Let $X$ be a smooth complex projective variety. We know that $\Omega^1_X$ has a non-zero section if and only if the abelianization of the fundamental group of X is infinite. This follows from Hodge theory essentially.
Suppose that $X$ is a simply connected smooth projective surface of general type. Then Hodge theory tells us that $\Omega^1_X$ has no non-zero section.
What else does Hodge theory (in its most general sense) tell us?
What if stick to minimal surfaces, i.e., those with ample cotangent line bundle?