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Title: Kähler-Einstein metrics and normalized volumes of valuations
Authors: Liu, Yuchen
Advisors: Kollár, János
Contributors: Mathematics Department
Keywords: Fano varieties
Kähler-Einstein metrics
normalized volume of valuations
Subjects: Mathematics
Issue Date: 2017
Publisher: Princeton, NJ : Princeton University
Abstract: In this thesis we study questions relating "global volumes" of Kähler-Einstein Fano varieties and "local volumes" at a singularity. The "local volumes" is interpreted as the normalized volume of real valuations centered at a klt singularity recently introduced by Li. In the first part, we show that the volume of a Kähler-Einstein Fano variety is bounded from above by the normalized volume of any valuation centered at a closed point. This refines a recent result of Fujita. In the second part, we study the minimization problem of normalized volumes over cone singularities. We show that a Fano manifold is K-semistable if and only if the normalized volume function over its affine cone is minimized at the canonical valuation. This confirms a conjecture of Li.
Alternate format: The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog:
Type of Material: Academic dissertations (Ph.D.)
Language: en
Appears in Collections:Mathematics

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