Cluster category and birational geometry
摘要： A fundamental problem in birational geometry, in particular in minimal model program, is to classify contractible rational curves. In this talk, we will build a relation between it and the theory of cluster category. To be more specific, we associate to each 3-dimensional flopping contraction a cluster category in the sense of Aimot. Based on our previous work on noncommutative Mather-Yau theorem, we show that the cluster category is essentially determined by its cluster tilting algebra. On the other hand, we formulate certain necessary condition on the associated cluster category for a rigid rational curve to be contractible, which is conjecturally to be also sufficient. The talk is based on a joint work with Guisong Zhou 1803.06128, and the work in progress joint with Bernhard Keller.