Let $S=E\times C$ be a product of two curves, where $E$ is an elliptic curve and $C$ is a curve of genus at least two. Consider a foliation on $S$ generated by a global holomorphic 1-form $p\_1^\*(\omega\_1)+p\_2^\*(\omega\_2)$, where $p\_i$ is a projection map and $\omega\_1$ and $\omega\_2$ are nonzero holomorphic 1-forms on $E$ and $C$ respectively. Is it possible to say something about algebraicity of leaves of this foliation?



algebraic leaves of foliation on a product of two curves
The foliation $\mathcal F$ defined by $p\_1^\* \omega\_1 + p\_2^\* \omega\_2$ is everywhere transverse to the fibration $p\_2 : S \to C$. One can therefore lift paths
from $C$ to leaves of $\mathcal F$ in order to obtain a representation
$$
\rho: \pi\_1(C,b) \to \mathrm{Aut}(p\_2^{-1}(p)) \simeq \mathrm{Aut}(E) \, . 
$$
Since the foliation is defined by a closed $1$-form it is a simple matter
to realize that the image of this representation is contained in the subgroup of 
translations of $\mathrm{Aut}(E)$. It is then clear that the leaves of $\mathcal F$ are algebraic if, and only if, the image of $\rho$ is contained in a torsion subgroup of $E$. 


Notice that the representation in question is rather simple. 
If we identify $E$ with $\mathbb C / \Gamma$ where $\Gamma$ is a lattice in $\mathbb C$ then $\omega\_1$ can be identified with $\lambda [dz]$. Therefore 
$$
\rho(\gamma) = \lambda^{-1} \cdot \int\_{\gamma} \omega\_2 \in (\mathbb C / \Gamma,+) \subset \mathrm{Aut}(E) \, 
$$
for any $\gamma \in \pi\_1(C,b)$.


When $\mathcal F$ has algebraic leaves then Poincaré irreducibility Theorem (see for example Debarre's book Complex Tori and Abelian Varieties) implies that the Jacobian of $C$ is isogeneous to $E \times A$ where $A$ is an abelian variety of dimension $g(C)-1$. Reciprocally when the Jacobian of $C$ has this property we can arguee as Pierre did in other answer to this question to produce a foliation having all its leaves algebraic.


