The derived contraction algebra

TL;DR

This paper introduces the derived contraction algebra, an enhanced invariant that captures the geometry and deformation theory of contractions of rational curves, extending previous algebraic invariants to a derived setting.

Contribution

It develops the derived contraction algebra using dg singularity categories, controlling deformations and autoequivalences in a broader geometric context.

Findings

Derived contraction algebra recovers local geometry of contractions.

Controls derived noncommutative deformations of rational curves.

Extends noncommutative autoequivalence understanding beyond simple flops.

Abstract

Using Braun-Chuang-Lazarev's derived quotient, we enhance the contraction algebra of Donovan-Wemyss to an invariant valued in differential graded algebras. Given an isolated contraction $X \to X_\mathrm{con}$ of an irreducible rational curve $C$ to a point $p$, we show that its derived contraction algebra controls the derived noncommutative deformations of $C$. We use dg singularity categories to prove that, when $X$ is smooth, the derived contraction algebra recovers the geometry of $X_\mathrm{con}$ complete locally around $p$, establishing a positive answer to a derived version of a conjecture of Donovan and Wemyss. When $X \to X_\mathrm{con}$ is a simple threefold flopping contraction, it is known that the Bridgeland-Chen flop-flop autoequivalence of $D^b(X)$ is a `noncommutative twist' around the contraction algebra. We show that the derived contraction algebra controls an analogous autoequivalence in more general settings, and in particular for partial resolutions of Kleinian singularities.