--- created: 2023-04-17T14:46 updated: 2024-09-02T16:31 aliases: - K3 - K3s - K3 surface - K3 surfaces --- # K3 Surfaces - [[hyperkahler]] manifolds: higher dimensional generalizations of K3s - Interesting questions: $\Aut(X)$ and rigidity of surface with fixed configurations of curves # Moduli - $F_{2d}$: primitively[^1] polarized K3s. - $F_{2d}$ is an open subspace of the [Shimura variety](Unsorted/Shimura%20variety.md) associated to $\SO_{2, 19}$. - $\mcm_{2d}$: polarized K3s # Facts - $\Pic(X)$ is torsionfree. - If $M$ is torsion, check $h^{0}(S, M)-h^{1}(S, M)+h^{0}\left(S, M^{-1}\right)=\chi(S, M)=\chi\left(S, \mathscr{O}_{S}\right)=2$, so either $M$ or $M\inv$ has sections. But if $M^{\otimes N}$ is trivial, then there is a nonzero section $s^N$ vanishing nowhere, so $s\in H^0(M)$ vanishes nowhere, so $M$ is trivial. - $c_2(X) = 24$ and $\beta_2(X) = 22$ by Noether's formula $12\chi(\OO_X) = c_1^2 + c_2 = c_2$ and $\chi(\OO_X) = 2$ by the above. - $c_2 = \chi(X) = \beta_2 + 2$. - The index of $\lkt$ is given by Hirzebruch's formula $\tau(S)=\frac{1}{3}\left(c_{1}^{2}(S)-2 c_{2}(S)\right)=-16$ - $\chi(\K3, \OO_{\K3}) = 1 + (1 + 20 + 1) + 1 = 24 = \dim_\CC H^*(X; \CC)$. - ![](attachments/2023-03-12Hodge.png) - Full $H^2$: - $H^2(X; \ZZ) = U\sumpower 3 \oplus E_8(-1)\sumpower 2 \da \lkt$ - $\sgn H^2(X; \ZZ) = (3,19)$. - $\Pic$: - $\Pic(X) \cong \NS(X) \cong H^2(X;\ZZ)_\alg \da S_X = H^2(X;\ZZ)\intersect H^{1, 1}(X) = \gens{\omega + \bar\omega}^\perp \intersect \lkt$ varies algebraically with $\rho \da \rank_\ZZ \Pic(X) \in \ts{0, \cdots, 20}$ with all cases occurring. - $T_X \da S_X^\perp$. - $\sgn S_X = (1, \rho - 1)$, not generally unimodular. - $\sgn T_X = (2, 20-\rho)$ - $L_d$: the lattice $v^\perp\subset \lkt$ where $v\in H^2(X;\ZZ)_\alg$ is associated to $\mcl$ a polarization of degree $2d$. - Period domains: - $\rank_\ZZ \lkt =22 \implies \dim \PP(\lkt\tensor\CC) = 21$ and $\dim \PP(L_d\tensor \CC) = 20$ - The most general period domain: $$\Omega(\lkt) \da \ts{x\in \PP(\lkt\tensor \CC) \st x^2 = 0, \,\,\norm x > 0} \subset \PP(\lkt \tensor \CC) \cong \PP(\CC^{22}) \cong \PP^{21}$$ but $$\mcm^{\mathrm{naive} }_{\K3}\da \dcosetl{\Orth(\lkt)}{\Omega(\lkt)}$$is not separated. - For $F_{2d}$, get a nicer stack: set $$\Lambda_{2d}\da c_1(\mcl)^\perp = E_8(-1)\sumpower 2 \oplus U\sumpower 2 \oplus \gens{2-2g}$$ where $c_1(\mcl)^2 = 2g-2$ and $\sgn \Lambda_{2d} = (2,19)$. - The period domain for $F_{2d}$: $$\Omega(\Lambda_{2d})\da \Omega(\lkt)\intersect \PP(\Lambda_{2d}) = \ts{x\in \PP(\Lambda_{2d}\tensor \CC) \st x^2 = 0,\,\norm x > 0} \subset \PP(\CC^{21}) \cong \PP^{20}$$ - Moduli space $$F_{2d} \da \dcosetl{\Orth(\Lambda_{2d})}{ \Omega(\Lambda_{2d})}$$ - General $X\in F_{2d}$ has $\Pic(X) = \gens{c_1(\mcl)}_\ZZ$. - $\omega\in H^{2, 0}(X)$ determines the Hodge structure and isomorphism class by Torelli. - Degenerations: - **Type I**: Smooth. - **Type II**: $\mcx_0 = R_1 \union E_1 \union E_2 \cdots \union E_n \union R_2$ with $R_i$ rational and $E_i$ elliptic ruled, glued by (possibly nodal) elliptic curves ![](attachments/2023-03-13kulikovii.png) - **Type III**: $\mcx_0 = \Union_i V_i$ with each $V_i$ a rational surface and each $V_{ij}$ a cycle of (at worst nodal) rational curves (or empty). # Definitions - **Definition**: Let $k$ be a field. A non-singular, proper surface $X$ over $k$ is called a K3 surface if $\Omega_{X / k}^2 \cong \mathcal{O}_X$ and $H^1\left(X, \mathcal{O}_X\right)=0$. - Such a K3 surface is automatically projective - Defining condition: trivial [canonical bundle](canonical%20bundle.md) - Special case ($\dim_\CC = 2$) of [Calabi-Yau](Calabi-Yau.md) manifolds - See [polarization](Unsorted/polarization.md); if $2d = L^2$ then $h_L(t) = dt^2 + 2$. - **Theorem**: $L\in \Pic(\K3)^\amp$ then... - $L$ effective, $\tau_{\geq 1} H^*(L) = 0$, $L^{\geq 2}$ is globally generated, and $L^{\geq 3}$ is very ample. ![](attachments/Pasted%20image%2020220806205030.png) ![](attachments/Pasted%20image%2020220806225013.png) ![](attachments/Pasted%20image%2020220806225042.png) ![](attachments/Pasted%20image%2020220806225128.png) Relation to [formal groups](Unsorted/Formal%20group.md): ![](attachments/Pasted%20image%2020220806225947.png) ![](attachments/Pasted%20image%2020220807160042.png) To compute these Hodge numbers, see See [Hodge structure](Unsorted/pure%20Hodge%20structure.md) # Examples - Sextics: - Let $S$ be a non-singular sextic curve in $\mathbb{P}_k^2$ where $k$ is a field and consider a double cover i.e., a finite generically étale morphism, $\pi: X \rightarrow \mathbb{P}_k^2$ which is ramified along $S$. Then $X$ is a K3 surface. - Complete intersections: - Let $X$ be a smooth surface which is a complete intersection of $n$ hypersurfaces of degree $d_1, \ldots, d_n$ in $\mathbb{P}^{n+2}$ over a field $k$. The adjunction formula shows that $\Omega_{X / k}^2 \cong \mathcal{O}_X\left(d_1+\cdots+d_n-n-3\right)$. So a necessary condition for $X$ to be a K3 surface is $d_1+\cdots+d_n=n+3$. The first three possibilities are: $$\begin{array}{cc} n=1 & d_1=4 \\ n=2 & d_1=2, d_2=3 \\ n=3 & d_1=d_2=d_3=2 \end{array}$$ - For a complete intersection $M$ of dimension $n$ one has that $H^i\left(M, \mathcal{O}_M(m)\right)=0$ for all $m \in \mathbb{Z}$ and $1 \leq i \leq n-1$. Hence in those three cases we have $H^1\left(X, \mathcal{O}_X\right)=0$ and therefore $X$ is a $\mathrm{K} 3$ surface. - Kummers: - Let $A$ be an abelian surface over a field $k$ of characteristic different from 2. Let $A[2]$ be the kernel of the multiplication by-2-map, let $\pi: \tilde{A} \rightarrow A$ be the blow-up of $A[2]$ and let $\tilde{E}$ be the exceptional divisor. - The automorphism $[-1]_A$ lifts to an involution $[-1]_{\tilde{A}}$ on $\tilde{A}$. Let $X$ be the quotient variety of $\tilde{A}$ by the group of automorphisms $\left\{\operatorname{id}_{\tilde{A}},[-1]_{\tilde{A}}\right\}$ and denote by $\iota: \tilde{A} \rightarrow X$ the quotient morphism. It is a finite map of degree 2. - We have the following diagram of morphisms over $k$; the variety $X$ is a K3 surface and it is called the **Kummer surface** associated to $A$: ![](attachments/2023-03-10diag.png) ![](attachments/Pasted%20image%2020220806205056.png) ![](attachments/Pasted%20image%2020220806205358.png) ![](attachments/Pasted%20image%2020210627224900.png) ![](attachments/Pasted%20image%2020210627224917.png) ![](attachments/Pasted%20image%2020210630223310.png) ![](attachments/Pasted%20image%2020220319203907.png) ![](attachments/Pasted%20image%2020220325220758.png) ![](attachments/Pasted%20image%2020220806215941.png) ![](attachments/Pasted%20image%2020220806220056.png) # Taubes ![](attachments/Pasted%20image%2020220325220934.png) ![](attachments/Pasted%20image%2020220325220949.png) # Misc ![](attachments/Pasted%20image%2020220410153834.png) ![](attachments/Pasted%20image%2020220411221850.png) ![](attachments/Pasted%20image%2020220411221910.png) # Examples The [Kummer surface](Kummer%20surface): ![](attachments/Pasted%20image%2020220411221942.png) The [Fermat quartic](Unsorted/Fermat%20quartic.md): ![](attachments/Pasted%20image%2020220411221951.png) ![](attachments/Pasted%20image%2020220411222015.png) ![](attachments/Pasted%20image%2020220523003700.png) ![](attachments/Pasted%20image%2020220807000006.png) # Pencils ![](attachments/Pasted%20image%2020220516171318.png) # Neron-Severi ![](attachments/Pasted%20image%2020220807000615.png) # Weyl group ![](attachments/Pasted%20image%2020220807000804.png) ![](attachments/Pasted%20image%2020220807162647.png) ![](attachments/Pasted%20image%2020220807162702.png) # Kahler cone ![](attachments/Pasted%20image%2020220807000846.png) # Cohomology ![](attachments/Pasted%20image%2020220807154205.png) ![](attachments/Pasted%20image%2020220807154317.png) # Picard ![](attachments/Pasted%20image%2020220807154354.png) #open/problems ![](attachments/Pasted%20image%2020220807171346.png) ## Neron-Severi Using [crystalline cohomology](Unsorted/crystalline%20cohomology.md): ![](attachments/Pasted%20image%2020220807154524.png) # Polarization ![](attachments/Pasted%20image%2020220807154746.png) ![](attachments/Pasted%20image%2020220807161813.png) # The DM Stack ![](attachments/Pasted%20image%2020220807154940.png) Deformations: ![](attachments/Pasted%20image%2020220807155048.png) ![](attachments/Pasted%20image%2020220807155206.png) ![](attachments/Pasted%20image%2020220807155217.png) ## Using periods ![](attachments/Pasted%20image%2020220807155359.png) See [level structure](Unsorted/level%20structure.md) ![](attachments/Pasted%20image%2020220807155637.png) [Torelli](Unsorted/Torelli.md) map: ![](attachments/Pasted%20image%2020220807162010.png) ## As a complex manifold ![](attachments/Pasted%20image%2020220807165231.png) ![](attachments/Pasted%20image%2020220807165304.png) # Stability ![](attachments/Pasted%20image%2020220807161746.png) # Automorphic forms ![](attachments/Pasted%20image%2020220807162056.png) # Heights See [F-crystal](Unsorted/Hodge%20F-crystal.md). ![](attachments/Pasted%20image%2020220807165855.png) # Lifts ![](attachments/Pasted%20image%2020220807171017.png) # Crystalline cohomology ![](attachments/Pasted%20image%2020220807171144.png) # Torsion ![](attachments/2023-02-25torsion.png) # Mirror lattices ![](attachments/2023-02-26mirror1.png) ![](attachments/2023-02-26mirror2.png) # Computations ![](attachments/2023-03-05comp.png) [^1]: Primitive: not a positive power of another line bundle.