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?

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    $\begingroup$ actually, the canonical bundle is usually only big and nef, not ample (it is ample on the canonical model). One obvious answer: since $b_1=0$ (first Betti number), we know that the Albanese variety is trivial, i.e., every morphism to an Abelian variety/a torus is trivial. There are plenty of minimal surfaces of general type, and as far as I know, we do not have a good picture at all, even over the simply connected ones. However, being a simply connected 4-manifold, its homeomorphism type is determined by its intersection form (Freedman's theorem), but this has nothing to do with Hodge theory. $\endgroup$ – Christian Liedtke Jun 8 '13 at 12:19

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