Hodge ideals and microlocal V-filtration

TL;DR

This paper establishes a close relationship between Hodge ideals and microlocal V-filtration, revealing that certain invariants like log-canonicity are determined by microlocal Bernstein-Sato polynomials, thus connecting Hodge theory with microlocal analysis.

Contribution

It demonstrates that Hodge ideals coincide with parts of the microlocal V-filtration modulo the ideal of the divisor, linking Hodge-theoretic and microlocal invariants.

Findings

Hodge ideals match the positive part of the microlocal V-filtration modulo the divisor ideal.

The j-log-canonicity is characterized by the microlocal log-canonical threshold.

The microlocal log-canonical threshold equals the maximal root of the microlocal Bernstein-Sato polynomial.

Abstract

We show that the Hodge ideals in the sense of Mustata and Popa are quite closely related to the induced microlocal V-filtration on the structure sheaf, defined by using the microlocalization of the V-filtration of Kashiwara and Malgrange. More precisely the former coincide, module the ideal of the divisor, with the part of the latter indexed by positive integers, although they are different without modulo the ideal in general. This coincidence implies that the $j$-log-canonicity in their sense is determined by the microlocal log-canonical threshold of the divisor, which coincides with the maximal root of the reduced (or microlocal) Bernstein-Sato polynomial up to a sign.