nLab intermediate Jacobian

Idea

Traditionally the $(k+1)$st intermediate Jacobian variety $J^{k+1}(\Sigma)$ of a complex analytic space $\Sigma$ is the quotient of its ordinary cohomology in degree $2k+1$ with real number coefficients by that with integer coefficients

$$J^{k+1}(\Sigma) \coloneqq H^{2k+1}(\Sigma, \mathbb{R}) / H^{2k+1}(\Sigma, \mathbb{Z}) \,.$$

This space naturally carries the structure of a complex manifold (in fact two such structures, named after Griffiths and after Weil) and this complex analytic space, which is in fact a complex torus, is properly what is called the $(k+1)$st intermediate Jacobian variety of $\Sigma$. This terminology derives from the term Jacobian variety which is the (historically earlier) special case for $k = 0$ and $dim_{\mathbb{C}}(\Sigma) = 1$.

Notice that, conceptually, we may (cf. at Deligne cohomology the exact sequences and generally at differential cohomology hexagon):

• think of $H^{2k+1}(\Sigma,\mathbb{R})$ as the space of those $n$-form connections on $\Sigma$ which are both flat and have trivial underlying line $n$-bundle;

• think of $H^{2k + 1}(\Sigma,\mathbb{Z})$ as the group of “large” (i.e.: not connected to the identity) higher gauge transformations acting on these gauge fields;

• hence understand $J^{k+1}(\Sigma)$ as the moduli space of flat $n$-form connections on trivial underlying line $n$-bundles.

This turns out to be a natural and useful perspective on intermediate Jacobians: Deligne’s theorem (discussed as theorem below) characterizes the intermediate Jacobians as subgroups of the relevant Deligne cohomology group of $\Sigma$, where Deligne cohomology is a model for ordinary differential cohomology that classifies these line n-bundles with connection. Moreover, as discussed in Rem. below, Deligne’s theorem in the formulation of Esnault & Viehweg (1988) may be rephrased such as to manifestly give a formal incarnation of the the statement that $J^{k+1}(\Sigma)$ is just that subgroup of the Deligne complex given by line $n$-connections with trivial curvature and trivial underlying line $n$-bundle.

This formulation, in turn, has an evident generalization from ordinary differential cohomology to general (namely Whitehead-generalized) differential cohomology, which we discuss further below.

This perspective on intermediate Jacobians from higher gauge theory also faithfully reflects their role in fundamental physics (in quantum field theory and string theory) [Witten (1996), Hopkins & Singer (2002)]. Here higher dimensional Chern-Simons theory has as fields certain higher gauge fields specified by some type of differential cohomology, and the connected components of its phase space (of solutions to the equations of motion) is precisely the corresponding intermediate Jacobian. Moreover the transgression of the higher Chern-Simons action functional produces a line bundle on the intermediate Jacobian, which is the prequantum line bundle of the theory. By geometric quantization one has to choose a Kähler polarization for this line bundle and the Weil complex structure on $J^{k+1}(\Sigma)$ is precisely that. In terms of complex geometry this state of affairs directly translates into the statement that the Weil intermediate Jacobians are polarized varieties. In fact they are principally polarized, which on the physics side corresponds to the metaplectic correction of the Kähler polarization used for geometric quantization. The holomorphic section of the resulting Theta characteristic on the intermediate Jacobian is physically the partition function of self-dual higher gauge theory on $\Sigma$ (see there for more) which mathematically is the corresponding theta function.

By way of these deep relations intermediate Jacobians play an important role in (higher) geometry.

Definition

For differential ordinary cohomology

The underlying real manifold

Let $\Sigma$ be a projective smooth complex variety (see at GAGA).

Definition

For $k \in \mathbb{N}$ the $k$th intermediate Jacobian of $\Sigma$ is, as a real manifold, the quotient

$$J^{k+1}(\Sigma) \coloneqq H^{2k+1}(\Sigma,\mathbb{R})/H^{2k+1}(\Sigma,\mathbb{Z})$$

of the ordinary cohomology groups of $X$ with coefficients in the abelian groups of real numbers and of integers, respectively, induced by the canonical inclusion $\mathbb{Z} \hookrightarrow \mathbb{R}$.

Here $H^{2k+1}(\Sigma,\mathbb{R})$ is naturally a vector space over the real numbers and this is what induces the smooth manifold-structure on the quotient.

For the purpose of eventually equipping this with the structure of a complex manifold one may realizes it as the quotient of the complex vector space of complex ordinary cohomology, as follows:

Remark

A real differential form

$$\alpha \;\in\; \Omega^{2k+1}_{\mathbb{R}}(\Sigma)$$

is, by the Hodge theorem, a sum of complex differential forms in homogeneous Dolbeault bidegree of the form

$$\alpha \;=\; \alpha^{2k+1,0}+ \alpha^{2k,1} + \cdots + \alpha^{k+1,k} + \overline{\alpha^{k+1,k}} + \cdots + \overline{\alpha^{2k,1}} + \overline{\alpha^{2k+1,0}} \,,$$

where

$$\overline{(-)} \;\colon\; \Omega^{p,q}(\Sigma)\longrightarrow \Omega^{q,p}(\Sigma)$$

is the antilinear function on complex differential forms given by complex conjugation.

It follows with the de Rham theorem that:

Proposition

There is a canonical isomorphism of real vector spaces

$$H^{2k+1}(\Sigma, \mathbb{R}) \;\simeq\; H^{2k+1}(\Sigma,\mathbb{C})/(F^{k+1} H^{2k+1}(\Sigma,\mathbb{C})) \,,$$

where

$$F^{k+1} H^{2k+1}(\Sigma,\mathbb{C}) \;\coloneqq\; \underset{p \geq k+1}{\oplus} H^{p,2k+1-p}(\Sigma)$$

is the $(k+1)$st stage in the Hodge filtration of $H^{2k+1}(\Sigma,\mathbb{C})$.

Hence an equivalent way of writing the intermediate Jacobian (still as a real manifold) is as the quotient space of the real manifold underlying a complex vector space, as follows:

Proposition

The intermediate Jacobian of def. is equivalently

$$J^{k+1}(\Sigma) \;\simeq\; H^{2k+1}(\Sigma,\mathbb{C}) / \big( F^{k+1} H^{2k+1}(\Sigma, \mathbb{C}) \,\oplus\, H^{2k+1}(\Sigma,\mathbb{Z}) \big) \,.$$

Yet one more reformulation is useful when properly working in complex analytic geometry/GAGA:

Definition

Write $\big(\mathbf{B}^k \mathbb{G}_a\big)_{conn}$ for the abelian sheaf of chain complexes on site of complex manifolds which assigns the truncated de Rham complexes of holomorphic differential forms:

$$\big(\mathbf{B}^k \mathbb{G}_a\big)_{conn} \;\coloneqq\; \left( \mathcal{O} \stackrel{\partial }{\to} \Omega^{1} \stackrel{\partial}{\to} \cdots \stackrel{\partial}{\to} \Omega^{k} \right)$$

regarded as sitting in degrees $k$ to 0.

(e.g. Esnault & Viehweg (1988), top of p. 14)

Proposition

The quotient in prop. is equivalently the abelian sheaf hypercohomology with coefficients in $\mathbf{B}^{2k}\mathbb{G}_a$ of def. :

$$H^{2k+1}\big( \Sigma ,\, \mathbb{C} \big) / F^{k+1} H^{2k+1}(\Sigma, \mathbb{C}) \;\simeq\; \Big[ \Sigma ,\, \mathbf{B}^k\big(\mathbf{B}^k \mathbb{G}_a\big)_conn \Big] \,.$$

(e.g. Esnault & Viehweg (1988), 2.5 b)).

There are two canonical ways of equipping $H^{2k+1}(\Sigma,\mathbb{C})$, and hence the above quotient, with the structure of a complex manifold. Sometimes these agree, but in general they do not, and hence they go by different names:

1. Griffith intermediate Jacobian,

2. Weil intermediate Jacobian.

Characterization as Hodge-trivial Deligne cohomology

A theorem due to Pierre Deligne says that the intermediate Jacobian $J^k(\Sigma)$ is characterised as being the fiber of a canonical map from (complex analytic) Deligne cohomology to the $k$th Hodge filtration of integral cohomology.

Definition

The group $Hdg^{k+1}(\Sigma)$ of Hodge cohomology classes is the subgroup of $\mathbb{Z}(k+1)$-cohomology classes whose image in complex cohomology is in the $(k+1)$st stage of the Hodge filtration, hence the group sitting in the following pullback diagram

$$\array{ Hdg^{k+1}(\Sigma) &\longrightarrow& F^{k+1} H^{2k+2}(\Sigma;\,\mathbb{C}) \\ \Big\downarrow &{}^{{}_{(pb)}}& \Big\downarrow \\ H^{2k+2}\big(\Sigma;\,\mathbb{Z}(k+1)\big) &\longrightarrow& H^{2k+2}(\Sigma;\,\mathbb{C}) \mathrlap{\,.} }$$

The following says this in a complex analytic-way that generalizes:

Proposition

Equivalently, the Hodge cohomology classes of def. are given by the pullback

$$\array{ Hdg^{k+1}(\Sigma) &\longrightarrow& H^{2k+2}\big(\Sigma;\, \Omega^{\bullet \geq k+1}\big) \\ \Big\downarrow && \Big\downarrow \\ H^{2k+2}\big(\Sigma;\,\mathbb{Z}(k+1)\big) &\longrightarrow& H^{2k+2}(\Sigma;\,\mathbb{C}) } \,,$$

where now in the top right we have the abelian sheaf hypercohomology with coefficients in the holomorphic de Rham complex, truncated (but otherwise unshifted) as indicated.

(Esnault & Viehweg (1988), section 7.8)

Theorem

(Deligne)

As an abelian group the intermediate Jacobian $J^k(\Sigma)$, def. , is the fiber of the canonical map from Deligne cohomology to Hodge cohomology classes, has as fitting into a short exact sequence of the following form:

$$0 \to J^{k+1}(\Sigma) \longrightarrow H^{2k+2}\big(\Sigma;\, \mathbb{Z}(k+1)_{D}\big) \longrightarrow Hdg^{k+1}(\Sigma) \to 0 \,.$$

(e.g. Esnault & Viehweg (1988), (7.9); Peters & Steenbrink (2008), lemma 7.20)

Proof

Use prop. and the characteristic long exact sequences of ordinary differential cohomology.

Remark

The fiber product-incarnation of $Hdg^{k+1}(\Sigma)$ in prop. is noteworthy in that it is analogous to the homotopy fiber-characterization of the holomorphic Deligne complex itself.

Consider the following diagram of sheaves of chain complexes on the site $SteinSp$ of Stein manifolds (see at complex analytic ∞-groupoid for more on this):

$$\array{ \mathbb{Z}(p)[-2k-2] && && \Omega^{\bullet \geq k+1}[-2k-2] \\ & \searrow && \swarrow \\ && \mathbb{C}[-2k-2] } \,.$$

This is just of the form as discussed in some detail at circle n-bundle with connection and also at differential cohomology diagram in the section on Deligne coefficients. In particular the homotopy limit over this diagram – hence the homotopy fiber of the two maps – is a version of the Deligne complex:

$$\array{ &&\mathbb{Z}(p)_D[-2k-2] \\ && = \\ &&(\mathbb{Z}(p) \to \mathcal{O} \to \Omega^1 \to \cdots \to \Omega^{k} \to 0 \to \cdots)[-2k-2] \\ & \swarrow && \searrow \\ \mathbb{Z}(p)[-2k-2] && (hpb) && \Omega^{\bullet \geq k+1}[-2k-2] \\ & \searrow && \swarrow \\ && \mathbb{C}[-2k-2] } \,.$$

Since homotopy pullbacks are preserved by foming mapping spaces into them, this statement holds true after evaluating on $\Sigma$ (which produces the Cech-Deligne complexes). Forming the 0-truncation $\tau_0$ of the result gives the differential cohomology group $H^{2k+2}(\Sigma, \mathbb{Z}(k+1)_{D})$ appearing in theorem .

Alternatively, first passing to the 0-truncation of the diagram and then producing the pullback yields the Hodge cocycle group of prop. .

Accordingly, the statement of theorem may equivalently be rephrased in the following more suggestive way:

the intermediate Jacobian $J^{k+1}(\Sigma)$ is the fiber in

$$J^{k+1}(\Sigma) \longrightarrow \tau_0\left( [\Sigma,\mathbb{Z}(p)[-2k-2]] \underset{[\Sigma, \mathbb{C}[-2k-2]]}{\times} \Omega^{\bullet \geq k+1}[-2k-2] \right) \longrightarrow \tau_0 [\Sigma,\mathbb{Z}(p)[-2k-2]] \underset{\tau_0 [\Sigma, \mathbb{C}[-2k-2]]}{\times} \tau_0 [\Sigma, \Omega^{\bullet \geq k+1}[-2k-2]] \,.$$

This formulation of the intermediate Jacobian has a straightforward generalization from ordinary differential cohomology to differential Whitehead-generalized cohomology. This we turn to below.

The Griffith complex structure

The isomorphism

$$H^{2k+1}(\Sigma , \mathbb{C}) \simeq H^{2k+1}(\Sigma , \mathbb{R})\otimes_{\mathbb{R}} (\mathbb{C})$$

induces a complex manifold structure on $H^{2k+1}(\Sigma , \mathbb{C})$ and hence the structure of a complex torus on $k$th intermediate Jacobian as defined above. This is the structure originally defined in (Griffiths 68a, Griffiths 68b) and hence called the Griffith intermediate Jacobian. Reviews include (Walls (2012), Esnault & Viehweg (1988), section 7.8).

The Weil complex structure

There is another natural complex structure on $H^{2k-1}(X, \mathbb{R})/H^{2k-1}(X, \mathbb{Z})$, equipped with that it is called the Weil intermediate Jacobian.

Let as before $n \,\coloneqq\, dim_{\mathbb{C}}(\Sigma)$. Choose a Hermitian manifold structure on $\Sigma$. Then Serre duality on forms of total odd degree

$$\bar \star \;\colon\; \Omega^{p,2k+1-p}(\Sigma) \longrightarrow \Omega^{n-p-2k-1,p}(\Sigma)$$

is an antilinear function which squares to -1. Therefore

$$i \bar \star \;\colon\; H^{2k+1}(\Sigma,\mathbb{C}) \to H^{2k+1}(\Sigma,\mathbb{C})$$

is a real structure on $H^{2k+1}(\Sigma,\mathbb{C})$. This hence defines a complex manifold structure on $H^{2k+1}(\Sigma,\mathbb{C})$ and hence on the above quotient which is the intermediate Jacobian $J^{k+1}(\Sigma)$. As such this is the Weil intermediate Jacobian.

The polarized mid-dimensional Weil (Lazzeri) intermediate Jacobian

The Weil intermediate Jacobian is particularly interesting in mid degree, hence if

$$n =dim_{\mathbb{C}}(\Sigma) = 2k+1$$

then for $J^{k+1}(\Sigma)$. This case is also known as Lazzeri’s Jacobians see (Rubei 98).

In this case the intersection pairing

$$(\alpha, \beta) \mapsto \int_{\Sigma}\alpha \wedge \beta$$

defines a symplectic form, for which the Hodge star operator is a compatible complex structure and hence the Serre duality-pairing

$$(\alpha, \beta) \mapsto \int_{\Sigma}\alpha \wedge \star \beta$$

is the corresponding Kähler. This makes the Weil intermediate Jacobian a polarized variety.

Notice that the holomorphic coordinates in

$$ker \tfrac{1}{2}( 1 + i \bar \star ) \in H^{2k+1}(\Sigma, \mathbb{C})$$

may be thought of as the mid-degree self-dual higher gauge fields on $\Sigma$. From this point of view the above is the Kähler polarization of the prequantum line bundle on higher dimensional Chern-Simons theory in dimension $4k+3$.

The intermediate Jacobian of a Hodge structure

By prop. above the intermediate Jacobian is defined by the canonical Hodge filtering on complex ordinary cohomology. The definition obtained this way directly generalizes to other Hodge structures $H$ and hence one speaks more generally of the intermediate Jacobian

$$J(H)= H_{\mathbb{C}}/(H_\mathbb{Z}\oplus F^{k+1})$$

if $H$ has weight $2k+1$.

(e.g. Peters-Steenbrink 08, example 3.30, section 7.1.2)

For differential generalized cohomology

General construction

under construction, see also (Hopkins-Quick 12)…

The formulation of the traditional intermediate Jacobian by remark above suggest the following generalization.

(We use notation from differential cohomology hexagon).

Given any differential cohomology spectrum $\hat E$ (hence a spectrum object) in a cohesive (∞,1)-topos $\mathbf{H}$, it sits in its differential cohomology hexagon part of which is the homotopy pullback

$$\array{ && \flat_{dR} \hat E \\ & {}^{\mathllap{\theta_{\hat E}}}\nearrow && \searrow \\ \hat E && && \Pi \flat_{dR} \hat E \\ & \searrow && \nearrow_{\mathrlap{ch_{\hat E}}} \\ && \Pi \hat E } \,.$$

Hence for any $\Sigma$ also the mapping spectra

$$\array{ && [\Sigma, \flat_{dR} \hat E] \\ & {}^{\mathllap{\theta_{\hat E}}}\nearrow && \searrow \\ [\Sigma, \hat E] && && [\Sigma, \Pi \flat_{dR} \hat E] \\ & \searrow && \nearrow_{\mathrlap{ch_{\hat E}}} \\ && [\Sigma, \Pi \hat E] } \,.$$

Write $\tau_0 \colon \mathbf{H}\to \mathbf{H}$ for the 0-truncation map and consider the fiber product $Hdg(\Sigma,E)$ in

$$\array{ && \tau_0 [\Sigma, \flat_{dR} \hat E] \\ & {}^{\mathllap{\theta_{\hat E}}}\nearrow && \searrow \\ Hdg(\Sigma,\hat E) && && \tau_0 [\Sigma, \Pi \flat_{dR} \hat E] \\ & \searrow && \nearrow_{\mathrlap{ch_{\hat E}}} \\ && \tau_0[\Sigma, \Pi \hat E] } \,.$$

We may call this the Hodge cohomology of $\Sigma$ with coefficients in $\hat E$. The evident morphism of diagrams induces a morphism

$$[\Sigma, \hat E]\longrightarrow Hdg(\Sigma, \hat E)$$

and the homotopy fiber of that

$$\mathbf{J}(\Sigma,\hat E) \longrightarrow [\Sigma, \hat E]\longrightarrow Hdg(\Sigma, \hat E)$$

we may call the intermediate Jacobian $\infty$-stack of $\Sigma$ with coefficients in $\hat E$.

Notice that by commutativity of homotopy pullbacks with homotopy fibers, this is equivalently the homotopy pullback in

$$\array{ && ker([\Sigma,\flat_{dR}\hat E] \to \tau_0[\Sigma,\flat_{dR}\hat E] ) \\ & \nearrow && \searrow \\ \mathbf{J}(\Sigma,\hat E) && && ker([\Sigma,\Pi \flat_{dR}\hat E] \to \tau_0[\Sigma,\Pi \flat_{dR}\hat E] ) \\ & \searrow && \nearrow \\ && ker([\Sigma,\Pi\hat E] \to \tau_0[\Sigma,\Pi\hat E] ) } \,.$$

In this form this manifestly says that $\mathbf{J}(\Sigma,\hat E)$ is precisely the differential cohomology theory of $\Sigma$ obtained from $\hat E$ by restricting to trivial curvature and trivial underlying $\Pi(E)$-cohomology.

For complex K-theory

Intermediate Jacobians of K-theory classes were considered in the physics-style literature in (Witten 99, section 4.3, Moore-Witten 99, section 3, DMW 00, section 7.1, Belov-Moore 06b, section 5) as a means for quantization of the RR-field in type II superstring theory as a self-dual higher gauge theory (see there at Examples – RR-fields in 10d). A mathematical discussion inspired by this is in (MPS 11).

Properties

Relation between the Griffiths and the Weil complex structure

While the Griffiths complex structure on the intermediate Jacobian is not Kähler/not an algebraic polarization as the Weil complex structure is, it still has an p-convex polarization and there is a symplectomorphism which is an isomorphism between the Griffiths and the Weil intermediate Jacobians as real symplectic manifolds

$$(J^{k+1}(X), \omega_{Griffiths}) \simeq (J^{k+1}(X), \omega_{Weil}) \,.$$

This is due to (Griffiths 68b), recalled as Griffiths 12 (2.6)

(…)

Cycle map / Abel-Jacobi map

The intermediate Jacobians receive canonical maps from cycles (…) See at Abel-Jacobi map.

Polarization

For a Hodge manifold the intermediate Jacobian canonically inherits the structure of a polarized variety. (…)

Theta-characteristics

A certain square root of the canonical bundle on intermediate Jacobians – hence a Theta characteristic – in dimension $2k+1$ thought of as moduli spaces of (flat) circle (2k+1)-bundles with connection yields the partition function of self-dual higher gauge theory. (Witten 96, Hopkins-Singer 02).

Relation to Artin-Mazur formal groups

By theorem the formal geometry of intermediate Jacobians around their canonical point is equivalently the deformation theory of Deligne cohomology/line n-bundles with connection. This is given by (when it exists) the Artin-Mazur formal group for deformations of Deligne cohomology (see there).

Examples

The two extreme cases of intermediate Jacobians $J^k(\Sigma)$ with minimal $k = 0$ and maximal $k = dim_{\mathbb{C}}(\Sigma)= 1$ go by special names, the

• Picard variety

• Albanese variety

respectively.

Of special interest are also the intermediate Jacobian

• of Calabi-Yau varieties.

$k = 0$: the Picard variety $J^1(\Sigma)$

Proposition

The intermediate Jacobian $J^1(\Sigma)$, def. , of a complex curve ($dim_{\mathbb{C}}(\Sigma) = 1$) coincides with the connected component $Pic_0(\Sigma)$ of the Picard variety $Pic(\Sigma)$ of $\Sigma$, hence with the Jacobian variety $Jac(\Sigma)$:

$$J^1(\Sigma) = Pic_0(\Sigma) = Jac(\Sigma) \,.$$

First consider the elementary proof by direct inspection (e.g. Polishchuk 03, section 16.4):

Proof

Notice that the canonical map

$$H^1(\Sigma,\mathbb{R}) \hookrightarrow H^1(\Sigma, \mathbb{C}) \to H^{0,1}(\Sigma) \stackrel{\simeq}{\to} H^1(\Sigma, \mathcal{O}_{\Sigma})$$

is an isomorphism. The first map is induced by the splitting $H^1(\Sigma, \mathbb{C}) \simeq H^1(\Sigma,\mathbb{R})\oplus i H^1(\Sigma,\mathbb{R})$ given by complexification and the second by the splitting $H^1(\Sigma,\mathbb{C}) \simeq H^{0,1}(\Sigma)\oplus H^{1,0}(\Sigma)$ of Dolbeault cohomology, the last map is the Dolbeault isomorphism.

Therefore by the long exact sequence in cohomology of the exponential exact sequence we have that

$$\begin{aligned} J^1(\Sigma) & \coloneqq H^1(\Sigma, \mathbb{R})/H^1(\Sigma, \mathbb{Z}) \\ & \simeq H^1(\Sigma,\mathcal{O}_{\Sigma})/H^1(\Sigma, \mathbb{Z}) \\ & \simeq ker(H^1(\Sigma, \mathcal{O}^\times_{\Sigma})\to H^2(\Sigma, \mathbb{Z})) \\ & = Pic_0(\Sigma) \end{aligned}$$

is the connected component of the Picard variety of $\Sigma$.

Alternatively, prop. derives from theorem as follows:

Proof

Since $k = 0$ then $\mathbf{B}^2\mathbb{Z}(k+1)_D\simeq \mathbf{B}\mathbb{G}_m$ is just the universal moduli stack of line bundles without connection and so $H^{2k+2}(\Sigma, \mathbb{Z}(k+1)_D ) \simeq H(\Sigma,\mathbf{B}\mathbb{G}_m)$ is the full Picard variety. The fiber in the exact sequence in theorem then restricts this to the elements which have trivial first Chern class, hence the Jacobian variety.

Remark

There is a non-abelian generalization of this statement that the moduli space of real bundles with flat connections is equivalently a moduli space of complex-analytic bundles, but without connection. This is a corollary of the Narasimhan-Seshadri theorem (for $dim_{\mathbb{C}}\Sigma = 1$) or of the Donaldson-Uhlenbeck-Yau theorem (for Kähler manifolds $\Sigma$) and generally of the Kobayashi-Hitchin correspondence (for arbitrary complex $\Sigma$), stated for instance as (Scheinost-Schottenloher 96, corollary 1.16):

the moduli space of flat SU(n)-principal connections on $\Sigma$ is equivalently the moduli space of SL(n,C)-holomorphic vector bundles which have vanishing Chern classes and are semi-stable.

$k = n-1$: Albanese variety

For $\Sigma$ any space of complex dimension $n \coloneqq dim_{\mathbb{C}}(\Sigma)$ then with $k = n-1$ the $(k+1)$st intermediate Jacobian is built from cohomology in degree one less than the real dimension of $\Sigma$:

$$J^{n-1}(\Sigma) = H^{2k-1}(\Sigma,\mathbb{R})/H^{2k-1}(\Sigma, \mathbb{Z}) \,.$$

This $(n-1)$st intermediate Jacobian is known as the Albanese variety of $\Sigma$.

Of Calabi-Yau varieties

A review of intermediate Jacobians of Calabi-Yau varieties of (complex) dimension 3 is in (Baarsma 11, section 2).

The (real) dimensional of the intermediate Jacobian of a CY3 $X$ is

$$dim (J(X)) = 2(1+ h^{1,2})$$

(e.g. Baarsma 11, (2.21))

Hence the intermediate Jacobian of a rigid CY3 (with $h^{1,2} = 0$) is an elliptic curve (e.g. BKNPP 09, (1.8)).

$n = 3$: supergravity C-field

For the moment see at 7d Chern-Simons theory and at M5-brane.

$n = 3$: type II 3-form

For the RR-field component in degree 4 of type IIA superstring theory: (Morrison 95)

Related concepts

moduli spaces of line n-bundles with connection on $n$-dimensional $X$

$n$	Calabi-Yau n-fold	line n-bundle	moduli of line n-bundles	moduli of flat/degree-0 n-bundles	Artin-Mazur formal group of deformation moduli of line n-bundles	complex oriented cohomology theory	modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$		unit in structure sheaf	multiplicative group/group of units		formal multiplicative group	complex K-theory
$n = 1$	elliptic curve	line bundle	Picard group/Picard scheme	Jacobian	formal Picard group	elliptic cohomology	3d Chern-Simons theory/WZW model
$n = 2$	K3 surface	line 2-bundle	Brauer group	intermediate Jacobian	formal Brauer group	K3 cohomology
$n = 3$	Calabi-Yau 3-fold	line 3-bundle		intermediate Jacobian		CY3 cohomology	7d Chern-Simons theory/M5-brane
$n$				intermediate Jacobian

References

In ordinary differential cohomology

General

The definition of the Griffith intermediate Jacobian is due to

• Phillip Griffiths, Periods of integrals on algebraic manifolds. I Construction and properties of the modular varieties“, American Journal of Mathematics 90 (2): 568–626, (1968) [doi:10.2307/2373545, ISSN 0002-9327, JSTOR 2373545, MR 0229641]

• Phillip Griffiths, Periods of integrals on algebraic manifolds. II Local study of the period mapping“, American Journal of Mathematics 90 (3): 805–865 (1968) [doi:10.2307/2373485, ISSN 0002-9327, JSTOR 2373485, summary:pdf, MR 0233825 ]

Review:

• Phillip Griffiths, section 1 of Some results on algebraic cycles on algebraic manifolds, Proceedings of the International Conference on Algebraic Geometry, Tata Institute (Bombay), 2012 (web, pdf)

• Haus (2022), Ch 1.

For $k = 0$ but with generalization to non-abelian moduli space of flat connections the Grifiths-like follows also with the Donaldson-Uhlenbeck-Yau theorem as discussed in

• Peter Scheinost, Martin Schottenloher, pp. 154 (11 of 76) of Metaplectic quantization of the moduli spaces of flat and parabolic bundles, J. reine angew. Mathematik, 466 (1996) (web)

The mid-dimensional case was discussed in unpublished work by Lazzeri, see

• Elena Rubei, Lazzeri’s Jacobian of oriented compact riemannian manifolds (arXiv:math/9812110)

The relation of the intermediate Jacobian to Deligne cohomology (Deligne’s theorem) due to Pierre Deligne is discussed in

• Hélène Esnault, Eckart Viehweg, Section 7 of: Deligne-Beilinson cohomology, in: Michael Rapoport, Norbert Schappacher, Peter Schneider (eds.), Beilinson's Conjectures on Special Values of L-Functions Perspectives in Mathematic 4, Academic Press, Inc. (1988) [ISBN:978-0-12-581120-0, pdf]

Reviews and surveys include

• eom, Intermediate Jacobian

• Wikipedia, Intermediate Jacobian

• Patrick Walls, Intermediate Jacobians and Abel-Jacobi maps, 2012 (pdf)

• Jean-Luc Brylinski, around theorem I 1.5.11 of Loop Spaces, Characteristic Classes and Geometric Quantization Springer, 2007

• Arnaud Beauville, Variétés de Prym et jacobiennes intermédiaire, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 10 no. 3 (1977), p. 309-391 (pdf) (jstor)

• Valentin Zakharevich, Mixed Intermediate Jacobians, 2012 (pdf)

• Alexander Polishchuk, Abelian varieties, Theta functions and the Fourier transform, Cambridge University Press (2003) (pdf)

Discussion of the generalization to Hodge structures includes

• Chris Peters, Jozef Steenbrink, Mixed Hodge Structures, Ergebisse der Mathematik (2008) (pdf)

• pdf

For Calabi-Yau 3-folds

Discussion of intermediate Jacobians of Calabi-Yau 3-folds includes

• C. Herbert Clemens, Phillip Griffith, The intermediate Jacobian of the cubic threefold, Annals of Mathematics Second Series, Vol. 95, No. 2 (Mar., 1972), pp. 281-356 (JSTOR)

• Claire Voisin (pdf)

• Andreas Höring, Minimal classes on the intermediate Jacobian of a generic cubic threefold, 2008 (pdf)

In positive characteristic:

• Gerard van der Geer, T. Katsura, On the height of Calabi-Yau varieties in positive characteristic (arXiv:math/0302023)

Applications in string theory:

• David Morrison, section 4 of Mirror Symmetry and the Type II String, Nucl.Phys.Proc.Suppl. 46 (1996) 146-155 (arXiv:hep-th/9512016)

• Diaconescu, Ron Donagi, Tony Pantev, Intermediate Jacobians and ADE Hitchin Systems (arXiv:hep-th/0607159)

• Ling Bao, Axel Kleinschmidt, Bengt E. W. Nilsson, Daniel Persson, Boris Pioline, Instanton Corrections to the Universal Hypermultiplet and Automorphic Forms on SU(2,1), Commun. Num. Theor. Phys. 4 (1), 187-266 (2010) (arXiv:0909.4299)

• A. Baarsma, The hypermultiplet moduli space of compactified type IIA string theory, Master Thesis, Utrecht 2011 (web)

The relation of Theta characteristics on intermediate Jacobians to self-dual higher gauge theory was first recognized in

• Edward Witten, Five-Brane Effective Action In M-Theory, J. Geom. Phys. 22 (1997) 103-133 [arXiv:hep-th/9610234]

and the argument there was made rigorous in

• Mike Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology, and M-Theory (2005)

For generalized cohomology

Intermediate Jacobians of K-theory classes were discussed in the physics literature context of self-dual higher gauge theory for RR-fields in

• Edward Witten, Duality Relations Among Topological Effects In String Theory, JHEP 0005:031,2000 (arXiv:hep-th/9912086)

• Gregory Moore, Edward Witten, Self-Duality, Ramond-Ramond Fields, and K-Theory, JHEP 0005:032,2000 (arXiv:hep-th/9912279)

• D. Diaconescu, Gregory Moore, Edward Witten, $E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory, Adv.Theor.Math.Phys.6:1031-1134,2003 (arXiv:hep-th/0005090), summarised in A Derivation of K-Theory from M-Theory (arXiv:hep-th/0005091)

• Dmitriy Belov, Greg Moore, Type II Actions from 11-Dimensional Chern-Simons Theories (arXiv:hep-th/0611020)

A mathematical discussion inspired by this is in

• Stefan Müller-Stach, Chris Peters, Vasudevan Srinivas, Abelian varieties and theta functions associated to compact Riemannian manifolds; constructions inspired by superstring theory (arXiv:1105.4108, pdf slides)

Discussion of intermediate Jacobians in generalized Hodge-filtered differential cohomology,:

• Michael Hopkins, Gereon Quick, Hodge filtered complex bordism (arXiv:1212.2173)

specifically in Hodge-filtered differential cobordism cohomology:

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

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