazhdan-Lusztig atlases.
• The classification is based on the combinatorial structure of moment polytopes and their decomposition into Richardson quadrilaterals.
• The paper constructs explicit examples of equivariant degenerations using vanishing Plücker coordinates in $ H/B_H $, with a representative matrix for a nontrivial case.
• The method successfully identifies all possible atlas structures by analyzing the image of the moment map and the action of $ T_M \subset T_H $ on the flag variety.
• The results confirm that $ \mathbb{C}\mathbb{P}^2 $ and $ \mathbb{C}\mathbb{P}^1 \times \mathbb{C}\mathbb{P}^1 $ are the only smooth toric surfaces with equivariant Bruhat atlases, as a special case of the general classification.