$\DeclareMathOperator\Alb{Alb}\DeclareMathOperator\CH{CH}$I'm wondering how the Albanese functor interacts with correspondences. Specifically if $X$, $Y$ are say smooth projective schemes over an algebraically closed field $k$ and $Z$ a correspondence between them. Does this induce a morphism of abelian varieties $$\Alb(X)\to \Alb(Y)?$$

If $X$, $Y$ are curves this is the case as the Albanese and Jacobian agree, and certainly there exists a morphism between their $k$-points (at the level of sets) since the Albanese map $\CH_0(X)^0\to \Alb(X)(k)$ is a bijection (well conditional on Bloch–Beilinson for $k=\overline{\mathbb{Q}}$).

$\DeclareMathOperator\Alb{Alb}\DeclareMathOperator\CH{CH}$I'm wondering how the Albanese functor interacts with correspondences. Specifically if $X$, $Y$ are say smooth projective schemes over an algebraically closed field $k$ and $Z$ a correspondence between them. Does this induce a morphism of abelian varieties $$\Alb(X)\to \Alb(Y)?$$

If $X$, $Y$ are curves this is the case as the Albanese and Jacobian agree, then one can quote say Cor 6.3 of Milne's notes on Jacobian Varieties, which says that morphisms on Jacobians of curves are in bijection to those divisors (or line bundles) which are trivial after pulling back to either factor.

I'm wondering how the Albanese functor interacts with correspondences. Specifically if $X,Y$ are say smooth projective schemes over an algebraically closed field $k$ and $Z$ a correspondence between them. Does this induce a morphism of abelian varieties $$Alb(X)\to Alb(Y)?$$

If $X,Y$ are curves this is the case as the Albanese and Jacobian agree, and certainly there exists a morphism between their $k$-points (at the level of sets) since the Albanese map $CH_0(X)^0\to Alb(X)(k)$ is a bijection (well condidtional on Bloch-Beilinson for $k=\overline{\mathbb{Q}}$).

$\DeclareMathOperator\Alb{Alb}\DeclareMathOperator\CH{CH}$I'm wondering how the Albanese functor interacts with correspondences. Specifically if $X$, $Y$ are say smooth projective schemes over an algebraically closed field $k$ and $Z$ a correspondence between them. Does this induce a morphism of abelian varieties $$\Alb(X)\to \Alb(Y)?$$

If $X$, $Y$ are curves this is the case as the Albanese and Jacobian agree, and certainly there exists a morphism between their $k$-points (at the level of sets) since the Albanese map $\CH_0(X)^0\to \Alb(X)(k)$ is a bijection (well conditional on Bloch–Beilinson for $k=\overline{\mathbb{Q}}$).