Hodge–de Rham spectral sequence

Hodge–de Rham spectral sequence
Name	Hodge–de Rham spectral sequence
Field	Algebraic geometry, Complex geometry, Differential geometry
Introduced by	W. V. D. Hodge, Georges de Rham
First published	1930s–1950s

Hodge–de Rham spectral sequence The Hodge–de Rham spectral sequence is a computational and conceptual tool connecting the sheaf cohomology of holomorphic differential forms on a complex manifold or algebraic variety to its de Rham cohomology, drawing on ideas from W. V. D. Hodge, Georges de Rham, André Weil, Jean-Pierre Serre, and Alexander Grothendieck. It appears in the interaction of analytic, algebraic, and topological methods, relating the Dolbeault bicomplex, the Frölicher spectral sequence, and Hodge theory as developed by Pierre Deligne and Kunihiko Kodaira. The sequence plays a central role in the comparison theorems of Hodge theory, algebraic de Rham cohomology, and mixed Hodge structures studied by Phillip Griffiths and Wilfried Schmid.

Introduction

The Hodge–de Rham spectral sequence arises from the bicomplex of sheaves or differential forms on a complex manifold or smooth proper algebraic variety over C; its construction ties together the Dolbeault resolution used by Henri Cartan, the analytic methods of Kiyoshi Oka, and algebraic techniques refined by Alexander Grothendieck and Jean-Louis Verdier. Historically it provided a bridge between de Rham’s theorem, as expounded by Georges de Rham, and Hodge’s decomposition, developed by W. V. D. Hodge in studies influenced by F. R. Neumann and contemporaries in the Cambridge school. Subsequent generalizations were shaped by work of Pierre Deligne, John Tate, and researchers in the Institute for Advanced Study.

Construction of the Hodge–de Rham Spectral Sequence

One begins with the bicomplex (the Dolbeault complex) of sheaves Ω^{p,q} of smooth (p,q)-forms on a complex manifold X, invoking resolutions and hypercohomology techniques popularized by Jean-Pierre Serre and Alexander Grothendieck. Filtering the total complex by the p-degree yields a spectral sequence whose E_1-page is H^q(X, Ω^p) — sheaf cohomology groups classically computed using Čech methods of Oswald Veblen and analytic techniques refined by Kiyoshi Oka. The differentials on successive pages reflect operators related to the ∂ and ∂̄ operators studied by Kunihiko Kodaira and Kōsaku Yosida; functorial properties rely on pushforward and pullback formalism of Grothendieck and the derived category language introduced by Alexander Grothendieck and Jean-Louis Verdier.

Convergence and Degeneration

Under hypotheses such as compact Kähler structure (studied by Shing-Tung Yau and Andreotti–Vesentini), the spectral sequence degenerates at the E_1-page, yielding the classical Hodge decomposition proven in the foundational work of W. V. D. Hodge and modernized by P. A. Griffiths. For smooth proper algebraic varieties over C, the degeneration statement is implied by Hodge theory and by the comparison isomorphisms of Deligne connecting algebraic de Rham cohomology and Betti cohomology, with proofs invoking the hard Lefschetz theorem of Solomon Lefschetz and its extensions by Pierre Deligne. In non-Kähler settings, counterexamples constructed by Michel Kervaire and pathologies investigated by Henri Cartan show nondegeneration, captured by the Frölicher spectral sequence examined by Alfred Frölicher.

Relation to Hodge Theory and Hodge Decomposition

When the spectral sequence degenerates at E_1 for a compact Kähler manifold, one obtains the Hodge decomposition H^n(X, C) ≅ ⊕_{p+q=n} H^q(X, Ω^p), a cornerstone of classical Hodge theory developed by W. V. D. Hodge and expanded by Phillip Griffiths and Wilfried Schmid. This decomposition underpins the theory of Hodge structures studied by Jean-Pierre Serre, John Tate, and Pierre Deligne, and it interacts with the polarizations given by the Hodge–Riemann bilinear relations used by Carl Ludwig Siegel and in proofs by Shing-Tung Yau. The spectral sequence framework also informs the study of mixed Hodge structures on singular varieties and degenerations, central to work of Pierre Deligne and Richard Hain.

Examples and Computations

Classic computations include compact complex tori (analyzed by André Weil and Hermann Weyl), projective spaces (treated in foundational algebraic geometry by Oscar Zariski and André Weil), and complex K3 surfaces (intensively studied by K3 surfaces researcher Piatetski-Shapiro and Igor Shafarevich). For a smooth projective curve (Riemann surface) the E_1-degeneration yields the classical Hodge splitting tied to work of Bernhard Riemann and Friedrich Schottky. Non-Kähler examples exhibiting nondegeneration were constructed by Kodaira and further analyzed by Phillip Griffiths and Daniele Angella in modern computational studies using spectral sequence packages inspired by algorithms of Jean-Pierre Serre.

Applications in Algebraic and Complex Geometry

The Hodge–de Rham spectral sequence is instrumental in comparison theorems between algebraic de Rham cohomology and singular cohomology as formalized by Grothendieck and Deligne, with consequences for the study of period maps developed by Phillip Griffiths and for Torelli-type theorems explored by Igor Shafarevich. It informs the theory of variations of Hodge structure investigated by Wilfried Schmid and the study of moduli spaces pursued by David Mumford and Michael Artin. In arithmetic geometry, its algebraic incarnations influence p-adic Hodge theory advanced by Jean-Marc Fontaine and Peter Scholze, and connect with the étale cohomology framework of Alexander Grothendieck and Jean-Pierre Serre.

Variants and Generalizations

Generalizations include the Frölicher spectral sequence of Alfred Frölicher, algebraic de Rham spectral sequences in the work of Alexander Grothendieck and Pierre Deligne, p-adic analogues appearing in the research of Jean-Marc Fontaine and Kiran Kedlaya, and mixed Hodge theoretic spectral sequences developed by Pierre Deligne and Richard Hain. Derived and homotopical extensions incorporate techniques from the derived category approach of Alexander Grothendieck and Amnon Neeman as well as newer perspectives from Jacob Lurie’s work in higher algebra and spectral algebraic geometry.

Category:Algebraic geometry