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There's a very interesting text by Cumrun Vafa called Geometric Physics.

Here I'm particularly interested in Chapter 4, where we take a Calabi-Yau manifold presented as a degenerating fibration:

The appearance of the Dynkin structure for the K3 singularities appears mathematically as purely “accidental”. However this accident gets explained in this duality context: One identifies the singular K3 geometries with A-D-E singularities with the points on the heterotic side with enhanced A-D-E gauge symmetry...

I look at the pictures (p.15) and I have a very simple question:

  • Is this vanishing K3 obtainable as a vanishing/nearby cycle functor for the cohomology of the fibration?

If it is, I will finally have an example of the abovementioned functor. If not, how to describe this K3 from a math point of view? One possible way would be to vary Kahler parameters and get a true, finite-size K3. Other descriptions?

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2 Answers 2

I am not completely sure what the question is here so let me say something about the picture in general. There is no "vanishing K3" in Vafa's description. The singular K3 geometry that Vafa is talking about is a K3 surface with some ADE singularity. These surfaces are of finite size from the metric point of view - both their volume and their diameter are finite. The only difference between them and their resolutions is that the resolutions have additional Kahler parameters - the volumes of the exceptional spheres of the resolution. In other words, the singular K3 surfaces are parametrized by special loci in the interior of the Kahler moduli - the equations of these loci are given by setting the volumes of the exceptional spheres to zero.

You can also describe these singular K3s as complex (rather than Kahler) degenerations of smooth K3s, and I suspect that this is what you are after with your question. K3 surfaces with ADE singularities appear naturally as special fibers in one parameter families of K3 surfaces where the total space of the family and the general fiber are smooth. The typical picture is that you have a smooth complex threefold $X$ and a proper holomorphic map $f : X \to D$ to a one dimensional complex disk centered at zero, so that the fibers $X_{t} = f^{-1}(t)$ corresponding to $t \neq 0$ are all smooth K3 surfaces, and the special fiber $X_{0} = f^{-1}(0)$ is a K3 with say a single ADE singularity. Are you asking for specific models of this setup. There are many ways to get these, e.g. by looking at double covers of rational surfaces, or by looking at families of elliptic K3s. The easiest example is to look at a pencil of smooth quartics in $\mathbb{P}^{3}$ degenerating to a quartic with an isolated ordinary double point (which is in the ADE nomenclature is just an $A_{1}$ singularity).

Now for the family $f : X \to D$ you can form the nearby and vanishing cycle sheaves at $0 \in D$. They are very easy to describe. For instance the sheaf of vanishing cycles will be supported at the singularity $x \in X_{0}$ of $X_{0}$ and its stalk at $x$ will be the root lattice of the ADE group.

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"These surfaces are of finite size from the metric point of view" -- thanks, I missed that! "You can also describe these singular K3s as complex ... " -- I was thinking, maybe there's a better way, but that's fine too! "s stalk at x will be the root lattice of the ADE group." -- sounds interesting, can it be seen from H^2(singular K3)? –  Ilya Nikokoshev Nov 10 '09 at 22:00
    
Well, you can't just see it in terms of the singular K3 but you can see it as the kernel of the specialization map $H_{2}(X_{t}) \to H_{2}(X_{0})$. This specialization map is induced from the retraction of the tubular neighborhood of $X_{0}$ onto $X_{0}$. –  Tony Pantev Nov 16 '09 at 1:19

by the way, "singular k3 surface" has a quite different meaning it the literature: it refers to a smooth K3 surface with Picard number 20. reference: Shioda, "singular" K3 surfaces

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