K-stability is an algebraic notion that characterizes when a smooth Fano variety admits a Kahler-Einstein metric. A key motivation for understanding K-stability is to construct moduli spaces for Fano varieties. In this talk, I will explain that a K-polystable degeneration of a family of Q-Fano varieties is necessarily unique. The result is a key step in the program to construct a proper moduli space parameterizing K-polystable Q-Fano varieties and essentially verifies the separateness of said moduli space. This is joint work with Chenyang Xu. |

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