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Push-forward of smooth measures and strong Thom stratifications

Avraham Aizenbud, Nir Avni, Shahar Carmeli

Abstract

We study the collection of measures obtained via push-forward along a map between smooth varieties over p-adic fields. We investigate when the stalks of this collection are finite-dimensional. We provide an algebro-geometric criterion ensuring this property. This criterion is formulated in terms of a canonical subvariety of the cotangent bundle of the source of the map.

Push-forward of smooth measures and strong Thom stratifications

Abstract

We study the collection of measures obtained via push-forward along a map between smooth varieties over p-adic fields. We investigate when the stalks of this collection are finite-dimensional. We provide an algebro-geometric criterion ensuring this property. This criterion is formulated in terms of a canonical subvariety of the cotangent bundle of the source of the map.
Paper Structure (20 sections, 28 theorems, 61 equations)

This paper contains 20 sections, 28 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Background and Motivation
  4. Outline of the Proofs
  5. \ref{['thm:intro.thom.crit']}
  6. \ref{['thm:intro.main']}
  7. Questions
  8. Acknowledgments
  9. Stratifications
  10. Quasi-transitive morphisms and strong Thom Stratifications
  11. Construction of a Functorial Stratification
  12. Existence of coarse regular stratification
  13. Quasi-transitive morphisms admit strong Thom Stratifications
  14. Local structure theorem for strongly Thom stratified morphisms
  15. Idea of the proof
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\phi\colon X\tilde{o} Y$ be a morphism of smooth algebraic varieties defined over a $p$-adic field $F$. Assume that for every point $(x,v)\in \mathfrak{B}_{\phi}$ we have Then the stalks of $\phi_*\mathcal{M}_c(X)$ are finite-dimensional.

Theorems & Definitions (81)

  • Definition 1.2: the scheme $\mathfrak{B}_\phi$
  • Theorem 1: \ref{['thm:main']}
  • Theorem 2: \ref{['thm:trans_iff_thom']}
  • Example 1.3
  • Theorem 3: \ref{['thm:structure.Thom']}
  • Definition 1.4
  • Remark 1.8
  • Definition 2.1: Stratification
  • Remark 2.2
  • Definition 2.3
  • ...and 71 more