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Characteristic cycles of real and complex constructible sheaves, revisited

Ren Fernandes, Kazuki Kudomi, Kiyoshi Takeuchi

Abstract

For a smooth morphism $f: X \longrightarrow Σ$ of real analytic manifolds and an $\mathbb{R}$-constructible sheaf $F$ on $X$ satisfying some condition, we define a family of Lagrangian cycles parameterized by $Σ$ that we call the relative characteristic cycle of $F$ for $f$. In this way, the theory of characteristic cycles due to Kashiwara and Schapira is naturally extended to the relative setting. Based on it, we then prove a formula for the characteristic cycles of real nearby cycle sheaves. This leads us to obtain also formulas for the characteristic cycles of various constructible sheaves, such as specialization, microlocalization, and complex nearby and vanishing cycle sheaves, in a unified manner. In fact, our methods allow us to calculate not only their characteristic cycles but also their microlocal types in many situations. We will illustrate it by various examples.

Characteristic cycles of real and complex constructible sheaves, revisited

Abstract

For a smooth morphism of real analytic manifolds and an -constructible sheaf on satisfying some condition, we define a family of Lagrangian cycles parameterized by that we call the relative characteristic cycle of for . In this way, the theory of characteristic cycles due to Kashiwara and Schapira is naturally extended to the relative setting. Based on it, we then prove a formula for the characteristic cycles of real nearby cycle sheaves. This leads us to obtain also formulas for the characteristic cycles of various constructible sheaves, such as specialization, microlocalization, and complex nearby and vanishing cycle sheaves, in a unified manner. In fact, our methods allow us to calculate not only their characteristic cycles but also their microlocal types in many situations. We will illustrate it by various examples.
Paper Structure (10 sections, 28 theorems, 231 equations, 1 figure)

This paper contains 10 sections, 28 theorems, 231 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary notions and results
  3. Borel-Moore homology cycles in the subanalytic setting
  4. Limits of Borel-Moore homology cycles and their properties
  5. Intersection numbers of Borel-Moore homology cycles and limits
  6. Specialization and microlocalization functors
  7. Definition of relative characteristic cycles
  8. Real nearby cycle sheaves and their characteristic cycles
  9. Characteristic cycles of complex nearby and vanishing cycle sheaves
  10. Some results on commutative diagrams for the six operations

Key Result

Theorem 1.1

In the situation as above, we have in $T^\ast X_0$, where this equality holds true only over each relatively compact open subset of $X_0$.

Figures (1)

  • Figure :

Theorems & Definitions (50)

  • Theorem 1.1: see Theorem \ref{['thm-limformula']}
  • Proposition 2.1
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6: Schmid-Vilonen SV96
  • ...and 40 more