Log canonical pairs with conjecturally minimal volume

TL;DR

This paper constructs and analyzes log canonical pairs with minimal volume, conjecturing these are the smallest in each dimension, and extends known results about minimal log discrepancies to all dimensions.

Contribution

It provides new examples of log canonical pairs with minimal volume and proves properties of Esser's minimal mld example in all dimensions.

Findings

Constructed log canonical pairs with smallest known volume

Confirmed the minimal volume conjecture in dimension 2

Extended properties of Esser's minimal mld example to all dimensions

Abstract

We construct log canonical pairs $(X,B)$ with $B$ a nonzero reduced divisor and $K_X+B$ 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 $1/13$ in dimension 2 and $1/311$ 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).