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By Hacon-McKernan-Xu, there is a positive lower bound in each dimension for the volumes of all klt varieties with ample canonical class. We show that these bounds must go to zero extremely fast as the dimension increases, by constructing a klt n-fold with ample canonical class whose volume is less than 1/2^{2^n}. These examples should be close to optimal. We also construct, for every n, a klt Fano variety of dimension n such that the space of sections of the mth power of the anticanonical bundle is zero for all m from 1 to about 2^{2^n}. Here again there is some bound in each dimension, by Birkar’s theorem on boundedness of complements, and we are showing that the bound must increase extremely fast with the dimension.
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Klt varieties with conjecturally minimal volume
We construct klt projective varieties with ample canonical class and the smallest known volume. We also find exceptional klt Fano varieties with the smallest known anti-canonical volume. We conjecture that our examples have the smallest volume in every dimension, and we give low-dimensional evidence for that. In order to improve on earlier examples, we are forced to consider weighted hypersurfaces that are not quasi-smooth. We show that our Fano varieties are exceptional by computing their global log canonical threshold (or alpha-invariant) exactly; it is extremely large, roughly 2^{2^n} in dimension n. These examples give improved lower bounds in Birkar’s theorem on boundedness of complements for Fano varieties.
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- Award ID(s):
- 2054553
- PAR ID:
- 10519767
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2024
- Issue:
- 1
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 462 to 491
- Subject(s) / Keyword(s):
- Variety of general type Fano variety klt singularities alpha invariant
- 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/10.1093/imrn/rnad047
