In an exercise of Voisin book, says:


Let $j:C\rightarrow S$ the inclusion of a smooth curve on a smooth connected projective surface. Set
$H=ker(j\_\*:H^1(C,\mathbb{Z})\rightarrow H^3(S,\mathbb{Z}))$.


We also write $A\subset J(C)$ for the Abelian subvariety corresponding to the Hodge substructure $H$.


I do not understand how the correspondence between Abelian subvarieties and Hodge substructures goes, so I will be grateful if someone can suggest to me a reference to learn it.
Thank you



Abelian varieties corresponding to Hodge substructures
Hodge substructures of $H^1(C,\mathbb Z)$ and Abelian subvarieties of $J(C)$ ar essentially the same thing:


In general, let $V,\omega$ be a free $\mathbb Z$-module of even rank with an integral symplectic form. A *polarized Hodge structure of weight 1* on $V$ is the data of a decomposition
$V\_{\mathbb C} := V\otimes\_{\mathbb Z} \mathbb C = V^{1,0} \oplus V^{0,1}$
with $V^{0,1} = \overline{V^{1,0}}$ and a positivity condition with respect to $\omega$. This is canonically equivalent to the data of a linear complex structure $I$ on $V\_{\mathbb R} := V\otimes\_{\mathbb Z} \mathbb R$ for which $\omega$ is of positive. The associated abelian variety is $V\_{\mathbb R} / V\_{\mathbb Z}$, with this linear complex structure.


A Hodge substructure is a $\mathbb Z$-submodule $W\subset V$ such that $W\otimes \mathbb C$ is compatible with the Hodge decomposition, i.e.
$$W\_\mathbb C:= W \otimes \mathbb C = W\_\mathbb C \cap V^{1,0} \oplus W \_\mathbb C \cap V^{0,1}~.$$
Equivalently, $W\_\mathbb R := W\otimes \mathbb R$ is $I$-invariant. Then $W\_\mathbb R/W$ is an abelian subvariety of $V\_\mathbb R / V$. Conversely, every abelian subvariety is a the quotient of a $I$-invariant subspace.


In your case, $V= H^1(C,\mathbb Z)$, $V^{1,0} = H^{1,0}(C)$, the associated abelian variety is $J(C)$ and $H= W$. What remains to understand is that $H$ is a Hodge substructure, and this is because $j\_\*$ is compatible with the Hodge decomposition (it maps $H^{1,0}$ to $H^{2,1}$ and $H^{0,1}$ to $H^{1,2}$).


