3634 index index.xml Deligne's "Hodge Theory I, II, and III" TO-DO: search for all todo TO-DO: make subsections appear in PDF (use chunk environment) TO-DO: check PDF output TO-DO: name all proof trees something like proof-of-hodge-theory-iii-X.Y.Z? This is an English translation of P. Deligne's three papers on Hodge theory: "Théorie de Hodge I", Actes du Congrès intern. math. 1 (1970) pp. 425–430. [PDF] "Théorie de Hodge II", Pub. Math. de l'IHÉS 40 (1971) pp. 5–58. [PDF] "Théorie de Hodge III", Pub. Math. de l'IHÉS 44 (1974) pp. 5–77. [PDF] The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein. Any notes by the translator are in italics and prefixed with "[Trans.]". You can view the entire source code of this translation (and contribute or submit corrections) in the GitHub repository. Corrections and comments welcome. Hodge Theory I Hodge Theory II Introduction Filtrations Filtered objects Opposite filtrations The two filtrations lemma Hypercohomology of filtered complexes Hodge structures Pure structures Hodge theory Mixed structures Hodge theory of non-singular algebraic varieties Logarithmic poles and residues Mixed Hodge theory Applications and supplements The fixed set theorem The semi-simplicity theorem Supplement to [D1968] Homomorphisms of abelian schemes Hodge Theory III Introduction Terminology and notation Cohomological descent Simplicial topological spaces Cohomology of simplicial topological spaces Cohomological descent Examples of simplicial topological spaces Classifying spaces Construction of hypercovers Relative cohomology Multi-simplicial spaces Supplements to §1 Filtered derived category Supplements to the two-filtrations lemma Hodge theory of algebraic spaces Hodge complexes Separated algebraic spaces Hodge theory of simplicial schemes Examples and applications Cohomology of groups and of classifying spaces Hodge theory of smooth hypersurfaces, following Griffiths Construction of complexes of first-order differential operators Hodge theory in level 1-motives 1-motives and bi-extensions Algebraic interpretation of the mixed : the case of curves Translation of a theorem of Picard Bibliography