nLab Hodge-filtered differential cohomology



Differential cohomology

Complex geometry



What is called Hodge-filtered cohomogy [Hopkins & Quick (2014)] is the variant of differential generalized cohomology obtained by passing from real differential geometry to complex geometry:

Where differential cohomology by default pairs a given Whitehead-generalized cohomology theory EE of underlying topological spaces with the degree-filtered E E_\bullet \otimes \mathbb{R}-valued de Rham complexes of differential forms on real smooth manifolds, the Hodge-filtered variant pairs instead with the Hodge filtered E E_\bullet \otimes \mathbb{C}-valued Dolbeault complexes [Haus & Quick (2022), p. 3], hence equivalently (see there) with the degree-filtered holomorphic de Rham complexes [Hopkins & Quick (2014), Def. 4.2].

Concretely, for pp \in \mathbb{Z}, the Hodge-filtered EE-cohomology E 𝒟 (p)(𝒳)E^\bullet_{\mathcal{D}}(p)(\mathcal{X}) of a complex manifold 𝒳\mathcal{X}, or more generally of an \infty -stack over the site Mfd Mfd_{\mathbb{C}} of all complex manifolds (with open covers), is the cohomology in the \infty -category of ( , 1 ) (\infty,1) -sheaves of spectra Sp(Sh (Mfd ))Sp\big(Sh_\infty(Mfd_{\mathbb{C}})\big) which is represented by the homotopy fiber product-spectrum

(1)E 𝒟(p)E×Ω p(-;π (E))Ω (-;π (E)), E_{\mathcal{D}}(p) \;\; \coloneqq \;\; E \underset{ \Omega^{\geq p}(\text{-};\pi_{\bullet}(E)\otimes \mathbb{C}) }{\times} \Omega^{\bullet}(\text{-};\pi_{\bullet}(E)\otimes \mathbb{C}) \,,


This is Hopkins & Quick (2014), Def. 4.2, being the direct holomorphic analog of the respective definition of differential cohomology (cf. the differential cohomology hexagon) in Hopkins Singer (2005) (except for that conventional rescaling by (2πi) /2(2\pi \mathrm{i})^{\bullet/2}.)

In the special case where EHE \,\coloneqq\, H\mathbb{Z} is integral ordinary cohomology the above homotopy pullback (1) reproduces the Deligne complex in its original form (see the details spelled out there; but the key observation may be recognized already in the classical review of Esnault & Viehweg (1988), Def. 2.6), whence the subscript “𝒟\mathcal{D}” in the above definition may be read as being for “generalized Deligne cohomology”.



The general concept of Hodge-filtered differential cohomology and introducing the special case of Hodge-filtered complex cobordism cohomology:

Hodge-filtered ordinary cohomology

The case of Hodge-filtered integral\;ordinary cohomology is [cf. Haus (2022), §3.2] the original definition of Deligne cohomology, see there for references.

Hodge-filtered topological K-theory

A Hodge-filtered form of complex topological K-theory appears (cf. Quick (2016), p. 2) in:

  • Max Karoubi, Théorie générale des classes caractéristiques secondaires, K-Theory 4 1 (1990) 55-87 [doi:10.1007/BF00534193, pdf]

  • Max Karoubi, Classes Caractéristiques de Fibrés Feuilletés, Holomorphes ou Algébriques, in: Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part II (Antwerp 1992), K-Theory 8 2 (1994) 153-211 [doi:10.1007/BF00961455]

Hodge-filtered complex cobordism

The case of Hodge filtered differential MU-cobordism cohomology theory

Introduction and survey:

Refinement of the Abel-Jacobi map to Hodge filtered differential MU-cobordism cohomology theory:

A geometric cocycle model by actual cobordism-classes:

On Umkehr maps in this context:

Last revised on June 10, 2023 at 09:39:20. See the history of this page for a list of all contributions to it.