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Hodge Theory and Complex Algebraic Geometry II

Book Description

The second volume of this modern account of Kaehlerian geometry and Hodge theory starts with the topology of families of algebraic varieties. Proofs of the Lefschetz theorem on hyperplane sections, the Picard-Lefschetz study of Lefschetz pencils, and Deligne theorems on the degeneration of the Leray spectral sequence and the global invariant cycles follow. The main results of the second part are the generalized Noether-Lefschetz theorems, the generic triviality of the Abel-Jacobi maps, and most importantly Nori's connectivity theorem, which generalizes the above. The last part of the book is devoted to the relationships between Hodge theory and algebraic cycles. The book concludes with the example of cycles on abelian varieties, where some results of Bloch and Beauville, for example, are expounded. The text is complemented by exercises giving useful results in complex algebraic geometry. It will be welcomed by researchers in both algebraic and differential geometry.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Introduction
  7. I: The Topology of Algebraic Varieties
    1. 1. The Lefschetz Theorem on Hyperplane Sections
      1. 1.1 Morse theory
        1. 1.1.1 Morse’s lemma
        2. 1.1.2 Local study of the level set
        3. 1.1.3 Globalisation
      2. 1.2 Application to affine varieties
        1. 1.2.1 Index of the square of the distance function
        2. 1.2.2 Lefschetz theorem on hyperplane sections
        3. 1.2.3 Applications
      3. 1.3 Vanishing theorems and Lefschetz’ theorem
      4. Exercises
    2. 2. Lefschetz Pencils
      1. 2.1 Lefschetz pencils
        1. 2.1.1 Existence
        2. 2.1.2 The holomorphic Morse lemma
      2. 2.2 Lefschetz degeneration
        1. 2.2.1 Vanishing spheres
        2. 2.2.2 An application of Morse theory
      3. 2.3 Application to Lefschetz pencils
        1. 2.3.1 Blowup of the base locus
        2. 2.3.2 The Lefschetz theorem
        3. 2.3.3 Vanishing cohomology and primitive cohomology
        4. 2.3.4 Cones over vanishing cycles
      4. Exercises
    3. 3. Monodromy
      1. 3.1 The monodromy action
        1. 3.1.1 Local systems and representations of π1
        2. 3.1.2 Local systems associated to a fibration
        3. 3.1.3 Monodromy and variation of Hodge structure
      2. 3.2 The case of Lefschetz pencils
        1. 3.2.1 The Picard-Lefschetz formula
        2. 3.2.2 Zariski’s theorem
        3. 3.2.3 Irreducibility of the monodromy action
      3. 3.3 Application: the Noether-Lefschetz theorem
        1. 3.3.1 The Noether-Lefschetz locus
        2. 3.3.2 The Noether-Lefschetz theorem
      4. Exercises
    4. 4. The Leray Spectral Sequence
      1. 4.1 Definition of the spectral sequence
        1. 4.1.1 The hypercohomology spectral sequence
        2. 4.1.2 Spectral sequence of a composed functor
        3. 4.1.3 The Leray spectral sequence
      2. 4.2 Deligne’s theorem
        1. 4.2.1 The cup-product and spectral sequences
        2. 4.2.2 The relative Lefschetz decomposition
        3. 4.2.3 Degeneration of the spectral sequence
      3. 4.3 The invariant cycles theorem
        1. 4.3.1 Application of the degeneracy of the Leray-spectral sequence
        2. 4.3.2 Some background on mixed Hodge theory
        3. 4.3.3 The global invariant cycles theorem
      4. Exercises
  8. II: Variations of Hodge Structure
    1. 5. Transversality and Applications
      1. 5.1 Complexes associated to IVHS
        1. 5.1.1 The de Rham complex of a flat bundle
        2. 5.1.2 Transversality
        3. 5.1.3 Construction of the complexes kl,r
      2. 5.2 The holomorphic Leray spectral sequence
        1. 5.2.1 The Leray filtration on Ωpx and the complexes kp,q
        2. 5.2.2 Infinitesimal invariants
      3. 5.3 Local study of Hodge loci
        1. 5.3.1 General properties
        2. 5.3.2 Infinitesimal study
        3. 5.3.3 The Noether-Lefschetz locus
        4. 5.3.4 A density criterion
      4. Exercises
    2. 6. Hodge Filtration of Hypersurfaces
      1. 6.1 Filtration by the order of the pole
        1. 6.1.1 Logarithmic complexes
        2. 6.1.2 Hodge filtration and filtration by the order of the pole
        3. 6.1.3 The case of hypersurfaces of Pn
      2. 6.2 IVHS of hypersurfaces
        1. 6.2.1 Computation of ∇
        2. 6.2.2 Macaulay’s theorem
        3. 6.2.3 The symmetriser lemma
      3. 6.3 First applications
        1. 6.3.1 Hodge loci for families of hypersurfaces
        2. 6.3.2 The generic Torelli theorem
      4. Exercises
    3. 7. Normal Functions and Infinitesimal Invariants
      1. 7.1 The Jacobian fibration
        1. 7.1.1 Holomorphic structure
        2. 7.1.2 Normal functions
        3. 7.1.3 Infinitesimal invariants
      2. 7.2 The Abel-Jacobi map
        1. 7.2.1 General properties
        2. 7.2.2 Geometric interpretation of the infinitesimal invariant
      3. 7.3 The case of hypersurfaces of high degree in Pn
        1. 7.3.1 Application of the symmetriser lemma
        2. 7.3.2 Generic triviality of the Abel-Jacobi map
      4. Exercises
    4. 8. Nori’s Work
      1. 8.1 The connectivity theorem
        1. 8.1.1 Statement of the theorem
        2. 8.1.2 Algebraic translation
        3. 8.1.3 The case of hypersurfaces of projective space
      2. 8.2 Algebraic equivalence
        1. 8.2.1 General properties
        2. 8.2.2 The Hodge class of a normal function
        3. 8.2.3 Griffiths’ theorem
      3. 8.3 Application of the connectivity theorem
        1. 8.3.1 The Nori equivalence
        2. 8.3.2 Nori’s theorem
      4. Exercises
  9. III: Algebraic Cycles
    1. 9. Chow Groups
      1. 9.1 Construction
        1. 9.1.1 Rational equivalence
        2. 9.1.2 Functoriality: proper morphisms and flat morphisms
        3. 9.1.3 Localisation
      2. 9.2 Intersection and cycle classes
        1. 9.2.1 Intersection
        2. 9.2.2 Correspondences
        3. 9.2.3 Cycle classes
        4. 9.2.4 Compatibilities
      3. 9.3 Examples
        1. 9.3.1 Chow groups of curves
        2. 9.3.2 Chow groups of projective bundles
        3. 9.3.3 Chow groups of blowups
        4. 9.3.4 Chow groups of hypersurfaces of small degree
      4. Exercises
    2. 10. Mumford’s Theorem and its Generalisations
      1. 10.1 Varieties with representable CH0
        1. 10.1.1 Representability
        2. 10.1.2 Roitman’s theorem
        3. 10.1.3 Statement of Mumford’s theorem
      2. 10.2 The Bloch-Srinivas construction
        1. 10.2.1 Decomposition of the diagonal
        2. 10.2.2 Proof of Mumford’s theorem
        3. 10.2.3 Other applications
      3. 10.3 Generalisation
        1. 10.3.1 Generalised decomposition of the diagonal
        2. 10.3.2 An application
      4. Exercises
    3. 11. The Bloch Conjecture and its Generalisations
      1. 11.1 Surfaces with pg = 0
        1. 11.1.1 Statement of the conjecture
        2. 11.1.2 Classification
        3. 11.1.3 Bloch’s conjecture for surfaces which are not of general type
        4. 11.1.4 Godeaux surfaces
      2. 11.2 Filtrations on Chow groups
        1. 11.2.1 The generalised Bloch conjecture
        2. 11.2.2 Conjectural filtration on the Chow groups
        3. 11.2.3 The Saito filtration
      3. 11.3 The case of abelian varieties
        1. 11.3.1 The Pontryagin product
        2. 11.3.2 Results of Bloch
        3. 11.3.3 Fourier transform
        4. 11.3.4 Results of Beauville
      4. Exercises
  10. References
  11. Index