The derived contraction algebra

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.

54 pages. v2: fixed mistakes in 5.2.12 and 6.6.1, added some material, exposition improved, cosmetic changes

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• Sur les A-infini catégories
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• A model category structure on the category of simplicial categories