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

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39 d consisting of vectors (X, π(g)X) for X ∈ V . Hence, there is a map ϕ L from the group G to the Grassmannian G(d, 2d) of d-dimensional subspaces in a 2d-dimensional vector space ϕ L : G → G(d, 2d); ϕ L : g ↦→ Λ g . Note that the restriction to ϕ L (G) ≃ G of the tautological quotient vector bundle over G(d, 2d) is isomorphic to L. Remark 3.9. This construction repeats the following well-known construction (see [17]). Let M be a smooth variety, S be a vector bundle of rank d over M, and Γ(S) be a subspace of dimension N in the space of all global sections of S. Suppose that at each point x ∈ M the sections of Γ(S) span the fiber of S at the point x. Then one can map M to the Grassmannian G(N − d, N) by assigning to each point x ∈ M the subspace of all sections from Γ(L) that vanish at x. Clearly, the vector bundle S coincides with the pull-back of the tautological quotient vector bundle over the Grassmannian G(N − d, N). Denote by X L the closure of ϕ L (G) in G(d, 2d). This is a compact projective variety. It is equivariant under an action of G × G coming from its representation π ⊕ π in the space V ⊕ V . Namely, an element (g 1 , g 2 ) ∈ G × G takes a subspace Λ ⊂ V ⊕ V to a subspace (π(g 1 ), π(g 2 ))(Λ). The open dense G × G-orbit in X L is clearly isomorphic to π(G) ≃ G. Example 1. Demazure embedding. Let G be a group of adjoint type, and let π be its adjoint representation on the Lie algebra g. Then as I already noted the corresponding vector bundle L coincides with the tangent bundle of G. The corresponding embedding ϕ L : G → G(n, g ⊕ g) coincides with the one constructed by Demazure [4]. Demazure’s construction is as follows. An element x ∈ G goes to the Lie algebra of the stabilizer of x under the standard action of G×G. E.g. the identity element gets mapped to the Lie algebra g ⊂ g ⊕ g embedded diagonally. Then the
40 conjugation by an element (g 1 , g 2 ) ∈ G × G maps the Lie algebra g ⊂ g ⊕ g to the Lie algebra of the stabilizer of an element g 1 g −1 2 . It is now easy to see that the embedding ϕ L and Demazure embedding are the same. The compactification X L in this case is isomorphic to the wonderful compactification of G [4]. Example 2. a) Let G be GL(V ) and let π be its tautological representation on a space V of dimension d. Then ϕ L is an embedding of GL(V ) into the Grassmannian G(d, 2d). Notice that the dimensions of both varieties are the same. Hence, the compactification X L coincides with G(d, 2d). b) Take SL(V ) instead of GL(V ) in the previous example. Its compactification X L is a hypersurface in the Grassmannian G(d, 2d) which can be described as follows. Consider the Plücker embedding p : G(d, 2d) → P(Λ d (V 1 ⊕ V 2 )) (V 1 and V 2 are two copies of V ). Then p(X L ) is a special hyperplane section of p(G(d, 2d)). Namely, the decomposition V 1 ⊕ V 2 yields a decomposition of Λ d (V 1 ⊕ V 2 ) into the direct sum. This sum contains two one-dimensional components p(V 1 ) and p(V 2 ) (which are considered as lines in Λ d (V 1 ⊕ V 2 )). In particular, for any vector in Λ d (V 1 ⊕ V 2 ) it makes sense to speak of its projections to p(V 1 ) and p(V 2 ). On V 1 and V 2 there are two special n−forms, preserved by SL(V ). These forms give rise to two 1-forms l 1 and l 2 on p(V 1 ) and p(V 2 ), respectively. Consider a hyperplane H in Λ d (V 1 ⊕ V 2 ) consisting of all vectors v such that the functionals l 1 , l 2 take the same values on the projections of v to p(V 1 ) and p(V 2 ), respectively. Then it is easy to check that p(X L ) = p(G(d, 2d)) ∩ P(H). Denote by L X a restriction to X L of the tautological quotient vector bundle U d over the Grassmannian G(d, 2d). Assume that X L is smooth. Let c 1 , . . . , c d ∈ H ∗ (X L ) be the subvarieties dual to the Chern classes of L X . I will also call them Chern classes. Proposition 3.10. The homology class of the closure of ϕ L (S i (L)) in X L coincides
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: 38 then the subvariety h 1 S i (U d
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

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