K3 Surface - The Period Map

The Period Map

There is a coarse moduli space for marked complex K3 surfaces, a non-Hausdorff smooth analytic space of dimension 20. There is a period mapping and Torelli theorem for complex K3 surfaces.

If M is the set of pairs consisting of a complex K3 surface S and a Kähler class of H1,1(S,R) then M is in a natural way a real analytic manifold of dimension 60. There is a refined period map from M to a space KΩ0 that is an isomorphism.The space of periods can be described explicitly as follows:

• L is the even unimodular lattice II_3,19
• Ω is the Hermitian symmetric space consisting of the elements of the complex projective space of L⊗C that are represented by elements ω with (ω,ω)=0, (ω,ω^*)>0.
• KΩ is the set of pairs (κ, ) in (L⊗R, Ω) with (κ,E(ω))=0, (κ,κ)>0
• KΩ0 is the set of elements (κ, ) of KΩ such that (κd) ≠ 0 for every d in L with (d,d)=−2, (ω,d)=0.