Lectures on K3 Surfaces

Chapter 1. Basic definitions 7
1. Algebraic K3 surfaces 7
2. Classical invariants 9
3. Complex K3 surfaces 13
4. More examples 18
Chapter 2. Linear systems 21
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Chapter 1. Basic definitions 7
1. Algebraic K3 surfaces 7
2. Classical invariants 9
3. Complex K3 surfaces 13
4. More examples 18
Chapter 2. Linear systems 21
1. General results: linear systems, curves, vanishing 21
2. Smooth curves on K3 surfaces 25
3. Vanishing and global generation on K3 surfaces 27
4. Existence of K3 surfaces 33
Chapter 3. Hodge structures I 37
1. Abstract notions 37
2. Weight one and two 43
3. Geometric examples 45
4. Transcendental lattices and endomorphism fields 48
Chapter 4. The Kuga–Satake construction 55
1. Recollections: Clifford algebra and Spin-group 55
2. From weight two to weight one 57
3. Examples 63
Chapter 5. Moduli spaces of polarized K3 surfaces 67
1. Moduli functor 67
2. Via Hilbert schemes 69
3. Local structure of the Hilbert scheme 75
4. As Deligne–Mumford stack 77
Chapter 6. Periods 83
1. Period domains 83
2. Local period map 87
3. Global period map 92
4. Moduli spaces of polarized K3 surfaces, again 96
Chapter 7. Surjectivity of the period map and Global Torelli 101
1. Deformation equivalence of K3 surfaces 101
2. Moduli space of marked K3 surfaces 103
3. Twistor lines 105
4. Local and global surjectivity of the period map 109
5. Global Torelli theorem 110
6. Other approaches 115
Chapter 8. Ample cone and Kähler cone 117
1. Ample and nef cone 117
2. Chambers and walls 120
3. Effective cone 125
4. Cone conjecture 132
5. Kähler cone 135
Chapter 9. Special vector bundles on K3 surfaces 141
1. Basic techniques and first examples 141
2. Simple vector bundles and Brill–Noether general curves 145
3. Stability of special bundles 148
4. Stability of the tangent bundle 151
Chapter 10. Moduli spaces of sheaves on K3 surfaces 157
1. General theory 157
2. On K3 surfaces 163
3. Some moduli spaces 167
Chapter 11. Elliptic K3 surfaces 173
1. Singular fibres 173
2. Weierstrass equation 179
3. Mordell–Weil group 184
4. Jacobian fibration 189
5. Tate–Šafarevič group 194
Chapter 12. Chow ring and Grothendieck group 203
1. Generalities on CH⇤ (X ) and K (X ) 203
2. Chow groups: Mumford and Bloch–Be ̆ılinson 208
3. Beauville–Voisin ring 214
Chapter 13. Rational curves on K3 surfaces 219
1. Existence results 222
2. A glimpse of deformation theory of curves on K3 surfaces 227
3. Arithmetic approach 230
4. Counting of rational curves 234
5. Density results 235
Chapter 14. Lattices 239
1. Existence, uniqueness, and embeddings of lattices 244
2. Orthogonal group 249
3. Embeddings of Picard, transcendental, and Kummer lattices 251
4. Niemeier lattices 258
Chapter 15. Automorphisms 263
1. Symplectic automorphisms 263
2. Automorphisms via periods 269
3. Finite groups of symplectic automorphisms 274
4. Special cases 281
Index
Bibliography 291
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