Hodge Theory III

Deligne, P. “Théorie de Hodge III”.

Publications Mathématiques de l’IHÉS 44 (1974) pp. 5–77.

publications.ias.edu/node/373

Table of contents

• Introduction
• Terminology and notation

5. Cohomological descent
   • 5.1 Simplicial topological spaces
   • 5.2 Cohomology of simplicial topological spaces
   • 5.3 Cohomological descent
6. Examples of simplicial topological spaces
   • 6.1 Classifying spaces
   • 6.2 Construction of hypercovers
   • 6.3 Relative cohomology
   • 6.4 Multi-simplicial spaces
7. Supplements to §1
   • 7.1 Filtered derived category
   • 7.2 Supplements to the two-filtrations lemma
8. Hodge theory of algebraic spaces
   • 8.1 Hodge complexes
   • 8.2 Separated algebraic spaces
   • 8.3 Hodge theory of simplicial schemes
9. Examples and applications
   • 9.1 Cohomology of groups and of classifying spaces
   • 9.2 Hodge theory of smooth hypersurfaces, following Griffiths
   • 9.3 Construction of complexes of first-order differential operators
10. Hodge theory in level ≤1
    • 10.1 1-motives
    • 10.2 1-motives and bi-extensions
    • 10.3 Algebraic interpretation of the mixed \mathrm{H}^1: the case of curves
    • 10.4 Translations of a theorem of Picard