A Course in Hodge Theory (With Emphasis on Multiple Integrals)

Introduction
What is Hodge Theory?
The main purposes of the present book
Prerequisites and how to read the book?
Synopsis of the contents of this book
Further reading
Origins of Hodge theory
Origins of singular homology
Origins of de Rham cohomology
The Legendre relation: a first manifestation of algebraic cycles
Multiple integrals
Tame polynomials and Hodge cycles
Exercises
Origins of the Hodge conjecture
Lefschetz's puzzle: Picard's 0-formula
Hyperelliptic surfaces
Computing Picard's 0
Exercises
Homology theory
Early history of topology
Before getting into axiomatic approach
Eilenberg-Steenrod axioms of homology
Singular homology
Some consequences of the axioms
Leray-Thom-Gysin isomorphism
Exercises
Lefschetz theorems
Introduction
Main theorem
Some consequences of the main theorem
Lefschetz theorem on hyperplane sections
Topology of smooth complete intersections
Hard Lefschetz theorem
Lefschetz decomposition
Lefschetz theorems in cohomology
Intersection form
Cycles at infinity and torsions
Exercises
Picard-Lefschetz theory
Introduction
Ehresmann's fibration theorem
Monodromy
Vanishing cycles
The case of isolated singularities
Picard-Lefschetz formula
Vanishing cycles as generators
Lefschetz pencil
Proof of the main theorem
Vanishing cycles are all conjugate by monodromy
Global invariant cycle theorem
Exercises
Topology of tame polynomials
Introduction
Vanishing cycles and orientation
Tame polynomials
Picard-Lefschetz theory of tame polynomials
Distinguished set of vanishing cycles
Monodromy of zero dimensional varieties
Monodromy of families of elliptic curves
Join of topological spaces
Direct sum of polynomials
Intersection of topological cycles
Exercises
Hodge conjecture
Introduction
De Rham cohomology
Integration
Hodge decomposition
Polarization
Hodge conjecture I
Hodge conjecture II
Computational Hodge conjecture
The Z-module of algebraic cycles
Hodge index theorem
Exercises
Lefschetz (1,1) theorem
Introduction
Lefschetz (1,1) theorem
Picard and Neron-Severi groups
Exercises
De Rham cohomology and Brieskorn module
Introduction
The base ring
Differential forms
Vector fields
Cohen-Macaulay rings
Tame polynomials
Examples of tame polynomials
De Rham lemma
The discriminant of a polynomial
The double discriminant of a tame polynomial
De Rham cohomology and Brieskorn modules
Main theorem
Proof of the main theorem for a homogeneous tame polynomial
Proof of the main theorem for an arbitrary tame polynomial
Exercises
Hodge filtrations and Mixed Hodge structures
Introduction
Gauss-Manin system M
Two filtrations of M
Homogeneous tame polynomials
Weighted projective spaces
De Rham cohomology of hypersurfaces
Residue map
A basis of Brieskorn module
Exercises
Gauss-Manin connection for tame polynomials
Introduction
Gauss-Manin connection
Picard-Fuchs equations
Gauss-Manin connection matrix
Calculating Gauss-Manin connection
R[] structure of H''
Gauss-Manin system
Griffiths transversality
Tame polynomials with zero discriminant
Exercises
Multiple integrals and periods
Introduction
Integrals
Integrals and Gauss-Manin connections
Period matrix
Period matrix and Picard-Fuchs equations
Homogeneous polynomials
Residues and integrals
Integration over joint cycles
Taylor series of periods
Periods of projective hypersurfaces
Exercises
Noether-Lefschetz theorem
Introduction
Hilbert schemes
Topological proof of Noether-Lefschetz theorem
Proof of Noether-Lefschetz theorem using integrals
Exercises
Fermat varieties
A brief history
Multiple integrals for Fermat varieties
De Rham cohomology of affine Fermat varieties
Hodge numbers
Hodge conjecture for the Fermat variety
Hodge cycles of the Fermat variety, I
Intersection form
Hodge cycles of the Fermat variety, II
Period matrix
Computing Hodge cycles of the Fermat variety
Dimension of the space of Hodge cycles
A table of Picard numbers for Fermat surfaces
Rank of elliptic curves over functions field
Computing a basis of Hodge cycles using Gaussian elimination
Choosing a basis of homology
An explicit Hodge cycle
Exercises
Periods of Hodge cycles of Fermat varieties
Introduction
Computing periods of Hodge cycles
The beta factors
An invariant of Hodge cycles
The geometric interpretation of
-invariant of
Linear Hodge cycles
General Hodge cycles
Periods of join Hodge cycles
Computing periods of hypersurfaces (by Emre Sertöz)
Exercises
Algebraic cycles of the Fermat variety
Introduction
Trivial algebraic cycles
Automorphism group of the Fermat variety
Linear projective cycles
Aoki-Shioda algebraic cycles
Adjunction formula
Hodge conjecture for Fermat variety
Some conjectural components of the Hodge locus
Intersections of components of the Hodge loci
Exercises
Why should one compute periods of algebraic cycles?
Introduction
Periods of algebraic cycles
Sum of two linear cycles
Smooth and reduced Hodge loci
The creation of a formula
Uniqueness of components of the Hodge locus
Semi-irreducible algebraic cycles
How to to deal with Conjecture 18.1?
Final comments
Hodge cycles for cubic hypersurfaces
Abstract
Introduction
Smaller deformation space
Infinitesimal Hodge loci
Proof of Theorem 19.1
Finding algebraic cycles
Learning from cubic fourfolds
Quartic hypersurfaces
Online supplemental items
Introduction
How to start?
Tame polynomials
Generalized Fermat variety
Hodge cycles and periods
De Rham cohomology of Fermat varieties
Hodge cycles supported in linear cycles
General Hodge cycles
The Hodge cycle
Codimension of the components of the Hodge locus
Hodge locus and linear cycles
Join Hodge cycles
Exercises
Some mathematical olympiad problems
Introduction
Some problems on the configuration of lines
A very big matrix: how to compute its rank?
Determinant of a matrix
Some modular arithmetic
Last step in the proof of ...
References
Index