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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*{Abel-Jacobi map} \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{for_curves}{for curves}\dotfill \pageref*{for_curves} \linebreak \noindent\hyperlink{on_deligne_cohomology}{on Deligne cohomology}\dotfill \pageref*{on_deligne_cohomology} \linebreak \noindent\hyperlink{on_higher_chow_groups}{on higher Chow groups}\dotfill \pageref*{on_higher_chow_groups} \linebreak \noindent\hyperlink{via_extensions_of_mixed_hodge_structures}{via extensions of mixed Hodge structures}\dotfill \pageref*{via_extensions_of_mixed_hodge_structures} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The Abel-Jacobi map refers to various homomorphisms from certain groups of [[algebraic cycles]] to some sorts of [[Jacobian]]s or generalized Jacobians. Such maps generalize the classical Abel-Jacobi map from points of a complex algebraic curve to its [[Jacobian]], which answers the question of which divisors of degree zero arise from [[meromorphic functions]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{for_curves}{}\subsubsection*{{for curves}}\label{for_curves} Let $X$ be a smooth projective complex curve. Recall that a [[Weil divisor]] on $X$ is a formal linear combination of closed points. Classically, the Abel-Jacobi map \begin{displaymath} \alpha : \Div^0(X) \longrightarrow J(X), \end{displaymath} on the group of Weil divisors of degree zero, is defined by integration. According to Abel's theorem, its kernel consists of the principal divisors, i.e. the ones coming from meromorphic functions. \hypertarget{on_deligne_cohomology}{}\subsubsection*{{on Deligne cohomology}}\label{on_deligne_cohomology} The cycle map to [[de Rham cohomology]] due to (\hyperlink{ZeinZucker81}{Zein-Zucker 81}) is discussed in (\hyperlink{EsnaultViehweg88}{Esnault-Viehweg 88, section 6}). The refinement to [[Deligne cohomology]] in (\hyperlink{EsnaultViehweg88}{Esnault-Viehweg 88, section 6}). By the characterization of [[intermediate Jacobians]] as a subgroup of the [[Deligne complex]] (see \href{intermediate+Jacobian#CharacterizationAsHodgeTrivialDeligneCohomology}{intermediate Jacobian -- characterization as Hodge-trivial Deligne cohomology} this induces a map from cycles to [[intermediate Jacobians]]. This is the Abel-Jacobi map (\hyperlink{EsnaultViehweg88}{Esnault-Viehweg 88, theorem 7.11}). \hypertarget{on_higher_chow_groups}{}\subsubsection*{{on higher Chow groups}}\label{on_higher_chow_groups} An Abel-Jacobi map on [[higher Chow groups]] is discussed in \hyperlink{KLMS04}{K-L-MS 04}. \hypertarget{via_extensions_of_mixed_hodge_structures}{}\subsubsection*{{via extensions of mixed Hodge structures}}\label{via_extensions_of_mixed_hodge_structures} An alternate construction of the Abel-Jacobi map, via [[Hodge theory]], is due to Arapura-Oh. By a theorem of Carlson, the [[Jacobian]] is identified with the following group of [[extensions]] in the [[abelian category]] of [[mixed Hodge structures]]: \begin{displaymath} J(X) = Ext^1_{MHS}(\mathbf{Z}(-1), H^1(X, \mathbf{Z})) \end{displaymath} where $\mathbf{Z}(-1)$ is the Tate Hodge structure. Given a [[divisor]] $D$ of degree zero, one can associate to it a certain class in the above extension group. This gives a map \begin{displaymath} \alpha : Div^0(X) \longrightarrow J(X) \end{displaymath} which is called the Abel-Jacobi map. The Abel theorem says that its [[kernel]] is precisely the subgroup of [[principal divisors]], i.e. divisors which come from invertible rational functions. See (\hyperlink{ArapuraOh97}{Arapura-Oh, 1997}) for details of this construction. \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item Fouad El Zein and Steven Zucker, \emph{Extendability of normal functions associated to algebraic cycles}, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), Ann. of Math. Stud., vol. 106, Princeton Univ. Press, Princeton, NJ, 1984, pp. 269--288. \href{http://www.ams.org/mathscinet-getitem?mr=756857}{MR 756857} \item [[Hélène Esnault]], [[Eckart Viehweg]], \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}) \item [[Claire Voisin]], section 12 of \emph{[[Hodge theory and Complex algebraic geometry]] I,II}, Cambridge Stud. in Adv. Math. \textbf{76, 77}, 2002/3 \item Donu Arapura, Kyungho Oh. \emph{On the Abel-Jacobi map for non-compact varieties}. Osaka Journal of Mathematics 34 (1997), no. 4, 769--781. \href{http://projecteuclid.org/euclid.ojm/1200787781}{Project Euclid}. \item Matt Kerr, James Lewis, Stefan M\"u{}ller-Stach, \emph{The Abel-Jacobi map for higher Chow groups}, 2004, \href{http://arxiv.org/abs/math/0409116}{arXiv:0409116}. \item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Abel–Jacobi_map}{Abel-Jacobi map}} \end{itemize} Remarks on generalization to the more general context of [[anabelian geometry]] are in \begin{itemize}% \item [[Minhyong Kim]], \emph{Galois Theory and Diophantine geometry, 2009 (\href{http://www.ucl.ac.uk/~ucahmki/cambridgews.pdf}{pdf})} \end{itemize} [[!redirects Abel-Jacobi maps]] [[!redirects Abel-Jacobi theorem]] [[!redirects Abel-Jacobi theorems]] \end{document}