Cluster categories and rational curves

Abstract

We study rational curves on smooth complex Calabi-Yau 3-folds via noncommutative 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 €. The curve is called nc rigid if H0Г is finite-dimensional. When C is contractible, H0Г is isomorphic to the contraction algebra defined by Donovan and Wemyss. More generally, one can show that there exists a Г pro-representing the (derived) multipointed deformation (defined by Kawamata) of a collection of rational curves C1..... Ct with dim (HomY(ҨCi, ҨCj)) δij. The collection is called nc rigid if H0Г is finite-dimensional. We prove that Г is a homologically smooth bimodule 3-CY algebra. As a consequence, we define a (2-CY) cluster category ҼГ for such a collection of rational curves in Y. It has finite-dimensional morphism spaces if and only if the collection is nc rigid. When Uti=1 Ci is (formally) contractible by a morphism Ŷ→Х then lГ isequivalenttothesingularitycategory of (formula presented) and thus categorifies the contraction algebra of Donovan and Wemyss. The Calabi-Yau structure on Y determines a canonical class [w] (defined up to right equivalence) in the zeroth Hochschild homology of H0Г. Using our previous work on the noncommutative Mather-Yau theorem and singular Hochschild cohomology, we prove that the singularities underlying a 3-dimensional smooth flopping contraction are classified by the derived equivalence class of the pair.H0Г; [w]. We also give a new necessary condition for contractibility of rational curves in terms of Г.