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date: 2023-02-09 09:38
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# Background
## Lattices
### Even unimodular lattices
- Definition of a lattice in $V \in \mods{\RR}$?
- $\Lambda \subseteq V$ a discrete cocompact subgroup, or a free $\ZZ\dash$submodule of $\GG_a(V)$.
- Usually equipped with $\beta: \Sym^2 V\to R$.
- What is the dual of a lattice? ✅ 2023-02-07
- $L\dual \da \zmod(L\to \ZZ) \cong \ts{q\in L_\QQ \st (q, \Lambda) \subseteq \ZZ}$.
- What is a nondegenerate pairing on $\Lambda$?
- The linear extension of $\beta$ to $V = \Lambda\tensor_\ZZ \RR$ induces an isomorphism $V\iso V\dual$.
- Equivalently $\disc \Lambda \neq 0$.
- What is the signature of a lattice?
- $\sgn \Lambda = (n_+, n_-, n_0)$
- What is a hyperbolic lattice?
- Any lattice of signature $(1, n)$.
- What is $\disc(\Lambda)$ for a lattice $(\Lambda, \beta)$?
- $\disc(\Lambda) \da \det G(\beta)$ where $G(\beta)$ is the Gram matrix.
- Interpret as $\disc(\Lambda) = \covol(\Lambda)^2 \da \vol(V/\Lambda)^2$.
- When integral, $\disc(\Lambda ) = \size(\Lambda\dual/\Lambda)$.
- Can compute by taking $M$ a generator matrix for $\Lambda$, then $\disc \Lambda = M^tM = (\det M)^2$.
- What is a **unimodular** lattice?
- Integral with $\disc \Lambda = +1$.
- Equivalently, $\covol(\Lambda) \da \vol(V/\Lambda) = 1$.
- Equivalently, $\Lambda = \Lambda\dual$.
- Equivalently $D_\Lambda = \ts{0}$.
- More generally, $I\dash$modular if $\Lambda = I\Lambda\dual$.
- What is an even unimodular lattices?
- $\Lambda = \Lambda\dual$ and $Q(\Lambda) \subseteq R$, or equivalently $\beta(x,x)\in 2R$ for all $x\in \Lambda$.
- Equivalently the Gram matrix is in $\GL_n(\ZZ)$ with even diagonal entries.
- **Theorem (Rohklin)**
- If $X$ is a smooth closed spin 4-manifold, $H^2(X)$ is an even lattice of signature $(n_+, n_-)$ and $n_+ - n_- \equiv 0 \mod 16$.
- Converse: if $H^2(X)$ is even and $\pi_1 X = 1$, then $X$ is spin.
- Counterexample if $\pi_1 X\neq 1$: the Enriques surface has $\pi_1 X = C_2$ and $H^2 = E_8 \oplus H$ which has signature 8.
- **Theorem**: All even indefinite unimodular lattices of rank $n$ satisfy $n\equiv 0 \mod 8$ and $p-q \equiv 0 \mod 8$.
- If so, there is a unique one: $\Pi_{p, q}$: vectors $\tv{v_1,\cdots, v_{p+q}}$ where $v_i\in \ZZ$ or ${1\over 2}\ZZ$ and $\sum v_i$ is even.
- $\Pi_{8, 0} = E_8$.
- **Theorem**: classification of **even**, **unimodular**, **definite** integral lattices
- Definite means losing uniqueness, compared to indefinite.
- $\dim = 8: E_8$
- $\dim = 16: E_8\sumpower{2}, \Lambda_{16}$.
- $\dim = 24$: 24 Neimeier lattices,
- Including the Leech lattice: uniquely has no roots.
- All obtained as primitive sublattices of $\Pi_{1, 25} =U \oplus E_8(-1)\sumpower 3$ by specifying a primitive isotropic vector.
- Conway's group $\Orth(N)$ the isometries of the Leech lattice, contains the Mathieu group $M_{23}$
- $\dim = 32$: billions.
- $\dim = 40: > 10^{80}$??
- Why are 8 and 24 special?
- 8 is the smallest dimension of an even unimodular attice, namely $E_8$
- 24 is the smallest dimension of an even unimodular lattice with no roots (check $\theta$ functions)
- $0^2 + 1^2 + \cdot + 24^2 = 70^2$ and this is the only $n$ for which the sum of the first $n$ squares is a square.
- [ ] What is the discriminant group? Why care?
- [ ] $D_\Lambda \da \Lambda\dual/\Lambda$.
- [ ] $\Lambda$ not unimodular $\implies D_\Lambda \neq 0$.
- [ ] What is the discriminant form?
- [ ] For an even lattice, $$\begin{align*}q_\Lambda: D_\Lambda &\to \QQ/2\ZZ \\ x + \Lambda &\mapsto \beta(x,x) +2\ZZ\end{align*}$$
- [ ] Obtained by extending $\beta$ to $\Lambda\dual$ and then mapping to the quotient.
- [ ] What is the twist $\Lambda(n)$ of a lattice $\Lambda$?
- [ ] $\beta_{\Lambda(n)}(x,y) = n\beta_\Lambda(x, y)$.
- [ ] What is a 2-elementary lattice?
- [ ] $\Lambda$ with discriminant group $D_\Lambda\cong C_2^r$ for some $r$.
- [ ] What is a primitive sublattice?
- [ ] $R \leq \Lambda$ a co-torsionfree sublattice, ie $\Lambda/R$ is torsionfree.
- [ ] Yields a split SES $R\injects \Lambda \surjects \Lambda/R$ so $\Lambda \cong R \oplus \Lambda/R$, not necessarily orthogonal wrt $\beta$.
- [ ] Examples: any $R \leq \Lambda$ of the form $R = S^\perp$ for some $S\leq \Lambda$ is always primitive.
## Modular Forms
# K3s
## Definitions of K3s
- What is a lattice isometry?
- A map $f: (\Lambda_1, \beta_1) \to (\Lambda_2, \beta_2)$ where $\beta_2(f(x), f(y)) = \beta_1(x,y)$.
- Forms a group $\Orth(\Lambda)$.
- Nonsymplectic automorphisms
- For $G \leq \Aut(X)$ finite order $n \geq 2$, $G\actson H^{2, 0}(X) = \CC\omega_X$ gives a character and thus $$G_0 \injects G \surjects \mu_n$$
- Nonsymplectic if $G\neq G_0$, so $\alpha\neq 1$. Means $\sigma(\omega_X) = \zeta_n \omega_X$ for some primitive root of unity.
- Why care? Admitting such an automorphism $\implies$ projective automatically.
- Possibilities for orders are in $\ts{n \st \phi(n) \leq 20}\smts{60}$ for the totient function.
## Moduli
- 
- [x] What is the moduli space of all K3s vs projective K3s? ✅ 2023-01-28
- [ ] All K3s: $\dcosetr{\Omega}{\Orth(\lkt)}$, 20-dimensional.
- [ ] Projective: 19-dimensonal, since one has to vary periods in the orthogonal complement of an ample class.
## Degenerations
## Automorphisms of K3s
- Mukai's theorem
- The Mathieu group
- Symplect vs non-symplectic automorphisms
## Moduli of K3s
### Background: curves
### K3s