a gauss-bonnet theorem, chern classes and an adjunction formula

9
a smooth complex variety X one can define the following geometric invariants: the
global Euler characteristic χ(X, F) and the characteristic cycle CC(F).
The global Euler characteristic χ(X, F) is defined as the the Euler characteristic
of the complex of cohomology groups H i (X, F). The latter are obtained by taking
the derived functor of global sections of F (see [21]).
There exists a morphism from the derived category of constructible complexes on a
smooth complex variety X to the group of Lagrangian cycles in the cotangent bundle
T ∗ X (see [21] Section 9.4). It has nice functorial properties. The characteristic cycle
CC(F) of a constructible complex F is the image of F under this morphism.
A constructible complex F on X is called a perverse sheaf if it satisfies the following
two conditions. First, the local cohomology H i (F x ) are supported on a subset of
dimension at most −i. Second, the local cohomology H i c(F x ) with compact support
are supported on a subset of dimension at most i ([21, 18, 31]).
We now formulate the main results. Let G be a complex reductive group, and
let F be a constructible complex on G.
The characteristic cycle of F is a linear
combination of Lagrangian subvarieties CC(F) = ∑ c α T ∗ X α
G. Here and in the sequel
T ∗ XG denotes the closure of the conormal bundle to the smooth locus of a subvariety
X ⊂ G. With X one can associate a nonnegative number gdeg(X) called the Gaussian
degree of X. It is equal to the number of zeros of a generic left-invariant differential
1-form on G restricted to X. The precise definitions of the Gaussian degree and of
the Gauss map are given in section 2.2.
Theorem 2.1. If F is equivariant under the adjoint action of G, then its Euler
characteristic can be computed in terms of the characteristic cycle by the following
formula
χ(G, F) = ∑ c α gdeg(X α ).
For a perverse sheaf the multiplicities c α of its characteristic cycle are nonnegative