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

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## 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?

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