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Contents

Idea

What is called Hodge-filtered cohomogy [[Hopkins & Quick (2014)](#HopkinsQuick14)] 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](HausQuick22)], hence equivalently (see there) with the degree-filtered holomorphic de Rham complexes [[Hopkins & Quick (2014), Def. 4.2](#HopkinsQuick14)].

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

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](https://arxiv.org/abs/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 ofDeligne 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](http://dx.doi.org/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](http://dx.doi.org/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](https://arxiv.org/abs/1503.08449)]

A geometric cocycle model by actual cobordism-classes:

• Knut Bjarte Haus, Geometric Hodge filtered complex cobordism, PhD thesis (2022) [[ntnuopen:3017489](https://ntnuopen.ntnu.no/ntnu-xmlui/handle/11250/3017489)]

• Knut Bjarte Haus, Gereon Quick, Geometric Hodge filtered complex cobordism [[arXiv:2210.13259](https://arxiv.org/abs/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](https://arxiv.org/abs/2303.15899)]