Characteristic cycles of constructible sheaves

Abstract

In his paper [K], Kashiwara introduced the notion of characteristic cycle for complexes of constructible sheaves on manifolds: let X be a real analytic manifold, and F a complex of sheaves of C-vector spaces on X , whose cohomology is constructible with respect to a subanalytic strati cation; the characteristic cycle CC(F) is a subanalytic, Lagrangian cycle (with in nite support, and with values in the orientation sheaf of X ) in the cotangent bundle T ∗X . The de nition of CC(F) is Morse-theoretic. Heuristically, CC(F) encodes the in nitesimal change of the Euler characteristic of the stalks of F along the various directions in X . It tends to be di cult in practice to calculate CC(F) explicitly for all but the simplest complexes F; on the other hand, the characteristic cycle construction has good functorial properties. The behavior of CC(F) with respect to the operations of proper direct image, Verdier duality, and non-characteristic inverse image of F is well understood [KS]. In this paper, we describe the e ect of the operation of direct image by an open embedding. Combining our result with those that were previously known, we obtain descriptions of CC(Rf∗F) and CC(f∗F) – analogous to those in [KS] – for arbitrary morphisms f : X → Y in the semi-algebraic category, and complexes F with semi-algebraically constructible cohomology. In e ect, this provides an axiomatic characterization of the functor CC, at least in the semi-algebraic context. Our arguments do apply more generally in the subanalytic case, but because statements become quite convoluted, we shall not strive for the greatest degree of generality. As a concrete application, we consider the case of the ag manifold X of a complex semisimple Lie algebra g. Here the Weyl group W of g operates,