I’m reading Daniel Huybrechts’ book on K3 surfaces now (because I miss doing mathematics), and so I thought I could write some short blog posts about it as I read through it.

We’ll start off with the definition. I’ll assume we’re working over $\mathbb C$ unless I say something else. Then a K3 surface is a complete non-singular variety $X$ of dimension $2$ such that $\Omega_{X/\mathbb C}^2 \simeq \mathcal O_X$ and $H^1(X, \mathcal O_X) = 0$ (the last condition is to separate K3 surfaces from Abelian surfaces). In a sense, they are Calabi-Yau varieties of dimension 2.

## The easiest example

The by far easiest example of a K3 surface is a smooth quartic in $\mathbb P^3$. Indeed:

[ 0 \to \mathcal O_{\mathbb P^3} (-4) \to \mathcal O_{\mathbb P^3} \to \mathcal O_X \to 0, ]

which gives us $H^1(X, \mathcal O_X) = 0$ by chasing some long exact sequences. The triviality of the canonial bundle can be found by the adjunction formula: $\omega_X = \omega_{\mathbb P^3} \otimes \mathcal O(4) \simeq \mathcal O(-4) \otimes \mathcal O(4) \simeq \mathcal O_X$.

An interesting special case is given by the Fermat quartic defined by $x_0^4+x_1^4+x_2^4+x_3^4=0$. It has a lot of discrete symmetries, given by scaling the coordinates by fourth roots of unity.

## Another example, not so easy

Another class of examples comes from double coverings of $\mathbb P^2$, branched along a curve of degree $6$. We can be very explicit: let $\mathbb P = \mathbb P(1,1,1,3)$ be a weighted projective space (this is defined by dividing out by $\lambda \cdot (a,b,c,d) \sim (\lambda a, \lambda b, \lambda c, \lambda^3 d)$). Let $f \in k[x,y,z]$ be a degree $6$ polynomial. Then let $X$ be the zeroes of $w^2=f(x,y,z)$ in $\mathbb P$, where we view $f$ as a polynomial in $k[x,y,z,w]$ in the natural grading.

This is a homogeneous polynomial in the grading on $\mathbb P$, so $X$ is well-defined as a variety in $\mathbb P$. We have a natural map $\pi: X \to \mathbb P^2$ by forgetting the last coordinate. It is easy to see that $X$ is smooth as long as $C=Z(f) \subset \mathbb P^2$ is smooth.

### Little aside…

Here Huybrechts uses the following fact:

If $\pi: X \to Y$ is an affine morphism and $\mathscr F$ is a coherent sheaf on $X$, then $H^i(X, \mathscr F) \simeq H^i(Y, \pi_\ast \mathscr F)$.

I have seen this many times, but never seen a proof. Looking it up, it is (of course) an exercise in Hartshorne. Turns out the proof is easy, so here’s a sketch: cover $Y$ by affines $U_i$. Then $\pi^{-1}(U_i)$ is an affine covering of $X$ since $\pi$ is an affine morphism. Then one can compare the Čech complexes of $\mathscr F$ and $\pi_\ast \mathscr F$ using this covering, and observe that they are in fact equal. Then since $\mathscr F$ is quasi-coherent, Čech cohomology computes sheaf cohomoogy, and so the the statement follows.

### Back to the example…

Now a local computation (or by some other means?) we see that $\pi_\ast \mathcal O_X \simeq \mathcal O_{\mathbb P^2} \oplus \mathcal O_{\mathbb P^2}(-3)$ (one way is to notice that the projective coordinate ring of $X$ is $k[x,y,z,w]/(f(x,y,z)-w^2)$, which is equal to $k[x,y,z] \oplus k[x , y ,z] (-3)$ as a $k[x,y,z]$-module. But I’m not positive that this is a valid argument).

But now by the aside above, this lets us see that $H^1(X,\mathcal O_X) \simeq H^1(\mathbb P^2, \mathcal O_{\mathbb P^2} \oplus \mathcal O_{\mathbb P^2}(-3)) = 0$.

Finally, we must argue that the canonical bundle on $X$ is trivial. This follows from Hurwitz’ formula for branched coverings, which says that $K_X=\pi^\ast(K_Y) + R$, where $K_X$ is the class of the canononical line bundle on $X$, and $R$ is the ramification divisor on $X$. The map ramifies along the curve defined by $f=0$. On $X$, this is defined by the zeros of $w=0$. But $w$ has degree 3 in the graded coordinate ring of $X$, so $\mathscr O_X(-R)=\mathscr O_X(-3)$. On the other hand $K_Y=-3H$, and it follows that $K_X=-3H’ + 3H’ = 0$, where $H’$ is the class of a hyperplane in $X$.

Now we have two examples of K3 surfaces. As I progress further in the book, I hope to write more blog posts explaining other interesting stuff about these fascinating creatures.