[Paper Review] Equivariant $K$-theory of smooth projective spherical varieties

Soumya Banerjee, Mahir Bilen Can|arXiv (Cornell University)|Mar 16, 2016

Advanced Algebra and Geometry3 references3 citations

TL;DR

This paper provides an integral equivariant K-theory description for smooth projective spherical varieties, generalizing Brion's equivariant Chow cohomology results. It establishes a GKM-type presentation using congruence conditions on torus fixed points, with structure constants derived from Schubert classes in wonderful compactifications of minimal rank symmetric varieties, extending prior work on adjoint groups to broader spherical varieties.

ABSTRACT

We present a description of the equivariant $K$-theory of a smooth projective spherical variety. This provides an integral $K$-theory version of Brion's calculation of equivariant Chow-cohomology of such varieties. We consider the equivariant $K$-theory of wonderful compactifications of minimal rank symmetric varieties. We obtain a formula for their structure constants in terms of certain lower dimensional Schubert classes. This generalizes results of Uma on equivariant compactifications of adjoint groups.

Motivation & Objective

• To extend integral equivariant K-theory computations beyond adjoint groups to general smooth projective spherical varieties.
• To provide a GKM-type presentation for equivariant K-theory analogous to Brion’s rational Chow cohomology results.
• To resolve the challenge of non-T-skeletal spherical varieties with positive-dimensional families of torus-invariant curves.
• To generalize Uma’s results on wonderful compactifications of adjoint groups to minimal rank symmetric varieties.
• To establish a co-base change framework using toric geometry to reduce higher-dimensional problems to rank-one spherical SL₂ compactifications.

Proposed method

• The method reduces the equivariant K-theory computation to rank-one spherical SL₂ compactifications via classification results from Ahiezer and Brion.
• It uses a co-base change theorem (Proposition 2.7) to lift K-theory data from toric varieties to general spherical varieties.
• Congruence conditions on fixed-point tuples encode the K-theory ring: f_x - f_y ≡ 0 mod (1 - χ) for T-stable curves with weight χ.
• Higher-order congruences account for components isomorphic to ℙ², ℙ¹×ℙ¹, and ruled surfaces ℱₙ, with Weyl group symmetries encoded in the conditions.
• The Weyl group action is used to define G-equivariant K-theory as W-invariants of the T-equivariant theory.
• A commensurability argument links K-theory and Chow-theory presentations via completion and the equivariant Riemann-Roch isomorphism.

Experimental results

Research questions

• RQ1 How can one describe the integral equivariant K-theory of smooth projective spherical varieties beyond the group case?
• RQ2 What are the precise congruence conditions that define the equivariant K-theory ring for such varieties?
• RQ3 How do structure constants in the K-theory ring relate to Schubert classes in wonderful compactifications of minimal rank symmetric varieties?
• RQ4 Can the GKM-type framework be extended to non-T-skeletal spherical varieties with positive-dimensional families of torus-invariant curves?
• RQ5 To what extent does the equivariant Riemann-Roch isomorphism preserve commensurability between K-theory and Chow-theory presentations?

Key findings

• The T-equivariant K-theory of a smooth projective spherical G-variety X is isomorphic to the subring of tuples (f_x) in ∏_{x∈X^T} K_*(k)⊗R(T) satisfying specific congruence conditions on fixed points.
• The congruence conditions are determined by T-stable curves, ℙ², ℙ¹×ℙ¹, and ruled surfaces ℱₙ components in X^{Ker(χ)}, with Weyl group symmetries encoding the geometry.
• Structure constants in the K-theory ring of wonderful compactifications of minimal rank symmetric varieties are expressed in terms of lower-dimensional Schubert classes.
• The G-equivariant K-theory is obtained as the W-invariant subring of the T-equivariant K-theory, reflecting the Weyl group action on fixed points.
• A commensurable K-theory presentation implies a commensurable Chow-theory presentation, with finitely many relations determining the associated graded ring.
• The results generalize Uma’s work on adjoint groups and extend the GKM-type framework to all smooth projective spherical varieties, including non-T-skeletal cases.