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

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43 • Vanishing. If i > d then the Chern class [S i (L)] vanishes. As follows from Lemma 3.11 even more precise statement is true. If i > d(π), then the Chern class [S i (L)] vanishes. • Dimensions. If i ≤ d(π), then the i-th Chern class [S i (L)] has the codimension i (see Lemma 3.11). In compact situation Definition 1 would rather be a definition of homology cycles dual to the usual Chern classes. The class [S i (L)] can also be viewed as a linear functional on C i (G): on each cycle Y ∈ C ∗ (G) of dimension i the Chern class S i (L) takes the value (S i (L), Y ). The remaining properties follow directly from the description of the Chern classes given in the proof of Lemma 3.6. • Pull-back. Let ϕ : G 1 → G 2 be a homomorphism of two reductive groups G 1 and G 2 , and let L be an equivariant vector bundle on G 2 corresponding to a representation π. The pull-back ϕ ∗ L is an equivariant fiber bundle on G 1 corresponding to the representation π ◦ ϕ. The Chern classes of L and of ϕ ∗ L are related as follows: [S i (ϕ ∗ L)] = ϕ −1 [S i (L)]. In particular, let us apply this formula to the homomorphism π : G → π(G). We immediately get that if the representation π : G → End(V ) corresponding to a vector bundle L has a nontrivial kernel, then S i (L) are invariant under left and right multiplications by elements of the kernel. • Line bundles. Suppose that L has rank 1. Then L corresponds to a character π : G → C ∗ . The first Chern class of L coincides with the class of a hypersurface {π(g) = 1}.
44 3.4 Proof of Theorem 3.1 (an adjunction formula) In this section, I will use only the Chern classes S i = S i (TG) of the tangent bundle. The proof is a modification of Bernstein’s proof of formula (1) in the torus case. I will use the same ideas. Denote by f a linear functional which defines the hyperplane H, i.e. H = {x ∈ End(V ) : f(x) = C} for some constant C. It follows from noncompact Morse theory [18] that (−1) n−1 χ(π) is equal to the number of critical points of f restricted to π(G) (see [22] for the proof of exactly this statement by various methods). Let us express the number of the critical points in terms of the degrees of π(S i ). Recall that vector fields v 1 , . . . , v n on G were given by the formula v i (g) = X i g − gY i . Note that for any representation π of the group G the direct images π ∗ v i are well-defined (they are given by the formula dπ(X i )π(g) − π(g)dπ(Y i ), where dπ : g → End(V ) is the derived map) and can be straightforwardly extended to the linear vector fields on the whole End(V ). Namely, for x ∈ End(V ) set π ∗ v i (x) = dπ(X i )x−xdπ(Y i ). Abusing notation I will denote vector fields π ∗ v i again by v i and will write S i instead of π(S i ) during the proof. Since S 1 has a codimension 1 in G one can always choose a hyperplane H in such a way that all critical points of f restricted to G lie outside S 1 . Clearly, each critical point x satisfy n linear equations of the form f(v i (x)) = 0. The converse is also true for points outside S 1 , since at these points vector fields v 1 , . . . , v n span the tangent space to G. Denote by V n (f) a vector subspace given by the equations f(v i (x)) = 0 for i = 1, . . . , n. It has codimension n. Denote by S(f) the set of the critical points of the function f restricted to G. We get that S(f) = G ∩ V n (f) \ S 1 ∩ V n (f). It turns out that if f is generic, then all the intersection points of G ∩ V n (f) are transverse, and their number equals to the degree of G. Moreover, the following statement is true.
Page 1 and 2: A GAUSS-BONNET THEOREM, CHERN CLASS
Page 3 and 4: a group action to extend to noncomp
Page 5 and 6: Table of Contents Acknowledgement i
Page 7 and 8: 2 Theorem 1.1. [26] If F is equivar
Page 9 and 10: 4 a subvariety X along the stratum
Page 11 and 12: 6 subvarieties of the group G (see
Page 13 and 14: Chapter 2 A Gauss-Bonnet theorem fo
Page 15 and 16: 10 [15]. The Gaussian degrees of X
Page 17 and 18: 12 2.2 Preliminaries Gaussian degre
Page 19 and 20: 14 Proposition 2.5. If X is a compl
Page 21 and 22: 16 The numbers e(α, β) are useful
Page 23 and 24: 18 Any reductive group G admits an
Page 25 and 26: 20 where gtg −1 = x. By calculati
Page 27 and 28: 22 l −1 (ε) ∩ X β of the stra
Page 29 and 30: 24 C X the constant sheaf on X exte
Page 31 and 32: 26 of the subvariety π(G) in End(V
Page 33 and 34: 28 because of commutativity). The a
Page 35 and 36: 30 group acting on the faces of the
Page 37 and 38: 32 dense subset of H such that for
Page 39 and 40: 34 is trivial). However, they prese
Page 41 and 42: 36 bundle L. Proof. The idea of the
Page 43 and 44: 38 then the subvariety h 1 S i (U d
Page 45 and 46: 40 conjugation by an element (g 1 ,
Page 47: 42 to the degree of D. If i > d(π)
Page 51 and 52: 46 Proposition 3.14. Suppose that t
Page 53 and 54: 48 C π (S i+1 ) in C π (S i ) is
Page 55 and 56: 50 elements of π are polynomials i
Page 57 and 58: 52 [12] I.M. Gelfand, M.M. Kapranov

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

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