ag.algebraic geometry - Classification of simply connected smooth projective varieties?

The best kind of classification of smooth projective varieties would be a list of deformation types. We have this for curves, and something close to it for surfaces of Kodaira dimension less than 2. We do not have such a list for smooth projective surfaces of Kodaira dimension 2, not even for the simply connected ones, and the situation in higher dimensions is worse.

The next best kind might be a complete invariant, algorithmically computable from the defining equations. So far as I know, we don't have this either. For instance, one might ask whether the canonical ring $bigoplus_n{H^0(K^n)}$ is algorithmically computable. A recent triumph of the minimal model program has been to prove that this is finitely generated for non-singular varieties of general type; its Proj is then the canonical model of the variety. I presume that it's not effectively computable at present; perhaps someone else can comment on positive or negative results in this direction (say for surfaces?).

A still weaker request would be for a theorem which says "there exists an algorithm to decide whether these two varieties are deformation-equivalent". I think we probably do have this, by virtue of Grothendieck's theorem that, after you fix the Hilbert polynomial, there's a proper Hilbert scheme, which in particular has only finitely many connected components. As a matter of logic (logicians, please correct me if necessary!), if there are only finitely many possibilities, an algorithm exists to test which one you have - because there exists a finite list of those possibilities, encoded as numbers, and you just have to check yours against each of them. What we don't have is a practical method to produce that list.

This assertion does have content; by contrast, there's no algorithm to decide diffeomorphism of compact smooth manifolds (given as real semi-analytic sets, say) because one can't compute $pi_1$. There is an algorithm to check diffeomorphism of simply connected manifolds of dimension $>4$ (compute the cohomology groups, compare the finite list of $k$-invariants specifying the Postnikov tower and hence the homotopy type, compare the Pontryagin classes, appeal to a finiteness result from surgery theory - see Nabutovsky-Weinberger, "Algorithmic aspects of homeomorphism problems", MR1707346).