The $G$-Noncommutative Minimal Model Program

TL;DR

This paper develops a $G$-equivariant noncommutative minimal model program to construct stability condition paths on derived categories with group actions, extending previous frameworks to equivariant settings.

Contribution

It introduces the $G$-NMMP framework, employs induction for finite groups, and defines $ ext{T}$-stability conditions to handle algebraic group actions.

Findings

Constructed quasi-convergent paths for finite groups using induction.

Reformulated stability conditions via $ ext{T}$-stability for algebraic groups.

Built paths for equivariant projective spaces from small quantum cohomology.

Abstract

In this paper, we study the $G$-equivariant noncommutative minimal model program ($G$-NMMP), as an equivariant generalization of the framework introduced in arXiv:2301.13168. The aim of this program is to construct quasi-convergent paths in the spaces of Bridgeland stability conditions on derived categories of $G$-equivariant coherent sheaves. For finite groups, we employ induction techniques to construct such paths from the non-equivariant setting. In the setting of algebraic group actions, we introduce the notion of $\mathbb T$-stability conditions to reformulate the proposal, and then we construct quasi-convergent paths for equivariant projective spaces from small quantum cohomology.