nLab Hodge cycle

Idea

On a Kähler manifold $X$ a Hodge cycle is a $2p$-cycle in integral ordinary homology which corresponds to an element in Dolbeault homology of bidegree $(p,p)$.

Dually, a Hodge cocycle on $X$ is an integral ordinary cohomology class of even degree $2p$ which corresponds to an element in the Dolbeault cohomology group $H^{p,p}(X)$.

Definition

For Kähler manifolds

Definition

For $X$ be a Kähler manifold with Dolbeault cohomology groups $H^{p,q}(X)$, then for any $p \in \mathbb{N}$ the group of Hodge cohomology classes represented by Hodge cocycles is the fiber product

$$Hdg^p(X) \coloneqq H^{2p}(X,\mathbb{Z}) \underset{H^{2p}(X,\mathbb{C})}{\times} H^{p,p}(X)$$

in

$$\array{ && Hdg^p(X) \\ & \swarrow && \searrow \\ H^{2p}(X,\mathbb{Z}) && && H^{p,p}(X) \\ & {}_{\mathllap{H^{2p}(X,\mathbb{Z} \hookrightarrow \mathbb{C})}}\searrow && \swarrow_{\mathrlap{\iota}_{p,p}} \\ && H^{2p}(X,\mathbb{C}) } \,$$

where

• $\iota_{p,p}$ is the inclusion of the direct summand given by the Hodge decomposition;

• $H^{2p}(X,\mathbb{Z} \hookrightarrow \mathbb{C})$ is the morphism on cohomology groups induced by the canonical inclusion of coefficients (integers into complex numbers) as indicated.

For general complex analytic spaces

Def. has the following equivalent reformulation, which generalizes the definition to complex analytic spaces and to schemes over the complex numbers.

Since the left arrow here factors through real cohomology, this is in fact the same as the pullback

Definition

For $X$ a complex analytic space with Hodge filtration $F^\bullet H^{2p}(X,\mathbb{C})$ on its complex cohomology, then the group of Hodge cohomology classes represented by Hodge cocycles is the fiber product

$$Hdg^p(X) \coloneqq H^{2p}(X,\mathbb{Z}) \underset{H^{2p}(X,\mathbb{C})}{\times} F^p H^{2p}(X,\mathbb{X})$$

in

$$\array{ && Hdg^p(X) \\ & \swarrow && \searrow \\ H^{2p}(X,\mathbb{Z}) && && F^p H^{2p}(X,\mathbb{C}) \\ & \searrow && \swarrow \\ && H^{2p}(X,\mathbb{C}) } \,.$$

Proposition

If the complex analytic space $X$ happens to carry the structure of a Kähler manifold, then definition reduces to definition .

(e.g. Esnault-Viehweg 88, section 7.8)

Proof

The map from integral cohomology to complex cohomology factors through that of real cohomology via

$$\mathbb{Z} \hookrightarrow \mathbb{R}\hookrightarrow \mathbb{C}$$

and by Hodge symmetry the real cohomology classes are precisely thos represented by elements of the form $\alpha + \overline{\alpha}$, where

$$\overline{(-)} \;\colon\; H^{\bullet_1, \bullet_2}(X) \to H^{\bullet_2, \bullet_1}(X)$$

is complex conjugation. But the mid-dimensional Hodge filtration is

$$F^p H^{2p}(X,\mathbb{C}) = H^{p,p}(X) \oplus H^{p+1,p-1}(X) \oplus H^{p+2,p-2}(X) \oplus \cdots \oplus H^{2p,0}(X)$$

and hence the only elements invariant under complex conjugation – hence the only real cohomology classes – that it includes are those in $H^{p,p}(X)$, which is the subgroup appearing in def. .

Remark

In the form of def. the definition of Hodge cocycles generalizes to any context in which there is a Hodge structure. Notably it generalizes to a natural concept of Hodge cocycles in generalized (Eilenberg-Steenrod) cohomology (e.g. Hopkins-Quick 12, below remark 4.12)

Properties

Relation to intermediate Jacobians

There is a canonical map from holomorphic Deligne cohomology (ordinary differential cohomology) $H^{2k+2}(X, \mathbb{Z}\to \Omega^0 \to \cdots\to \Omega^{k}\to 0 \to \cdots)$ to Hodge cocycles. The fiber of that map is the intermediate Jacobian $J^{k+1}(X)$. See there for details.

References

The general formulation in terms of Hodge filtration for ordinary cohomology is noted for instance in

• Hélène Esnault, Eckart Viehweg, section 7.8 of Deligne-Beilinson cohomology in Rapoport, Schappacher, Schneider (eds.) Beilinson’s Conjectures on Special Values of L-Functions . Perspectives in Math. 4, Academic Press (1988) 43 - 91 (pdf)

in the context of discussion of intermediate Jacobians.

The general formulation in terms of Hodge filtration for generalized (Eilenberg-Steenrod) cohomology is noted for instance in

• Michael Hopkins, Gereon Quick, below remark 4.12 in Hodge filtered complex bordism (arXiv:1212.2173)

See also

• Wikipedia, Hodge cycles