In dimensions 1 and 2 there is only one, respectively 2, compact Kaehler manifolds with zero first Chern class, up to diffeomorphism. However, it is an open problem whether or not the number of topological types of such manifolds of dimension 3 (Calabi-Yau threefolds) is bounded. I would like to ask what is known in this direction. In particular, is it known that the Euler characteristic or the total Betti number of Calabi-Yau threefolds can't be arbitrarily large? Are there any mathematical (or physical?) reasons to expect either answer?


As a side question: I remember having heard several times something like "Calabi-Yau 3-folds parametrize (some kind of) vacua in string theory" but was never able to make precise sense of this. So any comments on this point or references accessible to mathematicians would be very welcome.



Topologically distinct Calabi-Yau threefolds
This is a very good question, and I would really love to know the answer since its current state seems to be quite obscure. Below is just a collection of remarks, surely not the full answer by any means. I would like to argue that for the moment there is no any deep mathematical reason to think that the Euler number of CY 3-folds is bounded. I don't believe either that there is any physical intuition on this matter. But there is some empirical information, and I will describe it now, starting by speaking about "how many topological types of CY manifolds we know for the moment". 


As far as I understand for today the construction of Calabi-Yau 3-folds, that brought by far the largest amount of examples is the construction of Batyrev. He starts with a reflexive polytope in dimension 4, takes the corresponding toric 4-fold, takes a generic anti-canonical section and obtains this way a Calabi-Yau orbifold. There is always a crepant resolution. So you get a smooth Calabi-Yau. Reflexive polytopes in dimension 4 are classified the number is 473,800,776. I guess, this number let Miles Reid to say in his article "Updates on 3-folds" in 2002 <http://arxiv.org/PS_cache/math/pdf/0206/0206157v3.pdf> , page 519 


"This gives some 500,000,000 families of CY 3-folds, so much more impressive than a mere infinity (see the website <http://tph16.tuwien.ac.at/~kreuzer/CY/>). There are certainly many more; I believe there are infinitely many families, but the contrary opinion is widespread"


A problem with the number 500,000,000 in this phrase is that it seems more related to the number of CY orbifolds, rather than to the number of CY manifolds obtained by resolving them. Namely, the singularities that appear in these CY orbifolds can be quite involved and they have a lot of resolutions (I guess at least thousands sometimes), so the meaning of 500,000,000 is not very clear here. 


This summer I asked Maximillian Kreuzer (one of the persons who actually got this number 473,800,776 of polytopes), a question similar to what you ask here. And he said that he can guarantee that there exist at least 30108 topological types of CY 3-folds. Why? Because for all these examples you can calculate Hodge numbers $h^{1,1}$ and $h^{2,1}$, and you get 30108 different values. Much less that 473,800,776. As for more refined topological invariants (like multiplication in cohomology) according to him, this was not really studied, so unfortunately 30108 seems to be the maximal number guarantied for today. But I would really love to know that I am making a mistake here, and there is some other information.


Now, it seems to me that the reason, that some people say, that the Euler characteristics of CY 3-folds could be bounded is purely empirical. Namely, the search for CY 3-folds is going for 20 years already. Since then a lot of new families were found. We know that mirror symmetry started with this symmetric table of numbers "($h^{1,1}, h^{2,1})$", and the curious fact is that, according to Maximillian, what happened to this table in 20 years -- it has not got any wider in 20 years, it just got denser. The famous picture can be found on page 9 of the following notes of Dominic Joyce <http://people.maths.ox.ac.uk/~joyce/SympGeom2009/SGlect13+14.pdf> . So, this means that we do find new families of CY manifolds, all the time. But the values of their Hodge numbers for some reason stay in the same region. Of course this could easily mean that we are just lacking a good construction. 


Final remark is that in the first version of this question it was proposed to consider complex analytic manifolds with $c\_1=0$. If we don't impose condition of been Kahler, then already in complex dimension 2 there is infinite number of topological types, given by Kodaira surfaces, they are elliptic bundles over elliptic curve. In complex dimension 3 Tian have shown that for every $n>1$ there is a holomorphic structure on the connected sum of n copies of $S^3\times S^3$, with a non-vanishing holomorphic form. Surely these manifolds are non-Kahler. So if you want to speak about any finiteness, you need to discuss say, Kahler 3-folds with non-vanishing holomorphic volume form, but not all complex analytic ones.


