---
date: 2022-05-14 20:29
modification date: Saturday 14th May 2022 20:29:51
title: "Abel-Jacobi map"
aliases: [Abel-Jacobi map, Abel theorem, universal divisor]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---

- Tags: 
	- #AG 
- Refs:
	- #todo/add-references
- Links:
	- [theta divisor](theta%20divisor)
	- [abelian variety](Unsorted/abelian%20variety.md)
	- [Jacobian](Unsorted/Jacobian.md)

---

# Abel-Jacobi map

- Let $X \in \smooth\Var\slice{\CC}^{\proj, \dim = n}$
- Let $Z^i_h(X)$ be the nullhomologous $i\dash$cycles.
- Let $J^\ell(X)$ be ???
- Define the [Chow group](Unsorted/Chow%20ring.md).
- Define the **Griffiths Abel-Jacobi map**:
$$
\Phi_X: Z^i_h(X) \to J^{2i-1}(X)
$$
- Define
$$
\begin{align*}
G: H_{2 n-2 i+1, B}(X, \mathbb{Z}) 
&\rightarrow \Fil^{n-i+1} H_{B}^{2 n-2 i+1}(X, \mathbb{C})\dual \\
\alpha &\mapsto \int_\alpha
\end{align*},
$$
and $\im G$ as the **group of periods**.
- If $z \in Z^{n-i}_h(X)$, so of real dimension $2n-2i$, there is a $\Gamma \in \Fil^{n-i+1} H_{B}^{2 n-2 i+1}(X, \mathbb{C})\dual$ of dimension $2n-2i+1$ with $\bd \Gamma = z$, and $\int_\Gamma$ is well-defined even if $\Gamma$ is not a closed cycle (key idea: choose representatives of closed forms and apply the $\del\delbar$ 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 
	  $$\Phi_X(z) \da \int_\gamma \in {\Fil^{n-i+1} H_{B}^{2 n-2 i+1}(X, \mathbb{C})\dual \over \im G}, \quad \in J^{2i-1}(X)?$$
	- The **Abel-Jacobi** map factors through rational equivalence, yielding
$$
\Phi_X: \Ch^i_h(X) \to J^{2i-1}(X)
$$


## $\ell\dash$adic variant

- Pick $x\in Z^r(X\slice k)$ of dimension $r$ where $x\in Z^r_h(X\slice{\kbar})$, so a cycle on $X$ which becomes nullhomologous after base change to $\bar X \da X\tensor_k \kbar$.
- This determines a class
$$
e_{Z} \in H^{1}\left(G_{K}, H_{\text {èt }}^{2 r-1}\left(X \otimes \bar{K}, \mathbb{Z}_{\ell}(r)\right)\right)
$$
which only depends on the rational equivalence class of $z$ and the image of $z$ under the Abel-Jacobi map
$$
C H_{h}^{r}(X) \rightarrow H^{1}\left(G_{K};\,  H_{\et}^{2 r-1}\left(X \otimes \bar{K}, \mathbb{Z}_{\ell}(r)\right)\right)
$$
	- This map is defined on rational equivalence classes of homologically trivial codimension $r$ cycles on $X$


# Abel's theorem

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

# Universal divisors

- If $X \in \smooth\Var\slice{\CC}^{\proj}$ then $J^1(X) \in \Ab\Var$ 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 $P\in \Pic(J^1(X) \times X)$ 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 $X\in \Ab\Var$.
	- Note: it is not unique, but can be normalized by fixing $x\in X$ and imposing that the restriction $\ro{P}{H^1(X)\times \ts x}$ is trivial.