Let $X$ be a simply connected complex manifold. How can one show that there exists no holomorphic map to a curve $C$ of genus $g\ge 1$?
This shows up in a paper I am reading now. I thought this can be seen by the induced map on the fundamental groups $\pi_1(X)\rightarrow \pi_1(C)$, but even if this map is trivial, there may be a non-trivial map.
If X maps to an elliptic curve, then it's Albanese variety is nontrivial. In particular, its fundamental group is nontrivial. So simply connected varieties do not map nontrivially to elliptic curves.
If a variety maps to an elliptic curve, then use pushforward and pullback to see that the elliptic curve is isogenous to an abelian subvariety of the Albanese.