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Cluster categories and rational curves

Given a semi-simple collection of rational curves on a smooth quasi-projective 3-fold, its multipointed noncommutative deformation is represented by a negatively graded DGA \Gamma. The finite dimensionality of the cohomology of \Gamma seems to relate to contractibility of the collection of rational curves. For CY 3-folds, \Gamma is a bimodule 3CY DG algebra. If we further assume contractibility then H^0\Gamma is isomorphic to the contraction algebra of Donovan and Wemyss. And the cluster category of \Gamma is dg-equivalent to the singularity category of the contracted space. In some sense the CY algebra \Gamma links the deformation theory of the exceptional fibres and the singularity theory of the contracted space. In this talk I will present a joint work with Bernhard Keller, where we prove that the derived Morita type of the contraction algebra together with a canonical class in its 0-th Hochschild homology defined via CY structure determines the analytic type of the singularity of the contracted space.

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On symmetric quivers and their degenerations

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