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The $G$-Noncommutative Minimal Model Program

Dongjian Wu, Nantao Zhang

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.

The $G$-Noncommutative Minimal Model Program

Abstract

In this paper, we study the -equivariant noncommutative minimal model program (-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 -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 -stability conditions to reformulate the proposal, and then we construct quasi-convergent paths for equivariant projective spaces from small quantum cohomology.
Paper Structure (24 sections, 30 theorems, 236 equations)

This paper contains 24 sections, 30 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main proposal
  4. Main results
  5. Contents
  6. Preliminaries
  7. Bridgeland stability conditions
  8. G-equivarinat triangulated categories
  9. Semiorthogonal decompositions and $G$-exceptional collections
  10. $G$-noncommutative minimal model program
  11. Quasi-convergent paths
  12. Equivariant quantum cohomology
  13. Main proposal
  14. Inducing quasi-convergent paths
  15. Inducing theorem
  16. ...and 9 more sections

Key Result

Theorem 1.4

Let $X$ be a smooth projective variety with an action of a finite group $G$. Let $\sigma_{t}=(Z_t,\mathcal{P}_t)$ for $t\in[t_0,\infty)$ be a quasi-convergent path in $\mathrm{Stab}(X)^G$ that solves the non-equivariant intro:proposal via a fundamental solution $\Phi_t$ of eq: qde. Then it induces a

Theorems & Definitions (79)

  • Definition 1.1: \ref{['def:T-stab']}
  • Remark 1.3
  • Theorem 1.4: \ref{['thm: inducing solutions']}
  • Theorem 1.5: \ref{['thm:tncmmp']}
  • Theorem 2.1: MR2373143
  • Example 2.2
  • Definition 2.3: Cotti2019EquivariantQD
  • Definition 3.1
  • Definition 3.2: halpernleistner2024stabilityconditionssemiorthogonaldecompositions
  • Definition 3.3: halpernleistner2024stabilityconditionssemiorthogonaldecompositions
  • ...and 69 more