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. Contents......Page 0003-0006.pbm.djvu Introduction......Page 0006-0012.pbm.djvu I. Preliminaries......Page 0015-0010.pbm.djvu 1. Holomorphic Functions of Many Variables......Page 0016-0012.pbm.djvu 1.1.1. Definition and basic properties......Page 0017-0013.pbm.djvu 1.1.2. Background on Stokes' formula......Page 0018-0015.pbm.djvu 1.1.3. Cauchy's formula......Page 0019-0018.pbm.djvu 1.2.1. Cauchy's formula and analyticity......Page 0020-0019.pbm.djvu 1.2.2. Applications of Cauchy's formula......Page 0021-0001.pbm.djvu 1.3. The equation ∂g/∂z̅ = f......Page 0023-0006.pbm.djvu Exercises......Page 0024-0008.pbm.djvu 2. Complex Manifolds......Page 0025-0009.pbm.djvu 2.1.1. Definitions......Page 0025-0010.pbm.djvu 2.1.2. The tangent bundle......Page 0026-0012.pbm.djvu 2.1.3. Complex manifolds......Page 0027-0014.pbm.djvu 2.2.1. Tangent bundle of a complex manifold......Page 0028-0015.pbm.djvu 2.2.2. The Frobenius theorem......Page 0029-0017.pbm.djvu 2.2.3. The Newlander-Nirenberg theorem......Page 0031-0001.pbm.djvu 2.3.1. Definition......Page 0032-0004.pbm.djvu 2.3.2. Local exactness......Page 0033-0006.pbm.djvu 2.3.3. Dolbeault complex of a holomorphic bundle......Page 0034-0008.pbm.djvu 2.4. Examples of complex manifolds......Page 0035-0010.pbm.djvu Exercises......Page 0036-0012.pbm.djvu 3. Kähler Metrics......Page 0037-0014.pbm.djvu 3.1.1. Hermitian geometry......Page 0038-0015.pbm.djvu 3.1.2. Hermitian and Kähler metrics......Page 0039-0017.pbm.djvu 3.1.3. Basic properties......Page 0039-0018.pbm.djvu 3.2.1. Background on connections......Page 0040-0020.pbm.djvu 3.2.2. Kähler metrics and connections......Page 0041-0002.pbm.djvu 3.3.1. Chern form of line bundles......Page 0043-0006.pbm.djvu 3.3.2. Fubini-Study metric......Page 0044-0007.pbm.djvu 3.3.3. Blowups......Page 0045-0009.pbm.djvu Exercises......Page 0047-0013.pbm.djvu 4. Sheaves and Cohomology......Page 0047-0014.pbm.djvu 4.1.1. Definitions, examples......Page 0048-0016.pbm.djvu 4.1.2. Stalks, kernels, images......Page 0050-0020.pbm.djvu 4.1.3. Resolutions......Page 0051-0002.pbm.djvu 4.2.1. Abelian categories......Page 0053-0006.pbm.djvu 4.2.2. Injective resolutions......Page 0054-0007.pbm.djvu 4.2.3. Derived functors......Page 0055-0010.pbm.djvu 4.3. Sheaf cohomology......Page 0057-0013.pbm.djvu 4.3.1. Acyclic resolutions......Page 0057-0014.pbm.djvu 4.3.2. The de Rham theorems......Page 0060-0019.pbm.djvu 4.3.3. Interpretations of the group H1......Page 0061-0001.pbm.djvu Exercises......Page 0062-0004.pbm.djvu II. The Hodge Deeornposition......Page 0063-0006.pbm.djvu 5. Harmonic Forms and Cohomology......Page 0064-0008.pbm.djvu 5.1.1. The L2 metric......Page 0065-0010.pbm.djvu 5.1.3. Adjoints of the operators ∂ ̄......Page 0066-0012.pbm.djvu 5.1.4. Laplacians......Page 0068-0015.pbm.djvu 5.2.1. Symbols of differential operators......Page 0068-0016.pbm.djvu 5.2.2. Symbol of the Laplacian......Page 0069-0017.pbm.djvu 5.2.3. The fundamental theorem......Page 0070-0019.pbm.djvu 5.3.1. Cohomology and harmonic forms......Page 0070-0020.pbm.djvu 5.3.2. Duality theorems......Page 0071-0001.pbm.djvu Exercises......Page 0074-0007.pbm.djvu 6. The Case of Kähler Manifolds......Page 0074-0008.pbm.djvu 6.1.1. Kähler identities......Page 0075-0010.pbm.djvu 6.1.2. Comparison of the Laplacians......Page 0076-0012.pbm.djvu 6.1.3. Other applications......Page 0077-0013.pbm.djvu 6.2.1. Commutators......Page 0078-0015.pbm.djvu 6.2.2. Lefschetz decomposition on forms......Page 0079-0017.pbm.djvu 6.2.3. Lefschetz decomposition on the cohomology......Page 0080-0019.pbm.djvu 6.3.1. Other Hermitian identities......Page 0081-0001.pbm.djvu 6.3.2. The Hodge index theorem......Page 0082-0003.pbm.djvu Exercises......Page 0083-0005.pbm.djvu 7. Hodge Structures and Polarisations......Page 0084-0007.pbm.djvu 7.1.1. Hodge structure......Page 0084-0008.pbm.djvu 7.1.2. Polarisation......Page 0086-0011.pbm.djvu 7.1.3. Polarised varieties......Page 0086-0012.pbm.djvu 7.2.1. Projective space......Page 0089-0018.pbm.djvu 7.2.2. Hodge structures of weight 1 and abelian varieties......Page 0090-0019.pbm.djvu 7.2.3. Hodge structures of weight 2......Page 0091-0001.pbm.djvu 7.3.1. Morphisms of Hodge structures......Page 0093-0005.pbm.djvu 7.3.2. The pullback and the Gysin morphism......Page 0094-0007.pbm.djvu 7.3.3. Hodge structure of a blowup......Page 0096-0011.pbm.djvu Exercises......Page 0097-0013.pbm.djvu 8. Holomorphic de Rham Complexes and Spectral Sequences......Page 0098-0015.pbm.djvu 8.1.1. Resolutions of complexes......Page 0099-0017.pbm.djvu 8.1.2. Derived functors......Page 0100-0020.pbm.djvu 8.1.3. Composed functors......Page 0103-0005.pbm.djvu 8.2.1. Holomorphic de Rham resolutions......Page 0104-0007.pbm.djvu 8.2.2. The logarithmic case......Page 0104-0008.pbm.djvu 8.2.3. Cohomology of the logarithmic complex......Page 0105-0009.pbm.djvu 8.3.1. Filtered complexes......Page 0106-0011.pbm.djvu 8.3.2. Spectral sequences......Page 0106-0012.pbm.djvu 8.3.3. The Frölicher spectral sequence......Page 0108-0015.pbm.djvu 8.4.1. Filtrations on the logarithmic complex......Page 0109-0018.pbm.djvu 8.4.2. First terms of the spectral sequence......Page 0110-0019.pbm.djvu 8.4.3. Deligne's theorem......Page 0112-0004.pbm.djvu Exercises......Page 0113-0005.pbm.djvu III. Variations of Hodge Structure......Page 0114-0008.pbm.djvu 9. Families and Deformations......Page 0115-0010.pbm.djvu 9.1.1. Trivialisations......Page 0116-0011.pbm.djvu 9.1.2. The Kodaira-Spencer map......Page 0117-0014.pbm.djvu 9.2.1. Local systems and flat connections......Page 0120-0019.pbm.djvu 9.2.2. The Cartan-Lie formula......Page 0121-0002.pbm.djvu 9.3.1. Semicontinuity theorems......Page 0122-0003.pbm.djvu 9.3.2. The Hodge numbers are constant......Page 0123-0006.pbm.djvu 9.3.3. Stability of Kähler manifolds......Page 0124-0007.pbm.djvu 10. Variations of Hodge Structure......Page 0125-0010.pbm.djvu 10.1.1. Grassmannians......Page 0126-0011.pbm.djvu 10.1.2. The period map......Page 0127-0014.pbm.djvu 10.1.3. The period domain......Page 0129-0017.pbm.djvu 10.2.1. Hodge bundles......Page 0130-0020.pbm.djvu 10.2.2. Transversality......Page 0131-0001.pbm.djvu 10.2.3. Computation of the differential......Page 0131-0002.pbm.djvu 10.3.1. Curves......Page 0133-0005.pbm.djvu 10.3.2. Calabi-Yau manifolds......Page 0135-0009.pbm.djvu Exercises......Page 0135-0010.pbm.djvu IV. Cycles and Cycle Classes......Page 0136-0012.pbm.djvu 11. Hodge Classes......Page 0137-0014.pbm.djvu 11.1.1. Analytic subsets......Page 0138-0015.pbm.djvu 11.1.2. Cohomology class......Page 0140-0020.pbm.djvu 11.1.3. The Kähler case......Page 0142-0004.pbm.djvu 11.1.4. Other approaches......Page 0143-0006.pbm.djvu 11.2.1. Construction......Page 0144-0007.pbm.djvu 11.3.1. Definitions and examples......Page 0145-0010.pbm.djvu 11.3.2. The Hodge conjecture......Page 0148-0015.pbm.djvu 11.3.3. Correspondences......Page 0148-0016.pbm.djvu Exercises......Page 0149-0018.pbm.djvu 12. Deligne-Beilinson Cohomology and the Abel-Jacobi Map......Page 0151-0001.pbm.djvu 12.1.1. Intermediate Jacobians......Page 0151-0002.pbm.djvu 12.1.2. The Abel-Jacobi map......Page 0152-0003.pbm.djvu 12.1.3. Picard and Albanese varieties......Page 0154-0007.pbm.djvu 12.2.1. Correspondences......Page 0156-0011.pbm.djvu 12.2.2. Some results......Page 0107-0013.pbm.djvu 12.3.1. The Deligne complex......Page 0158-0015.pbm.djvu 12.3.2. Differential characters......Page 0159-0017.pbm.djvu 12.3.3. Cycle class......Page 0161-0001.pbm.djvu Exercises......Page 0162-0004.pbm.djvu Bibliography......Page 0163-0006.pbm.djvu Index......Page 0165-0010.pbm.djvu 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 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.