Let $S$ be a connected smooth projective surface.
Let $C$ a smooth curve on $S$
In page 9 of the paper "https://arxiv.org/abs/1704.04187v1" a read the following:
Let \begin{equation*} r: C\rightarrow S \end{equation*} be the closed embedding of the curve $C$ into $S$.
Let \begin{equation*} {r}_*:H^1(C,\mathbb{Q})\rightarrow H^3(S,\mathbb{Q}), \end{equation*} the Gysin homomorphism in cohomology groups whose kernel is $H^1(C,\mathbb{Q})_{van}$, the group of vanishing cycles; and let $B$ be the subvariety in $J=J(C)$, the jacobian of the curve $C$, corresponding o the $\mathbb{Q}$-vector subspace $H^1(C,\mathbb{Q})_{van}$ of $H^1(C,\mathbb{Q})$.
I would like to know what does it mean "$B$ be the subvariety in $J$ corresponding o the $\mathbb{Q}$-vector subspace $H^1(C,\mathbb{Q})_{van}$"? I mean how $B$ and $H^1(C,\mathbb{Q})_{van}$ are related?
I already know the relation between subvarieties of the Jacobian $J$ and Hodge substructures of $H^1(C,\mathbb{Z})$. So, maybe in order to obtain the $\mathbb{Q}$-vector subspace corresponding to $B$, I need first to have its corresponding Hodge substructure in $H^1(C,\mathbb{Z})$ and then just tensor it with $\mathbb{Q}$? Am I right?
