HODGE IDEALS FOR -DIVISORS, -FILTRATION, AND MINIMAL EXPONENT

Mircea Mustaţă,Mihnea Popa

doi:10.1017/fms.2020.18

Mircea Mustaţă, Mihnea Popa

Open Access

https://doi.org/10.1017/fms.2020.18

Copy DOI

Abstract

We compute the Hodge ideals of $\mathbb{Q}$ -divisors in terms of the $V$ -filtration induced by a local defining equation, inspired by a result of Saito in the reduced case. We deduce basic properties of Hodge ideals in this generality, and relate them to Bernstein–Sato polynomials. As a consequence of our study we establish general properties of the minimal exponent, a refined version of the log canonical threshold, and bound it in terms of discrepancies on log resolutions, addressing a question of Lichtin and Kollár.

Highlights

This paper establishes a connection between the Hodge ideals of a Q-divisor, as defined in [MP19b], and the V -filtration along an appropriately chosen hypersurface
It is inspired by Saito’s [Sai16], which explained such a connection expressed in terms of the microlocal V -filtration, in the case of the Hodge ideals of reduced divisors studied in [MP19a]
In [MP19b], we observe that it carries a natural filtration FpM( f β), with p 0, which makes it a filtered direct summand in a D-module underlying a mixed Hodge module

Summary

Introduction

This paper establishes a connection between the Hodge ideals of a Q-divisor, as defined in [MP19b], and the V -filtration along an appropriately chosen hypersurface. Following [Sai16], we denote by αZ the negative of the largest root of bZ (s)/(s + 1) (with the convention that this is ∞ if bZ (s) = s + 1) This invariant is called the minimal exponent of Z , and is a refined version of the log canonical threshold of the pair (X, Z ), which is equal to min{αZ , 1}; see Section 6 for a discussion. Using further properties of Hodge ideals of Q-divisors proved in [MP19b], we deduce some general properties of the minimal exponent αD for any effective divisor D, extending important features of the log canonical threshold. Under a suitable log-canonicity hypothesis, the jumping coefficients of higher Hodge ideals in the same interval, in the sense of Corollary B(3), lead to further such roots This follows quickly from results proved in the final two sections of the paper

Main results

Consequences