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

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Theorem 1.1. [26] If F is equivariant under the adjoint action of G, then its Euler
characteristic can be computed in terms of its characteristic cycle ∑ c α T ∗ X α
G by the
following formula
χ(G, F) = ∑ c α gdeg(X α ),
where coefficients c α are the multiplicities of the characteristic cycle.
In other words, the Euler characteristic of a sheaf is equal to the intersection
index of the characteristic cycle with the graph of a generic left-invariant 1-form on
the group G. The Gauss map provides a definition of such intersection index in the
noncompact situation. In this form, Theorem 1.1 is very similar to the well-known
Kashiwara’s index theorem, which holds for compact manifolds.
Theorem 1.1 relates two geometric invariants of a sheaf: the Euler characteristic
and the characteristic cycle. In some cases this relation can be used to estimate one
of these invariants if the other is known (see Corollary 1.3). This works especially
well if the sheaf F is perverse (see Corollary 1.2). In this case the multiplicities c α of
the characteristic cycle are nonnegative.
Theorem 1.1 gives a tool to classify adjoint equivariant perverse sheaves with a
given Euler characteristic. E.g. if the Euler characteristic is small then the characteristic
cycle cannot have components with large Gaussian degrees. This gives some
essential restrictions on the characteristic cycle. One of my future plans is to explore
the geometry of subvarieties with a given Gaussian degree and to give a geometric
description of perverse sheaves with a given Euler characteristic. I am especially
interested in perverse sheaves with Euler characteristic 1. When G is a complex
torus such sheaves (if irreducible) are identified with complexes of solutions of the
generalized hypergeometric systems [11]. Irreducible hypersurfaces in (C ∗ ) n of the
Gaussian degree 1 are identified with discriminantal hypersurfaces of the generalized
hypergeometric functions [19].