Cluster categories and rational curves

Hua, Zheng¹ ; Keller, Bernhard²

¹ Department of Mathematics, The University of Hong Kong, Hong Kong
² Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, Paris, France

Geometry & topology, Tome 28 (2024) no. 6, pp. 2569-2634.

Résumé

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 $\Gamma$. The curve is called nc rigid if $H^0\Gamma$ is finite-dimensional. When $C$ is contractible, $H^0\Gamma$ is isomorphic to the contraction algebra defined by Donovan and Wemyss. More generally, one can show that there exists a $\Gamma$ pro-representing the (derived) multipointed deformation (defined by Kawamata) of a collection of rational curves $C_1,\dots ,C_t$ with $\dim ({\mathrm{Hom}}_Y({\mathcal{𝒪}}_{C_i},{\mathcal{𝒪}}_{C_j}))={\delta }_{ij}$. The collection is called nc rigid if $H^0\Gamma$ is finite-dimensional. We prove that $\Gamma$ is a homologically smooth bimodule 3–CY algebra. As a consequence, we define a (2–CY) cluster category ${\mathcal{𝒞}}_{\Gamma }$ 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 ${\bigcup }_{i=1}^t{C}_i$ is (formally) contractible by a morphism $Ŷ\to \widehat{X}$, then ${\mathcal{𝒞}}_{\Gamma }$ is equivalent to the singularity category of $\widehat{X}$ 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  $H^0\Gamma$. 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 $(H^0\Gamma ,[w])$. We also give a new necessary condition for contractibility of rational curves in terms of $\Gamma$.

Détail

DOI : 10.2140/gt.2024.28.2569

Classification : 14A22
Keywords: contractible curves, cluster categories, noncommutative deformations, quivers with potentials

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