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created: 2023-03-29T11:30
updated: 2023-03-29T11:30
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# Questions

- Moduli of quasipolarized K3s: not separated?
- Moduli of polarized K3s: no fine space?
- How much of this theory goes through for an arbitrary Hodge structure of weight 2? E.g. replace $\NS(X)$ with $L^{1, 1}\intersect (L\tensor_\ZZ \CC)$ for $L\in \zmod$ a lattice with a HS, and $T \da \NS(X)^\perp$.
- Fibers vs stalks of a local system?
- Specialization and generization?
- Research questions/directions:
	- Semitoroidal compactifications of moduli of sheaves on K3s? With involution?
	- Torelli theorems for polarized K3s with an algebra of commuting automorphisms?

# Reminders of some basic facts

- $h_{\K3}(u, v) = \sum h^{p, q}u^p v^q = 1 + (u^2 + 20 uv + v^2) +u^2v^2$
- So $p_{\K3}(t) = \sum \beta_k t^k = h_{\K3}(t,t) = 1 + 22 t^2 + t^4$.
- $(H^2, \cupprod) \cong \E_8(-1)\sumpower{2} \oplus U\sumpower{3}$.
-  Lefschetz $(1,1) \implies \Pic(X) \cong H^2(X;\ZZ)\intersect H^{1, 1}(X)$.
- Regular morphisms: flat and regular pointwise, so minimal number of generators of $\mfm_x$ is equal to $\dim \OO_{X, x}$.s
- nef: non-negative degree on every curve.

# Moduli Setup

- Moduli of polarized K3s
	- Functor $\mcm_{2d}$, no fine space, coarse space $M_{2d}$.
	- Easy to show $M_{2d}$ exists as a separated algebraic space, harder to show exists as a quasiprojective space (Hilbert scheme or GIT)
	- Defining the moduli functor: or a Noetherian base $S$, $\mcm_{2d}: \opcat{\Sch\slice S^\ft}\to \Grpd$ sends $T$ to $( f:X\to T, L)$ with $f$ smooth proper and $L\in \Pic_{X/T}(T)$ such that
		- Every geometric base change yields $X_k$ a K3 with $L_{X_k}$ primitive ample and $L_{X_k}^2 = 2d$
		- Thus the fibers are polarized K3s of degree $2d$.
	- Note $\Pic_{X/T}(A) \da \Pic(X_T)/q^* \Pic(T)$ where $q: X_T\to T$.
	- $\mcm_{2d}$ is a separated DM stack, not smooth over $\ZZ$, smooth over $\ZZ\invert{2d}$.
- Construct as an open subvariety in the quasiprojective variety $\tilde \Orth(\Lambda_{2d})/\DD_{2d}$
  
  ## General BB
  
- Here $\DD = \ts{x\in \PP(\lkt\tensor \CC) \st x^2 = 0, \abs{x} > 0} \subseteq \PP(\lkt\tensor \CC)$ is a subset of a nondegenerate quadric in $\PP^{22}$.
	- $\dim(\lkt \tensor \CC) = h^2 = 1 + 22 + 1 = 24$.
- $\T_x \PP(\lkt \tensor \CC) = \Hom(\ell_x, \lkt \tensor \CC / \ell_x)$ where $\ell_x$ is the line corresponding to $x\in \lkt \tensor\CC$.
- Differentiate the $x^2 = 0$ condition to get $\T_x \DD = \Hom(\ell_x,\ell_x^\perp/\ell_x)$, maps $\ell_x\to \lkt/\ell_x$ with image orthogonal to $\ell_x$.
- Properties of $\DD$:
	- Special case of Griffiths period domains parameterizing polarized Hodge structures of arbitrary weight. 
	- $\pi_0 \DD = 1$ for $n_+ \geq 3$ and $\pi_0 \DD = C_2$ for $n_+ = 2$.
	- $\DD$ bijects with Hodge structures of K3 type on $\lkt$ where $\forall \sigma\in (\lkt)^{2, 0}$, one has $\sigma^2 = 0, \abs{\sigma} > 0, \sigma\in (\lkt^{1, 1})^\perp$.
- Alternate descriptions
	- **Grassmannian**/homogeneous/symmetric space description: $$\DD \iso \Gr^{po}_2(\lkt\tensor \RR)\iso \dcosetr{\Orth(n_+, n_-)}{\SO_2\times \Orth(n_+-2,\, n_-)}$$
		- Here $\Gr_2^{po}(\cdots)$ is the orientation double cover of $\Gr_2^p(\cdots)$ the planes where the restriction $\inp{\wait}\wait\mid_P$ is positive definite.
	- **Tube domain** description: $\lkt \tensor \RR = U_\RR \oplus W$ and define the tube domain $\mch \da \ts{z\in W\tensor \CC\st \Im(z)^2 > 0}$.
		- Get an isomorphism $\DD\iso \mch$ where $z\mapsto \tv{1: -z^2 : z\sqrt{2} }$.
- $\Orth(\lkt)\actson \DD$ is only nice for $n_+ = 2$, generally not Hausdorff otherwise.
	- Can have infinite stabilizers: if $\size \Aut(X) = \infty$ and the Hodge structure is determined by a point $x\in \DD$, then $\Stab_{\Orth(\lkt)}(x) \contains \Aut(X)$.
	- How to construct: an elliptic K3 with a non-torsion section
- Fix $n_+ = 2$ for the remaining discussion.
- **Prop**: Every arithmetic subgroup $\Gamma$ admits a finite index torsionfree normal subgroup $\Gamma'\leq \Gamma$.
	- **Arithmetic** subgroups: $\Gamma\leq \Orth(\lkt)$ of finite index, e.g. for $G\leq \GL_n(\QQ)$, $\Gamma\leq G(\QQ)$ is arithmetic $\iff$ commensurable with $G(\QQ) \intersect \GL_n(\ZZ)$.
	- $\Gamma_1, \Gamma_2$ **commensurable** if $\Gamma_1\intersect \Gamma_2$ has finite index.
	- Can realize $\Gamma' =\ts{g\in \Gamma \st g \equiv \id \mod \ell}$ for some large $\ell$, so $\Gamma' = \ker(\Gamma \to \Gamma \tensor_\ZZ \FF_p)$.
		- E.g. $\Gamma(p) \da \ts{M\in \SL_2(\ZZ) \st M\equiv \id \mod p} \leq \SL_2(\ZZ)$, so $\Gamma(p) = \ker(\SL_2(\ZZ) \to \SL_2(\FF_p))$.
- **Prop**: If $\Gamma \leq \Orth(\lkt)$ finite index and torsionfree $\implies \dcosetl{\Gamma}{\DD}$ is a complex manifold (free and properly discontinuous action).  Equivalently, $\DD \to \dcosetl{\Gamma}{\DD}$ is a covering space.
	- Yields an automorphic space $$\dcosetl{\Gamma}{\DD}\iso \dcoset{\Gamma}{\Orth(n_+, n_-)}{\SO_2\times \Orth(n_+-2,\, n_-)}$$
	- For $n_+ = 2$, $$\dcosetl{\Gamma}{\DD}\iso \dcoset{\Gamma}{\Orth(2, n_-)}{\SO_2\times \Orth(0,\, n_-)}$$
	- **Proof**: $\Gamma$ acts properly discontinuously for $n_+ = 2 \implies$ finite stabilizers.
	- $\Gamma$ torsionfree $\implies$ trivial stabilizers.
- **Thm (BB):** If $\Gamma\leq \Orth(\lkt)$ is torsionfree (dropping finite index) then $\dcosetl{\Gamma}{\DD}$ is a smooth quasiprojective variety.
	- If $\Gamma$ is not torsionfree, the quotient is a non-smooth normal quasiprojective variety: take $\Gamma' \leq \Gamma$ finite index torsionfree, then $\dcosetl{\Gamma'}{\DD} \to \dcosetl{\Gamma}{\DD}$ is a finite quotient.
	- *To ask*: do quasiprojective varieties always admit a compactification by taking the closure of the image...?

## Periods

- **Local Torelli**: $\Def(X) \injects \DD$ bijects onto its (open) image via the local period map where $\Def(X)$ is the universal deformation space.
	- Thus deformations of K3s are controlled by deformations of their Hodge structures.
- **Noether-Lefschetz** locus $\mathrm{NL} \subset \Def(X)$ are $X'$ with $\Pic(X')\neq 1$.
	- $\mathrm{NL}$ is dense and decomposes as $\mathrm{NL} = \Union_i \mathrm{NL}_i$ where $\codim \mathrm{NL}_i = 1$ (hyersurfaces) are smooth
- Isotrivial: not all fibers isomorphic.
- By the monodromy representation, $$\RR^2 f_* \ul \ZZ\mapstofrom \Rep(\pi_1(S, 0), \GL\qty{ H^2(X_0; \ZZ)) } \mapstofrom \Bun^\flat_{\GL}(S)$$ 
	- The former has "fibers" $(H^2(X_t; \ZZ)$?
	- If $\pi_1 S = 1$ then $\RR^2 f_* \ul \ZZ =\ul{V}$ canonically where $V\da H^2(X_0; \ZZ)$ is the stalk.
	- The flat vector bundle is $\RR^2 f_* \ul \ZZ \tensor_{\ZZ} \OO_S$ has fibers $H^2(X_t; \CC)$.
- **Prop (VHS)**: $f_* \Omega^2{X/S}\injects \RR^2 f_* \ul \ZZ \tensor_{\ZZ} \OO_S$ which fiberwise restricts to $H^{2, 0}(X_t) \injects H^2(X_t; \CC)$.
- Constructing the period map:
	- Let $E\leq \OO_S\sumpower{N+1}$ by a holomorphic rank $r$ sub-bundle over $S$
	- Classified by a map $\phi_E: S\to \Gr_r(\AA^{N+1})$.
	- For $r=1$, $\phi_E: S\to \PP^N$ and $E\cong \phi_E^* \OO_{\PP^N}(-1)$ where $\OO_{\PP^N}(-1) \leq \OO\sumpower{N+1}$.
	- Explicitly $\phi_E(t)$ is the line given by the fiber $E(t) \subset \OO\sumpower{N+1}(t) \cong \AA^{N+1}$.
	- For $\pi_1(S) = 1$ and after fixing a marking $\psi: H^2(X_t; \ZZ) \iso \lkt$, $$\begin{align*}P: S &\to \PP(\lkt\tensor \CC) \\ t &\mapsto [\psi(H^{2, 0}(X_t))]\end{align*}$$
	- This factors through $\DD \injects \PP(\lkt \tensor \CC)$ since $\int \omega\wedge \omega = 0$ and $\int \omega\wedge \bar\omega > 0$.
- **Theorem: Griffiths Tranversality**:
	- Write $\ell \da H^{2, 0}(X_0)$, then $$dP:\T_0 S\to\T_{P(0)} \DD \iso\Hom(\ell, \ell^\perp/\ell)$$
	- Can decompose $$dP = \T_0 S \mapsvia{\mathrm{KS}} H^1(X_0, \T_{X_0})\isovia{\text{contract } \omega_{X_0} } H^1(X_0; \Omega_{X_0}) \iso H^{2, 0}(X)^\perp/H^{2, 0}(X)$$
	- Here $\mathrm{KS}$ comes from: dualize $f^* \Omega_S\injects \Omega_X \surjects \Omega_{X/S}$ to get $\T_{X_0} \injects \T_{X}\mid_{X_0} \surjects f^* \T_S \mid_{X_0}$ and take the boundary map, using $f^* \T_S\mid_{X_0} = \T_0 S\tensor_\CC \OO_S$.
- General GT: $\nabla(\Fil^p)\subset \Fil^{p-1}\tensor \Omega_S$.

# Lefschetz Data

- Abstract setup:
	- $k$ a number field
	- $H\in \grmods{k}$ which vanishes in odd degree.
	- $\beta\in \gr\Sym^2(H)$, so $\beta(\wait, \wait): H\tensor_k H\to k$ with $\beta(H^i, H^j) = 0$ when $i\neq j$.
	- $V\in \mods{k}$ with $\alpha: \gr\Sym^*(V[-2])\to \Endo_{\grmods{k}}(H)$, i.e. commuting endomorphism $\alpha(v): H\to H[-2]$ depending linearly on $v\in V$; satisfying $\beta(\alpha(v)(h_1), h_2) = \beta(h_1,\alpha(v)(h_2))$.
	- An open convex cone $V^\amp \subseteq V$.
	- Hard Lefschetz: $v\in V^\amp \implies \alpha(v)^i: H^{-i} \iso H^i$ for all $i$.
	- Primitive subspaces: $P_v^{-i}\da \ker\qty{H^i \mapsvia{\alpha(v)^{i+1}} H^{i+2}} \leq H^{-i}$.
	- An isomorphism of $k[v]\dash$modules $\bigoplus_{i\geq 0} k[v]/v^{i+1}\tensor_k P_v^{-i}\iso H$.

- For varieties
	- $\dim X = n, H\subseteq \PP^N$ a generic hyperplane $\implies \omega \da X\intersect H \in A^2(X)$ is codimension 2 $\implies \omega\in H^2(X; \CC)$ the class of a hyperplane section.
	- Theorem: $\omega^{n-i}: H^i \iso H^{2n-i}$.
	- $\implies \ts{\beta_{2i}}, \ts{\beta_{2i+1}}$ are symmetric and unimodal.
- Alternative grading convention?
	- $L(\wait) \da (\wait)\cupprod \omega$ for $\omega$ a Kahler class.
	- $P^i(X) \da \ker\qty{H^i \mapsvia{L^{n-i+1}} H^{2n-i+2}}$ 
	- $L^i: H^{n-i} \iso H^{n+i}$, so $\beta_{n+i} = \beta_{n-i}$.
	- $H^i = \bigoplus_{j=0}^{i/2} L^j P^{i-2j}(X)$.s
	- $\beta_{i-2}\leq \beta_i$  since $L: H^{i-2} \injects H^i$.
- Hodge-Riemann bilinear relations:
	- $Q\in \Sym^2 H^i(X; \CC)$ where $$Q(\alpha\tensor \beta) = (-1)^{i(i-1)\over 2}\int_X \alpha\cupprod \beta\cupprod \omega^{n-i}$$
	- $\alpha\in P^{p+q} \intersect H^{p, q}\implies \sqrt{-1}^{p-q} Q(\alpha\tensor \bar\alpha) > 0$
	- $\tilde Q(a\tensor b)\da Q(a\tensor C b)$ defines a positive definite Hermitian form on $P^i(X)$ where $C: H^i\selfmap$ acts by $\cdot \sqrt{-1}^{p-q}$ on $H^{p, q}$.
- How to prove Hard Lefschetz:
	- Check $[\Lambda, L] = H, [H, L] = -2 L, [H, \Lambda] = 2\Lambda$
	- Check that if $V\in \mods{\liesl_2(\CC)}$ then $H$ is diagonalizable with integer eigenvalues, $L^i: V_\alpha \iso V_{-\alpha}$ induces an isomorphism on weight spaces, $V\cong \bigoplus_{i\geq 0} L^i \ker(\Lambda)$, and $L^{i+1}(\ker(\Lambda) \intersect V_i) = 0$.
	- Use $\liesl_2(\CC)$ modules are semisimple, so direct sum of simples, and there is a unique simple module of dimension $N+1$ for every $N\geq 0$: $\Sym^n(\CC^2)$, which decomposes into weight spaces: ![](attachments/2023-02-20weightspace.png)