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

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Acknowledgement I am very grateful to my scientific advisors Askold Khovanskii and Mikhail Kapranov for many useful discussions. I am also grateful to the external examiner Alexander Braverman for reading my thesis and writing the report. I appreciate the discussions with Kiumars Kaveh on the topics of my thesis. I would like to thank the graduate secretary Ida Bulat for her constant help and the University of Toronto for financial support during my PhD studies. iv
Table of Contents Acknowledgement iv 1 Introduction and main results 1 2 A Gauss-Bonnet theorem for constructible sheaves on reductive groups 8 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Euler characteristic of invariant subvarieties . . . . . . . . . . . . . . 16 2.4 Gaussian degree of invariant subvarieties . . . . . . . . . . . . . . . . 17 2.5 The Euler characteristic of the complex link . . . . . . . . . . . . . . 20 2.6 Proof of Theorem 2.1 (a Gauss-Bonnet theorem) . . . . . . . . . . . . 22 2.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Chern classes of reductive groups and an adjunction formula 25 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Equivariant compactifications and the ring of conditions . . . . . . . 28 3.3 Chern classes with the values in the ring of conditions . . . . . . . . . 33 3.4 Proof of Theorem 3.1 (an adjunction formula) . . . . . . . . . . . . . 44 3.5 Degree of the first and of the last Chern classes . . . . . . . . . . . . 48 3.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Bibliography 51 v
Page 1 and 2: A GAUSS-BONNET THEOREM, CHERN CLASS
Page 3: a group action to extend to noncomp
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 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
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a gauss-bonnet theorem, chern classes and an adjunction formula

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