---
created: 2023-03-29T11:25
updated: 2023-03-29T11:25
---

References:

- Compactifying period domains Laza-Zhang <https://www.math.stonybrook.edu/~rlaza/vancouver_lect_final.pdf#page=1&zoom=180,-45,796>

# Attempted Setup

- Let $\sigma_1, \sigma_2\in \Aut(X)$ nonsymplectic with $[\sigma_1, \sigma_2] = 0$ and consider $\Sigma \da G_1 G_2 = \gens{\sigma_1, \sigma_2}$ so $\size \Sigma = 4 \implies \Sigma \cong C_4$ or $C_2^2$.
- Always have $\sigma_i$ preserves $S_X$ and $T_X$.
- Note $G_i\actson V\da H^2(X; \ZZ)$ decomposes $V\cong V^{-1} \oplus V^{1}$ which coincides with $V = T_X \oplus S_X$ into $\NS(X)$ and the transcendental lattice, so $G_i^*\actson S_X$ by $-1$.
	- Generally: $\mfh \actson V$ is a representation of $\mfh$; simultaneously diagonalize to get a weight space decomposition $V\cong \bigoplus_{\alpha\in \mfh\dual} V_{\alpha}$ where $V_\alpha \da \ts{v\in V \st h.x = \alpha(h)x\, \forall h\in \mfh}$ for $\alpha\in \mfh\dual$.
	- Writing $\mfh = \ts{e,a,b,c \st a^2 = b^2 = c^2 = e, c=ab}$, a representation is determined by $e, x \mapsto 1$ and the rest $\mapsto -1 = \zeta_2$ for one of $x=a,b,c$ for 4 total irreducible representations.
- Define $A_i = Z_{\Aut(X)} G_i = \ts{f\in \Aut(X) \st f \sigma_i = \sigma_i f}$ the centralizer of $G_i$ in $\Aut(X)$.
	- Generally: $A \da Z_{\Aut(X)} G = \ts{f\in \Aut(X) \st [f, \sigma_1] = [f, \sigma_2] = 0}$. 
- $\ts{(X, \sigma_i) }/\Def \mapstofrom \ts{S\injects \lkt \text{ primitive 2-elementary hyperbolic}}\modiso \mapstofrom \ts{(g,k,\delta)}$.

## Defining the moduli space

- Fix $R \da \gens{\rho_1,\rho_2} \leq \Orth(\lkt)$ define an $R\dash$marking of $(X, \Sigma)$ as $\phi: H^2(X; \ZZ) \to \lkt$ such that $\sigma_i^* = \phi\inv \rho_i \phi$: ![](attachments/2023-02-25-diag.png)
- Familes of $R\dash$marked surfaces: $(\mcx, \Sigma_B)$: smooth morpism $\mcx \to B$ where $\Sigma_B \in \Aut_B(\mcx)$ and $\phi_S: \RR^2 f_* \ul{\ZZ} \iso \lkt \tensor \ul{\ZZ}_B$ with fibers $R\dash$marked surfaces.
- $(\lkt\tensor \CC)^\Sigma = \ts{x\in \lkt\tensor \CC \st \rho(x) = \alpha(s) x \, \forall s\in \Sigma}$ where $\alpha: \Sigma\to \CC\units$ is a fixed character.
- Change of markings group: $$\Gamma_R \da Z_{\Orth(\lkt)}(R) = \ts{\gamma\in \Orth(\lkt) \st [\gamma, \rho] = 0 \, \forall \rho \in R}$$
- $\DD_R = \PP( (\lkt \tensor \CC)^\Sigma ) \intersect \DD$.
- Generic transcendental lattice $T_R = (\lkt\tensor\CC)^\prim  \intersect \lkt$; intersect $\lkt$ with sum of primitive eigenspaces of $R$.
- Generic Picard lattice: $S_R \da T_R^\perp$.
- $L^R = \Fix_{S_R}(R) \subseteq S_R$.
- Need $\sgn T_R = (2, n)$...
- What type of domain is $\DD_\rho$?
- Define a quotient stack $F_R \da \mcm_R/ \Gamma_R$.
	- $\mcm_R$ an open subset of $\pi\inv(D_R/\Delta_R)$ restricted from $\pi: \mcm\to \DD$.

## Setup from the end (generalization)

- Fix $\sigma: \Sigma \injects \Aut(X)$ finite, 
	- Note $\Sigma\actson H^{2, 0}(X) \cong \CC$ yields $\Sigma_0 \injects \Sigma \surjectsvia{\alpha} \CC^*$
	- So $\alpha: \Sigma\to \CC\units$ is a fixed character
	- Nonsymplectic $\implies \Sigma_0 \neq 0 \iff \alpha\neq 1$.
- Fix $\rho: \Sigma\injects \Orth(\lkt)$ and $G \da H^2(\Sigma)$ where $g\mapsto g^*$.
- Fix $\chi: G\to \CC\units$
- Vary pairs $(X, \sigma)$.
- Define $(\rho, \chi)\dash$markings as $\phi:H^2(X; \ZZ)\to \lkt$ such that [see link here](https://q.uiver.app/?q=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) ![](attachments/2023-02-25diag22.png)
	- And induced character $\alpha:
- Families: $(f: \mcx\to B, \sigma_B: G\to \Aut_B(\mcx), \phi_B:\RR^2 f_* \ul{\ZZ} \iso \lkt \tensor \ul{\ZZ}_B)$ where $\mcx\to B$ is a smooth family where every fiber is $(\rho, \chi)$ marked.
- Define $\mcm_{\rho, \chi}$
- Define $(\lkt\tensor \CC)^{\rho, \chi} = \ts{v\in \lkt\tensor \CC\st \rho(g)(x) = \chi(g)\cdot x}$ the common eigenspace of $G$.
- Define $\DD_{\rho, \chi} = \ts{\omega \in \PP(\lkt\tensor \CC)\st \abs{\omega} > 0 }$.
	- $\size \im \chi = 2\implies$ Type IV domain
	- $\size \im \chi > 2\implies$ Type I domain, complex ball.
- $\lkt^G \da \ts{v \in \lkt \st \rho(g)(v) = v}$ is the fixed sublattice
- $$\Delta_{\rho, \chi} = \Union_{\delta \in (\lkt^G)^\perp \text{ roots}} \delta^\perp \intersect \Delta_{\rho}$$?
- $\Gamma_\rho \da \ts{\gamma\in \Orth(\lkt) \st [\gamma, \rho] = 0}$.
- Theorem:
	- $\exists \mcm_{\rho, \chi}$ a fine moduli space with an etale period map $\pi_\rho: \mcm_{\rho, \chi} \to \DD_{\rho, \chi}\sm \Delta_{\rho, \chi}$
	- Fibers $\pi_\rho\inv(v)$ are torsors over $\Gamma_\rho\intersect(W_v \intersect C_2)$.
	- Moduli stack $\mcm_{\rho, \chi}  = [F_{\rho, \chi} / \Gamma_\rho]$
	- Coarse space $$M_{\rho ,\chi} \iso \dcosetr{ \qty{\DD_{\rho, \chi}\sm \Delta_{\rho,\chi}}}{ \Gamma_\rho}$$
- Prove lemma 2.6 and theorem 2.9.2

See [commuting automorphisms](Unsorted/commuting%20automorphisms.md).


# Notes

![](attachments/2023-02-27invariantlattice.png)

![](attachments/2023-02-27abcd.png)

![](attachments/2023-02-27-bb.png)

![](attachments/2023-02-27bb2.png)

Orbits of isotropic lines corresponding to 0-cusps/points $Q_\ell$ and isotropic planes corresponding to 1-cusps/curves $X_\pi$.
![](attachments/2023-02-27adasdsa.png)

Toric compactifications:
![](attachments/2023-02-27toriccpt.png)

Perp: ![](attachments/2023-02-27perp-1.png)

Toroidal compactifications:
![](attachments/2023-02-27redix.png)

Quasi-polarized K3s
![](attachments/2023-02-27quasi.png)

The disadvantage of working with the moduli functor of quasi-polarised $\mathrm{K}_3$ surfaces is that it is not separated; therefore it is more convenient to treat the moduli functor of polarised $\mathrm{K}_3$ surfaces with $\mathrm{ADE}$ singularities, which is separated and whose coarse moduli space can be identified with $\mathcal{F}_{2 d}$.

See [level structure](Unsorted/level%20structure.md)

Symmetric space: covariant derivative of curvature tensor vanishes (so curvature tensor is invariant under parallel transport)

A bounded domain $\Omega$ in a complex vector space is said to be a **bounded symmetric domain** if for every $x$ in $\Omega$, there is an involutive biholomorphism $\sigma_x$ of $\Omega$ for which $x$ is an isolated fixed point.

[[Hermitian symmetric domains]]

Period domains:

![](attachments/2023-02-27perdom.png)

![](attachments/2023-02-27perdm2.png)

![](attachments/2023-02-27never1.png)

Curves as a ball quotient:

![](attachments/2023-02-27ballquot.png)

[congruence subgroup](Unsorted/congruence%20subgroup.md)

![](attachments/2023-02-27congsubgr.png)


$\DD \cong \Hol^+(\DD)/K$ for $K$ a maximal compact $\implies$ cusps correspond to rational maximal parabolic subgroups of $H$.

For $\Ag$:
![](attachments/2023-02-27ag-1.png)

Modular varieties of orthogonal type:

![](attachments/2023-02-27orthtype.png)

![](attachments/2023-02-27laz.png)


Isotropic sublattices:

![](attachments/2023-03-01-isotrop.png)

Symplectic actions

![](attachments/2023-03-01sympact.png)

# Video notes

$G\actson X$ nonsymplectically implies $G\actson T(X)$ trivially and $D_{\NS(X)}$ trivially, so invariant subspaces are contained in $\NS(X)$.

$G\actson \K3$ symplectically iff $G\leq M_{23}$ with at least 5 orbits under $M_{23}\actson [24]$.

See Artin invariant.