Abel-Jacobi map

Let $X \in {\mathsf{sm}}{\mathsf{Var}}_{/ {{\mathbf{C}}}} ^{\mathop{\mathrm{proj}}, \dim = n}$
Let $Z^i_h(X)$ be the nullhomologous $i{\hbox{-}}$cycles.
Let $J^\ell(X)$ be ???
Define the Chow group.
Define the Griffiths Abel-Jacobi map: \begin{align*} \Phi_X: Z^i_h(X) \to J^{2i-1}(X) \end{align*}
Define \begin{align*} \begin{align*} G: H_{2 n-2 i+1, B}(X, \mathbb{Z}) &\rightarrow {\operatorname{Fil}}^{n-i+1} H_{B}^{2 n-2 i+1}(X, \mathbb{C}) {}^{ \vee }\\ \alpha &\mapsto \int_\alpha \end{align*}, \end{align*} and $\operatorname{im}G$ as the group of periods.
If $z \in Z^{n-i}_h(X)$, so of real dimension $2n-2i$, there is a $\Gamma \in {\operatorname{Fil}}^{n-i+1} H_{B}^{2 n-2 i+1}(X, \mathbb{C}) {}^{ \vee }$ of dimension $2n-2i+1$ with ${{\partial}}\Gamma = z$, and $\int_\Gamma$ is well-defined even if $\Gamma$ is not a closed cycle (key idea: choose representatives of closed forms and apply the ${\partial}{ \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu}$ lemma). Then $\Gamma$ is determined up to addition of closed chains, making $\int_\Gamma$ defined up to elements of the form $\int_\alpha$.
- Thus there is a well-defined map \begin{align*}\Phi_X(z) \coloneqq\int_\gamma \in {{\operatorname{Fil}}^{n-i+1} H_{B}^{2 n-2 i+1}(X, \mathbb{C}) {}^{ \vee }\over \operatorname{im}G}, \quad \in J^{2i-1}(X)?\end{align*}
- The Abel-Jacobi map factors through rational equivalence, yielding \begin{align*} \Phi_X: \mathsf{Ch}^i_h(X) \to J^{2i-1}(X) \end{align*}

$\ell{\hbox{-}}$adic variant

Pick $x\in Z^r(X_{/ {k}} )$ of dimension $r$ where $x\in Z^r_h(X_{/ { { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } }} )$, so a cycle on $X$ which becomes nullhomologous after base change to $\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu \coloneqq X\otimes_k { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } $.
This determines a class
\begin{align*} e_{Z} \in H^{1}\left(G_{K}, H_{\text {èt }}^{2 r-1}\left(X \otimes \mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu, \mathbb{Z}_{\ell}(r)\right)\right) \end{align*} which only depends on the rational equivalence class of $z$ and the image of $z$ under the Abel-Jacobi map
\begin{align*} C H_{h}^{r}(X) \rightarrow H^{1}\left(G_{K};\, H_{\text{ét}}^{2 r-1}\left(X \otimes \mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu, \mathbb{Z}_{\ell}(r)\right)\right) \end{align*}

This map is defined on rational equivalence classes of homologically trivial codimension $r$ cycles on $X$

Statement for codimension 1 cycles: the Abel Jacobi map is an isomorphism \begin{align*} \Phi_{X}: \mathrm{CH}_h^{1}(X) { \, \xrightarrow{\sim}\, }J^{1}(X) \end{align*}
Proof:
- Look at LES for exponential exact sequence, identigy ${\operatorname{CH}}^1(X) = \operatorname{Pic}(X) = \operatorname{Pic}(X^{\mathrm{an}})$ and $\mathsf{Ch}^1_h(X) = \ker c_1 \subseteq \operatorname{Pic}(X^{\mathrm{an}})$ to get \begin{align*} 0 \rightarrow H^{1}(X, \mathbb{Z}) \rightarrow H^{1}\left(X, \mathcal{O}_{X}\right) \rightarrow \mathrm{CH}^{1}(X)_{\text {h }} \rightarrow 0\end{align*}
- Yields $H^{0, 1}(X) = H^1(X; {\mathcal{O}}_X)$ and $J^1(X) { \, \xrightarrow{\sim}\, }{\operatorname{CH}}^1_h(X)$; show it is the inverse of $\Phi_X$.

If $X \in {\mathsf{sm}}{\mathsf{Var}}_{/ {{\mathbf{C}}}} ^{\mathop{\mathrm{proj}}}$ then $J^1(X) \in {\mathsf{Ab}}{\mathsf{Var}}$ and is a moduli of nullhomologous divisors on $X$ modulo linear equivalence, or equivalently topologically trivial line bundles on $X$.
Theorem: there is a line bundle $P\in \operatorname{Pic}(J^1(X) \times X)$ such that for any $t\in J^1(X)$ the associated divisor $D_t$ is nullhomologous and $\Phi_X(D_t) = t$. Such a divisor $D_t$ is called a universal divisor or Poincare divisor when $X\in {\mathsf{Ab}}{\mathsf{Var}}$.
- Note: it is not unique, but can be normalized by fixing $x\in X$ and imposing that the restriction ${ \left.{{P}} \right|_{{H^1(X)\times \left\{{x}\right\}}} }$ is trivial.