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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 surface over the complex numbers.

The algebro-geometric definition

In algebraic geometry a surface is a surface if it is smooth, projective, with trivial canonical bundle, and such that . In this case one automatically gets: .

Examples

  • If is a smooth curve of degree and is the double cover of branched along , then surface; indeed in the Picard group of we have . A similar claim hods even if the curve is singular; the modification is that now one has to consider the normalization of the branch double cover. Specifically if the curve is a six lines tangent to a conic, then on recovers for the double cover model of a Kummer surface.
  • A quartic surface in
  • A complete intersection of a quadric and a cubic hyper-surfaces in
  • A complete intersection of three quadric hypersurfaces in

In the last three examples one may verify that the canonical bundle is trivial using adjunction formula

Polarization

Complex definition

In complex geometry a surface is complete smooth simply connected surface with trivial canonical class.

The Hodge diamond

The period map and the Torelli theorem

Complex algebraic K3 surfaces

Moduli