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Derived autoequivalences of length 2 flops via GIT

Aporva Varshney

Abstract

We obtain the derived autoequivalences of a flopping rational curve of length 2 using GIT and the theory of windows applied to the universal length 2 flop. We show that the stringy Kähler moduli space (SKMS) associated to the GIT problem, as constructed by Halpern-Leistner--Sam, matches the description of the space obtained for length 2 threefolds by Hirano--Wemyss as a quotient of a Bridgeland stability manifold. Furthermore, we show that its fundamental group acts via contraction algebra and fibre algebra twists, hence recovering the monodromy action described by Donovan--Wemyss. In particular, this shows that the two approaches to building the SKMS agree in this setting.

Derived autoequivalences of length 2 flops via GIT

Abstract

We obtain the derived autoequivalences of a flopping rational curve of length 2 using GIT and the theory of windows applied to the universal length 2 flop. We show that the stringy Kähler moduli space (SKMS) associated to the GIT problem, as constructed by Halpern-Leistner--Sam, matches the description of the space obtained for length 2 threefolds by Hirano--Wemyss as a quotient of a Bridgeland stability manifold. Furthermore, we show that its fundamental group acts via contraction algebra and fibre algebra twists, hence recovering the monodromy action described by Donovan--Wemyss. In particular, this shows that the two approaches to building the SKMS agree in this setting.

Paper Structure

This paper contains 20 sections, 27 theorems, 115 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The stringy Kähler moduli space
  3. Monodromy action
  4. Universal flops
  5. SKMS of a GIT Quotient
  6. Results and structure
  7. Preliminaries
  8. Contraction and fibre algebras
  9. The universal flop of length 2
  10. Quasi-symmetric linear GIT problem
  11. Windows and the stringy Kähler moduli space
  12. Autoequivalences
  13. Constructing spherical functors
  14. Window shift autoequivalences
  15. Orthogonal subcategories
  16. ...and 5 more sections

Key Result

Proposition 1.1

The universal flop of length $2$ is a GIT quotient of the stack where $V$ is a two dimensional vector space.

Figures (3)

  • Figure 1: The polytope $\overline{\nabla}$ in the character lattice of the maximal torus, with key representations placed on their highest weights. Red line denotes the Weyl invariant line.
  • Figure 2: The resulting hyperplane arrangement in blue and green with translates omitted, and Weyl invariant line shown in red.
  • Figure 3: One-parameter subgroups corresponding to each face of $\overline{\nabla}$.

Theorems & Definitions (51)

  • Proposition 1.1: =\ref{['prop:git-problem']}
  • Proposition 1.2: § \ref{['subsec:fibre']}
  • Proposition 1.3: § \ref{['sec:contractiontwist']}
  • Proposition 1.4: =\ref{['prop:smooth-algebra']}, \ref{['prop:equivalent-twists']}
  • Example 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 4.1
  • ...and 41 more