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 II3,19
- 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.
Read more about this topic: K3 Surface
Famous quotes containing the words period and/or map:
“There is a period near the beginning of every man’s life when he has little to cling to except his unmanageable dream, little to support him except good health, and nowhere to go but all over the place.”
—E.B. (Elwyn Brooks)
“When I had mapped the pond ... I laid a rule on the map lengthwise, and then breadthwise, and found, to my surprise, that the line of greatest length intersected the line of greatest breadth exactly at the point of greatest depth.”
—Henry David Thoreau (1817–1862)