K3Moduli
Polarized K3 surfaces
ample L or semi-ample or pseudo-ample or nef and big. But all primitive
smooth or ADE singularity
polarized and morphism and GIT moduli
e.g.
Cohomology etc
K3 lattice obsidian K3 lattice
Local Torelli theorem
Period space and coarse moduli space
coarse moduli space for polarized K3 obsidian
coarse moduli space of polarized K3 obsidian
Baily-Borel
Toy example:
Period map
period map
$\mathfrak{p}$
Hodge Stratification
4 types of K3, and stratification of moduli spaces
Stratification of $\mathfrak{M}^{IV}_4$obsidian
resolve period map deg 2
deg 2 example
Looijenga
Arrangement
Looijenga has devised a comparison framework that applies to locally symmetric varieties associated to type IV or I1,n Hermitian symmetric domains
Stratification of $\Delta$ obsidian
Stratification of $\mathcal{H}$ obsidian
Flip center prediction obsidian
Period space side (Looijenga)
deg 2 example
cubic threefold
More motivation of flips
- HK program
- VGIT
“Roughly, the space of all possible linearizations is divided into finitely many polyhedral chambers within which the quotient is constant (2.3), (2.4), and when a wall between two chambers is crossed, the quotient undergoes a birational transformation which, under mild conditions, is a flip in the sense of Mori (3.3)” (
Laza O’Grandy Predictions and Evidence
Locally symmetric variety of type IV
period point for K3 in F(19) obsidian
Hodge bundle
Heegner divisors
3 types of vectors in $\Lambda$ obsidian
3 types of Heegner divisorsobsidian
Period point in Hyperelliptic Unigonal divisor obsidian
D-tower
Picture of D-tower [obsidian](/wiki/zotero/Picture-of-D-tower-Article-J26RYRSS
$\mathcal{F}(19)$
BB boundary of $\mathcal{F}_4$obsidian
divisor relations
stratification of $\Delta$ deg 4
precise stratification of centersobsidian
VGIT
Def of VGIT for hyperellptic K3 obsidian
Iso for VGIT and models of quadrtic K3 obsidian
K-stability
Hyperellptic deg 4
Iso between Kmoduli and VGIT obsidian