Cluster categories and rational curves
Zheng Hua, Bernhard Keller
Abstract
We study rational curves on smooth complex Calabi--Yau threefolds 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 $Γ$. The curve is called nc rigid if $H^0Γ$ is finite dimensional. When $C$ is contractible, $H^0Γ$ 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 $C_1,\ldots,C_t$ so that ${\mathrm{dim}}({\rm{Hom}}_Y({\mathcal{O}}_{C_i},{\mathcal{O}}_{C_j}))=δ_{ij}$. The collection is called nc rigid if $H^0Γ$ is finite dimensional. We prove that $Γ$ is a homologically smooth bimodule 3CY algebra. As a consequence, we define a (2CY) cluster category ${\mathcal{C}}_Γ$ for such a collection of rational curves in $Y$. It has finite-dimensional morphism spaces iff the collection is nc rigid. When $\bigcup_{i=1}^tC_i$ is (formally) contractible by a morphism $\hat{Y}\to \hat{X}$, ${\mathcal{C}}_Γ$ is equivalent to the singularity category of $\hat{X}$ 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 $H^0Γ$. 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 $(H^0Γ, [w])$. We also give a new necessary condition for contractibility of rational curves in terms of $Γ$.