Deligne cohomology

Deligne cohomology is a cohomology theory developed by Pierre Deligne that blends sheaf cohomology, Hodge theory, and homological algebra to encode arithmetic, topological, and differential-geometric information about complex algebraic varieties and smooth manifolds. It provides a bridge between the theories of Alexander Grothendieck, Jean-Pierre Serre, Armand Borel, Hermann Weyl, and John Tate by packaging regulator maps, intermediate Jacobians, and Chern classes into a single formalism. The theory has influenced work by Alexander Beilinson, Barry Mazur, Kazuya Kato, Vladimir Voevodsky, and Maxim Kontsevich and appears in interactions among the Hodge conjecture, the Beilinson conjectures, and the Atiyah–Singer index theorem.

Definition and construction

Deligne cohomology is defined for a smooth complex algebraic variety or a smooth manifold using a complex of sheaves introduced by Pierre Deligne; this construction combines the constant sheaf, the de Rham complex, and truncations related to the Hodge filtration familiar from Jean Leray and Henri Cartan; the resulting hypercohomology groups are the Deligne cohomology groups. The standard model uses the Deligne complex built from the constant sheaf Z(p) and the sheaves of holomorphic differential forms, reflecting ideas of André Weil, Oscar Zariski, Kurt Gödel's contemporaries in formal structures, and later enhancements by Grothendieck–Verdier duality frameworks associated with Alexander Grothendieck and Jean-Louis Verdier. Concretely, for an integer p one forms a mapping cone combining Z(p), Ω^0 → Ω^1 → ... → Ω^{p-1} with the Hodge filtration tied to constructions used by Wilhelm Killing and processed through derived functor techniques influenced by Henri Cartan and Jean-Pierre Serre.

Basic properties and functoriality

Deligne cohomology is contravariantly functorial for morphisms of varieties and smooth maps of manifolds, satisfying exact sequences that relate it to singular cohomology with coefficients in Z(p) and to filtered de Rham cohomology appearing in the work of Phillip Griffiths and Wilhelm Wirtinger. It admits a product structure compatible with cup products of Élie Cartan and Hyman Bass-style operations, and it fits into long exact sequences that compare motivic cohomology as formulated by Spencer Bloch and regulator maps studied by Don Zagier. Poincaré duality and purity statements in Deligne cohomology reflect the duality theories of Jean-Pierre Serre and Grothendieck duality, and cycle class maps compare algebraic cycles of Grothendieck and André Weil to classes in Deligne cohomology as in the work of Stephen Bloch and Vladimir Voevodsky.

Relation to other cohomology theories

Deligne cohomology sits at the crossroads of singular cohomology, de Rham cohomology, Hodge theory, and motivic cohomology developed by Alexander Beilinson and Spencer Bloch, and it gives a concrete realization of regulator maps to K-theory of schemes explored by Daniel Quillen and Charles Weibel. It refines Betti cohomology as in the classical theorems of Henri Poincaré and compares to crystalline cohomology studied by Pierre Berthelot and to étale cohomology as in the work of Michael Artin and Gerd Faltings; in particular it links the Hodge filtration of Phillip Griffiths with the Galois-module structures appearing in John Tate's theory of Tate modules. The relation to intermediate Jacobians of Griffiths and to analytic Abel–Jacobi maps used by Abel and Jacobi clarifies the connection between algebraic cycles and transcendental invariants central to the Hodge conjecture.

Examples and computations

Basic computations include Deligne cohomology of projective spaces and complex tori, where Deligne groups recover Chern classes of holomorphic line bundles and the Picard variety studied by André Weil and David Mumford. For a compact Riemann surface—classical objects in the work of Riemann and Bernhard Riemann—Deligne cohomology in degree two classifies holomorphic line bundles with connection and relates to the Jacobian varieties of Jacobi and to the Theta divisor as in Mumford's work. For higher-dimensional Calabi–Yau varieties central to Maxim Kontsevich and Shing-Tung Yau, Deligne cohomology computations enter mirror symmetry contexts explored by Kontsevich and Cumrun Vafa, while explicit regulator computations have been performed for modular curves examined by Pierre Deligne and Yuri Manin.

Applications in arithmetic geometry and physics

In arithmetic geometry Deligne cohomology underlies constructions of regulators and special value formulas appearing in the Birch and Swinnerton-Dyer conjecture and the Beilinson conjectures, interacting with the arithmetic of L-functions studied by Andrew Wiles, Richard Taylor, and Gerd Faltings. It provides a framework for secondary invariants such as heights in Arakelov theory developed by Henri Gillet and Christophe Soulé, and it informs the study of motives and periods central to Grothendieck's program and to subsequent advances by Yves André and Uwe Jannsen. In mathematical physics Deligne cohomology classifies higher gauge fields and Chern–Simons invariants in topological quantum field theories pioneered by Edward Witten, Michael Atiyah, and Graeme Segal and appears in string theory contexts investigated by Joseph Polchinski and Cumrun Vafa; it encodes holonomy of gerbes as used by Jean-Michel Bismut and Simons-type constructions relevant to anomaly cancellation studied by Alvarez-Gaumé and Ed Witten.

Category:Algebraic topology