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Title: Birational superrigidity and K-stability
Authors: Zhuang, Ziquan
Advisors: Kollár, János
Contributors: Mathematics Department
Keywords: Birational superrigidity
Fano variety
Kähler–Einstein metric
Subjects: Mathematics
Issue Date: 2019
Publisher: Princeton, NJ : Princeton University
Abstract: We consider two different notions on Fano varieties: birational superrigidity, coming from the study of rationality, and K-stability, which is related to the existence of K\"ahler-Einstein metrics. In the first part, we show that birationally superrigid Fano varieties are also K-stable as long as their alpha invariants are at least $\frac{1}{2}$, partially confirming a conjecture of Odaka-Okada and Kim-Okada-Won. In the second part, we prove the folklore prediction that smooth Fano complete intersections of Fano index one are birationally superrigid and K-stable when the dimension is large. In the third part, we introduce an inductive argument to study the birational superrigidity and K-stability of singular complete intersections and in particular prove an optimal result on the birational superrigidity and K-stability of hypersurfaces of Fano index one with isolated ordinary singularities in large dimensions. Finally we provide an explicit example to show that in general birational superrigidity is not a locally closed property in families of Fano varieties, giving a negative answer to a question of Corti.
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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