Hodge theory and complex algebraic geometry 2

Contents......Page 0003-0006.pbm.djvu
Introduction......Page 0006-0012.pbm.djvu
I. The Topology of Algebraic Varieties......Page 0014-0008.pbm.djvu
1. The Lefschetz Theorem on Hyperplane Sections......Page 0015-0010.pbm.djvu
1.1.1. Morse\'s lemma......Page 0016-0011.pbm.djvu
1.1.2. Local study of the level set......Page 0017-0014.pbm.djvu
1.1.3. Globalisation......Page 0019-0018.pbm.djvu
1.2.1. Index of the square of the distance function......Page 0020-0019.pbm.djvu
1.2.2. Lefschetz theorem on hyperplane sections......Page 0021-0002.pbm.djvu
1.2.3. Applications......Page 0023-0005.pbm.djvu
1.3. Vanishing theorems and Lefschetz\' theorem......Page 0024-0007.pbm.djvu
Exercises......Page 0025-0010.pbm.djvu
2 Lefschetz Pencils......Page 0026-0012.pbm.djvu
2.1.1. Existence......Page 0027-0013.pbm.djvu
2.1.2. The holomorphic Morse lemma......Page 0029-0017.pbm.djvu
2.2.1. Vanishing spheres......Page 0029-0018.pbm.djvu
2.2.2. An application of Morse theory......Page 0030-0019.pbm.djvu
2.3.1. Blowup of the base locus......Page 0032-0004.pbm.djvu
2.3.2. The Lefschetz theorem......Page 0033-0005.pbm.djvu
2.3.3. Vanishing cohomology and primitive cohomology......Page 0034-0008.pbm.djvu
2.3.4. Cones over vanishing cycles......Page 0036-0011.pbm.djvu
Exercises......Page 0037-0013.pbm.djvu
3. Monodromy......Page 0039-0018.pbm.djvu
3.1.1. Local systems and representations of π₁......Page 0040-0020.pbm.djvu
3.1.2. Local systems associated to a fibration......Page 0042-0004.pbm.djvu
3.1.3. Monodromy and variation of Hodge structure......Page 0043-0005.pbm.djvu
3.2.1. The Picard-Lefschetz formula......Page 0044-0008.pbm.djvu
3.2.2. Zariski\'s theorem......Page 0048-0016.pbm.djvu
3.2.3. Irreducibility of the monodromy action......Page 0049-0018.pbm.djvu
3.3.1. The Noether-Lefschetz locus......Page 0050-0020.pbm.djvu
3.3.2. The Noether-Lefschetz theorem......Page 0052-0004.pbm.djvu
Exercises......Page 0053-0005.pbm.djvu
4. The Leray Spectral Sequence......Page 0055-0009.pbm.djvu
4.1.1. The hypercohomology spectral sequence......Page 0056-0011.pbm.djvu
4.1.2. Spectral sequence of a composed functor......Page 0059-0018.pbm.djvu
4.1.3. The Leray spectral sequence......Page 0060-0020.pbm.djvu
4.2.1. The cup-product and spectral sequences......Page 0062-0004.pbm.djvu
4.2.2. The relative Lefschetz decomposition......Page 0063-0006.pbm.djvu
4.2.3. Degeneration of the spectral sequence......Page 0064-0008.pbm.djvu
4.3.1. Application of the degeneracy of the Leray-spectral sequence......Page 0065-0009.pbm.djvu
4.3.2. Some background on mixed Hodge theory......Page 0065-0010.pbm.djvu
4.3.3. The global invariant cycles theorem......Page 0067-0014.pbm.djvu
Exercises......Page 0068-0015.pbm.djvu
II. Variations of Hodge Structure......Page 0069-0018.pbm.djvu
5. Transversality and Applications......Page 0070-0020.pbm.djvu
5.1.1. The de Rham complex of a flat bundle......Page 0071-0001.pbm.djvu
5.1.2. Transversality......Page 0072-0004.pbm.djvu
5.1.3. Construction of the complexes K_lr......Page 0074-0008.pbm.djvu
5.2.1. The Leray filtration on Ω^p_X and the complexes K_pq......Page 0075-0009.pbm.djvu
5.2.2. Infinitesimal invariants......Page 0076-0012.pbm.djvu
5.3.1. General properties......Page 0077-0014.pbm.djvu
5.3.2. Infinitesimal study......Page 0079-0017.pbm.djvu
5.3.3. The Noether-Lefschetz locus......Page 0080-0019.pbm.djvu
5.3.4. A density criterion......Page 0081-0002.pbm.djvu
Exercises......Page 0082-0004.pbm.djvu
6. Hodge Filtration of Hypersurfaces......Page 0084-0007.pbm.djvu
6.1.1. Logarithmic complexes......Page 0085-0009.pbm.djvu
6.1.2. Hodge filtration and filtration by the order of the pole......Page 0086-0011.pbm.djvu
6.1.3. The case of hypersurfaces of ℙⁿ......Page 0087-0014.pbm.djvu
6.2.1. Computation of ∇¯......Page 0089-0018.pbm.djvu
6.2.2. Macaulay\'s theorem......Page 0091-0002.pbm.djvu
6.2.3. The symmetriser lemma......Page 0093-0006.pbm.djvu
6.3.1. Hodge loci for families of hypersurfaces......Page 0094-0008.pbm.djvu
6.3.2. The generic Torelli theorem......Page 0095-0010.pbm.djvu
Exercises......Page 0098-0015.pbm.djvu
7. Normal Functions and Infinitesimal Invariants......Page 0100-0019.pbm.djvu
7.1.1. Holomorphic structure......Page 0100-0020.pbm.djvu
7.1.2. Normal functions......Page 0101-0002.pbm.djvu
7.1.3. Infinitesimal invariants......Page 0102-0003.pbm.djvu
7.2.1. General properties......Page 0102-0004.pbm.djvu
7.2.2. Geometric interpretation of the infinitesimal invariant......Page 0104-0008.pbm.djvu
7.3.1. Application of the symmetriser lemma......Page 0108-0016.pbm.djvu
7.3.2. Generic triviality of the Abel-Jacobi map......Page 0109-0018.pbm.djvu
Exercises......Page 0112-0003.pbm.djvu
8. Nori\'s Work......Page 0113-0006.pbm.djvu
8.1.1. Statement of the theorem......Page 0114-0008.pbm.djvu
8.1.2. Algebraic translation......Page 0115-0009.pbm.djvu
8.1.3. The case of hypersurfaces of projective space......Page 0117-0014.pbm.djvu
8.2.1. General properties......Page 0120-0019.pbm.djvu
8.2.2. The Hodge class of a normal function......Page 0120-0020.pbm.djvu
8.2.3. Griffiths\' theorem......Page 0122-0004.pbm.djvu
8.3.1. The Nori equivalence......Page 0123-0006.pbm.djvu
8.3.2. Nori\'s theorem......Page 0124-0008.pbm.djvu
Exercises......Page 0126-0011.pbm.djvu
III. Algebraic Cycles......Page 0127-0014.pbm.djvu
9. Chow Groups......Page 0128-0016.pbm.djvu
9.1.1. Rational equivalence......Page 0129-0018.pbm.djvu
9.1.2. Functoriality: proper morphisms and flat morphisms......Page 0130-0019.pbm.djvu
9.1.3. Localisation......Page 0133-0005.pbm.djvu
9.2.1. Intersection......Page 0134-0007.pbm.djvu
9.2.2. Correspondences......Page 0135-0010.pbm.djvu
9.2.3. Cycle classes......Page 0136-0012.pbm.djvu
9.2.4. Compatibilities......Page 0137-0014.pbm.djvu
9.3.2. Chow groups of projective bundles......Page 0140-0020.pbm.djvu
9.3.3. Chow groups of blowups......Page 0141-0002.pbm.djvu
9.3.4. Chow groups of hypersurfaces of small degree......Page 0142-0004.pbm.djvu
Exercises......Page 0143-0006.pbm.djvu
10. Mumford\'s Theorem and its Generalisations......Page 0145-0009.pbm.djvu
10.1.1. Representability......Page 0146-0011.pbm.djvu
10.1.2. Roitman\'s theorem......Page 0148-0015.pbm.djvu
10.1.3. Statement of Mumford\'s theorem......Page 0150-0020.pbm.djvu
10.2.1. Decomposition of the diagonal......Page 0151-0002.pbm.djvu
10.2.2. Proof of Mumford\'s theorem......Page 0153-0005.pbm.djvu
10.2.3. Other applications......Page 0155-0009.pbm.djvu
10.3.1. Generalised decomposition of the diagonal......Page 0156-0012.pbm.djvu
10.3.2. An application......Page 0157-0014.pbm.djvu
Exercises......Page 0158-0015.pbm.djvu
11. The Bloch Conjecture and its Generalisations......Page 0159-0018.pbm.djvu
11.1.1. Statement of the conjecture......Page 0160-0019.pbm.djvu
11.1.2. Classification......Page 0161-0001.pbm.djvu
11.1.3. Bloch\'s conjecture for surfaces which are not of general type......Page 0162-0004.pbm.djvu
11.1.4. Godeaux surfaces......Page 0163-0006.pbm.djvu
11.2.1. The generalised Bloch conjecture......Page 0167-0013.pbm.djvu
11.2.2. Conjectural filtration on the Chow groups......Page 0168-0015.pbm.djvu
11.2.3. The Saito filtration......Page 0169-0018.pbm.djvu
11.3.1. The Pontryagin product......Page 0170-0019.pbm.djvu
11.3.2. Results of Bloch......Page 0170-0020.pbm.djvu
11.3.3. Fourier transform......Page 0174-0007.pbm.djvu
11.3.4. Results of Beauville......Page 0175-0010.pbm.djvu
Exercises......Page 0176-0011.pbm.djvu
References......Page 0177-0014.pbm.djvu
Index......Page 0180-0019.pbm.djvu