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An algorithm for Hodge ideals

Guillem Blanco

Abstract

We present an algorithm to compute the Hodge ideals of $\mathbb{Q}$-divisors associated to any reduced effective divisor $D$. The computation of the Hodge ideals is based on an algorithm to compute parts of the $V$-filtration of Malgrange and Kashiwara on $ι_{+}\mathscr{O}_X(*D)$ and the characterization of the Hodge ideals in terms of this $V$-filtration. In particular, this gives a new algorithm to compute the multiplier ideals and the jumping numbers of any effective divisor.

An algorithm for Hodge ideals

Abstract

We present an algorithm to compute the Hodge ideals of -divisors associated to any reduced effective divisor . The computation of the Hodge ideals is based on an algorithm to compute parts of the -filtration of Malgrange and Kashiwara on and the characterization of the Hodge ideals in terms of this -filtration. In particular, this gives a new algorithm to compute the multiplier ideals and the jumping numbers of any effective divisor.

Paper Structure

This paper contains 11 sections, 9 theorems, 44 equations, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The V-filtration of Kashiwara and Malgrange
  3. The smooth case
  4. The graph embedding
  5. Hodge ideals and the V-filtration
  6. The Bernstein-Sato polynomial
  7. The algorithm
  8. The s-parametric annihilator
  9. Modulo operation
  10. The algorithm
  11. Examples

Key Result

Theorem 1.1

Let $\mathcal{M}$ be a regular holonomic $\mathscr{D}_X$-module with quasi-unipotent monodromy around $Z$. Then, $\mathcal{M}$ admits a $V$-filtration along $Z$.

Theorems & Definitions (15)

  • Theorem 1.1: Kas83Mal83
  • Theorem 3.1: MP20b
  • Theorem 3.2: BS05
  • Proposition 4.1: MP20b
  • Proposition 4.2: Ber72
  • Lemma 4.3: Sai88
  • Theorem 7.1
  • proof
  • Corollary 7.2
  • Remark 3.1
  • ...and 5 more