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Introduction to the monodromy conjecture

Willem Veys

Abstract

The monodromy conjecture is a mysterious open problem in singularity theory. Its original version relates arithmetic and topological/geometric properties of a multivariate polynomial $f$ over the integers, more precisely, poles of the $p$-adic Igusa zeta function of $f$ should induce monodromy eigenvalues of $f$. The case of interest is when the zero set of $f$ has singular points. We first present some history and motivation. Then we expose a proof in the case of two variables, and partial results in higher dimension, together with geometric theorems of independent interest inspired by the conjecture. We conclude with several possible generalizations.

Introduction to the monodromy conjecture

Abstract

The monodromy conjecture is a mysterious open problem in singularity theory. Its original version relates arithmetic and topological/geometric properties of a multivariate polynomial over the integers, more precisely, poles of the -adic Igusa zeta function of should induce monodromy eigenvalues of . The case of interest is when the zero set of has singular points. We first present some history and motivation. Then we expose a proof in the case of two variables, and partial results in higher dimension, together with geometric theorems of independent interest inspired by the conjecture. We conclude with several possible generalizations.
Paper Structure (23 sections, 23 theorems, 68 equations, 3 figures)

This paper contains 23 sections, 23 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. History
  3. Archimedean zeta functions
  4. Relation with monodromy
  5. $p$-adic zeta functions
  6. The conjecture; $p$-adic version
  7. More ' Igusa type' zeta functions and formulas via resolution
  8. Denef's formula
  9. The topological zeta function
  10. The motivic zeta function
  11. Monodromy and b-function
  12. The case of plane curves ($n=2$)
  13. Non-contribution
  14. Proof of the conjecture
  15. Structure of resolution graph and determination of all poles
  16. ...and 8 more sections

Key Result

theorem 1

Let $f \in {\mathbb R} [x_1, \dots, x_n] \setminus {\mathbb R}$ and $\varphi: {\mathbb R}^n \to {\mathbb C}$ a $C^\infty$ function with compact support. Then $Z(f,\varphi;s)$ has a meromorphic continuation to ${\mathbb C}$. Moreover, if $h$ is an embedded resolution of $\{f=0\}$ with numerical data

Theorems & Definitions (56)

  • theorem 1
  • theorem 2: Bernstein
  • theorem 3
  • remark 1
  • proposition 1
  • theorem 4
  • corollary 1
  • definition 1
  • remark 2
  • remark 3
  • ...and 46 more