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On the monodromy conjecture, holomorphy conjecture, and embedded Nash problem for Pfaffian ideals

Yifan Chen, Quan Shi, Yongxin Xu, Huaiqing Zuo

TL;DR

The authors analyze Pfaffian ideals within the skew-symmetric matrix space $\mathcal{M}$ and prove the monodromy conjecture, holomorphy conjecture, and embedded Nash problem for the $m_0$-Pfaffian ideal $\mathcal{P}_{m_0}$ across all admissible $m_0$. They develop a detailed combinatorial description of contact loci via partitions $\bm{\lambda}$, establish a canonical log resolution, and compute both motivic/topological zeta functions and Verdier monodromy eigenvalues, linking poles to eigenvalues. The results yield explicit pole data and eigenvalues, resolve the embedded Nash problem in this Pfaffian setting, and confirm the conjectures in full generality for these ideals. This advances the understanding of singularity invariants for Pfaffian ideals and provides a complete picture of the monodromy/holomorphy landscape in this context.

Abstract

We resolve the monodromy conjecture, holomorphy conjecture, and embedded Nash problem for Pfaffian ideals.

On the monodromy conjecture, holomorphy conjecture, and embedded Nash problem for Pfaffian ideals

TL;DR

The authors analyze Pfaffian ideals within the skew-symmetric matrix space and prove the monodromy conjecture, holomorphy conjecture, and embedded Nash problem for the -Pfaffian ideal across all admissible . They develop a detailed combinatorial description of contact loci via partitions , establish a canonical log resolution, and compute both motivic/topological zeta functions and Verdier monodromy eigenvalues, linking poles to eigenvalues. The results yield explicit pole data and eigenvalues, resolve the embedded Nash problem in this Pfaffian setting, and confirm the conjectures in full generality for these ideals. This advances the understanding of singularity invariants for Pfaffian ideals and provides a complete picture of the monodromy/holomorphy landscape in this context.

Abstract

We resolve the monodromy conjecture, holomorphy conjecture, and embedded Nash problem for Pfaffian ideals.

Paper Structure

This paper contains 4 sections, 13 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Contact loci
  4. Monodromy conjecture

Key Result

Theorem 1.1

Let $\mathcal{M}=M_m^{\mathrm{skew}}(\mathbb C)$ be the space of $m\times m$ skew-symmetric matrices and $\mathcal{P}_{m_0}$ be the $m_{0}$-th Pfaffian ideal for every $1\le m_{0}\le \lfloor m/2 \rfloor$, then: In particular, the monodromy conjecture and holomorphy conjecture hold for $(\mathcal{M},\mathcal{P}_{m_0})$.

Theorems & Definitions (28)

  • Theorem 1.1: Monodromy conjecture and holomorphy conjecture for Pfaffian ideals
  • Remark 1.2
  • Theorem 1.3: Embedded Nash problem for Pfaffian ideals
  • Theorem 2.1: ELM04Cohomology_of_Contact_Loci
  • Definition 2.2: Grothendieck ring
  • Theorem 2.3: PV10
  • Conjecture 2.4: Monodromy conjecture, DL98PV10
  • Conjecture 2.5: Holomorphy conjecture
  • Definition 3.1
  • Proposition 3.2
  • ...and 18 more