nLab Hodge-filtered differential cohomology

Idea

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 $E$ of underlying topological spaces with the degree-filtered $E_\bullet \otimes \mathbb{R}$-valued de Rham complexes of differential forms on real smooth manifolds, the Hodge-filtered variant pairs instead with the Hodge filtered $E_\bullet \otimes \mathbb{C}$-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 $p \in \mathbb{Z}$, the Hodge-filtered $E$-cohomology $E^\bullet_{\mathcal{D}}(p)(\mathcal{X})$ of a complex manifold $\mathcal{X}$, or more generally of an $\infty$-stack over the site $Mfd_{\mathbb{C}}$ of all complex manifolds (with open covers), is the cohomology in the $\infty$-category of $(\infty,1)$-sheaves of spectra $Sp\big(Sh_\infty(Mfd_{\mathbb{C}})\big)$ which is represented by the homotopy fiber product-spectrum

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

where

• $E$ denotes the spectrum representing the given Whitehead-generalized cohomology theory, regarded as a locally constant $(\infty,1)$-sheaf of spectra;

• $\Omega^\bullet(\text{-};A_\bullet)$ denotes the abelian sheaf of chain complexes given by the holomorphic de Rham complex with coefficients in a graded abelian group $A_\bullet$ and understood as an $(\infty,1)$-sheaf of Eilenberg-MacLane spectra via the stable Dold-Kan correspondence;

• $\Omega^{\geq p}(A_\bullet)$ denotes the subcomplex of $\geq p$-forms, similarly regarded,

• $E$ is regarded as fibered over $\Omega^\bullet\big(\text{-};A_\bullet\big)$ via the “complexification” map on $E$ given by smash product $E \simeq E \wedge \mathbb{S}\longrightarrow E \wedge H \mathbb{C}$ with the unit of the ring spectrum $H \mathbb{C}$ (what we may recognize as the Chern-Dold character) or rather (to be compatible with Pierre Deligne‘s original convention) that map followed by the automorphism which in degree $2n$ is given by multiplication with $(2 \pi \mathrm{i})^n \,\in\, \mathbb{C}$ .

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 $(2\pi \mathrm{i})^{\bullet/2}$.)

In the special case where $E \,\coloneqq\, H\mathbb{Z}$ 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 “$\mathcal{D}$” in the above definition may be read as being for “generalized Deligne cohomology”.

References

General

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

• Michael J. Hopkins, Gereon Quick, Hodge filtered complex bordism, Journal of Topology 8 1 (2014) 147-183 [arXiv:1212.2173, doi:10.1112/jtopol/jtu021]

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

• Hopkins & Quick (2014), §5

Introduction and survey:

• Gereon Quick, Geometric Hodge filtered complex cobordism, talk at CQTS (March 2023) [video:YT]

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

• Gereon Quick, An Abel-Jacobi invariant for cobordant cycles, Documenta Mathematica 21 (2016) 1645–1668 [arXiv:1503.08449]

A geometric cocycle model by actual cobordism-classes:

• Knut Bjarte Haus, Geometric Hodge filtered complex cobordism, PhD thesis (2022) [ntnuopen:3017489]

• Knut Bjarte Haus, Gereon Quick, Geometric Hodge filtered complex cobordism [arXiv:2210.13259]

On Umkehr maps in this context:

• Knut Bjarte Haus, Gereon Quick, Geometric pushforward in Hodge filtered complex cobordism and secondary invariants [arXiv:2303.15899]