Hodge Theory and Complex Algebraic Geometry I: Volume 1 (Cambridge Studies in Advanced Mathematics, Series Number 76)

The first of two volumes offering a modern introduction to Kaehlerian geometry and Hodge structure. The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more theoretical way than is usual in geometry. The author then proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The book culminates with the Hodge decomposition theorem. The meanings of these results are investigated in several directions. Completely self-contained, the book is ideal for students, while its content gives an account of Hodge theory and complex algebraic geometry as has been developed by P. Griffiths and his school, by P. Deligne, and by S. Bloch. The text is complemented by exercises which provide useful results in complex algebraic geometry. Half-title......Page 3 Series-title......Page 4 Title......Page 5 Copyright......Page 6 Contents......Page 7 0 Introduction......Page 13 Part I Preliminaries......Page 31 1 Holomorphic Functions of Many Variables......Page 33 1.1.1 Definition and basic properties......Page 34 1.1.2 Background on Stokes’ formula......Page 36 1.1.3 Cauchy’s formula......Page 39 1.2.1 Cauchy’s formula and analyticity......Page 40 1.2.2 Applications of Cauchy’s formula......Page 42 1.3 The equation.........Page 47 Exercises......Page 49 2 Complex Manifolds......Page 50 2.1.1 Definitions......Page 51 2.1.2 The tangent bundle......Page 53 2.1.3 Complex manifolds......Page 55 2.2.1 Tangent bundle of a complex manifold......Page 56 2.2.2 The Frobenius theorem......Page 58 2.2.3 The Newlander–Nirenberg theorem......Page 62 2.3.1 Definition......Page 65 2.3.2 Local exactness......Page 67 2.3.3 Dolbeault complex of a holomorphic bundle......Page 69 Riemann surfaces......Page 71 Complex projective space......Page 72 Exercises......Page 73 3 Kähler Metrics......Page 75 3.1.1 Hermitian geometry......Page 76 3.1.2 Hermitian and Kähler metrics......Page 78 Volume form......Page 79 Submanifolds......Page 80 3.2.1 Background on connections......Page 81 3.2.2 Kähler metrics and connections......Page 83 3.3.1 Chern form of line bundles......Page 87 3.3.2 Fubini–Study metric......Page 88 3.3.3 Blowups......Page 90 Exercises......Page 94 4 Sheaves and Cohomology......Page 95 4.1.1 Definitions, examples......Page 97 4.1.2 Stalks, kernels, images......Page 101 4.1.3 Resolutions......Page 103 The Cech resolution......Page 104 The de Rham resolution......Page 105 The Dolbeault resolution......Page 106 4.2.1 Abelian categories......Page 107 4.2.2 Injective resolutions......Page 108 4.2.3 Derived functors......Page 111 4.3 Sheaf cohomology......Page 114 4.3.1 Acyclic resolutions......Page 115 4.3.2 The de Rham theorems......Page 120 4.3.3 Interpretations of the group H......Page 122 Exercises......Page 125 Part II The Hodge Decomposition......Page 127 5 Harmonic Forms and Cohomology......Page 129 5.1.1 The L metric......Page 131 5.1.3 Adjoints of the operators.........Page 133 5.1.4 Laplacians......Page 136 5.2.1 Symbols of differential operators......Page 137 5.2.2 Symbol of the Laplacian......Page 138 5.2.3 The fundamental theorem......Page 140 5.3.1 Cohomology and harmonic forms......Page 141 5.3.2 Duality theorems......Page 142 Exercises......Page 148 6 The Case of Kähler Manifolds......Page 149 6.1.1 Kähler identities......Page 151 6.1.2 Comparison of the Laplacians......Page 153 6.1.3 Other applications......Page 154 6.2.1 Commutators......Page 156 6.2.2 Lefschetz decomposition on forms......Page 158 6.2.3 Lefschetz decomposition on the cohomology......Page 160 6.3.1 Other Hermitian identities......Page 162 6.3.2 The Hodge index theorem......Page 164 Exercises......Page 166 7 Hodge Structures and Polarisations......Page 168 7.1.1 Hodge structure......Page 169 7.1.2 Polarisation......Page 172 7.1.3 Polarised varieties......Page 173 7.2.1 Projective space......Page 179 7.2.2 Hodge structures of weight 1 and abelian varieties......Page 180 7.2.3 Hodge structures of weight 2......Page 182 7.3.1 Morphisms of Hodge structures......Page 186 7.3.2 The pullback and the Gysin morphism......Page 188 7.3.3 Hodge structure of a blowup......Page 192 Exercises......Page 194 8 Holomorphic de Rham Complexes and Spectral Sequences......Page 196 8.1.1 Resolutions of complexes......Page 198 8.1.2 Derived functors......Page 201 8.1.3 Composed functors......Page 206 Application: Proof of the Leray–Hirsch theorem 7.33......Page 207 8.2.1 Holomorphic de Rham resolutions......Page 208 8.2.2 The logarithmic case......Page 209 8.2.3 Cohomology of the logarithmic complex......Page 210 8.3.1 Filtered complexes......Page 212 8.3.2 Spectral sequences......Page 213 8.3.3 The Frölicher spectral sequence......Page 216 8.4.1 Filtrations on the logarithmic complex......Page 219 8.4.2 First terms of the spectral sequence......Page 220 8.4.3 Deligne’s theorem......Page 225 Exercises......Page 226 Part III Variations of Hodge Structure......Page 229 9 Families and Deformations......Page 231 9.1.1 Trivialisations......Page 232 9.1.2 The Kodaira–Spencer map......Page 235 9.2.1 Local systems and flat connections......Page 240 9.2.2 The Cartan–Lie formula......Page 243 9.3.1 Semicontinuity theorems......Page 244 9.3.2 The Hodge numbers are constant......Page 247 9.3.3 Stability of Kähler manifolds......Page 248 10 Variations of Hodge Structure......Page 251 10.1.1 Grassmannians......Page 252 10.1.2 The period map......Page 255 10.1.3 The period domain......Page 258 10.2.1 Hodge bundles......Page 261 10.2.2 Transversality......Page 262 10.2.3 Computation of the differential......Page 263 10.3.1 Curves......Page 266 10.3.2 Calabi–Yau manifolds......Page 270 Exercises......Page 271 Part IV Cycles and Cycle Classes......Page 273 11 Hodge Classes......Page 275 11.1.1 Analytic subsets......Page 276 11.1.2 Cohomology class......Page 281 11.1.3 The Kähler case......Page 285 11.1.4 Other approaches......Page 287 11.2.1 Construction......Page 288 11.3.1 Definitions and examples......Page 291 11.3.2 The Hodge conjecture......Page 296 11.3.3 Correspondences......Page 297 Exercises......Page 299 12 Deligne–Beilinson Cohomology and the Abel–Jacobi map......Page 302 12.1.1 Intermediate Jacobians......Page 303 12.1.2 The Abel–Jacobi map......Page 304 12.1.3 Picard and Albanese varieties......Page 308 12.2.1 Correspondences......Page 312 12.2.2 Some results......Page 314 12.3.1 The Deligne complex......Page 316 12.3.2 Differential characters......Page 318 12.3.3 Cycle class......Page 322 Exercises......Page 325 Bibliography......Page 327 Index......Page 331 This is a modern introduction to Kaehlerian geometry and Hodge structure. It starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more theoretical way than is usual in geometry. The book culminates with the Hodge decomposition theorem. In between, the author proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The second part of the book investigates the meaning of these results in several directions. The book is is completely self-contained and can be used by students, while its content gives an up-to-date account of Hodge theory and complex algebraic geometry as has been developed by P. Griffiths and his school, by P. Deligne, and by S. Bloch. The text is complemented by exercises which provide useful results in complex algebraic geometry Main subject categories: • Hodge theory • Algebraic geometry • Holomorphic functions • Complex manifolds • Kähler metrics • Sheaves and cohomology • Kähler manifolds • Holomorphic de Rham complexes • Spectral sequences • Hodge theory • Cycles and cycle classes • Deligne–Beilinson Cohomology and the Abel–Jacobi MapThis is a modern introduction to Kählerian geometry and Hodge structure. Coverage begins with variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory (with the latter being treated in a more theoretical way than is usual in geometry). The book culminates with the Hodge decomposition theorem. In between, the author proves the Kähler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The second part of the book investigates the meaning of these results in several directions. This is a modern introduction to Kaehlerian geometry and Hodge structure. Coverage begins with variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory (with the latter being treated in a more theoretical way than is usual in geometry). The book culminates with the Hodge decomposition theorem. In between, the author proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The second part of the book investigates the meaning of these results in several directions. This is a completely self-contained modern introduction to Kaehlerian geometry and Hodge structure. The author proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. Aimed at students, the text is complemented by exercises which provide useful results in complex algebraic geometry The goal of this first volume is to explain the existence of special structures on the cohomology of Kahler manifolds, namely, the Hodge decomposition and the Lefschetz decomposition, and to discuss their basic properties and consequences.