Characteristic cycles

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 ⁰_SS(F) (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 ^b_ℝ−c (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.