ag.algebraic geometry – Abelian subvarieties corresponding to vector subspaces Answer

plane geometry - Does the intersection of two moving curves sweep out a continuous line?

Hello dear visitor to our network We will offer you a solution to this question ag.algebraic geometry – Abelian subvarieties corresponding to vector subspaces ,and the answer will be typical through documented information sources, We welcome you and offer you new questions and answers, Many visitor are wondering about the answer to this question.

ag.algebraic geometry – Abelian subvarieties corresponding to vector subspaces

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 $B$ other $H^1(C,\mathbb{Q})_{van}$ 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?

we will offer you the solution to ag.algebraic geometry – Abelian subvarieties corresponding to vector subspaces question via our network which brings all the answers from multiple reliable sources.