Cluster categories and rational curves

Abstract

<p>We study rational curves on smooth complex Calabi–Yau threefolds via noncommuta- tive algebra. By the general theory of derived noncommutative deformations due to Efimov, Lunts and Orlov, the structure sheaf of a rational curve in a smooth CY 3-fold Y is pro-represented by a nonpositively graded dg algebra Γ. The curve is called nc rigid if H0Γ is finite dimensional. When C is contractible, H0Γ is isomorphic to the contraction algebra defined by Donovan and Wemyss. More generally, one can show that there exists a Γ pro-representing the (derived) multi- pointed deformation (defined by Kawamata) of a collection of rational curves C1, . . . , Ct with dim(HomY (OCi , OCj )) = δij . The collection is called nc rigid if H 0 Γ is finite dimensional. We prove that Γ is a homologically smooth bimodule 3CY algebra. As a consequence, we define a (2CY) cluster category CΓ for such a collection of rational curves in Y . It has finite-dimensional morphism spaces iff the collection is nc rigid. When 􏰷ti=1 Ci is (formally) contractible by a mor- phism Y􏱮 → X􏱮, then CΓ is equivalent to the singularity category of X􏱮 and thus categorifies the contraction algebra of Donovan and Wemyss. The Calabi-Yau structure on Y determines a canon- ical class [w] (defined up to right equivalence) in the zeroth Hochschild homology of H0Γ. Using our previous work on the noncommutative Mather–Yau theorem and singular Hochschild coho- mology, we prove that the singularities underlying a 3-dimensional smooth flopping contraction are classified by the derived equivalence class of the pair (H0Γ,[w]). We also give a new necessary condition for contractibility of rational curves in terms of Γ.</p>