Chapter
1.2 Application to affine varieties
1.2.1 Index of the square of the distance function
1.2.2 Lefschetz theorem on hyperplane sections
1.3 Vanishing theorems and Lefschetz’ theorem
2.1.2 The holomorphic Morse lemma
2.2 Lefschetz degeneration
2.2.2 An application of Morse theory
2.3 Application to Lefschetz pencils
2.3.1 Blowup of the base locus
2.3.2 The Lefschetz theorem
2.3.3 Vanishing cohomology and primitive cohomology
2.3.4 Cones over vanishing cycles
3.1.1 Local systems and representations of π1
3.1.2 Local systems associated to a fibration
3.1.3 Monodromy and variation of Hodge structure
3.2 The case of Lefschetz pencils
3.2.1 The Picard–Lefschetz formula
3.2.3 Irreducibility of the monodromy action
3.3 Application: the Noether–Lefschetz theorem
3.3.1 The Noether–Lefschetz locus
3.3.2 The Noether–Lefschetz theorem
4 The Leray Spectral Sequence
4.1 Definition of the spectral sequence
4.1.1 The hypercohomology spectral sequence
4.1.2 Spectral sequence of a composed functor
4.1.3 The Leray spectral sequence
4.2.1 The cup-product and spectral sequences
4.2.2 The relative Lefschetz decomposition
4.2.3 Degeneration of the spectral sequence
4.3 The invariant cycles theorem
4.3.1 Application of the degeneracy of the Leray–spectral
sequence
4.3.2 Some background on mixed Hodge theory
4.3.3 The global invariant cycles theorem
II Variations of Hodge Structure
5 Transversality and Applications
5.1 Complexes associated to IVHS
5.1.1 The de Rham complex of a flat bundle
5.1.3 Construction of the complexes Kl,r
5.2 The holomorphic Leray spectral sequence
5.2.1 The Leray filtration on ΩpX and the complexes
Kp,q
5.2.2 Infinitesimal invariants
5.3 Local study of Hodge loci
5.3.2 Infinitesimal study
5.3.3 The Noether–Lefschetz locus
5.3.4 A density criterion
6 Hodge Filtration of Hypersurfaces
6.1 Filtration by the order of the pole
6.1.1 Logarithmic complexes
6.1.2 Hodge filtration and filtration by the order of
the pole
6.1.3 The case of hypersurfaces of Pn
6.2 IVHS of hypersurfaces
6.2.1 Computation of ∇bar
6.2.3 The symmetriser lemma
6.3.1 Hodge loci for families of hypersurfaces
6.3.2 The generic Torelli theorem
7 Normal Functions and Infinitesimal Invariants
7.1 The Jacobian fibration
7.1.1 Holomorphic structure
7.1.3 Infinitesimal invariants
7.2.2 Geometric interpretation of the infinitesimal
invariant
7.3 The case of hypersurfaces of high degree in Pn
7.3.1 Application of the symmetriser lemma
7.3.2 Generic triviality of the Abel–Jacobi map
8.1 The connectivity theorem
8.1.1 Statement of the theorem
8.1.2 Algebraic translation
8.1.3 The case of hypersurfaces of projective
space
8.2 Algebraic equivalence
8.2.2 The Hodge class of a normal function
8.3 Application of the connectivity theorem
8.3.1 The Nori equivalence
9.1.1 Rational equivalence
9.1.2 Functoriality: proper morphisms and flat
morphisms
9.2 Intersection and cycle classes
9.3.1 Chow groups of curves
9.3.2 Chow groups of projective bundles
9.3.3 Chow groups of blowups
9.3.4 Chow groups of hypersurfaces of small degree
10 Mumford’s Theorem and its Generalisations
10.1 Varieties with representable CH0
10.1.3 Statement of Mumford’s theorem
10.2 The Bloch–Srinivas construction
10.2.1 Decomposition of the diagonal
10.2.2 Proof of Mumford’s theorem
10.2.3 Other applications
10.3.1 Generalised decomposition of the diagonal
11 The Bloch Conjecture and its Generalisations
11.1 Surfaces with pg = 0
11.1.1 Statement of the conjecture
11.1.3 Bloch’s conjecture for surfaces which are not
of general type
11.2 Filtrations on Chow groups
11.2.1 The generalised Bloch conjecture
11.2.2 Conjectural filtration on the Chow groups
11.2.3 The Saito filtration
11.3 The case of abelian varieties
11.3.1 The Pontryagin product
11.3.4 Results of Beauville
Hodge Theory and Complex Algebraic Geometry II: Volume 2
( Cambridge Studies in Advanced Mathematics )
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