Sheaves on Manifolds pp 360-410 | Cite as

# Characteristic cycles

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## Summary

On a real analytic manifold *X*, we first introduce the notions of subanalytic chains and subanalytic cycles, with the help of the dualizing complex *ω* _{ X }. Then we define the intersection of two cycles.

If *F* is an ℝ-constructible object of **D** ^{ b }(*X*), we construct the characteristic cycle of *F*, CC(*F*), as the natural image of id_{ F } ∈ Hom(*F, F*) in *H* _{SS(F)} ^{0} (*T***X*; *π* ^{−1} *ω* _{ X }. This is a “Lagrangian cycle”. For example, if *M* is a closed submanifold of *X* and if *V* ∈ Ob(**D** ^{ b }(Mod^{ f } (*k*))) (the base ring *A* is now a field *k* of characteristic zero) then CC(*V* _{ M }) = *m*[*T* _{ M } ^{*} *X*], where [*T* _{ M } ^{*} *X*] is the Lagrangian cycle associated to the conormal bundle *T* _{ M } ^{*} *X* to *M* in *X* and *m* = *χ*(*V*) = ∑_{ j }(−1)^{ j } dim *H* ^{ j }(*V*). We study some functorial properties of characteristic cycles, such as non-characteristic inverse images or proper direct images, and we prove in particular that if *F* has compact support, the Euler-Poincaré index *χ*(*X*; *F*) = ∑^{ j }(−1)^{ j } dim *H* ^{ j }(*X; F*) can be obtained as the intersection number of CC(*F*) and the cycle associated to the zero-section of *T***X*. We also give a local Euler-Poincaré index formula. These indices can be calculated using a “Morse function with respect to SS(*F*)”.

Next we study a version of the “Lefschetz fixed point formula”. If *f* is an endomorphism of *X*, and if a morphism *φ* ∈ Hom(*f* ^{−1} *F, F*) is given, one defines a characteristic class C(*φ*) ∈ *H* ^{0}(*X*; *ω* _{ X }) whose degree calculates the trace tr(*φ*) of *Γ*(*X*; *φ*) ∈ Hom(*RΓ*(*X*; *F*), *RΓ*(*X*, *F*)), (one assumes *F* has compact support). When *f* has only finitely many fixed points and is transversal to the identity, we show how to deduce tr(*φ*) from a local formula.

Finally we study the Grothendieck group of **D** _{ℝ−c } ^{ b } (*X*). We prove that this group is isomorphic to the group of Lagrangian cycles on *T***X* (the isomorphism being defined by *F* ↦ CC(*F*)) as well as to the group of ℝ-constructible functions on *X* (the isomorphism is defined by *F* ↦ *χ*(*F*)(*x*) = *χ*(*F* _{ x })). This gives rise to a new calculus on the algebra of constructible functions on *X*.

We keep Conventions 8.0. Moreover in Section 9.1, and from Section 9.4 until the end of the chapter, the base ring *A* will be a (commutative) field of characteristic zero, and we will denote it by *k*.

## Keywords

Exact Sequence Vector Bundle Commutative Diagram Real Analytic Function Grothendieck Group## Preview

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