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

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of the subvariety π(G) in End(V )
χ(π) = (−1) n−1 deg π(G). (1)
• For the degree deg π(G) there is an explicit formula proved by Koushnirenko.
However, when G is arbitrary reductive group, only the second fact survives.
B.Kazarnovskii found an explicit formula for the degree deg π(G) that generalizes
Koushnirenko’s formula [23]. Later M.Brion established an analogous result for all
spherical homogeneous spaces [2].
As for the first fact, it is already wrong for SL 2 (C). K.Kaveh in his thesis computed
explicitly χ(π) and deg π(G) for all representations π of SL 2 (C) . His computation
shows that, in general, there is a discrepancy between these two numbers.
Kaveh also listed some special representations of reductive groups, for which these
numbers still coincide [22].
In this chapter, I will present a formula that generalizes formula (1) to the case
of arbitrary reductive groups. To do this I will construct subvarieties S i ⊂ G, whose
degrees fill the gap between the Euler characteristic and the degree. My construction
reminds the construction of the Chern classes of a vector bundle. The subvarieties
S i can be thought of as the Chern classes of the tangent bundle of G. I will also
construct the Chern classes of more general equivariant vector bundles over G (section
3.3). These Chern classes are in many aspects similar to the usual Chern classes of
compact manifolds. There is an analog of cohomology ring of G, where the Chern
classes of equivariant bundles live. This analog is the ring of conditions constructed
by De Concini and Procesi [5, 3](see section 3.2 for a brief reminder). It is useful in
solving enumerative problems. The intersection index in this ring is well-defined. In
particular, it makes sense to speak of the degree of π(S i ) in End(V ).
Denote by n and k the dimension and the rank of G, respectively. In the case of the
tangent bundle, it turns out (see Lemma 3.11) that the subvarieties S i are nonempty