Informal Geometric Analysis Seminar | Yuchen Liu (Northwestern)

 Title: On the space of special valuations

Abstract: A valuation on a Fano variety is called special if it corresponds to a special R-test configuration, i.e. a generalized test configuration over a higher dimensional base such that the central fiber is klt. We prove that the specialness of valuations in the dual complex of an affine log Calabi-Yau variety U is independent of the choice of Fano compactifications. In particular, the space of special valuations is invariant under the action of the automorphism group of U. Moreover, we describe the space of special valuations for the affine log Calabi-Yau threefold complement to the Markov cubic surface by building connections with dynamics on the hyperbolic plane. 

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