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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{intermediate Jacobian} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{complex_geometry}{}\paragraph*{{Complex geometry}}\label{complex_geometry} [[!include complex geometry - contents]] \hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology} [[!include differential cohomology - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{InComplexGeometry}{For differential ordinary cohomology}\dotfill \pageref*{InComplexGeometry} \linebreak \noindent\hyperlink{the_underlying_real_manifold}{The underlying real manifold}\dotfill \pageref*{the_underlying_real_manifold} \linebreak \noindent\hyperlink{CharacterizationAsHodgeTrivialDeligneCohomology}{Characterization as Hodge-trivial Deligne cohomology}\dotfill \pageref*{CharacterizationAsHodgeTrivialDeligneCohomology} \linebreak \noindent\hyperlink{GriffithIntermediateJacobian}{The Griffith complex structure}\dotfill \pageref*{GriffithIntermediateJacobian} \linebreak \noindent\hyperlink{WeilIntermediateJacobian}{The Weil complex structure}\dotfill \pageref*{WeilIntermediateJacobian} \linebreak \noindent\hyperlink{MidDimensionalWeilIntermediateJacobian}{The polarized mid-dimensional Weil (Lazzeri) intermediate Jacobian}\dotfill \pageref*{MidDimensionalWeilIntermediateJacobian} \linebreak \noindent\hyperlink{the_intermediate_jacobian_of_a_hodge_structure}{The intermediate Jacobian of a Hodge structure}\dotfill \pageref*{the_intermediate_jacobian_of_a_hodge_structure} \linebreak \noindent\hyperlink{GeneralDescripion}{For differential generalized cohomology}\dotfill \pageref*{GeneralDescripion} \linebreak \noindent\hyperlink{general_construction}{General construction}\dotfill \pageref*{general_construction} \linebreak \noindent\hyperlink{ForComplexKTheory}{For complex K-theory}\dotfill \pageref*{ForComplexKTheory} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{RelationBetweenGriffithsAndWeilStructure}{Relation between the Griffiths and the Weil complex structure}\dotfill \pageref*{RelationBetweenGriffithsAndWeilStructure} \linebreak \noindent\hyperlink{cycle_map__abeljacobi_map}{Cycle map / Abel-Jacobi map}\dotfill \pageref*{cycle_map__abeljacobi_map} \linebreak \noindent\hyperlink{polarization}{Polarization}\dotfill \pageref*{polarization} \linebreak \noindent\hyperlink{thetacharacteristics}{Theta-characteristics}\dotfill \pageref*{thetacharacteristics} \linebreak \noindent\hyperlink{RelationToArtinMazurFormalGroups}{Relation to Artin-Mazur formal groups}\dotfill \pageref*{RelationToArtinMazurFormalGroups} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{ExamplePicard}{$k = 0$: the Picard variety $J^1(\Sigma)$}\dotfill \pageref*{ExamplePicard} \linebreak \noindent\hyperlink{ExampleAlbanese}{$k = n-1$: Albanese variety}\dotfill \pageref*{ExampleAlbanese} \linebreak \noindent\hyperlink{ExampleCY}{Of Calabi-Yau varieties}\dotfill \pageref*{ExampleCY} \linebreak \noindent\hyperlink{_supergravity_cfield}{$n = 3$: supergravity C-field}\dotfill \pageref*{_supergravity_cfield} \linebreak \noindent\hyperlink{_type_ii_3form}{$n = 3$: type II 3-form}\dotfill \pageref*{_type_ii_3form} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{in_ordinary_differential_cohomology}{In ordinary differential cohomology}\dotfill \pageref*{in_ordinary_differential_cohomology} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{for_calabiyau_3folds}{For Calabi-Yau 3-folds}\dotfill \pageref*{for_calabiyau_3folds} \linebreak \noindent\hyperlink{for_generalized_cohomology}{For generalized cohomology}\dotfill \pageref*{for_generalized_cohomology} \linebreak \hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea} Traditionally the $(k+1)$st \emph{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 \begin{displaymath} J^{k+1}(\Sigma) \coloneqq H^{2k+1}(\Sigma, \mathbb{R}) / H^{2k+1}(\Sigma, \mathbb{Z}) \,. \end{displaymath} 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 \emph{[[complex torus]]}, is properly what is called the $(k+1)$st \emph{intermediate Jacobian variety} of $\Sigma$. This terminology derives from the term \emph{[[Jacobian variety]]} which is the (historically earlier) special case for $k = 0$ and $dim_{\mathbb{C}}(\Sigma) = 1$. Notice that conceptually we may \begin{itemize}% \item think of $H^{2k+1}(\Sigma,\mathbb{R})$ as the space of those [[circle n-bundle with connection|n-form connections]] on $\Sigma$ which are both [[flat infinity-connection|flat]] and have trivial underlying [[line n-bundle]]; \item think of $H^{2k + 1}(\Sigma,\mathbb{Z})$ as the group of ``large'' (not connected to the identity) [[higher gauge field|higher]] [[gauge transformations]] acting on these gauge fields; \item and hence understand $J^{k+1}(\Sigma)$ as the [[moduli space]] of flat $n$-form connections on trivial underlying line $n$-bundles. \end{itemize} This turns out to be a natural and useful perspective on intermediate Jacobians: \emph{Deligne's theorem} (discussed as theorem \ref{DeligneTheorem} 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 [[circle n-bundle with connection|line n-bundles with connection]]. Moreover, as discussed in prop. \ref{ReformulationOfDeligneTheorem} below, Deligne's theorem in the formulation of (\hyperlink{EsnaultViehweg88}{Esnault-Viehweg 88}) 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 [[circle n-bundle with connection|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 [[generalized (Eilenberg-Steenrod) cohomology|generalized Eilenberg-Steenrod type]]) [[differential cohomology]]. This we discuss further \hyperlink{GeneralDescripion}{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]]) (\hyperlink{Witten96}{Witten 96}, Hopkins-Singer 02). Here [[higher dimensional Chern-Simons theory]] has as [[field (physics)|fields]] types of [[higher gauge field]] specified by some type of [[differential cohomology]], and the connected components of its [[phase space]] 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 variety|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|higher]]) [[geometry]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{InComplexGeometry}{}\subsubsection*{{For differential ordinary cohomology}}\label{InComplexGeometry} \hypertarget{the_underlying_real_manifold}{}\paragraph*{{The underlying real manifold}}\label{the_underlying_real_manifold} Let $\Sigma$ be a [[projective variety|projective]] [[smooth variety|smooth]] [[complex variety]] (see at \emph{[[GAGA]]}). \begin{defn} \label{PlainIJacobian}\hypertarget{PlainIJacobian}{} For $k \in \mathbb{N}$ the \emph{$k$th intermediate Jacobian} of $\Sigma$ is, as a [[real manifold]], the [[quotient]] \begin{displaymath} J^{k+1}(\Sigma) \coloneqq H^{2k+1}(\Sigma,\mathbb{R})/H^{2k+1}(\Sigma,\mathbb{Z}) \end{displaymath} of the [[ordinary cohomology|ordinary]] [[cohomology groups]] of $X$ with [[coefficients]] in the [[abelian groups]] of [[real numbers]] and of [[integers]], respectively, induced by the canonical inclusion $\mathbb{Z} \hookrightarrow \mathbb{R}$. \end{defn} 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 eventually equipping this with the structure of a [[complex manifold]] one realizes this as the quotient of the [[complex vector space]] of cohomology with complex coefficients as follows: \begin{remark} \label{}\hypertarget{}{} A real [[differential form]] \begin{displaymath} \alpha \in \Omega^{2k+1}_{\mathbb{R}}(\Sigma) \end{displaymath} is, by the [[Hodge theorem]], a sum of complex differential forms in homogeneous [[Dolbeault complex|Dolbeault bidegree]] of the form \begin{displaymath} \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}} \,, \end{displaymath} where \begin{displaymath} \overline{(-)} \colon \Omega^{p,q}(\Sigma)\longrightarrow \Omega^{q,p}(\Sigma) \end{displaymath} is the [[antilinear function]] on complex differential forms given by [[complex conjugation]]. \end{remark} It follows with the [[de Rham theorem]] that: \begin{prop} \label{AsQuotientByHodgeFiltering}\hypertarget{AsQuotientByHodgeFiltering}{} There is a canonical [[isomorphism]] of real vector spaces \begin{displaymath} H^{2k+1}(\Sigma, \mathbb{R}) \simeq H^{2k+1}(\Sigma,\mathbb{C})/(F^{k+1} H^{2k+1}(\Sigma,\mathbb{C})) \,, \end{displaymath} where \begin{displaymath} F^{k+1} H^{2k+1}(\Sigma,\mathbb{C}) \coloneqq \underset{p \geq k+1}{\oplus} H^{p,2k+1-p}(\Sigma) \end{displaymath} is the $(k+1)$st stage in the [[Hodge filtration]] of $H^{2k+1}(\Sigma,\mathbb{C})$. \end{prop} Hence an equivalent way to write the intermediate Jacobian (still as a real manifold) as the quotient of the real manifold underlying a complex vector space is the following: \begin{prop} \label{}\hypertarget{}{} The intermediate Jacobian of def. \ref{PlainIJacobian} is equivalently \begin{displaymath} J^{k+1}(\Sigma) \simeq H^{2k+1}(\Sigma,\mathbb{C}) / (F^{k+1} H^{2k+1}(\Sigma, \mathbb{C}) \oplus H^{2k+1}(\Sigma,\mathbb{Z})) \,. \end{displaymath} \end{prop} Yet one more reformulation is useful when properly working in [[complex analytic geometry]]/[[GAGA]]: \begin{defn} \label{ModuliOfHolomorphickConnections}\hypertarget{ModuliOfHolomorphickConnections}{} Write $(\mathbf{B}^k \mathbb{G}_a)_{conn}$ for the chain complex of sheaves on the complex analytic site which assigns the truncated [[de Rham complex]] of [[holomorphic differential forms]] \begin{displaymath} (\mathbf{B}^k \mathbb{G}_a)_{conn} \coloneqq \left( \mathcal{O} \stackrel{\partial }{\to} \Omega^{1} \stackrel{\partial}{\to} \cdots \stackrel{\partial}{\to} \Omega^{k} \right) \end{displaymath} regarded as sitting in degrees $k$ to 0. \end{defn} e.g. (\hyperlink{EsnaultViehweg88}{Esnault-Viehweg 88, top of p. 14}) \begin{prop} \label{ReplacingHodgeFiltrationQuotientByHolomorphicConnections}\hypertarget{ReplacingHodgeFiltrationQuotientByHolomorphicConnections}{} The quotient in prop. \ref{AsQuotientByHodgeFiltering} is equivalently the [[abelian sheaf cohomology|abelian sheaf]] [[hypercohomology]] with [[coefficients]] in $\mathbf{B}^{2k}\mathbb{G}_a$ of def. \ref{ModuliOfHolomorphickConnections}: \begin{displaymath} H^{2k+1}(\Sigma,\mathbb{C})/F^{k+1} H^{2k+1}(\Sigma, \mathbb{C}) \simeq [\Sigma, \mathbf{B}^k(\mathbf{B}^k \mathbb{G}_a)_conn) ] \,. \end{displaymath} \end{prop} e.g. (\hyperlink{EsnaultViehweg88}{Esnault-Viehweg 88, 2.5 b)}). There are two canonical ways to equip $H^{2k+1}(\Sigma,\mathbb{C})$, 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,the \begin{itemize}% \item \hyperlink{GriffithIntermediateJacobian}{Griffith intermediate Jacobian} \end{itemize} and the \begin{itemize}% \item \hyperlink{WeilIntermediateJacobian}{Weil intermediate Jacobian} \end{itemize} \hypertarget{CharacterizationAsHodgeTrivialDeligneCohomology}{}\paragraph*{{Characterization as Hodge-trivial Deligne cohomology}}\label{CharacterizationAsHodgeTrivialDeligneCohomology} A theorem due to [[Pierre Deligne]] says that this $J^k(\Sigma)$ is characterised as being the [[fiber]] of a canonical map from ([[complex analytic geometry|complex analytic]]) [[Deligne cohomology]] to the $k$th [[Hodge filtration]] of [[integral cohomology]]. \begin{defn} \label{HodgeCohomologyClasses}\hypertarget{HodgeCohomologyClasses}{} The group $Hdg^{k+1}(\Sigma)$ of \emph{[[Hodge cocycle|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 a [[pullback]] diagram \begin{displaymath} \itexarray{ Hdg^{k+1}(\Sigma) &\longrightarrow& F^{k+1} H^{2k+2}(\Sigma,\mathbb{C}) \\ \downarrow && \downarrow \\ H^{2k+2}(\Sigma,\mathbb{Z}(k+1)) &\longrightarrow& H^{2k+2}(\Sigma,\mathbb{C}) } \,. \end{displaymath} \end{defn} The following says this in a [[complex analytic geometry|complex analytic]]-way that generalizes: \begin{prop} \label{HodgeCyclesAsPullbackOfHolomorphic}\hypertarget{HodgeCyclesAsPullbackOfHolomorphic}{} Equivalently the Hodge cohomology classes of def. \ref{HodgeCohomologyClasses} are given by the [[pullback]] \begin{displaymath} \itexarray{ Hdg^{k+1}(\Sigma) &\longrightarrow& H^{2k+2}(\Sigma, \Omega^{\bullet \geq k+1}) \\ \downarrow && \downarrow \\ H^{2k+2}(\Sigma,\mathbb{Z}(k+1)) &\longrightarrow& H^{2k+2}(\Sigma,\mathbb{C}) } \,, \end{displaymath} where now in the top right we have the [[abelian sheaf cohomology|abelian sheaf]] [[hypercohomology]] with [[coefficients]] in the [[holomorphic de Rham complex]], truncated (but otherwise unshifted) as indicated. \end{prop} (\hyperlink{EsnaultViehweg88}{Esnault-Viehweg 88, section 7.8}) \begin{theorem} \label{DeligneTheorem}\hypertarget{DeligneTheorem}{} \textbf{(Deligne)} As an [[abelian group]] the intermediate Jacobian $J^k(\Sigma)$, def. \ref{PlainIJacobian}, is characterized by fitting into the [[short exact sequence]] \begin{displaymath} 0 \to J^{k+1}(\Sigma)\to H^{2k+2}(\Sigma, \mathbb{Z}(k+1)_{D}) \to Hdg^{k+1}(\Sigma) \to 0 \,. \end{displaymath} \end{theorem} (e.g. \hyperlink{EsnaultViehweg88}{Esnault-Viehweg 88, (7.9)}, \hyperlink{PetersSteenbrink08}{Peters-Steenbrink 08, lemma 7.20}) . \begin{proof} Use prop. \ref{ReplacingHodgeFiltrationQuotientByHolomorphicConnections} and the \href{ordinary%20differential%20cohomology#CurvatureAndCharClass}{characteristic long exact sequences} of [[ordinary differential cohomology]]. \end{proof} \begin{remark} \label{AsFiberOf0TruncationOnFractureSquare}\hypertarget{AsFiberOf0TruncationOnFractureSquare}{} The [[fiber product]]-incarnation of $Hdg^{k+1}(\Sigma)$ in prop. \ref{HodgeCyclesAsPullbackOfHolomorphic} 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 \emph{[[complex analytic ∞-groupoid]]} for more on this): \begin{displaymath} \itexarray{ \mathbb{Z}(p)[-2k-2] && && \Omega^{\bullet \geq k+1}[-2k-2] \\ & \searrow && \swarrow \\ && \mathbb{C}[-2k-2] } \,. \end{displaymath} This is just of the form as discussed in some detail at \emph{[[circle n-bundle with connection]]} and also at \emph{[[differential cohomology diagram]]} in the section on \emph{\href{differential+cohomology%20diagram#DeligneCoefficients}{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]] \begin{displaymath} \itexarray{ &&\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] } \,. \end{displaymath} Since [[homotopy pullbacks]] are preserved by foming [[mapping spaces]] into them, this statement holds true after evaluating on $\Sigma$ (which produces the [[Cech cohomology|Cech-]][[Deligne complexes]]). Forming the [[n-truncated object of an (infinity,1)-category|0-truncation]] $\tau_0$ of the result gives the differential cohomology group $H^{2k+2}(\Sigma, \mathbb{Z}(k+1)_{D})$ appearing in theorem \ref{DeligneTheorem}. Alternatively, first passing the 0-truncation of the diagram and then producing the pullback yields the Hodge cocycle group of prop. \ref{HodgeCyclesAsPullbackOfHolomorphic}. Accordingly, the statement of theorem \ref{DeligneTheorem} may equivalently be rephrased in the following more suggestive way: the intermediate Jacobian $J^{k+1}(\Sigma)$ is the [[fiber]] in \begin{displaymath} 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]] \,. \end{displaymath} \end{remark} This formulation of the intermediate Jacobian has a straightforward generalization from [[ordinary differential cohomology]] to [[differential cohomology|differential]] [[generalized (Eilenberg-Steenrod) cohomology]]. This we turn to \hyperlink{GeneralDescripion}{below}. \hypertarget{GriffithIntermediateJacobian}{}\paragraph*{{The Griffith complex structure}}\label{GriffithIntermediateJacobian} The [[isomorphism]] \begin{displaymath} H^{2k+1}(\Sigma , \mathbb{C}) \simeq H^{2k+1}(\Sigma , \mathbb{R})\otimes_{\mathbb{R}} (\mathbb{C}) \end{displaymath} 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 (\hyperlink{Griffiths68a}{Griffiths 68a}, \hyperlink{Griffiths68b}{Griffiths 68b}) and hence called the \emph{Griffith intermediate Jacobian}. Reviews include (\hyperlink{Walls12}{Walls 12}, \hyperlink{EsnaultViehweg88}{Esnault-Viehweg 88, section 7.8}). \hypertarget{WeilIntermediateJacobian}{}\paragraph*{{The Weil complex structure}}\label{WeilIntermediateJacobian} 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 \emph{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 \begin{displaymath} \bar \star \;\colon\; \Omega^{p,2k+1-p}(\Sigma) \longrightarrow \Omega^{n-p-2k-1,p}(\Sigma) \end{displaymath} is an [[antilinear function]] which squares to -1. Therefore \begin{displaymath} i \bar \star \;\colon\; H^{2k+1}(\Sigma,\mathbb{C}) \to H^{2k+1}(\Sigma,\mathbb{C}) \end{displaymath} 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 \emph{Weil intermediate Jacobian}. \hypertarget{MidDimensionalWeilIntermediateJacobian}{}\paragraph*{{The polarized mid-dimensional Weil (Lazzeri) intermediate Jacobian}}\label{MidDimensionalWeilIntermediateJacobian} The Weil intermediate Jacobian is particularly interesting in mid degree, hence if \begin{displaymath} n =dim_{\mathbb{C}}(\Sigma) = 2k+1 \end{displaymath} then for $J^{k+1}(\Sigma)$. This case is also known as \emph{Lazzeri's Jacobians} see (\hyperlink{Rubei98}{Rubei 98}). In this case the [[intersection pairing]] \begin{displaymath} (\alpha, \beta) \mapsto \int_{\Sigma}\alpha \wedge \beta \end{displaymath} defines a [[symplectic form]], for which the [[Hodge star]] operator is a compatible [[complex structure]] and hence the [[Serre duality]]-pairing \begin{displaymath} (\alpha, \beta) \mapsto \int_{\Sigma}\alpha \wedge \star \beta \end{displaymath} is the corresponding [[Kähler manifold|Kähler]]. This makes the Weil intermediate Jacobian a [[polarized variety]]. Notice that the holomorphic coordinates in \begin{displaymath} ker \tfrac{1}{2}( 1 + i \bar \star ) \in H^{2k+1}(\Sigma, \mathbb{C}) \end{displaymath} 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 $4k+3$. \hypertarget{the_intermediate_jacobian_of_a_hodge_structure}{}\paragraph*{{The intermediate Jacobian of a Hodge structure}}\label{the_intermediate_jacobian_of_a_hodge_structure} By prop. \ref{AsQuotientByHodgeFiltering} 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 \begin{displaymath} J(H)= H_{\mathbb{C}}/(H_\mathbb{Z}\oplus F^{k+1}) \end{displaymath} if $H$ has weight $2k+1$. (e.g. \hyperlink{PetersSteenbrink08}{Peters-Steenbrink 08, example 3.30, section 7.1.2}) \hypertarget{GeneralDescripion}{}\subsubsection*{{For differential generalized cohomology}}\label{GeneralDescripion} \hypertarget{general_construction}{}\paragraph*{{General construction}}\label{general_construction} \begin{quote}% under construction, see also (\hyperlink{HopkinsQuick12}{Hopkins-Quick 12})\ldots{} \end{quote} The formulation of the traditional intermediate Jacobian by remark \ref{AsFiberOf0TruncationOnFractureSquare} above suggest the following generalization. (We use notation from \emph{[[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]] \begin{displaymath} \itexarray{ && \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 } \,. \end{displaymath} Hence for any $\Sigma$ also the [[mapping spectra]] \begin{displaymath} \itexarray{ && [\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] } \,. \end{displaymath} Write $\tau_0 \colon \mathbf{H}\to \mathbf{H}$ for the [[n-truncated object in an (infinity,1)-category|0-truncation]] map and consider the [[fiber product]] $Hdg(\Sigma,E)$ in \begin{displaymath} \itexarray{ && \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] } \,. \end{displaymath} We may call this the \emph{Hodge cohomology} of $\Sigma$ with coefficients in $\hat E$. The evident morphism of diagrams induces a morphism \begin{displaymath} [\Sigma, \hat E]\longrightarrow Hdg(\Sigma, \hat E) \end{displaymath} and the [[homotopy fiber]] of that \begin{displaymath} \mathbf{J}(\Sigma,\hat E) \longrightarrow [\Sigma, \hat E]\longrightarrow Hdg(\Sigma, \hat E) \end{displaymath} we may call the \emph{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 \begin{displaymath} \itexarray{ && 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] ) } \,. \end{displaymath} 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. \hypertarget{ForComplexKTheory}{}\paragraph*{{For complex K-theory}}\label{ForComplexKTheory} Intermediate Jacobians of [[K-theory]] classes were considered in the [[physics]]-style literature in (\hyperlink{Witten99}{Witten 99, section 4.3}, \hyperlink{MooreWitten99}{Moore-Witten 99, section 3}, \hyperlink{MW00}{DMW 00, section 7.1}, \hyperlink{BelovMooreII}{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 \emph{\href{self-dual%20higher%20gauge%20theory#RRFieldsin10d}{Examples -- RR-fields in 10d}}). A mathematical discussion inspired by this is in (\hyperlink{MPS11}{MPS 11}). \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{RelationBetweenGriffithsAndWeilStructure}{}\subsubsection*{{Relation between the Griffiths and the Weil complex structure}}\label{RelationBetweenGriffithsAndWeilStructure} While the Griffiths complex structure on the intermediate Jacobian is not K\"a{}hler/not an [[polarized variety|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 \begin{displaymath} (J^{k+1}(X), \omega_{Griffiths}) \simeq (J^{k+1}(X), \omega_{Weil}) \,. \end{displaymath} This is due to (\hyperlink{Griffiths68b}{Griffiths 68b}), recalled as \hyperlink{Griffiths12}{Griffiths 12 (2.6)} (\ldots{}) \hypertarget{cycle_map__abeljacobi_map}{}\subsubsection*{{Cycle map / Abel-Jacobi map}}\label{cycle_map__abeljacobi_map} The intermediate Jacobians receive canonical maps from cycles (\ldots{}) See at \emph{[[Abel-Jacobi map]]}. \hypertarget{polarization}{}\subsubsection*{{Polarization}}\label{polarization} For a [[Hodge manifold]] the intermediate Jacobian canonically inherits the structure of a [[polarized variety]]. (\ldots{}) \hypertarget{thetacharacteristics}{}\subsubsection*{{Theta-characteristics}}\label{thetacharacteristics} 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 n-bundles with connection|circle (2k+1)-bundles with connection]] yields the [[partition function]] of [[self-dual higher gauge theory]]. (\hyperlink{Witten96}{Witten 96}, \hyperlink{HopkinsSinger02}{Hopkins-Singer 02}). \hypertarget{RelationToArtinMazurFormalGroups}{}\subsubsection*{{Relation to Artin-Mazur formal groups}}\label{RelationToArtinMazurFormalGroups} By theorem \ref{DeligneTheorem} 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 \href{Artin-Mazur+formal+group#DeformationsOfDeligneCohomology}{deformations of Deligne cohomology} (see there). \hypertarget{examples}{}\subsection*{{Examples}}\label{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 \begin{itemize}% \item \hyperlink{ExamplePicard}{Picard variety} \item \hyperlink{ExampleAlbanese}{Albanese variety} \end{itemize} respectively. Of special interest are also the intermediate Jacobian \begin{itemize}% \item \hyperlink{ExampleCY}{of Calabi-Yau varieties}. \end{itemize} \hypertarget{ExamplePicard}{}\subsubsection*{{$k = 0$: the Picard variety $J^1(\Sigma)$}}\label{ExamplePicard} \begin{prop} \label{J1IsPicard}\hypertarget{J1IsPicard}{} The intermediate Jacobian $J^1(\Sigma)$, def. \ref{PlainIJacobian}, 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]] $Jac(\Sigma)$: \begin{displaymath} J^1(\Sigma) = Pic_0(\Sigma) = Jac(\Sigma) \,. \end{displaymath} \end{prop} First consider the elementary proof by direct inspection (e.g. \hyperlink{Polishchuk03}{Polishchuk 03, section 16.4}): \begin{proof} Notice that the canonical map \begin{displaymath} 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}) \end{displaymath} 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{displaymath} \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} \end{displaymath} is the connected component of the [[Picard variety]] of $\Sigma$. \end{proof} Alternatively, prop. \ref{J1IsPicard} derives from theorem \ref{DeligneTheorem} as follows: \begin{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 \ref{DeligneTheorem} then restricts this to the elements which have trivial [[first Chern class]], hence the [[Jacobian variety]]. \end{proof} \begin{remark} \label{GeneralizationToModuliOfGPrincipalConnections}\hypertarget{GeneralizationToModuliOfGPrincipalConnections}{} There is a [[non-abelian cohomology|non-abelian]] generalization of this statement that the [[moduli space of flat connections|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 (\hyperlink{ScheinostSchottenloher96}{Scheinost-Schottenloher 96, corollary 1.16}): the moduli space of [[flat connection|flat]] [[special unitary group|SU(n)]]-[[principal connections]] on $\Sigma$ is equivalently the moduli space of [[special linear group|SL(n,C)]]-[[holomorphic vector bundles]] which have vanishing [[Chern classes]] and are [[stable vector bundle|semi-stable]]. \end{remark} \hypertarget{ExampleAlbanese}{}\subsubsection*{{$k = n-1$: Albanese variety}}\label{ExampleAlbanese} 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 $\Sigma$: \begin{displaymath} J^{n-1}(\Sigma) = H^{2k-1}(\Sigma,\mathbb{R})/H^{2k-1}(\Sigma, \mathbb{Z}) \,. \end{displaymath} This $(n-1)$st intermediate Jacobian is known as the \emph{[[Albanese variety]]} of $\Sigma$. \hypertarget{ExampleCY}{}\subsubsection*{{Of Calabi-Yau varieties}}\label{ExampleCY} A review of intermediate Jacobians of [[Calabi-Yau varieties]] of ([[complex manifold|complex]]) [[dimension]] 3 is in (\hyperlink{Baarsma11}{Baarsma 11, section 2}). The (real) dimensional of the intermediate Jacobian of a CY3 $X$ is \begin{displaymath} dim (J(X)) = 2(1+ h^{1,2}) \end{displaymath} (e.g. \hyperlink{Baarsma11}{Baarsma 11, (2.21)}) Hence the intermediate Jacobian of a rigid CY3 (with $h^{1,2} = 0$) is an [[elliptic curve]] (e.g. \hyperlink{BKNPP09}{BKNPP 09, (1.8)}). \hypertarget{_supergravity_cfield}{}\subsubsection*{{$n = 3$: supergravity C-field}}\label{_supergravity_cfield} For the moment see at \emph{[[7d Chern-Simons theory]]} and at \emph{[[M5-brane]]}. \hypertarget{_type_ii_3form}{}\subsubsection*{{$n = 3$: type II 3-form}}\label{_type_ii_3form} For the [[RR-field]] component in degree 4 of [[type IIA superstring theory]]: (\hyperlink{Morrison95}{Morrison 95}) \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} [[!include moduli of higher lines -- table]] \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{in_ordinary_differential_cohomology}{}\subsubsection*{{In ordinary differential cohomology}}\label{in_ordinary_differential_cohomology} \hypertarget{general}{}\paragraph*{{General}}\label{general} The definition of the Griffith intermediate Jacobian is due to \begin{itemize}% \item [[Phillip Griffiths]], \emph{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 \item [[Phillip Griffiths]] \emph{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, MR 0233825 (summary [[GriffithsPeriodsSummary.pdf:file]]) \end{itemize} A review including also the Weil complex structure is in \begin{itemize}% \item [[Phillip Griffiths]], section 1 of \emph{Some results on algebraic cycles on algebraic manifolds}, Proceedings of the International Conference on Algebraic Geometry, Tata Institute (Bombay), 2012 (\href{http://publications.ias.edu/node/181}{web}, \href{http://publications.ias.edu/sites/default/files/someresultsonalg.pdf}{pdf}) \end{itemize} 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 \begin{itemize}% \item Peter Scheinost, [[Martin Schottenloher]], pp. 154 (11 of 76) of \emph{Metaplectic quantization of the moduli spaces of flat and parabolic bundles}, J. reine angew. Mathematik, 466 (1996) (\href{https://eudml.org/doc/153753}{web}) \end{itemize} The mid-dimensional case was discussed in unpublished work by Lazzeri, see \begin{itemize}% \item Elena Rubei, \emph{Lazzeri's Jacobian of oriented compact riemannian manifolds} (\href{http://arxiv.org/abs/math/9812110}{arXiv:math/9812110}) \end{itemize} The relation of the intermediate Jacobian to [[Deligne cohomology]] (Deligne's theorem) due to [[Pierre Deligne]] is discussed in \begin{itemize}% \item [[Hélène Esnault]], [[Eckart Viehweg]], section 7 of \emph{Deligne-Beilinson cohomology} in Rapoport, Schappacher, Schneider (eds.) \emph{Beilinson's Conjectures on Special Values of L-Functions} . Perspectives in Math. 4, Academic Press (1988) 43 - 91 (\href{http://www.uni-due.de/~mat903/preprints/ec/deligne_beilinson.pdf}{pdf}) \end{itemize} Reviews and surveys include \begin{itemize}% \item [[eom]], \emph{\href{http://www.encyclopediaofmath.org/index.php/Intermediate_Jacobian}{Intermediate Jacobian}} \item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Intermediate_Jacobian}{Intermediate Jacobian}} \item [[Patrick Walls]], \emph{Intermediate Jacobians and Abel-Jacobi maps}, 2012 ([[WallsJacobian.pdf:file]]) \item [[Jean-Luc Brylinski]], around theorem I 1.5.11 of \emph{Loop Spaces, Characteristic Classes and Geometric Quantization} Springer, 2007 \item [[Arnaud Beauville]], \emph{Vari\'e{}t\'e{}s de Prym et jacobiennes interm\'e{}diaire}, Annales scientifiques de l'\'E{}cole Normale Sup\'e{}rieure, S\'e{}r. 4, 10 no. 3 (1977), p. 309-391 (\href{http://math.unice.fr/~beauvill/pubs/prym.pdf}{pdf}) (\href{http://www.numdam.org/item?id=ASENS_1977_4_10_3_309_0}{jstor}) \item Valentin Zakharevich, \emph{Mixed Intermediate Jacobians}, 2012 (\href{http://www.algant.eu/documents/theses/zakharevich.pdf}{pdf}) \item [[Alexander Polishchuk]], \emph{Abelian varieties, Theta functions and the Fourier transform}, Cambridge University Press (2003) (\href{http://math1.unice.fr/~beauvill/pubs/poli.pdf}{pdf}) \end{itemize} Discussion of the generalization to [[Hodge structures]] includes \begin{itemize}% \item [[Chris Peters]], [[Jozef Steenbrink]], \emph{[[Mixed Hodge Structures]]}, Ergebisse der Mathematik (2008) (\href{http://www.arithgeo.ethz.ch/alpbach2012/Peters_Steenbrinck}{pdf}) \item \href{http://www-fourier.ujf-grenoble.fr/~peters/world.f/Torino.pdf}{pdf} \end{itemize} \hypertarget{for_calabiyau_3folds}{}\paragraph*{{For Calabi-Yau 3-folds}}\label{for_calabiyau_3folds} Discussion of intermediate Jacobians of [[Calabi-Yau 3-folds]] includes \begin{itemize}% \item C. Herbert Clemens, [[Phillip Griffith]], \emph{The intermediate Jacobian of the cubic threefold}, Annals of Mathematics Second Series, Vol. 95, No. 2 (Mar., 1972), pp. 281-356 (\href{http://www.jstor.org/stable/1970801}{JSTOR}) \item [[Claire Voisin]] (\href{http://www.math.polytechnique.fr/~voisin/Articlesweb/griffithsgroup.pdf}{pdf}) \item [[Andreas Höring]], \emph{Minimal classes on the intermediate Jacobian of a generic cubic threefold}, 2008 (\href{http://math.unice.fr/~hoering/articles/a5-intermediate.pdf}{pdf}) \end{itemize} In [[positive number|positive]] [[characteristic]]: \begin{itemize}% \item [[Gerard van der Geer]], T. Katsura, \emph{On the height of Calabi-Yau varieties in positive characteristic} (\href{http://arxiv.org/abs/math/0302023}{arXiv:math/0302023}) \end{itemize} Applications in [[string theory]]: \begin{itemize}% \item [[David Morrison]], section 4 of \emph{Mirror Symmetry and the Type II String}, Nucl.Phys.Proc.Suppl. 46 (1996) 146-155 (\href{http://arxiv.org/abs/hep-th/9512016}{arXiv:hep-th/9512016}) \item Diaconescu, [[Ron Donagi]], [[Tony Pantev]], \emph{Intermediate Jacobians and ADE Hitchin Systems} (\href{http://arxiv.org/abs/hep-th/0607159}{arXiv:hep-th/0607159}) \item Ling Bao, Axel Kleinschmidt, Bengt E. W. Nilsson, Daniel Persson, Boris Pioline, \emph{Instanton Corrections to the Universal Hypermultiplet and Automorphic Forms on SU(2,1)}, Commun. Num. Theor. Phys. 4 (1), 187-266 (2010) (\href{http://arxiv.org/abs/0909.4299}{arXiv:0909.4299}) \item A. Baarsma, \emph{The hypermultiplet moduli space of compactified type IIA string theory}, Master Thesis, Utrecht 2011 (\href{http://dspace.library.uu.nl/handle/1874/203182}{web}) \end{itemize} The relation of Theta characteristics on [[intermediate Jacobians]] to [[self-dual higher gauge theory]] was first recognized in \begin{itemize}% \item [[Edward Witten]], \emph{Five-Brane Effective Action In M-Theory}, J.Geom.Phys.22:103-133,1997 (\href{http://arxiv.org/abs/hep-th/9610234}{arXiv:hep-th/9610234}) \end{itemize} and the argument there was made rigorous in \begin{itemize}% \item [[Mike Hopkins]], [[Isadore Singer]], \emph{[[Quadratic Functions in Geometry, Topology, and M-Theory]]} \end{itemize} \hypertarget{for_generalized_cohomology}{}\subsubsection*{{For generalized cohomology}}\label{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 \begin{itemize}% \item [[Edward Witten]], \emph{Duality Relations Among Topological Effects In String Theory}, JHEP 0005:031,2000 (\href{http://arxiv.org/abs/hep-th/9912086}{arXiv:hep-th/9912086}) \item [[Gregory Moore]], [[Edward Witten]], \emph{Self-Duality, Ramond-Ramond Fields, and K-Theory}, JHEP 0005:032,2000 (\href{http://arxiv.org/abs/hep-th/9912279}{arXiv:hep-th/9912279}) \item D. Diaconescu, [[Gregory Moore]], [[Edward Witten]], \emph{$E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory}, Adv.Theor.Math.Phys.6:1031-1134,2003 (\href{http://arxiv.org/abs/hep-th/0005090}{arXiv:hep-th/0005090}), summarised in \emph{A Derivation of K-Theory from M-Theory} (\href{http://arxiv.org/abs/hep-th/0005091}{arXiv:hep-th/0005091}) \item Dmitriy Belov, [[Greg Moore]], \emph{Type II Actions from 11-Dimensional Chern-Simons Theories} (\href{http://arxiv.org/abs/hep-th/0611020}{arXiv:hep-th/0611020}) \end{itemize} A mathematical discussion inspired by this is in \begin{itemize}% \item [[Stefan Müller-Stach]], Chris Peters, Vasudevan Srinivas, \emph{Abelian varieties and theta functions associated to compact Riemannian manifolds; constructions inspired by superstring theory} (\href{http://arxiv.org/abs/1105.4108}{arXiv:1105.4108}, \href{http://www-fourier.ujf-grenoble.fr/~peters/world.f/Torino.pdf}{pdf slides}) \end{itemize} A discussion of intermediate Jacobians for any rationally periodic [[generalized (Eilenberg-Steenrod) cohomology]] theory is in \begin{itemize}% \item [[Michael Hopkins]], [[Gereon Quick]], \emph{Hodge filtered complex bordism} (\href{http://arxiv.org/abs/1212.2173}{arXiv:1212.2173}) \end{itemize} [[!redirects intermediate Jacobians]] [[!redirects higher Jacobian]] [[!redirects higher Jacobians]] [[!redirects Griffith intermediate Jacobian]] [[!redirects Griffith intermediate Jacobians]] [[!redirects Weil intermediate Jacobian]] [[!redirects Weil intermediate Jacobians]] \end{document}