Log canonical pairs with conjecturally minimal volume
Abstract
We construct log canonical pairs with a nonzero reduced divisor and ample that have the smallest known volume. We conjecture that our examples have the smallest volume in each dimension. The conjecture is true in dimension 2, by Liu and Shokurov. The examples are weighted projective hypersurfaces that are not quasi-smooth. We also develop an example for a related extremal problem. Esser constructed a klt Calabi-Yau variety which conjecturally has the smallest mld in each dimension (for example, mld in dimension 2 and in dimension 3). However, the example was only worked out completely in dimensions at most 18. We now prove the desired properties of Esser's example in all dimensions (in particular, determining its mld).
Keywords
Cite
@article{arxiv.2308.08034,
title = {Log canonical pairs with conjecturally minimal volume},
author = {Louis Esser and Burt Totaro},
journal= {arXiv preprint arXiv:2308.08034},
year = {2026}
}
Comments
29 pages; v4: the paper has been published and is left unchanged, but Conjecture 7.4 has been disproved by Jihao Liu (https://jihaoliu.org/notes/ET74.pdf). Namely, the klt Calabi-Yau variety of large index constructed by Esser-Totaro-Wang has smaller-than-expected index in dimension 159