K3Moduli-DLFQJ2XA

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.

3 type of K3 obsidian

Cohomology etc

K3 lattice obsidian K3 lattice

Local Torelli theorem

Strong Torelli obsidian

Period space and coarse moduli space

period space of K3 obsidian

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

def_stratification obsidian

4 types of K3, and stratification of moduli spaces

Kulikov Models types obsidian

Stratification of $\mathfrak{M}^{IV}_4$obsidian

resolve period map deg 2

deg 2 example

resolve p_2 obsidian

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

resolve p_2 obsidian

cubic threefold

More motivation of flips

  • HK program
  • VGIT

vgit and flips obsidian

“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

Hodge bundle obsidian

Heegner divisors

3 types of vectors in $\Lambda$ obsidian

3 types of Heegner divisorsobsidian

Period point in Hyperelliptic Unigonal divisor obsidian

Boundary 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

deg 4 K3

proved by K-stability obsidian