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 $l < g$?



Holomorphic/Symplectic embedding of Riemann surfaces
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).


