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I am wondering whether there is any sort of classification of simply connected smooth projective varieties, or any work in related directions.

The reason I am interested in this is because, by Deligne-Griffiths-Morgan-Sullivan, smooth projective varieties are formal. Thus, if we have a smooth projective variety that is simply connected, and if we know its cohomology ring, then we can compute its rational homotopy theory.

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Classification up to what? E.g. the classification of complex projective surfaces up to diffeomorphism is not known at the moment. –  algori Feb 19 '10 at 23:39
    
I have to second what algori said. Curves, well, we understand the topology of curves. But for surfaces, there's a LOT of difficulty that's swept up in the phrase "of general type." There MIGHT be something like this for Calabi-Yaus in low dimension, but I'm not certain how much is known there either. –  Charles Siegel Feb 19 '10 at 23:47
    
I do not think Calabi-Yau 3-folds are classified. As for formality, the fact that symmetric spaces are formal does not help to classify them. –  Igor Belegradek Feb 20 '10 at 0:18

3 Answers 3

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).

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There is a work in this direction, but it is clear it will go on for ages. Even in the case of surfaces the knowledge is quite tiny. For example, by Yau's theorem it is known that every surface homeomorphic to $\mathbb CP^2$ is biholomorphic to $CP^2$. But it is yet unknown if there is a surface of general type homeormorphic to $\mathbb CP^2$ blown up at $1$ point.

In general, there are not so many examples of topological types of surfaces for which we know the classification, here is one of examples, when the classification can be done http://arxiv.org/PS_cache/arxiv/pdf/0909/0909.1733v1.pdf (though it is very important here that the fundamental group is non-trivial).

There is a very nice article of Carlos Simpson, speaking about the huge gap between restrictions on that we know about algebraic (Kahler) manifolds, and actual examples that can be constructed. http://math.unice.fr/~carlos/preprints/kg7.pdf

On the positive side, it can be said, for example, that 3-dimensional Fano varieties were classified by 1980. At the same time the classification of 4 dimensional Fano still does not exit. Some people were speaking recently of possibility to use derived cathegories in order to predict this classification (in dim 4), but it is not at all clear how many years will pass before this will be done.

Final remark is that all simply-connected manifolds of dimension up to 6 are formal. And, again, it would not be a stretch to say that we don't know much about possible topological types. There are very few known restrictions coming mostly from Hodge theory (like parity of Betti numbers, or restrictions the on cohomology ring: see Voisin's "Hodge structures on cohomology algebras and geometry" http://arxiv.org/PS_cache/math/pdf/0703/0703523v3.pdf). There was also an explicit attempt to construct (collect) examples in (real) dimension 6, given in a nice article of Okonek and Van de Ven Cubic Forms and 3-folds. http://retro.seals.ch/cntmng;jsessionid=FF0F66FD1D82539A8CC10D78491EA4F4?type=pdf&rid=ensmat-001:1995:41::146&subp=hires

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I found Simpson's paper very interesting. Thanks for the reference! –  Don Stanley Feb 21 '10 at 17:10

That is way too ambitious, I think, considering what is known about classification of algebraic varieties. Ignoring the trivial, for these purposes, case of curves, the next and best studied case is algebraic surfaces. For them, the classification is classical and has been known for almost a century. So that is a good first case to consider.

So you are asking: what are the smooth projective surfaces $X$ with $\pi_1(X)=1$, and in particular with regularity $q=h^1(\mathcal O_X)=0$. Well, pick up your favorite text, Shafarevich etc. or Barth-(Hulek-)-Peters-van de Ven, or whatever, and go through the list.

Kodaira dimension $-\infty$: here you get all rational surfaces, and only these.

Kodaira dimension $0$: K3 surfaces.

Kodaira dimension $1$, elliptic surfaces $X\to C$ (a general fiber is elliptic). Clearly, $C$ must be $\mathbb P^1$. But actually getting $\pi_1(X)=1$ seems like not a completely trivial condition, something to think about.

Kodaira dimension $2$, i.e. surfaces of general type. Well, the examples of simply connected surfaces of general type are highly prized, especially if they have $p_g=h^0(\omega_X)=0$. Many such surfaces are known (e.g. http://en.wikipedia.org/wiki/Barlow_surface, some Godeaux surfaces, some Campedelli surfaces) but a complete classification? Not even close. Like I said, this is too ambitious for the present state of knowledge.

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