Klt Varieties With Conjecturally Minimal Volume

DOI10.1093/IMRN/RNAD047arXiv2210.11354OpenAlexW4360610608MaRDI QIDQ6142532

Burt Totaro

Publication date: 26 January 2024

Published in: IMRN. International Mathematics Research Notices (Search for Journal in Brave)

Abstract: 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 anticanonical 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.

Full work available at URL: https://arxiv.org/abs/2210.11354

Mathematics Subject Classification ID

Algebraic geometry (14-XX) Differential geometry (53-XX)

Related Items (2)

Fano varieties with conjecturally largest Fano index ⋮ Log canonical pairs with conjecturally minimal volume

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