Log canonical thresholds on varieties with bounded singularities

TL;DR

This paper proves that for varieties with bounded singularities and pairs with ideals having exponents in a set satisfying the DCC, the set of log canonical thresholds satisfies the ACC, confirming a conjecture of Shokurov.

Contribution

It establishes the Ascending Chain Condition for log canonical thresholds on varieties with bounded singularities, confirming Shokurov's conjecture in this context.

Findings

Set of log canonical thresholds satisfies ACC

Varieties with bounded singularities are formally locally subvarieties of bounded degree

Confirms Shokurov's conjecture for this class of varieties

Abstract

We consider pairs (X,A), where X is a variety with klt singularities and A is a formal product of ideals on X with exponents in a fixed set that satisfies the Descending Chain Condition. We also assume that X has (formally) bounded singularities, in the sense that it is, formally locally, a subvariety in a fixed affine space defined by equations of bounded degree. We prove in this context a conjecture of Shokurov, predicting that the set of log canonical thresholds for such pairs satisfies the Ascending Chain Condition.