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Abstract Prokhorov and Shramov proved that the BAB conjecture, which Birkar later proved, implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension.This property in particular gives an upper bound on the size of finite semi-simple groups (i.e., those with no nontrivial normal abelian subgroups) acting faithfully on 𝑛-dimensional complex Fano varieties, and this bound only depends on 𝑛.We investigate the geometric consequences of an action by a certain semi-simple group: the symmetric group.We give an effective upper bound for the maximal symmetric group action on an 𝑛-dimensional Fano variety.For certain classes of varieties – toric varieties and Fano weighted complete intersections – we obtain optimal upper bounds.Finally, we draw a connection between large symmetric actions and boundedness of varieties, by showing that the maximally symmetric Fano fourfolds form a bounded family.Along the way, we also show analogues of some of our results for Calabi–Yau varieties and log terminal singularities.
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Algebraicity of the metric tangent cones and equivariant K-stability
We prove two new results on the K K -polystability of Q \mathbb {Q} -Fano varieties based on purely algebro-geometric arguments. The first one says that any K K -semistable log Fano cone has a special degeneration to a uniquely determined K K -polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to complete the proof of Donaldson-Sun’s conjecture which says that the metric tangent cone of any point appearing on a Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds depends only on the algebraic structure of the singularity. The second result says that for any log Fano variety with the torus action, K K -polystability is equivalent to equivariant K K -polystability, that is, to check K K -polystability, it is sufficient to check special test configurations which are equivariant under the torus action.
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- Award ID(s):
- 2153115
- PAR ID:
- 10330263
- Date Published:
- Journal Name:
- Journal of the American Mathematical Society
- Volume:
- 34
- Issue:
- 4
- ISSN:
- 0894-0347
- Page Range / eLocation ID:
- 1175 to 1214
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/jams/974
