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Variation of Archimedean Zeta Function and $n/d$-Conjecture for Generic Multiplicities

Quan Shi, Huaiqing Zuo

TL;DR

The article develops the variation of the archimedean zeta function to study the Bernstein-Sato framework under multiplicative deformations $f_1^{a_1}\cdots f_r^{a_r}$, proving that poles persist generically and yield roots of the corresponding $b$-functions. It shows the $n/d$-conjecture holds for generic multiplicities in central essential indecomposable hyperplane arrangements, which in turn implies the strong monodromy conjecture in this generic regime. The construction hinges on a meromorphic extension of the variation zeta function to $\\mathbb C^{r+1}$ with a structured pole set and a self-scaling symmetry, together with a careful analysis of residues in the two-dimensional case. The results provide both a general, technique-driven route to generic monodromy phenomena for hyperplane arrangements and a concrete two-dimensional instance with explicit residue signs, illustrating the mechanism behind these conjectures and their interplay with log resolutions and dense edges.

Abstract

For $f_1,...,f_r\in \mathbb C[z_1,...,z_n]\setminus \mathbb C$, we introduce the variation of archimedean zeta function. As an application, we show that the $n/d$-conjecture, proposed by Budur, Mustaţă, and Teitler, holds for generic multiplicities. Consequently, strong monodromy conjecture holds for hyperplane arrangements with generic multiplicities as well.

Variation of Archimedean Zeta Function and $n/d$-Conjecture for Generic Multiplicities

TL;DR

The article develops the variation of the archimedean zeta function to study the Bernstein-Sato framework under multiplicative deformations , proving that poles persist generically and yield roots of the corresponding -functions. It shows the -conjecture holds for generic multiplicities in central essential indecomposable hyperplane arrangements, which in turn implies the strong monodromy conjecture in this generic regime. The construction hinges on a meromorphic extension of the variation zeta function to with a structured pole set and a self-scaling symmetry, together with a careful analysis of residues in the two-dimensional case. The results provide both a general, technique-driven route to generic monodromy phenomena for hyperplane arrangements and a concrete two-dimensional instance with explicit residue signs, illustrating the mechanism behind these conjectures and their interplay with log resolutions and dense edges.

Abstract

For , we introduce the variation of archimedean zeta function. As an application, we show that the -conjecture, proposed by Budur, Mustaţă, and Teitler, holds for generic multiplicities. Consequently, strong monodromy conjecture holds for hyperplane arrangements with generic multiplicities as well.

Paper Structure

This paper contains 9 sections, 23 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. $b-$function
  4. Archimedean zeta function
  5. Hyperplane arrangement and n/d-conjecture
  6. Variation of archimedean zeta function
  7. Definition and meromorphy
  8. Generic survival of poles and $n/d$-conjecture for generic multiplicities
  9. Dimension two case

Key Result

Theorem 1.2

For $f\in \mathbb C[z_1,...,z_n]\setminus \mathbb C$, let $\mu : Y \to {\mathbb C}^n$ be a log resolution of $f$. Suppose $\mu^*(\mathrm{div}\, f) = \sum_{i\in S} N_i D_i$ and $K_\mu = \sum_{i\in S}(\nu_i-1)D_i$, where $\{D_i\}_{i\in S}$ is a collection of simple normal crossings divisors and $K_\mu Here ${D_I^\circ = (\bigcap_{i\in I} D_i) \setminus (\bigcup_{j\in S\setminus I} D_j)}$, $[-]$ deno

Theorems & Definitions (63)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Conjecture 1.4: monodromy conjecture
  • Theorem 1.5
  • Conjecture 1.6: $n/d$-conjecture
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 53 more