Klt varieties of conjecturally minimal volume

TL;DR

This paper constructs new klt projective varieties with minimal known volume and exceptional Fano varieties with smallest anticanonical volume, providing evidence for conjectured minimality across dimensions.

Contribution

It introduces novel examples of klt varieties and Fano varieties with minimal volumes, using weighted hypersurfaces beyond quasi-smooth cases, and computes their log canonical thresholds exactly.

Findings

Constructed klt varieties with smallest known volume.

Found exceptional Fano varieties with minimal anticanonical volume.

Provided lower bounds supporting conjectured minimality in all dimensions.

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.