TITLE: When is a smooth projective variety a fibration
QUESTION [5 upvotes]: Let $X$ be a smooth projective variety. Is there a criterion (apart from the definition) for the existence of a projective curve $C$ and a proper surjective morphism $\pi:X \to C$?

REPLY [8 votes]: Another interesting theorem in this direction is Castelnuovo-de Franchis theorem. It says that if you have two linearly independent holomorphic 1-forms $\omega_1,\omega_2$ with $\omega_1\wedge\omega_2=0$ on $X$, then there exists a morphism $f:X\to C$ with $C$ a smooth curve of genus at least 2 and forms $\omega_i'$ on $C$ such that $\omega_i=f^*\omega_i'$.

REPLY [2 votes]: By a theorem of Gromov and Schoen if the fundamental group of X is a proper amalgamated product
or HNN extension then X maps surjectively to a curve .