2019/11/21 by Matt Booth, Booth, Matt
Computer Science · Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Logic, programming, and type systems #Numerical methods for differential equations #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1911.09626
openalex publication_date 2019/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Acon 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 Acon 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 Adercon, which I define to be Braun-Chuang-Lazarev's derived quotient of A by e. A priori, Adercon - 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 Adercon 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.