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Noncommutative varieties and stability conditions: an overview

Laura Pertusi

Abstract

Using the modern perspective of noncommutative algebraic geometry we survey some recent progress in the theory of stability conditions and moduli spaces with applications in hyperkähler geometry and classical algebraic geometry.

Noncommutative varieties and stability conditions: an overview

Abstract

Using the modern perspective of noncommutative algebraic geometry we survey some recent progress in the theory of stability conditions and moduli spaces with applications in hyperkähler geometry and classical algebraic geometry.
Paper Structure (23 sections, 27 theorems, 70 equations)

This paper contains 23 sections, 27 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Hints on stable $\infty$-categories
  3. Simplicial sets
  4. Horns and $\infty$-categories
  5. Homotopy category and stability
  6. Noncommutative schemes
  7. Linear categories
  8. Smooth and proper linear categories
  9. Examples
  10. Stability conditions on noncommutative schemes
  11. Bridgeland stability conditions
  12. Stability conditions in families
  13. Moduli spaces
  14. Results and open problems
  15. Stability conditions on $\mathrm{D}_{\mathrm{perf}}(X)$
  16. ...and 8 more sections

Key Result

Proposition 2.12

If $X$ is a topological space, then for every $n>0$, $0 \leq k \leq n$, every morphism of simplicial sets $f \colon \Lambda^n_k \to \mathop{\mathrm{Sing}}(X)$ extends to $\Delta^n \to \mathop{\mathrm{Sing}}(X)$ through the inclusion $\Lambda^n_k \hookrightarrow \Delta^n$: \xymatrix{ \Lambda^n_k \ar[

Theorems & Definitions (81)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Example 2.4
  • Remark 2.5
  • Definition 2.6: Riehl, Definitions 5.1, 5.2
  • Remark 2.7
  • Example 2.8
  • Remark 2.9
  • Definition 2.10
  • ...and 71 more