Characteristic cycles of real and complex constructible sheaves, revisited

TL;DR

This paper extends the theory of characteristic cycles to a relative setting for real and complex constructible sheaves, providing formulas and methods to compute their microlocal properties in various contexts.

Contribution

It introduces the concept of relative characteristic cycles for sheaves under smooth morphisms, generalizing existing theory and enabling unified calculations of characteristic cycles and microlocal types.

Findings

Defined relative characteristic cycles parameterized by the base manifold.

Derived formulas for characteristic cycles of real nearby cycle sheaves.

Unified approach for calculating characteristic cycles of various constructible sheaves.

Abstract

For a smooth morphism $f: X \longrightarrow \Sigma$ 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 $\Sigma$ 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.