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Motivic nearby cycles and their monodromy at a singular point

Ran Azouri

TL;DR

This work develops a motivic framework to refine classical invariants of singularities. It uses semistable reduction for (quasi-)homogeneous isolated singularities to compute nearby cycles and defines quadratic refinements of key invariants, such as the Euler characteristic and Milnor number, valued in the Grothendieck–Witt ring $\mathrm{GW}(k)$. It then introduces the motivic monodromy operator $N_f$ and a motivic Picard–Lefschetz theory within $\mathrm{DM}_{\mathbb{Q}}(-)$, providing explicit formulas and reductions to semistable models, and relating these to Denef–Loeser motivic fibers and recent equivalences between motives with monodromy and rigid analytic motives. Overall, the paper connects topology, algebraic geometry, and motivic homotopy theory to produce computable, refined invariants for singularities in characteristic zero, with potential broad impact on how singularities are analyzed motivically.

Abstract

In this survey, we explain how to compute both the quadratic Euler characteristic of nearby cycles, and the motivic monodromy, at a quasi-homogeneous singularity. This gives, for such singularity, a quadratic refinement to the Deligne--Milnor formula in characteristic zero, and an enhancement of the Picard--Lefschetz formula to Voevodsky motives with rational coefficients.

Motivic nearby cycles and their monodromy at a singular point

TL;DR

This work develops a motivic framework to refine classical invariants of singularities. It uses semistable reduction for (quasi-)homogeneous isolated singularities to compute nearby cycles and defines quadratic refinements of key invariants, such as the Euler characteristic and Milnor number, valued in the Grothendieck–Witt ring . It then introduces the motivic monodromy operator and a motivic Picard–Lefschetz theory within , providing explicit formulas and reductions to semistable models, and relating these to Denef–Loeser motivic fibers and recent equivalences between motives with monodromy and rigid analytic motives. Overall, the paper connects topology, algebraic geometry, and motivic homotopy theory to produce computable, refined invariants for singularities in characteristic zero, with potential broad impact on how singularities are analyzed motivically.

Abstract

In this survey, we explain how to compute both the quadratic Euler characteristic of nearby cycles, and the motivic monodromy, at a quasi-homogeneous singularity. This gives, for such singularity, a quadratic refinement to the Deligne--Milnor formula in characteristic zero, and an enhancement of the Picard--Lefschetz formula to Voevodsky motives with rational coefficients.

Paper Structure

This paper contains 17 sections, 13 theorems, 46 equations.

Table of Contents

  1. Acknowledgements
  2. The Milnor fiber and nearby cycles
  3. In topology - Milnor
  4. In algebraic geometry -- Deligne
  5. In motivic homotopy theory -- Ayoub
  6. Recent work
  7. Isolated singularities and semistable reduction
  8. Semistable reduction
  9. Homogeneous singularities
  10. Quasi-homogeneous singularities
  11. The quadratic Euler characteristic of nearby cycles
  12. Quadratic invariants
  13. The quadratic Euler characteristic
  14. The quadratic Milnor number
  15. The formulas
  16. ...and 2 more sections

Key Result

Proposition 2.2

Assume $f:X \rightarrow S$ admits a semistable reduction $Y \xrightarrow{\pi} X_r \xrightarrow{f_r} S'$ for some $r$. Let $r:X_r\to X$, and let $g=f_r \circ \pi$. Then

Theorems & Definitions (39)

  • Example 1.3: blowup
  • Definition 2.1
  • Proposition 2.2: Az
  • proof
  • Proposition 2.3: Ay07a, Az
  • proof
  • Definition 2.4
  • Proposition 2.5: Az, AJ
  • proof : Sketch of proof
  • Proposition 2.6: AJ for $\operatorname{DM}(-)$
  • ...and 29 more