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

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29 group, and let M be its homogeneous space. The action of G on M is called spherical, if a Borel subgroup of G has an open dense orbit in M. In this case, the homogeneous space M is also called spherical. An important and very useful property, which characterizes a spherical homogeneous space, is that any its compactification equivariant under the action of G contains only finite number of orbits [36, 30]. There is a natural action of the group G×G on G by left and right multiplications. Namely, an element (g 1 , g 2 ) ∈ G × G maps an element g ∈ G to g 1 gg −1 2 . This action is spherical as follows from the Bruhat decomposition of G with respect to some Borel subgroup. Thus the group G can be considered as a spherical homogeneous space of the doubled group G × G with respect to this action. For any representation π : G → End(V ) this action can be obviously extended to the action of π(G)×π(G) on the whole End(V ) by left and right multiplications. I will call such actions standard. Equivariant compactifications. With any representation π one can associate the following compactification of π(G). Take a cone over π(G) (consisting of all points x ∈ End(V ) such that λ·x belongs to π(G) for some λ ∈ C ∗ ), take its projectivization and then take its closure in P(End(V )). We obtain a compact projective variety X π ⊂ P(End(V )) with a natural action of G × G coming from the standard action of π(G) × π(G) on End(V ). Below I will list some important properties of this variety. Without loss of generality one can assume that π(G) is isomorphic to G. Fix a maximal torus T ⊂ G. Consider all weights of representation π, i.e. all characters of the maximal torus T occurring in π. Take their convex hull P π in the lattice of all characters of T . Then it is easy to see that P π is a polytope invariant under the action of the Weyl group of G. It is called the weight polytope of the representation π. The polytope P π contains a lot of information about the compactification X π . Theorem 3.2. 1) [20] The subvariety X π consists of the finite number of G × G- orbits. These orbits are in one-to-one correspondence with the orbits of the Weyl
30 group acting on the faces of the polytope P π . The codimension of an orbit in X π equals to the codimension of the corresponding face in P π . 2)[5] Let ρ be another representation of G. The subvarieties X π and X ρ are isomorphic if and only if the normal fans corresponding to the polytopes X π and X ρ coincide. If the first fan is a subdivision of the second, then there exists the equivariant map X π → X ρ . In particular, suppose that the group G is of adjoint type, i.e. it has a trivial center. Then G has one distinguished faithful representation, namely, the adjoint representation Ad : G → End(g); Ad(g)X = gXg −1 , X ∈ g. The corresponding compactification X Ad of the group G is called the wonderful compactification. It was introduced by De Concini and Procesi [5]. The wonderful compactification is smooth, and X Ad \ G is a divisor with normal crossings. There are k orbits O 1 , . . . , O k of codimension 1 in X Ad . The other orbits are obtained as the intersections of the closures O 1 , . . . , O k . More precisely, to any subset {i 1 , i 2 , . . . , i m } ⊂ {1, . . . , n} corresponds an orbit O i1 ∩ O i2 ∩ . . . ∩ O im of codimension m. So the number of orbits equals to 2 k . There is a unique closed orbit O 1 ∩ . . . ∩ O k , which is isomorphic to the product of two flag varieties G/B × G/B. Here B is a Borel subgroup of G. In fact, if one takes any irreducible representation π with the strictly dominant highest weight, then the corresponding compactification X π is isomorphic to the wonderful compactification. It is an immediate corollary from the second part of Theorem 3.2, since in this case, the normal fan of the weight polytope P π is the fan of Weyl chambers. Hence, it is the same as for the weight polytope P Ad . Equivariant vector bundles. Let L be a vector bundle over G of rank d. Denote by V g ⊂ L a fiber of L lying over an element g ∈ G. Assume that the standard
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: 28 because of commutativity). The a
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

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