Singularities and K-stability

singularities;log canonical thresholds;valuations;K-stability;Mathematics;Science;Mathematics

Blum, HaroldSmith, Karen E ;

University of Michigan

关键词: singularities;    log canonical thresholds;    valuations;    K-stability;    Mathematics;    Science;    Mathematics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/146044/blum_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

We study invariants of singularities that have arisen in connection withthe K-stability of Fano varieties. The first invariant we consider is Li;;s normalized volume function on the space ofvaluations over a klt singularitiy. Proving a conjecture of Li, we show that there always exists a valuation over a klt singularity with smallest normalized volume. Next, we present joint work with Mattias Jonsson on the log canonical and stability thresholds of a line bundle. The latter notion generalizes an invariant recently introduced by Fujita and Odaka, and can be used to characterize when a Fano variety is K-semistable or uniformly K-stable. We express the two thresholds as infima of certain functionals on the space of valuations and systematically study these invariants.