---
title: Geometry and Moduli of K3s
aliases:
  - Geometry and Moduli of K3s
annotation-target: https://arxiv.org/pdf/1501.04049.pdf
flashcard: Research::K3s
created: 2023-01-17T18:09
updated: 2024-01-05T21:48
---

# Geometry and Moduli of K3s

- [x] Discuss $F_g$. ✅ 2023-01-17
	- [ ] Irreducible coarse moduli space of polarized complex K3s of genus $g\geq 2$.
	- [ ] An open subset of a Shimura variety for $\SO_{2, 19}(\RR)$
	- [ ] Quasiprojective, $\dim F_g = 19$ for each $g$.
- [x] What is $\kappa(X)$ for $X$ a K3? ✅ 2023-01-17
	- [ ] $\kappa(X) = 0$, one of four such classes of minimal surfaces.
- [x] Where do K3s fit into the classification of surfaces? ✅ 2023-01-17
	- [ ] Positively curved: del Pezzo
	- [ ] Flat: K3s (generally CYs and hyperkahlers)
	- [ ] Negatively curved: general type.
- [x] What is the Hodge diamond of a K3? ✅ 2023-01-17
	- [ ] ![](attachments/2023-01-17-hodgeK3.png)
- [x] What are $q(X), p_g(X), \beta^+(X), \chi(X), c_1^2(X), c_2(X)$ for a K3? ✅ 2023-01-17
	- [ ] $q(X) = 0$
	- [ ] $p_g(X) = 1$
	- [ ] $\beta^+(X) = 3$
	- [ ] $\chi(X) = 2$
	- [ ] $c_1^2(X) = 0$
	- [ ] $c^2(X) =24$.
- [x] Describe the lattice $\Lambda_{\K3}$. ✅ 2023-01-17
	- [ ] $\Lambda_{\K3} \da H\sumpower{3} \oplus E_8\sumpower{2}$, nondegenerate, even, rank 22, signature $(3, 19)$.
- [x] Define $\NS(X)$. ✅ 2023-01-17
	- [ ] $$\NS(X) \da H^{1, 1}(X) \intersect H^{2}(X; \ZZ)\subseteq H^2(X; \CC)$$.
- [x] What is the transcendental sublattice? ✅ 2023-01-17
	- [ ] The smallest sublattice $T(X)$ of $H^2(X; \ZZ)$ containing a generator of $H^{2, 0}(X)$.
	- [ ] If $X$ is projective, $\NS(X)$ is nondegenerate, and $T(X) = \NS(X)^\perp$.
- [x] What is a marking on a K3? ✅ 2023-01-17
	- [ ] $\phi: H^2(X; \ZZ)\iso \Lambda_{\K3}$ an isometry.
- [x] Define the period domain $\DD$. What is its dimension? ✅ 2023-01-17
	- [ ] $$\DD \da \ts{[\omega]\in \PP(\Lambda_{\K3}\tensor_\ZZ \CC) \st \omega^2 = 0,\,\, \omega.\bar\omega > 0}$$
	- [ ] 20 dimensional $\CC\dash$manifold.
	- [ ] For a polarized K3 $(X, \phi)$, drops 1 dimension: $$\DD_{2k} \da \ts{[\omega]\in \DD \st \omega.h = 0}$$ where $h\in \Lambda_{\K3}$ is a fixed primitive class where $\phi\inv(h)\in \NS(X)^\amp$. This is a bounded symmetric domain of type IV.
- [x] Describe the period map of a single K3. ✅ 2023-01-17
	- [ ] Pick $\omega\in H^{2, 0}(X) \subseteq H^2(X;\CC)$ a nonvanishing holomorphic 2-form.
	- [ ] Check the cup product satisfies $\omega^2 = 0$ and $\omega.\bar\omega > 0$.
	- [ ] Extend a marking $\phi$ to $\phi_\CC: H^2(X;\CC)\to \Lambda_{\K3}\tensor_\ZZ \CC$.
	- [ ] Check $\im \phi_\CC = \spanof_\CC \ts{\phi_\CC(\omega)}$ is a line through the origin
	- [ ] Thus $\im \phi \in \DD$.
- [x] Why is the period domain used in moduli of K3s? ✅ 2023-01-17
	- [ ] Two K3s are isomorphic iff they admit markings defining the same point in $\DD$ by weak Torelli.
	- [ ] Weak Torelli: $X\iso X' \iff$ there is an isometry $H^2(X;\ZZ)\iso H^2(X';\ZZ)$ whose $\CC\dash$linear extension preserves Hodge decompositions on $H^2(X;\CC)$.
- [x] Describe the period map of a family of K3s. ✅ 2023-01-17
	- [ ] Let $f:\mcx \to B$ be a flat family with fiber $X$.
	- [ ] Choosing a marking $\phi: H^2(X;\ZZ)\iso \Lambda_{\K3}$ extends to a sheaf morphism $\phi_B: \RR^2 f_*\ul{\ZZ}\to\ul{\Lambda_{\K3}}$ where the latter is a constant sheaf.
	- [ ] Yields a holomorphic map $B\to \DD$.
- [x] How is $\DD$ related to moduli of K3s? ✅ 2023-01-17
	- [ ] Local Torelli: for a marked K3 $(X, \phi)$, the period map from the versal deformation space to $\DD$ is a local isomorphism.
	- [ ] Surjectivity: every $d\in \DD$ is a period point for some marked K3.
	- [ ] Thus almost a coarse moduli space: familes $\mcx\to B$ yield $B\to \DD$ and points correspond to iso classes, but quotienting by change-of-marking yields a non-Hausdorff quotient since it does not act properly discontinuously.
- [x] What are the two most widely studied classes of K3s? ✅ 2023-01-17
	- [ ] Sextic double planes: birational to $X\to \PP^1$ a degree 2 cover branched over a sextic curve (equivalently a sextic hypersurface in $\WP(1,1,1,3)$.)
	- [ ] Quartic hypersurfaces: 
		- [ ] $X\subseteq \PP^3$ a quartic hypersurface
		- [ ] $X\to \PP^1 \times \PP^1$ a double cover branched over a bidegree $(4, 4)$ curve.
		- [ ] $X\to C$ an elliptic fibration over the twisted cubic.
- [x] What is a polarized K3 of degree $2k$? ✅ 2023-01-17
	- [ ] A pair $(X, h \in \NS(X)^\amp)$ with $h$ primitive and $h^2 = 2k$.
- [x] What is the strong Torellli theorem for polarized K3s? ✅ 2023-01-17
	- [ ] Every Hodge isometry $\varphi: H^2(X;\ZZ)\iso H^2(X';\ZZ)$ is induced by a unique isomorphism $f:X\iso X'$ with $\phi = f^*$.
- [x] Discuss degenerations of K3 surfaces? ✅ 2023-01-17
	- [ ] Any degeneration can be converted to a Kulikov model after a base change and birational modification.
- [x] What is a Kulikov model? ✅ 2023-01-17
	- [ ] A semistable degeneration $\pi: \mcx\to \Delta$ with $\omega_{\mcx}\cong \OO_{\mcx}$.
- [x] What is a semistable degeneration? ✅ 2023-01-17
	- [ ] $\pi:\mcx \to \Delta$ with $\mcx$ smooth and central fiber reduced normal crossings.
- [x] What is the dual graph of a degeneration? ✅ 2023-01-17
	- [ ] Let $S_0=\bigcup V_i$ be the central fibre in a semistable degeneration. Define the dual graph $\Gamma$ of $S_0$ as follows: $\Gamma$ is a simplicial complex whose vertices $P_1, \ldots, P_r$ correspond to the components $V_1, \ldots, V_r$ of $S_0$; the $k$-simplex $\left\langle P_{i_0}, \ldots, P_{i_k}\right\rangle$ belongs to $\Gamma$ if and only if $V_{i_0} \cap \cdots \cap V_{i_k} \neq \emptyset$.
- [x] What is the classification of Kulikov models? ✅ 2023-01-25
	- [ ] Write $\pi: \mcx \to \Delta$ for a semistable degeneration with $\omega_{\mcx}\cong \OO_{\mcx}$ where all components of $\mcx_0$ are Kahler.
		- [ ] Type I: $\mcx_0$ is a smooth K3.
			- [ ] No double curves, no triple points.
		- [ ] Type II: $\mcx_0$ is a chain of elliptic ruled components with rational surfaces at each end where all double curves are smooth elliptic
			- [ ] Double curves but no triple points
		- [ ] Type III: $\mcx_0$ is rational surfaces meeting along rational curves forming cycles in each component and $\Gamma$ is a triangulation of $S^2$.
			- [ ] Triple points.
- [x] What is a Kummer surface? ✅ 2023-01-25
	- [ ] Suppose that $A$ is an abelian surface. The involution $\iota: A \rightarrow A$ given by $\iota(x)=-x$ (defined using the group law on $A$ ) has sixteen fixed points. If we quotient $A$ by this involution, we obtain a projective surface $\bar{S}$ with trivial dualizing sheaf and sixteen singularities of type $A_1$. Each of these singularities may be resolved by blowing up once, giving sixteen disjoint exceptional $(-2)$-curves $E_1, \ldots, E_{16}$. The resolution is a projective K3 surface $S$, called the Kummer surface associated to A.