Why does so much recent work involve K3 surfaces?

Question

I've been noticing that a whole lot of papers published to the Arxiv recently involve K3 surfaces. Can anyone give me (someone who, at this point, knows little more about K3 surfaces than their definition) an idea why they are coming up so often?

Some questions that might be relevant: Are there particular reasons that they are so important? Are there special techniques that are available for K3 surfaces, but not more generally, making them easier to study? Are they just "in vogue" at the moment? Are they more like a subject of research (e.g., people are carrying out some sort of program to better understand K3 surfaces) or a testing ground (people with ideas in all sorts of different areas end up working the ideas out over K3 surfaces, because more general versions are much more difficult)?

Answer 1

Projective algebraic surfaces are classified first by their Kodaira number $k(X)$. Surfaces with $k(X) = -1$ have been much studied, they are either rational or ruled. Surfaces with $k(X) = 2$ are of general type. Surfaces with $k(X) = 0$ are of several types (abelian, K3, Enriques, or hyperelliptic). Notice the rough analogy with curves, where we have genus 0 ($k(X)=-1$) are rational curves, genus 2 or greater ($k(X)=1$) are general type, and genus 1 ($k(X)=0$) are elliptic curves. So surfaces with $k(X)=0$ provide a testing ground for surface theory similar to the testing ground for curves provided by elliptic curves.

Among the $k(X)=0$ surfaces, certainly abelian surfaces have been the most studied. On the other hand, Enriques and hyperelliptic surfaces are rather special. That leave K3 surfaces as surfaces with $k(X)=0$ that do not have a group structure, yet exist in vast quantities. (The moduli space of algebraic K3 surfaces consists of a countable union of 19 dimensional varieties.) So presumably for geometers, K3 surfaces are a challenge because they have no group structure, yet are much easier than surfaces of general type.

As a number theorist, I look on K3 surfaces as providing a huge challenge to understand their arithmetic, e.g., the distribution of rational points, or the distribution of integral points on affine pieces. (Vojta's conjecture implies that the latter set is not Zariski dense, so this would be a great place to prove a piece of Vojta's conjecture that does not use an underlying group structure.) Another big conjecture (known in many cases) is that if a K3 surface $X$ is defined over a number field $K$, then there is a finite extension $L$ of $K$ such that $X(L)$ is Zariski dense in $X$.

[I know I omitted the $k(X)=1$ surfaces. They are elliptic surfaces, so also extremely interesting from both a geometric and an arithmetic perspective. But not relevant to the question about K3 surfaces.]

Answer 2

From the viewpoint of classical algebraic geometry, the reason is simple: they are easy to deal with and many things can be computed on them (e.g., their moduli, Picard lattices, automorphism groups, etc). For example, complete linear systems on projective K3 surfaces are particularly easy to study using results of Saint-Donat and Nikulin. Moreover, many special K3s turn up naturally in other problems from algebraic geometry e.g., as complete intersections (and this is not so much the case for other 'exotic' surfaces like Enriques surfaces). This makes K3 surfaces a nice testing ground for results in algebraic geometry.

Some examples:

-- Let $D$ be an effective divisor on a K3 surface. Then $|D|$ has no base-points outside its fixed components.

-- K3 surfaces have nice vanishing theorems: The vanishing of $h^2(X,D)=\dim H^2(X,O(D))$ for $D\neq 0$ effective is immediate by Serre duality: $h^2(X,D)=h^0(X,-D)=0$.

-- Let now $D$ be an effective nef divisor (so that $D.C\ge 0$ for every curve $C$). If $D^2>0$, then either $|D|$ is base-point free, then $h^1(X,D)=0$ and the generic member of $|D|$ is smooth and irreducible, or $|D|$ is not base-point free and then we have $D\sim kE+\Gamma$, where $k\geq2$, $E$ is an elliptic curve, $\Gamma$ is a rational curve and $E.\Gamma=1$. If $D^2=0$, then $| D|$ is composed with a pencil, i.e $D=kE$, where $E$ is an elliptic curve and $h^1(X,D)=k-1$. In sum, if you have a divisor that is free from fixed components, then you can calculate the dimensions of all the cohomology groups using Riemann-Roch.

-- Moreover, Saint-Donat gives precise results on the degrees of the generators and relations of the section ring $A=\bigoplus_{n\ge 0}H^0(X,nD)$. This means that one can easily study concrete projective models of K3 surfaces.

-- There are also strong results by Kovács on the effective cone of a K3 surface.

Answer 3

A famous instance of "K3 surfaces as proving ground" is:

Deligne, Pierre La conjecture de Weil pour les surfaces $K3$. (French) Invent. Math. 15 (1972), 206–226.

Compare with:

Deligne, Pierre La conjecture de Weil. I. (French) Inst. Hautes Études Sci. Publ. Math. No. 43 (1974), 273–307.

Answer 4

K3 surfaces are also interesting from the point of view of complex dynamics. To quote from Curtis T. McMullen's introduction to his paper ``Dynamics on K3 surfaces: Salem numbers and Siegel disks", Journal fur die Reine und Angewandte Mathematik 2002(545): 201–233,

"The first dynamically interesting automorphisms of compact complex manifolds arise on K3 surfaces. Indeed, automorphisms of curves are linear (genus 0 or 1) or of finite order (genus 2 or more). Similarly, automorphisms of most surfaces (including $\mathbb{P}^2$, surfaces of general type and ruled surfaces) are either linear, finite order or skew-products over automorphisms of curves. Only K3 surfaces, Enriques surfaces, complex tori and certain non-minimal rational surfaces admit automorphisms of positive topological entropy [Ca2]. The automorphisms of tori are linear, and the Enriques examples are double-covered by K3 examples."

In this paper McMullen gives examples of K3 surfaces admitting automorphisms with Siegel disks (i.e., domains on which the automorphism is conjugate to a rotation). There are countably many such surfaces, all of them non-projective. The citation [Ca2] is to the paper by S. Cantat, Dynamique des automorphismes des surfaces projectives complexes. CRAS Paris S ́er. I Math. 328 (1999), 901–906, which grew into a larger work: S. Cantat : Dynamique des automorphismes des surfaces K3 ; Acta Math. 187:1 (2001), 1--57

These papers are only a few examples (perhaps of landmark character); there has been a lot of study of dynamics on K3 surfaces going on indeed.

Answer 5

In the mathematical physics community, there was recently a resurgence of interest for K3-surfaces due to the observation of Egushi, Ooguri and Tashikawa that the elliptic genus of K3-surfaces seems to be built out of representations of the Mathieu group M24. This phenomenon has been dubbed the "Mathieu moonshine". There is still no definitive understanding of this relation, as far as I know.