nLab Hodge theory and Complex algebraic geometry

This page collects material related to the textbook

• Claire Voisin (translated by Leila Schneps),

  Hodge theory and Complex algebraic geometry

  Cambridge Stud. in Adv. Math.

  76, 77, 2002/3

  vol I: doi:10.1017/CBO9780511615344

  vol II: doi:10.1017/CBO9780511615177

on Hodge theory and GAGA.

Volume 1

I Preliminaries

1 Holomorphic functions of many variables

• holomorphic function

2. Complex manifold

• complex manifold

• Newlander-Nirenberg theorem

• Dolbeault complex

3. Kähler metric

• Kähler manifold

4. Sheaves and cohomology

• abelian sheaf cohomology

II The Hodge decomposition

5. Harmonic forms and cohomology

• harmonic differential form

• Hodge theorem

6. The case of Kähler manifolds

• hard Lefschetz theorem

• Lefschetz decomposition

7. Hodge structures and polarization

• Hodge structure

• polarized variety

8. Holomorphic de Rham complexes and spectral sequences

• hypercohomology

• holomorphic de Rham complex

• differential form with logarithmic singularities

• spectral sequence

• Frölicher spectral sequence

III Variations of Hodge structures

9. Families and deformations

• local system

• flat vector bundle

• Gauss-Manin connection

• twisted de Rham cohomology

10. Variations of Hodge structure

• variations of Hodge structure

IV. Cycles and cycle classes

11. Hodge classes

12. Deligne-Beilinson cohomology and the Abel-Jacobi map

• Abel-Jacobi map

• Deligne cohomology

Volume 2

(…)

category: reference