K3 surface

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In complex geometry and in algebraic geometry K3 surfaces are the 2-dimensional analog of elliptic curves. The complex and algebro-geometric definitions are slightly different, and coincide in the case where the surface is an algebraic $K3$ surface over the complex numbers.

1 The algebro-geometric definition
- 1.1 Examples
- 1.2 Polarization
2 Complex definition
- 2.1 The Hodge diamond
- 2.2 The period map and the Torelli theorem
3 Complex algebraic K3 surfaces
- 3.1 Moduli

The algebro-geometric definition

In algebraic geometry a surface $S$ is a $K3$ surface if it is smooth, projective, with trivial canonical bundle, and such that $h^1(O_S)=0$ . In this case one automatically gets: $h^2(O_S)=1$ .

Examples

If $C\subset\mathbb{P}^2$ is a smooth curve of degree $6$ and $p:S\to\mathbb{P}^1$ is the double cover of $\mathbb{P}^2$ branched along $C$ , then $S$ $K3$ surface; indeed in the Picard group of $S$ we have $K_S=p^*(K_{\mathbb{P}^2}-\frac{1}{2}[C])=p^*0=0$ . A similar claim hods even if the curve $C$ is singular; the modification is that now one has to consider the normalization of the branch double cover. Specifically if the curve $C$ is a six lines tangent to a conic, then on recovers for the double cover model of a Kummer surface.
A quartic surface in $\mathbb{P}^3$
A complete intersection of a quadric and a cubic hyper-surfaces in $\mathbb{P}^4$
A complete intersection of three quadric hypersurfaces in $\mathbb{P}^5$