UCLA

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 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.