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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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Varieties of general type with doubly exponential asymptotics
We construct smooth projective varieties of general type with the smallest known volume and others with the most known vanishing plurigenera in high dimensions. The optimal volume bound is expected to decay doubly exponentially with dimension, and our examples achieve this decay rate. We also consider the analogous questions for other types of varieties. For example, in every dimension we conjecture the terminal Fano variety of minimal volume, and the canonical Calabi-Yau variety of minimal volume. In each case, our examples exhibit doubly exponential behavior.
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- Award ID(s):
- 2054553
- PAR ID:
- 10425524
- Date Published:
- Journal Name:
- Transactions of the American Mathematical Society, Series B
- Volume:
- 10
- Issue:
- 10
- ISSN:
- 2330-0000
- Page Range / eLocation ID:
- 288 to 309
- 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/btran/125
