Categorical Hodge Correspondence

• Categorical Hodge correspondence is a framework that redefines classical Hodge theory by encoding cohomological data using derived categories and categorical invariants.
• It employs robust methodologies such as decategorification, noncommutative cyclic homology, and ∞-categorical tools to translate algebraic cycles into categorical Chern characters.
• This approach has broad applications—from addressing aspects of the Hodge conjecture to extending into nonabelian, p-adic, and logarithmic geometric settings—offering deep insights into modern geometric structures.

The literature on categorical Hodge correspondence treats Hodge theory not only as a theory of cohomology groups and filtrations, but also as a theory of categories, $K$-theory, Hochschild and cyclic invariants, and categorical symmetries. In one recurrent form, Hodge classes are identified with Chern characters of objects in a derived category; in another, a variety is assigned a stable $\infty$-category whose heart recovers variations of Hodge structure; in another, periodic cyclic homology of a dg-category carries the de Rham part of a noncommutative Hodge structure and reproduces geometric Hodge data (Mansour, 14 Jul 2025, Luo, 9 Sep 2025, Shklyarov, 2011, Vaintrob, 2017, Brown et al., 2024). The literature surveyed here suggests that the expression denotes a family of closely related programs rather than a single universally fixed formalism.

1. Conceptual architectures

A persistent template is a decategorification map from categorical invariants to classical Hodge-theoretic objects. In the derived-categorical approach to the Hodge conjecture, the basic bridge is the Chern character

$$\mathrm{ch}:K_0(X)\otimes \mathbb{Q}\to H^{2*}(X,\mathbb{Q}),$$

and, for smooth complete intersections, the key claim is

$$H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X)=\mathrm{ch}\big(K_0(D^b(X))\otimes\mathbb{Q}\big)_p.$$

This turns Hodge classes into categorical Chern characters of derived objects (Mansour, 14 Jul 2025).

A second template is explicitly functorial. For toroidal and logarithmic geometry, the categorified Hodge correspondence is defined as an $\infty$-functor

$$\mathrm{Hodge}:\mathbf{Var}_\mathbb{C}^{op}\to \mathbf{StCat}_\infty,$$

sending a smooth complex variety to a derived Hodge $\infty$-category whose heart is the category of polarizable variations of Hodge structure, and whose algebraic $K$-theory carries a categorical Chern character

$$\operatorname{ch}:K_0(\mathrm{Hodge}(X))\otimes\mathbb{Q}\to \bigoplus_p \mathrm{Hdg}^p(X).$$

The associated conjecture asserts that this map is an isomorphism (Luo, 9 Sep 2025).

A third template is noncommutative and homological. For a dg-category $\mathcal{C}$, a noncommutative Hodge filtration on $\infty$0 is defined by

$\infty$1

with rational structure supplied by the image of topological $\infty$2-theory under the topological Chern character. In that setting, Hodge classes are defined by

$\infty$3

and the noncommutative Hodge condition requires $\infty$4 (Brown et al., 2024).

These three architectures differ in language but share a common pattern: Hodge filtrations, Hodge classes, and period-type structures are reconstructed from categories and their invariants.

2. Derived categories, algebraic cycles, and Hodge conjectures

In the most direct algebro-geometric version, categorical Hodge correspondence is tied to the Hodge conjecture. For a smooth projective complex variety $\infty$5, the classical problem is

$\infty$6

A 2025 deformation-theoretic program proposes to attack this through derived categories and complete intersections. The strategy has two steps: first, prove the conjecture for smooth complete intersections using $\infty$7, Orlov-type semiorthogonal decompositions, and surjectivity of the Chern character onto Hodge classes; second, place an arbitrary smooth projective variety into a flat smooth family whose general fibers are smooth complete intersections and transport categorical Chern-character data across the family (Mansour, 14 Jul 2025).

In this framework, the derived category does not merely encode auxiliary structure. Semiorthogonal decompositions

$\infty$8

induce decompositions on $\infty$9, and hence on the Chern-character image in cohomology. The line-bundle part controls ambient algebraic classes, while the residual category $$\mathrm{ch}:K_0(X)\otimes \mathbb{Q}\to H^{2*}(X,\mathbb{Q}),$$0 is expected to control primitive contributions. The remaining obstruction in the general deformation argument is the “algebraicity of limits,” namely whether limits of algebraic classes along the family are again algebraic on the special fiber (Mansour, 14 Jul 2025).

A parallel categorical reformulation appears in the integral setting for CY2 categories. There topological $$\mathrm{ch}:K_0(X)\otimes \mathbb{Q}\to H^{2*}(X,\mathbb{Q}),$$1-theory and Hochschild homology supply a Mukai Hodge structure

$$\mathrm{ch}:K_0(X)\otimes \mathbb{Q}\to H^{2*}(X,\mathbb{Q}),$$2

and the categorical integral Hodge conjecture asks whether integral Hodge classes in this lattice come from $$\mathrm{ch}:K_0(X)\otimes \mathbb{Q}\to H^{2*}(X,\mathbb{Q}),$$3. For CY2 categories that deform to $$\mathrm{ch}:K_0(X)\otimes \mathbb{Q}\to H^{2*}(X,\mathbb{Q}),$$4 for a twisted K3 or abelian surface, integral Hodge classes of the specified Mukai square are proved algebraic, and this yields applications to cubic and Gushel–Mukai fourfolds (Perry, 2020).

For smooth proper 3-Calabi–Yau categories, the emphasis shifts from algebraic cycles to numerical Hodge data. Using a homological unit $$\mathrm{ch}:K_0(X)\otimes \mathbb{Q}\to H^{2*}(X,\mathbb{Q}),$$5 and Hochschild homology, categorical Hodge numbers $$\mathrm{ch}:K_0(X)\otimes \mathbb{Q}\to H^{2*}(X,\mathbb{Q}),$$6 are defined, and under a K-theoretic generation hypothesis they become intrinsic invariants of the category, independent of the chosen rank function. In the strict case $$\mathrm{ch}:K_0(X)\otimes \mathbb{Q}\to H^{2*}(X,\mathbb{Q}),$$7, a Hodge structure on $$\mathrm{ch}:K_0(X)\otimes \mathbb{Q}\to H^{2*}(X,\mathbb{Q}),$$8 is defined whose Hodge spaces have those categorical Hodge numbers as dimensions (Abuaf, 2018).

3. Cyclic homology, singularities, and logarithmic geometry

A distinct but closely related line of work identifies categorical Hodge data through cyclic homology. For a dg-algebra or dg-category, periodic cyclic homology is treated as a de Rham-type invariant equipped with a canonical $$\mathrm{ch}:K_0(X)\otimes \mathbb{Q}\to H^{2*}(X,\mathbb{Q}),$$9-connection. In the case of matrix factorizations $$H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X)=\mathrm{ch}\big(K_0(D^b(X))\otimes\mathbb{Q}\big)_p.$$0 of a polynomial with isolated singularity, the canonical noncommutative $$H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X)=\mathrm{ch}\big(K_0(D^b(X))\otimes\mathbb{Q}\big)_p.$$1-connection on $$H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X)=\mathrm{ch}\big(K_0(D^b(X))\otimes\mathbb{Q}\big)_p.$$2 is shown to coincide, up to shift, with the classical twisted de Rham connection

$$H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X)=\mathrm{ch}\big(K_0(D^b(X))\otimes\mathbb{Q}\big)_p.$$3

on the Brieskorn-lattice side (Shklyarov, 2011). This is one of the clearest instances where categorical and geometric Hodge data are proved to match at the level of bundles with connection.

The comparison is sharpened in categorical Saito theory. For an isolated hypersurface singularity $$H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X)=\mathrm{ch}\big(K_0(D^b(X))\otimes\mathbb{Q}\big)_p.$$4, an explicit cyclic minimal $$H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X)=\mathrm{ch}\big(K_0(D^b(X))\otimes\mathbb{Q}\big)_p.$$5-model for $$H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X)=\mathrm{ch}\big(K_0(D^b(X))\otimes\mathbb{Q}\big)_p.$$6 is constructed via Kontsevich deformation quantization, and the deformed Tsygan formality map yields an isomorphism between the categorical Variation of Semi-infinite Hodge Structure of $$H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X)=\mathrm{ch}\big(K_0(D^b(X))\otimes\mathbb{Q}\big)_p.$$7 and Saito’s geometric VSHS from primitive form theory. This identifies the categorical higher residue pairing and connections with their classical counterparts (Tu, 2019).

Logarithmic geometry provides another categorical realization. For a smooth open variety $$H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X)=\mathrm{ch}\big(K_0(D^b(X))\otimes\mathbb{Q}\big)_p.$$8 with toroidal compactification $$H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X)=\mathrm{ch}\big(K_0(D^b(X))\otimes\mathbb{Q}\big)_p.$$9, the category

$\infty$0

is introduced as a large Abelian symmetric monoidal category constructed from ramified towers, almost localization along the boundary, and étale sheafification. Its Hochschild homology recovers logarithmic forms,

$\infty$1

while its periodic cyclic homology recovers $\infty$2. The noncommutative Hodge-to-de Rham spectral sequence of the category is thus identified with the classical logarithmic Hodge-to-de Rham spectral sequence (Vaintrob, 2017).

For smooth projective hypersurfaces, the singularity category itself carries the relevant Hodge structure. If $\infty$3 with $\infty$4 homogeneous and $\infty$5 smooth of even dimension $\infty$6, then

$\infty$7

and this isomorphism identifies the noncommutative Hodge structure on $\infty$8 with the classical Hodge structure on $\infty$9. The resulting noncommutative Hodge condition for $$\mathrm{Hodge}:\mathbf{Var}_\mathbb{C}^{op}\to \mathbf{StCat}_\infty,$$0 is equivalent to the classical Hodge conjecture for $$\mathrm{Hodge}:\mathbf{Var}_\mathbb{C}^{op}\to \mathbf{StCat}_\infty,$$1 (Brown et al., 2024).

4. $$\mathrm{Hodge}:\mathbf{Var}_\mathbb{C}^{op}\to \mathbf{StCat}_\infty,$$2-categorical, toroidal, and $$\mathrm{Hodge}:\mathbf{Var}_\mathbb{C}^{op}\to \mathbf{StCat}_\infty,$$3-adic extensions

The toroidal/logarithmic program pushes categorical Hodge correspondence into stable $$\mathrm{Hodge}:\mathbf{Var}_\mathbb{C}^{op}\to \mathbf{StCat}_\infty,$$4-categories and obstruction theory. For a toroidal pair $$\mathrm{Hodge}:\mathbf{Var}_\mathbb{C}^{op}\to \mathbf{StCat}_\infty,$$5, logarithmic differential forms $$\mathrm{Hodge}:\mathbf{Var}_\mathbb{C}^{op}\to \mathbf{StCat}_\infty,$$6, Wei’s $$\mathrm{Hodge}:\mathbf{Var}_\mathbb{C}^{op}\to \mathbf{StCat}_\infty,$$7-degeneration, weighted toroidal structures, and an obstruction complex

$$\mathrm{Hodge}:\mathbf{Var}_\mathbb{C}^{op}\to \mathbf{StCat}_\infty,$$8

are assembled into a categorical package. The derived Hodge $$\mathrm{Hodge}:\mathbf{Var}_\mathbb{C}^{op}\to \mathbf{StCat}_\infty,$$9-category $\infty$0 is defined so that its heart is $\infty$1, its $\infty$2-theory decategorifies to Hodge classes, and a categorical obstruction theory is expected to lift long exact sequences in cohomology to long exact sequences in $\infty$3-theory (Luo, 9 Sep 2025).

In the weighted toroidal setting, rational weights are used to force vanishing of obstruction groups, so that weighted Hodge classes extend across the boundary and become absolute Hodge classes on the compactification. Under those hypotheses, all weighted Hodge classes are proved algebraic for projective toroidal varieties, and the heart of $\infty$4 is generated by algebraic cycles (Luo, 9 Sep 2025).

A technically different extension appears in Hodge–Iwasawa theory. There the stated aim is a serious unification of $\infty$5-adic Hodge theory and $\infty$6-adic Iwasawa theory by means of $\infty$7-categorical derived categories of inductive Banach modules and condensed solidification. Hodge modules are placed in $\infty$8-categorical derived categories of quasicoherent sheaves on analytic spectra of period rings, together with Frobenius-equivariant structures and homotopy limits or colimits over radii and intervals (Tong, 2023). This suggests that categorical Hodge correspondence is no longer confined to complex projective geometry, but can also be formulated over period-ring geometries and Iwasawa-theoretic deformations.

5. Nonabelian, wild, and representation-theoretic avatars

Another branch of the subject replaces algebraic cycles with moduli of bundles, local systems, or categorical representations. In nonabelian Hodge theory, the basic equivalence identifies the moduli space of polystable Higgs bundles of degree $\infty$9 with the character variety, via flat connections, harmonic metrics, and hyperkähler reduction (Thomas, 2022). In the parahoric setting on noncompact curves, this becomes an explicit equivalence of categories

$K$0

linking $K$1-stable logahoric Higgs torsors, logahoric connections, and filtered $K$2-local systems (Huang et al., 2022). Here the correspondence is already categorical in a literal sense.

Wild Hodge theory extends this picture to irregular singularities. Harmonic bundles with irregular behavior, wild Higgs bundles, and pure wild twistor $K$3-modules provide the irregular analogue of the Corlette–Simpson dictionary. This supplies the categorical framework used to prove Kashiwara’s generalized Hard Lefschetz theorem for simple holonomic $K$4-modules on smooth projective varieties (Sabbah, 2012).

Representation theory furnishes yet another meaning of “categorical Hodge.” In the Hecke category of Soergel bimodules, one finds global, relative, and local Hodge theories with graded symmetric forms, Lefschetz operators, primitive decompositions, hard Lefschetz, and Hodge–Riemann bilinear relations. These structures control Kazhdan–Lusztig polynomials, structure constants, and Jantzen filtrations (Williamson, 2016). Closely related is the study of Hodge modules on products of complementary Grassmannians: the associated graded of a Hodge-module pushforward is identified with a coherent kernel on a cotangent correspondence, and categorical $K$5-actions organize both Hodge and weight filtrations (Cautis et al., 2016).

These examples show that categorical Hodge correspondence is not limited to cycle-class questions. It also denotes the transfer of Lefschetz data, filtrations, and polarizations from geometry to monoidal categories, representation categories, and $K$6-module kernels.

6. Conjectures, limitations, and unifying themes

Several formulations are explicitly conjectural. The toroidal $K$7-categorical program proposes the categorical Hodge conjecture

$K$8

while its categorical obstruction theory remains conjectural at the $K$9-theoretic level (Luo, 9 Sep 2025). The deformation-theoretic reduction of the Hodge conjecture isolates algebraicity of limits as the remaining subtlety in passing from complete intersections to arbitrary smooth projective varieties (Mansour, 14 Jul 2025).

In noncommutative Hodge theory, the de Rham part is often more developed than the Betti part. For periodic cyclic homology of dg-categories, explicit $$\operatorname{ch}:K_0(\mathrm{Hodge}(X))\otimes\mathbb{Q}\to \bigoplus_p \mathrm{Hdg}^p(X).$$0-connections are available, and the matrix-factorization case matches twisted de Rham theory, but the Betti data are not treated in the foundational comparison for noncommutative Hodge structures (Shklyarov, 2011). This limits the extent to which one can speak of full noncommutative Hodge structures in the classical sense.

A common misconception is that categorical Hodge correspondence always means an equivalence between algebraic cycles and objects of a derived category. That is one important strand, but the surveyed literature also uses the term for $$\operatorname{ch}:K_0(\mathrm{Hodge}(X))\otimes\mathbb{Q}\to \bigoplus_p \mathrm{Hdg}^p(X).$$1-categorical realizations of variations of Hodge structure, for cyclic-homological reconstructions of de Rham or logarithmic Hodge theory, for nonabelian Hodge correspondences between moduli categories, and for Hodge-theoretic packages in purely categorical representation theory (Mansour, 14 Jul 2025, Vaintrob, 2017, Huang et al., 2022, Williamson, 2016).

The unifying theme is more stable than the terminology. Hodge-theoretic information—classes, filtrations, weights, residues, monodromy, primitive decompositions, Lefschetz operators, and period data—is repeatedly encoded in categories and then recovered through $$\operatorname{ch}:K_0(\mathrm{Hodge}(X))\otimes\mathbb{Q}\to \bigoplus_p \mathrm{Hdg}^p(X).$$2-theory, Hochschild or cyclic homology, topological $$\operatorname{ch}:K_0(\mathrm{Hodge}(X))\otimes\mathbb{Q}\to \bigoplus_p \mathrm{Hdg}^p(X).$$3-theory, or categorical symmetries. In that sense, categorical Hodge correspondence names a broad research program: to replace cohomological Hodge data by categorical objects whose decategorification reproduces the classical theory, and whose internal structure clarifies deformation, algebraicity, and functoriality across complex, logarithmic, noncommutative, and $$\operatorname{ch}:K_0(\mathrm{Hodge}(X))\otimes\mathbb{Q}\to \bigoplus_p \mathrm{Hdg}^p(X).$$4-adic settings.