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

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C X the constant sheaf on X extended by 0 to G. Let us compute the coefficients of the
characteristic cycle of C X . Fix some Whitney stratification S of X. Then Theorem
2.7 gives the following formula for the multiplicity of the characteristic cycle CC(C X )
along a stratum X α ∈ S
∑
c α (C X ) = (−1) dimX α+1
e(α, β)χ β (C X ).
X α ⊂X β
It is easy to see that the local Euler characteristic of C X at a point x is equal to the
Euler characteristic with compact support of a small open neighborhood of x. As
follows from the proof of Proposition 2.5, it equals to 1 for all x ∈ X. Thus we get
that c α coincides with (−1) dimXα+1 (−1 + e α ), where e α is the Euler characteristic of
the total complex link of X α .
Corollary 2.18. If X ⊂ G is a closed subvariety invariant under the adjoint action,
then the topological Euler characteristic of X can be computed as follows
χ(X) = ∑ (−1) dimXα (1 − e α )gdeg(X α ).
We now compute the Euler characteristic of sheaves with special characteristic
cycles. Namely, assume that the multiplicity c α of the characteristic cycle CC(F)
along a stratum X α is nonzero only if the stratum X α is nonsemisimple. Then by
Corollary 2.13 the Gaussian degree of X α is zero. Hence Theorem 2.1 immediately
implies the following corollary.
Corollary 2.19. If the characteristic cycle of F is supported on the set of nonsemisimple
elements of the group G, then the Euler characteristic of F vanishes.