McKaybook - McKay correspondence. Winter 2006/07 Igor V....

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Unformatted text preview: McKay correspondence. Winter 2006/07 Igor V. Dolgachev October 26, 2009 ii Contents 1 Kleinian surface singularities 1 1.1 Finite subgroups of SL(2 , C ) . . . . . . . . . . . . . . . . . . . . . 1 1.2 Grundformen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Algebras of invariants . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Intersection theory on surfaces 15 2.1 Intersection pairing . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Cartan matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Canonical class . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Geometry of graded algebras 29 3.1 Graded algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Ample invertible sheaves . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 Q-divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4 Cylinder constructions . . . . . . . . . . . . . . . . . . . . . . . . 47 3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4 Resolution of singularities 55 4.1 The blow-up schemes . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Cyclic quotient surface singularities . . . . . . . . . . . . . . . . . 59 4.3 The degree of an affine quasicone . . . . . . . . . . . . . . . . . . 63 4.4 Canonical sheaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.5 Finite group quotients . . . . . . . . . . . . . . . . . . . . . . . . 77 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5 McKay graphs 89 5.1 Linear representations of finite groups . . . . . . . . . . . . . . . 89 5.2 McKay graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 iii iv CONTENTS 6 Punctual Hilbert schemes 107 6.1 Hilbert schemes and symmetric products . . . . . . . . . . . . . . 107 6.2 G-Hilbert scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.3 Symplectic structure . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7 Quiver varieties 127 7.1 Quivers and their representations . . . . . . . . . . . . . . . . . . 127 7.2 Varieties of quiver representations . . . . . . . . . . . . . . . . . 130 7.3 McKay quivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.4 Preprojective algebras . . . . . . . . . . . . . . . . . . . . . . . . 135 7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 8 McKay correspondence 139 8.1 Semi-simple rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 8.2 Skew group algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.3 Quiver resolution of Klein surfaces . . . . . . . . . . . . . . . . . 147...
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McKaybook - McKay correspondence. Winter 2006/07 Igor V....

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