Let $A$ be an abelian variety, $g$ be a positive integer and $\mathcal{L}$ be an ample line boundle on $A$.

Question : Is there a real $r>0$ such that for all smooth curve $C$ of genus $g$ in $A$, we have $\deg_\mathcal{L} C\leq r$ ?

Thanks a lot.

Let $A$ be an abelian variety, $g$ be a positive integer and $\mathcal{L}$ be an ample line bundle on $A$.

Question : Is there a real $r>0$ such that, for all smooth curve $C$ of genus $g$ in $A$, we have $\deg_\mathcal{L} C\leq r$ ?

Thanks a lot.