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)?
Best Answer
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.]