The derived contraction algebra

TL;DR

This paper introduces the derived contraction algebra as a new, more refined invariant for flopping contractions, extending previous work on contraction algebras and their role in derived equivalences and deformation theory.

Contribution

It constructs the derived contraction algebra as a derived quotient, proving its properties and applications, and establishes a derived version of the Donovan-Wemyss conjecture.

Findings

The derived contraction algebra is a finer invariant than the classical contraction algebra.

It controls generalized flop-flop autoequivalences.

The derived quotient can be interpreted via derived deformation theory.

Abstract

A version of the Bondal-Orlov conjecture, proved by Bridgeland, states that if $X$ and $Y$ are smooth complex projective threefolds linked by a flop, then they are derived equivalent. Van den Bergh gave a new proof of Bridgeland's theorem using the notion of a NCCR, which is in particular a ring $A$ together with a derived equivalence between $X$ and $A$. This ring $A$ is constructed as an endomorphism ring of a decomposable module, and hence admits an idempotent $e$. Donovan and Wemyss define the contraction algebra $A_\mathrm{con}$ to be the quotient of $A$ by $e$; it is a finite-dimensional noncommutative algebra that is conjectured to completely recover the geometry of the base of the flop. They show that $A_\mathrm{con}$ represents the noncommutative deformation theory of the flopping curves, and also controls the Bridgeland-Chen flop-flop autoequivalence of the derived category of $X$. In this thesis, I construct and prove properties of a new invariant, the derived contraction algebra $A^\mathrm{der}_\mathrm{con}$, which I define to be Braun-Chuang-Lazarev's derived quotient of $A$ by $e$. A priori, $A^\mathrm{der}_\mathrm{con}$ - which is a dga, rather than just an algebra - is a finer invariant than the classical contraction algebra. I prove (using recent results of Hua and Keller) a derived version of the Donovan-Wemyss conjecture, a suitable phrasing of which is true in all dimensions. I prove that the derived quotient admits an interpretation in terms of derived deformation theory. I prove that $A^\mathrm{der}_\mathrm{con}$ controls a generalised flop-flop autoequivalence. These results both recover and extend Donovan-Wemyss's. I give concrete applications and computations in the case of partial resolutions of Kleinian singularities, where the classical contraction algebra becomes inadequate.