2
$\begingroup$

I would like to know what the following statement means:

"Let $B_t$ be the Abelian subvariety in $J_t$ corresponding to the $\mathbb{Q}$-vector subspace $H^1(C_t,\mathbb{Q})_{van}$ in the space $H^1(C_t,\mathbb{Q})\simeq H^1(J_t,\mathbb{Q})$".

Here $r_t:C_t\rightarrow S$ is the embedding of the smooth connected curve $C_t$ in the surface $S$, $J_t$ is the Jacobian of the curve $C_t$, and $(r_t)_*:H^1(C_t,\mathbb{Q})\rightarrow H^3(S,\mathbb{Q})$ is the Gysing homomorphism in cohomology groups associated to $r_t$.

It is in page 9 of the following paper: https://arxiv.org/abs/1704.04187v2

Improve this question
$\endgroup$
3
  • 2
    $\begingroup$ $r_t:H^1(C_t,\mathbb{Q})\rightarrow H^3(S,\mathbb{Q})$ is a morphism of Hodge structures, so its kernel $H^1(C_t,\mathbb{Q})_{van}$ carries a weight 1 Hodge structure, corresponding to an Abelian subvariety of $JC_t$. In more concrete term, it is the neutral component of the kernel of $(r_t)_{*}: JC_t\rightarrow \operatorname{Alb}(S) $. $\endgroup$
    – abx
    Commented Feb 12, 2021 at 5:29
  • $\begingroup$ Thank you @abx !!!. Any reference for "Hodge structure corresponding to an Abelian variety". $\endgroup$
    – Roxana
    Commented Feb 15, 2021 at 1:30
  • $\begingroup$ You may look for instance at Voisin's Hodge theory and complex algebraic geometry, § 7.2.2. $\endgroup$
    – abx
    Commented Feb 15, 2021 at 6:41

0

Reset to default

You must log in to answer this question.