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On the stability of vanishing cycles of étale sheaves in positive characteristic

Tong Zhou

Abstract

In positive characteristic, in contrast to the complex analytic case, vanishing cycles are highly sensitive to test functions (the maps to the henselian traits). We study this dependence and show that on a smooth surface, this dependence is generically only up to a finite jet of the test functions. We conjecture that this continues to hold in higher dimensions. We also study the class of sheaves whose vanishing cycles have the strongest stability. Among other things, we show that tame simple normal crossing sheaves belong to this class, and this class is stable under the Radon transform.

On the stability of vanishing cycles of étale sheaves in positive characteristic

Abstract

In positive characteristic, in contrast to the complex analytic case, vanishing cycles are highly sensitive to test functions (the maps to the henselian traits). We study this dependence and show that on a smooth surface, this dependence is generically only up to a finite jet of the test functions. We conjecture that this continues to hold in higher dimensions. We also study the class of sheaves whose vanishing cycles have the strongest stability. Among other things, we show that tame simple normal crossing sheaves belong to this class, and this class is stable under the Radon transform.
Paper Structure (14 sections, 32 theorems, 8 equations, 2 figures, 2 tables)

This paper contains 14 sections, 32 theorems, 8 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Review
  3. Complex analytic context
  4. Algebraic context
  5. The stability of vanishing cycles
  6. The stability of dimtot($\phi$)
  7. The high-jet stability of $\phi$
  8. Transverse test families
  9. Generic finite depth
  10. $\mu c$ sheaves
  11. Basic objects
  12. Properties
  13. Examples
  14. Appendix

Key Result

Theorem 1.3

(kashiwara_microlocal_1985, Theorem phistability/C) Let $X$ be a complex analytic manifold, $\mathcal{F}\in D(X)$, $(x, \xi)$ a smooth point of $SS\mathcal{F}$. Then: i) For two ttfun's $f, g$ of $\mathcal{F}$ at $(x, \xi)$, there exists a (noncanonical) isomorphism $\phi_f(\mathcal{F})_x\cong \phi_

Figures (2)

  • Figure 1: A 2-stage blowup sequence on a surface
  • Figure 2: Possible configuration changes of fixed points

Theorems & Definitions (79)

  • Definition 1.1: transverse test function
  • Definition 1.2: transverse test family
  • Theorem 1.3
  • Definition 1.5: depth of $\mathcal{F}$
  • Theorem 1.6: Theorems \ref{['thmstab']}, \ref{['thmstabbeau']}
  • Conjecture 1.7: Conjectures \ref{['finitedepth']}, \ref{['finitedepthrep']}
  • Theorem 1.8: Proposition \ref{['tamearemuc']}, Lemma \ref{['phiindepofttfun']} ii), Corollary \ref{['Radoncompat']}
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • ...and 69 more