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The Motivic Picard--Lefschetz Formula

Ran Azouri, Emil Jacobsen

TL;DR

This work provides a fully motivic enhancement of the Picard–Lefschetz formula by constructing and analyzing the motivic nearby cycles $\\Psi_f\\bm{1}$ and its monodromy within Ayoub’s motivic framework. The authors implement a sequence of reductions to convert a general quasi-homogeneous singularity into a semistable, ultimately one-dimensional, model, and they translate the monodromy problem into the study of the nearby Kummer motive $\\Psi_{\mathrm{id}}\mathcal{K}$, obtaining the key scalar $\\lambda=-1$. The resulting formula expresses $o^*\\Psi_f\\bm{1}$ and $o^!\\Psi_f\\bm{1}$ in terms of explicit motivic data from projective/quadratic hypersurface embeddings, enabling a concrete description in the quadratic case via Rost’s quadrics. Overall, the paper unifies motivic nearby cycles, monodromy, and singularity data into a coherent, realization-free framework with explicit quadratic refinements, enhancing classical PL theory in the motivic setting.

Abstract

We prove a motivic enhancement of the classical Picard--Lefschetz formula. Our proof is completely motivic, and yields a description of the motivic nearby cycles at a quasi-homogeneous singularity, as well as its monodromy, in terms of an embedding of projective hypersurfaces.

The Motivic Picard--Lefschetz Formula

TL;DR

This work provides a fully motivic enhancement of the Picard–Lefschetz formula by constructing and analyzing the motivic nearby cycles and its monodromy within Ayoub’s motivic framework. The authors implement a sequence of reductions to convert a general quasi-homogeneous singularity into a semistable, ultimately one-dimensional, model, and they translate the monodromy problem into the study of the nearby Kummer motive , obtaining the key scalar . The resulting formula expresses and in terms of explicit motivic data from projective/quadratic hypersurface embeddings, enabling a concrete description in the quadratic case via Rost’s quadrics. Overall, the paper unifies motivic nearby cycles, monodromy, and singularity data into a coherent, realization-free framework with explicit quadratic refinements, enhancing classical PL theory in the motivic setting.

Abstract

We prove a motivic enhancement of the classical Picard--Lefschetz formula. Our proof is completely motivic, and yields a description of the motivic nearby cycles at a quasi-homogeneous singularity, as well as its monodromy, in terms of an embedding of projective hypersurfaces.
Paper Structure (23 sections, 34 theorems, 114 equations)

This paper contains 23 sections, 34 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Proof strategy.
  3. Quadratic singularities.
  4. Historical context.
  5. Acknowledgements.
  6. Preliminaries
  7. Notations and conventions
  8. Singularities and semi-stable reduction
  9. Motives
  10. The Kummer motive
  11. Nearby cycles
  12. Monodromy and variation
  13. First reduction: to a semistable family
  14. The variation map and semistable reduction
  15. The monodromy of a special semistable family
  16. ...and 8 more sections

Key Result

Theorem 1.1

Assume the characteristic of $k$ is different from $2$. Let $X$ be regular and let $f \colon X \to \mathbb{A}^1$ be a flat, quasi-projective morphism. Assume $f$ is smooth except for an isolated homogeneous singularity $o \in X_0$, defined by a homogeneous polynomial $F$ of degree $r$. With notation such that the following diagram commutes: \begin{tikzcd} o^* \Psi_f \one \arrow[d, "\sim" {rotate=

Theorems & Definitions (83)

  • Theorem 1.1: The general Picard--Lefschetz formula, Thm. \ref{['thm:abstractPL']}
  • Remark 1.2
  • Theorem 1.3: The quadratic Picard--Lefschetz formula, Thm. \ref{['thm:quadratic_PL']}
  • Remark 1.4
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Proposition 2.7
  • ...and 73 more