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On derived equivalence for Abuaf flop: mutation of non-commutative crepant resolutions and spherical twists

Wahei Hara

Abstract

Recently, Segal constructed a derived equivalence for an interesting 5-fold flop that was provided by Abuaf. The aim of this article is to add some results for the derived equivalence for Abuaf's flop. Concretely, we study the equivalence for Abuaf's flop by using Toda-Uehara's tilting bundles and Iyama-Wemyss's mutation functors. In addition, we observe a "flop-flop=twist" result and a "multi-mutation=twist" result for Abuaf's flop.

On derived equivalence for Abuaf flop: mutation of non-commutative crepant resolutions and spherical twists

Abstract

Recently, Segal constructed a derived equivalence for an interesting 5-fold flop that was provided by Abuaf. The aim of this article is to add some results for the derived equivalence for Abuaf's flop. Concretely, we study the equivalence for Abuaf's flop by using Toda-Uehara's tilting bundles and Iyama-Wemyss's mutation functors. In addition, we observe a "flop-flop=twist" result and a "multi-mutation=twist" result for Abuaf's flop.

Paper Structure

This paper contains 31 sections, 39 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. NCCRs and Iyama-Wemyss's mutations
  4. Flop-Flop=Twist result
  5. Multi-mutation=twist result.
  6. Plan of the article
  7. Notations.
  8. Preliminaries
  9. Abuaf flop
  10. Non-commutative crepant resolution and tilting bundle
  11. Toda-Uehara's construction for tilting bundles and perverse hearts
  12. Iyama-Wemyss mutation
  13. Spherical twist
  14. Toda-Uehara's tilting bundles and Segal's tilting bundles
  15. Notations
  16. ...and 16 more sections

Key Result

Theorem 1.1

$Y$ and $Y'$ satisfy Toda-Uehara's assumptions.

Theorems & Definitions (79)

  • Theorem 1.1: see Section \ref{['subsec: TU assump']}
  • Theorem 1.3: = Theorem \ref{['mutation1']}, Theorem \ref{['mutation2']}
  • Theorem 1.4: = Theorem \ref{['thm flop-flop']}, Theorem \ref{['thm flop-flop2']}
  • Theorem 1.5: = Theorem \ref{['FM ker']}
  • Theorem 1.6: = Theorem \ref{['multimutationtwist']}
  • Definition 2.1
  • Conjecture 2.2: VdB04b, Conjecture 4.6
  • Definition 2.3
  • Example 2.4
  • Theorem 2.5
  • ...and 69 more