Derived contraction algebra

Abstract

All minimal models of a given variety are linked by special birational maps called flops, which are a type of codimension two surgery. 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 (i.e. their bounded derived categories of coherent sheaves are equivalent). Van den Bergh was able to give a new proof of Bridgeland’s theorem using the notion of a noncommutative crepant resolution, which is in particular a ring A together with a derived equivalence between X and A. The 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 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 it represents the noncommutative deformation theory of the flopping curves, as well as controlling the flop-flop autoequivalence of the derived category of X (which, for the algebraic model A, is the mutation-mutation autoequivalence). In this thesis, I construct and prove properties of a new invariant, the derived contraction algebra, which I define to be Braun–Chuang–Lazarev’s derived quotient of A by e. A priori, the derived contraction algebra – 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 a deformation-theoretic interpretation; the proof is purely homotopical algebra and relies at heart on a Koszul duality result. I moreover prove that in an appropriate sense, the derived contraction algebra controls the mutation-mutation 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.