Holomorphic/Symplectic embedding of Riemann surfaces

Question

Let $\Sigma_g$ denote a Riemann surface and let $X$ denote the complex surface $\Sigma_g \times \Sigma_g$. Then can there exist holomorphic embeddings of $\Sigma_l$ into $X$ for $l < g$?

What about in the symplectic category i.e if $\omega$ denotes the area 1 form on $\Sigma_g$ and we equip $X = \Sigma_g \times \Sigma_g$ with the form $\omega \oplus \omega$. Then does there exist a symplectic embedding $\Sigma_l$ into $X$ for $l < g$?

Answer 1

I am just posting my comment as one answer.

Lemma. For integers $\ell < g$, for every continuous map from $\Sigma_\ell$ to $\Sigma_g$, the pullback map on $H^2$ is the zero map.

Proof. The pullback map on $H^1$ has rank no greater than $2\ell$, since that is the rank of $H^1(\Sigma_\ell)$. Since $H^1(\Sigma_g)$ has rank $2g>2\ell$, there exists a nonzero element $\alpha$ in the kernel. Since the cup product pairing on $H^1(\Sigma_g)$ is nondegenerate, there is an element $\beta$ in $H^1(\Sigma_g)$ such that $\alpha\cup \beta$ is nonzero in $H^2(\Sigma_g)$. Since the pullback map on cohomology is a ring homomorphism, the pullback of $\alpha\cup \beta$ is zero. Since $H^2(\Sigma_\ell)$ is torsion-free, and since $\alpha\cup \beta$ is a nonzero multiple of the generator of $H^2(\Sigma_g)$, the pullback map on $H^2$ is zero. QED

Thus, for a generator $\omega$ of $H^2(\Sigma_g,\mathbb{R})$, for every continuous function from $\Sigma_\ell$ to $\Sigma_g\times \Sigma_g$, the pullback of $\omega\oplus \omega$ is zero in $H^2(\Sigma_\ell,\mathbb{R})$. Since a symplectic form on $\Sigma_\ell$ has nonzero cohomology class, there is no differentiable map from $\Sigma_\ell$ to $\Sigma_g\times \Sigma_g$ that pulls back $\omega\oplus \omega$ to a symplectic form on $\Sigma_\ell$.

As noted by @abx, there is no nonconstant holomorphic map from $\Sigma_\ell$ to $\Sigma_g$. The lemma shows that there is not even a map that pulls back $\omega$ to a differential form with nonzero cohomology class (every nonconstant holomorphic map pulls back $\omega$ to a differential form with nonzero cohomology class).