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Another example, added following the request of the OP to provide an example just involving complex geometry: If $X$ is a K3 surface, then from the Hodge structure on the primitive part of $H^2(X,\mathbb C)$, one can construct an abelian variety, the so-called Kuga--Satake abelian variety associated to $X$. The construction is made in terms of Hodge structures. One expects that in fact there should be a link between $X$ and its associated Kuga--Satake abelian variety provided by some correspondence, but this is not known in general. It would be implied by the Hodge conjecture. (This constructions is discussed in this answer and in the accompanying comments.)

Another example, added following the request of the OP to provide an example just involving complex geometry: If $X$ is a K3 surface, then from the Hodge structure on the primitive part of $H^2(X,\mathbb C)$, one can construct an abelian variety, the so-called Kuga--Satake abelian variety associated to $X$. The construction is made in terms of Hodge structures. One expects that in fact there should be a link between $X$ and its associated Kuga--Satake abelian variety provided by some correspondence, but this is not known in general. It would be implied by the Hodge conjecture. (This constructions is discussed in this answer and in the accompanying comments.)

Another example, added following the request of the OP to provide an example just involving complex geometry: If $X$ is a K3 surface, then from the Hodge structure on the primitive part of $H^2(X,\mathbb C)$, one can construct an abelian variety, the so-called Kuga--Satake abelian variety associated to $X$. The construction is made in terms of Hodge structures. One expects that in fact there should be a link between $X$ and its associated Kuga--Satake abelian variety provided by some correspondence, but this is not known in general. It would be implied by the Hodge conjecture. (This constructions is discussed in this answer and in the accompanying comments.)

In each case we also have a natural abelian (in fact Tannakian) category in play: in the complex case, the category of pure Hodge structures, and in the other cases, the category of $\ell$-adic representations of $G_K$ (the absolute Galois group of $K$) (for some prime $\ell$).

We may also consider the usual congruence subgroup $\Gamma_0(pq)$ consisting of matrices in $SL_2(\mathbb Z)$ that are upper triangular modulo $pq$, and form $X_0(N)$, the compacitifcation of $\Gamma_0(pq)\backslash \mathcal H$.

Now the theory of modular and automorphic forms and their associated Galois representations show that $X$ and $X_0(pq)$ are both naturally curves over $\mathbb Q$, and that there is an embedding of Galois representations $H^1(X) \to H^1(X_0(pq))$. Thus the Tate conjecture predicts that there is a correspondence between $X$ and $X_0(pq)$ inducing this embedding. Passing to Hodge structure, we would then find that the periods of holomorphic one-forms on $X$ should be among the periods of holomorphic one-forms on X_0(pq)$.

In each case we also have a natural abelian (in fact Tannakian) category in play: in the complex case, the category of pure Hodge structures, and in the other cases, the category of $\ell$-adic representations of $G_K$ (the absolute Galois group of $K$) (for some prime $\ell$, prime to the characteristic of $K$ in the case when $K$ is a finite field).

We may also consider the usual congruence subgroup $\Gamma_0(pq)$ consisting of matrices in $SL_2(\mathbb Z)$ that are upper triangular modulo $pq$, and form $X_0(pq)$, the compacitifcation of $\Gamma_0(pq)\backslash \mathcal H$.

Now the theory of modular and automorphic forms and their associated Galois representations show that $X$ and $X_0(pq)$ are both naturally curves over $\mathbb Q$, and that there is an embedding of Galois representations $H^1(X) \to H^1(X_0(pq))$. Thus the Tate conjecture predicts that there is a correspondence between $X$ and $X_0(pq)$ inducing this embedding. Passing to Hodge structure, we would then find that the periods of holomorphic one-forms on $X$ should be among the periods of holomorphic one-forms on $X_0(pq)$.