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created: 2023-03-12T18:15
updated: 2024-01-21T18:17
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# moduli of K3s

Recent work (2023) of Gross:

> **Abstract:** I will talk about work with Hacking, Keel and Siebert on using mirror constructions to provide partial compactifications of the moduli of K3 surfaces. Starting with a one-parameter maximally unipotent degeneration of Picard rank 19 K3 surfaces, we construct, using methods of myself and Siebert, a mirror family which is defined in a formal neighbourhood of a union of strata of a toric variety whose fan is defined, to first approximation, as the Mori fan of the original degeneration. This formal family may then be glued in to the moduli space of polarized K3 surfaces to obtain a partial compactification. Perhaps the most significant by-product of this construction is the existence of theta functions in this formal neighbourhood, certain canonical bases for sections of powers of the polarizing line bundle.
 
# Moduli stacks over Z

![](attachments/2023-03-12witt.png)
![](attachments/2023-03-12line.png)
![](attachments/2023-03-12formall.png)

![](attachments/2023-03-12mod.png)

One then proves that these are groupoids and stacks for the etale topology (prove that they're etale sheaves with effective descent data). See <https://arxiv.org/pdf/math/0506120.pdf#page=21&zoom=150,41,752>

Relative smoothness is shown here <https://arxiv.org/pdf/math/0506120.pdf#page=26&zoom=150,3,443>

# Periods

See <https://arxiv.org/pdf/math/0506120.pdf#page=25&zoom=150,-4,436>

For $F_{2d}$:

$$\Omega^\pm \da \ts{\omega\in \PP(\Lambda_{2d} \tensor \CC) \st \omega^2 = 0,\,\, \norm\omega > 0} \iso {\SO_{2, 19} \over \SO_2\times \SO_{19}}\,(\RR)$$

Yields a [bounded symmetric domain](Hermitian%20symmetric%20domains.md) of Type IV and dimension 19:

![](attachments/2023-03-12-3.png)

# Lattice polarization

- Setup: $\mcx \to B$ a family of K3s with $M\leq \lkt$ a fixed even sublattice.
- Primitive embeddings $M\injects \lkt$ with $\sgn M = (1, \rho - 1)$.
- Existence of $v\in M$ whose image in $\Pic(X_t)$ is the class of a hyperplane section of a projective embedding $X_t\injects \PP^N$.
- Markigns $\rho: \lkt \iso H^2(X_t;\ZZ)$ which restrict to $j_t: M\injects \Pic(X_t)$.
- $T \da M^\perp$ in $\lkt$.
- $[\omega_t]\in \Omega_T$ its corresponding period line.yields a map $B\to \Omega_T$
- Change of markings $\Gamma_T \da \ts{s\in \Orth(\lkt) \st s(T) \subseteq T} \cong \ker(\Orth(T) \to\Orth(D_T))$ the discriminant group
- A quotient that is too big: $\dcosetl{\Gamma_T}{\Omega_T}$.
- The discriminant locus: points which don't correspond to periods.
	- For $r\in T$ with $r^2 = -2$, take the hyperplane $H_r \da\ts{x\in T\tensor \CC \st rx=0}$.
	- Toward a contradiction, suppose $x$ is a period point.
	- $rx=0\implies x$ is an algebraic cycle, so $\pm x$ is effective; wlog $+x$.
	- Since $x\in (\im j_t)^\perp$ and $\im j_t$ contains the class of a hyperplane section of an embedding: contradiction.
- So set $$\Delta_T \da \Union_{r\in T, r^2=-2} H_r$$
- Yields a coarse space of $T\dash$polarized K3s: $$\dcosetl{\Gamma_T}{\Omega_T\sm \Delta_T}$$

# Misc

- $\Omega_1 = \HH$
- $\Omega_2 = \HH\times \HH$
- $\Omega_2 = \Sieg_2$.
- If $M_d \da \qty{U\sumpower 2\oplus (-2d)}^\perp$.