Mather discrepancy and the arc spaces

TL;DR

This paper introduces Mather discrepancy as a new invariant for classifying singularities, extending existing invariants to broader classes of varieties and establishing formulas and properties related to arc spaces.

Contribution

It proposes new invariants based on Mather discrepancy, generalizes log-canonical thresholds and minimal log-discrepancy, and proves related formulas and properties for arbitrary varieties.

Findings

Formulas for new log-canonical threshold in terms of arc spaces

Inversion of adjunction for wider class of singularities

Lower semicontinuity of new minimal log-discrepancy

Abstract

The goal of this paper is a classification theorem of the singularities according to a new invariant, Mather discrepancy. On the other hand, we show some evidences convincing us that Mather discrepancy is a considerable invariant: By introducing new log-canonical threshold and minimal log-discrepancy by means of Mather discrepancy instead of usual discrepancy of canonical divisors, we obtain the formulas of the new log-canonical threshold in terms of arc spaces, inversion of adjunction for wider class of singularities than the known one, lower seimicontinuity of the new minimal log-discrepancy and the affirmative answer to a conjecture of Shokurov type; One advantage of the new invariants is that these are defined for arbitrary varieties (without q-Gorenstein property); These results include the known results for usual log-canonical threshold and minimal log-discrepancy.