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

More documents

Recommendations

Info

17 coincides with O a ∩ T , we obtain that χ(O a ) = |W |/|Stab a|, where Stab a ⊂ W is the stabilizer of a in W . If a is not semisimple, then χ(O a ) = 0. Let F be a constructible complex on G. Proposition 2.9. Suppose that F is equivariant under the adjoint action of G. Let F T be a restriction of F onto T ⊂ G. Then the sheaves F and F T have the same Euler characteristic: χ(G, F) = χ(T, F T ). Proof. The sheaves F and F T have the same local Euler characteristic at a point x ∈ T , since F T is the restriction of F onto T . Thus the Euler characteristic χ(T, F T ) is equal to ∫ χ(F)dχ by Proposition 2.6. The function χ(F) is invariant under the T adjoint action of G, and Proposition 2.8 implies ∫ ∫ χ(F)dχ = χ(F)dχ. T G The last integral is equal to the Euler characteristic χ(G, F) by Proposition 2.6. 2.4 Gaussian degree of invariant subvarieties We now compare the Gaussian degrees of X in G and of X ∩ T in T . Clearly, the Gaussian degree of a k-dimensional subvariety is equal to the sum of the Gaussian degrees of its k-dimensional irreducible components. Thus we can assume that X is irreducible. There are two cases: the set of all nonsemisimple elements of X has codimension either 0 or at least 1. In what follows we prove that in the first case gdeg(X) = 0, and in the second case gdeg(X) = gdeg(X ∩ T ). In particular, the Gaussian degree of any orbit O a ⊂ G coincides with the Gaussian degree of its intersection with the maximal torus T .
18 Any reductive group G admits an embedding in GL N (C) for some N, such that the inner product tr(Y 1 Y 2 ) is nondegenerate on Lie algebra g. Let us fix such an embedding. Then g may be identified with the space of all left-invariant differential 1-forms on G: an element S ∈ g gives rise to a 1-form ω by the formula ω(Y ) = tr(x −1 Y S), (1) where x ∈ G and Y ∈ T x G. We will call such a form generic, if S is regular semisimple. Lemma 2.10. All generic left invariant 1-forms form an open dense subset in the space of all left-invariant 1-forms. Proof. All regular semisimple elements form an open dense subset in g. This implies the statement of the lemma. Proposition 2.11. The Gaussian degree of an orbit O a is equal to the number of the intersection points O a ∩ T . In particular, if a is a nonsemisimple element, then deg(O a ) = 0. Proof. Consider the map ϕ : G → O a ; ϕ : g ↦→ gag −1 . Since ϕ is smooth and surjective, the tangent space T x O a is the image of the induced map dϕ. A simple computation shows that T x O a = [g, x]. Let ω be a generic leftinvariant differential 1-form on G given by the formula (1). Then ω = 0 on T x O a is equivalent to tr(x −1 Y xS − Y S) = 0 for any Y ∈ g. Since the form tr(Y 1 Y 2 ) is Ad G-invariant, we have tr(x −1 Y xS −Y S) = tr(Y (xSx −1 −S)). The form tr(Y 1 Y 2 ) is nondegenerate on g. Thus x and S commute, and x belongs to some maximal torus T S that depends on S.
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: 16 The numbers e(α, β) are useful
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 and 48: 42 to the degree of D. If i > d(π)
Page 49 and 50: 44 3.4 Proof of Theorem 3.1 (an adj
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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?