--- created: 2023-03-26T11:58 updated: 2023-03-29T14:03 --- **Main problem**: For each cusp of the Baily-Borel compactification $\mcm^{\mathrm{BB}}$ (plus extra data), describe surfaces that appear as the central fibers $\mcx_0$ of Kulikov models (adapted to $R$ for stable pairs $(X, R)$), # References - Festi's talk at Banff: - Festi's paper: - Brandhorst-Hoffman, **Finite subgroups of automorphisms of K3s**: - **They might have solved the problem already, talk to Valery about this!** - They include code and a database: - Nikulin: - Morrison compactifications inspired by mirro symmetry: - Valery's papers: - Compact moduli of K3: - Mirror symmmetric nonsymplectic involution: - KSBA stuff: - Laza KSBA deg 2 K3 pairs - Luca KSBA purely nonsymp order 4 # Talk Notes: Dino Festi: K3 surfaces with two involutions and low Picard number - Motivation: find new examples of IHS manifolds. 2 big families and 2 sporadic examples. - Construction due to Joyce uses complex projective K3 surfaces: needs an anti-holomorphic involution $\sigma$ and a holomorphic involution $\iota$, plus a slope-stable line bundle. - Example of an anitholomorphic involution: $\sigma=$complex conjugation. Available when $X$ is defined over $\RR$. - If $\iota$ is defined over $\RR$ then $[\sigma \iota] = 1$ automatically. - Checking stability for $L$ is harder with higher Picard number. - If $X\slice \CC$ with $\Pic(X) = \gens{h}$ then $\Aut(X) = C_2 \iff h^2 = 2$ and is trivial otherwise. - Set $F = \Fix(\iota)$ for $\iota\in \Aut(X)$, then if $\iota$ is symplectic then $\Pic(X) \geq 9$, and if $\iota$ is nonsymplectic then either - $F = \emptyset$ and $\rho(X) \geq 10$ - $F = E_1\disjoint E_2$ for $E_i$ elliptic curves and $\rho(X) \geq 9$. - $F = C\disjoint E_1 \disjoint \cdots E_k$ with $E_i$ rational and $\rho(X) \geq 11- p_a(C) - k$. - Prop: if $\sgn N = (1,\rho-1)$ with $\rho \leq 10$ then $\exists X\slice \RR$ K3 with $\Pic X = \Pic X_\CC = N$. - Seemed to be known to Nikulin. - If $X$ is defined over $\RR$ then $\exists \tau$ with $\tau^*: H^2(X; \CC)\selfmap$ acts as complex conjugation: - $H^{2, 0} \mapstofrom H^{0, 2}$ - The restriction $\tau^*: H^2(X;\ZZ)\selfmap$ is an isometry - $A\mapsto -A$ for $A$ the ample cone. - $P\mapsto -P$ for $P$ the positive cone. # Valery's Paper Setup - $\bar F_S$ can be understood by looking at hyperbolic lattices $\bar T$ at $n\dash$cusps. - $V \da H^2(X; \ZZ)$ yields $V = V^+ \oplus V^-$ - Where $S \da V^+$, $\sgn S = (1, r)$, so hyperbolic - Yields $T \da V^-$, $\sgn T = (2, 20-r)$. - So $T = S^\perp$. - Get $A_S$ and $A_T$, $\iota$ acts by $+1$ on $A_S$ and $-1$ on $A_T$. - $\implies A_S = C_2^n$ for some $n$ $\implies A_S$ is $2\dash$elementary, classified by Nikulin (75 types). - For $F_S$: - Construct period domain $D_S = \ts{x\in \PP(S^\perp \tensor \CC ) \st x^2 = 0, \abs{x} > 0}$, note $S^\perp = T$. - Define $\Orth^*(S^\perp) \da \ts{\gamma\in \Orth(\lkt) \st \ro{\gamma}{S} = \id}$ - Coarse space $F_S \da \DD_S/\Orth^*(T)$; ADE singular surfaces $X$ with primitive embeddings $S\embeds \Pic X$. - $$\bd \overline{\DD_S /\Orth(S^\perp) }^{\mathrm{BB}} = \Disjoint_i L_i \Disjoint_j P_j$$ with $L_i$ corresponding to isotropic lines $\gens{e}_\ZZ \leq T$ and $P_j$ to isotropic planes $\gens{p_1, p_2}_\ZZ \leq T$. - Apply techniques of finding (orbits of) isotropic vectors in 2-elementary discriminant groups $A_T$. - At most 3 orbits (including zero) of isotropic vectors in $A_T$. - Set $\bar T \da e^\perp/e$ for $e\in T$ primitive isotropic, then $T = H \oplus \bar T$ where $e\in H$. - **Prop 5.5**: $e$ exists iff $\exists \bar T$ where if $\sgn T = (n_+, n_-)$ and $T\sim (r_T, a_T, \delta_T)$, $\sgn \bar T = (n_+ - 1, n_- - 1)$ and $\bar T\sim (r_{\bar T}, a_{\bar T}. \delta_{\bar T})$ where $r_{\bar T} = r_T - 2$, and either - $\delta_{\bar T} = \delta_T$ and - $a_{\bar T } = a_T$ - $a_{\bar T} = a_T - 2$ - $\delta_{\bar T} = 0$ and $\delta_T = 1$ and $a_{\bar T} = a_T - 2$. - Proof: study $\div(e)$ and $e^* \da e/\div(e)$. - **Bijection between mirror moves and primitive isotropic vectors:** - Given $e$, use $S\to \bar{T} \da e^\perp/e$, since $\sgn(S) = (1, r) \implies \sgn \bar T = (1, ?)$ is again hyperbolic. - Given an allowable mirror move, get existence of an isotropic vector $e$ by prop 5.5. - Mirror moves: $S\leadsto S^\perp \da T \ni e\leadsto \bar T\da e^\perp/e$. ## Our Setup - $G = \gens{\sigma, \iota}$ with $\sigma$ symplectic and $\iota$ non-symplectic - $G\actson V\da H^2(X; \ZZ)$ yields $V = V^{++} \oplus V^{+-} \oplus V^{-+} \oplus V^{--}$. - Set $S= V^{++}$ and $T = S^\perp$? # From new paper Recall if $G\actson X$ and $H\leq G$, - $\Stab_G(X) = \ts{x\in X \st gx = x}$. - $N_G(H) = \ts{g\in G \st [gH] = 1}$. - $Z_G(H) = \ts{g\in G \st [gh]=1 \,\, \forall h\in H}$ (pointwise, stronger). - The natural representation: - For $X \in \K3$, define the natural faithful representation $\rho_X: \Aut(X) \to \Orth(H^2(X;\ZZ))$ by $g\mapsto (g\inv)^*$. - Conjugate: - $(X, G)$ is conjugate to $(X', G')$ iff $\exists \phi: X\to X'$ with $\phi G\phi\inv = G'$. - Deformation equivalent: - $(X, G)$ is deformation equivalent to $(X', G')$ iff $\exist \mcx \to B$ with $\mcg\in \Aut(\mcx/B)$ with $\ro{(\mcx, \mcg)}{b_1}$ conjugate to $\ro{(\mcx, \mcg)}{b_2}$ for some $b_i\in B$. - $L\dash$marking: - $\eta: H^2(X;\ZZ)\to L$ with $\eta$ an isometry - $(X, \eta)$ is an $L\dash$marked surface. - $H\dash$marked surfaces: - Let $H\leq \Orth(L)$ be a finite subgroup. - $(X, \eta, G)$ is $H\dash$marked if $(X, \eta)$ is $L\dash$marked and $\eta \rho_X(G) \eta\inv = H$. - Effective $H$: - $H$ is effective iff there exists an $H\dash$marked K3. - Effective $\chi$: - $G\actson H^{2, 0}(X)$ induces $\chi: H\to \CC\units$, called an effective character. - $H_s \da \ker \chi$ - $$L_{\mathbb{C}}^\chi=\{x \in L \otimes \mathbb{C} \mid h(x)=\chi(h) \cdot x \text { for all } h \in H\}$$ - $$L_{\mathbb{R}}^{\chi+\chi}=\left\{x \in L_{\mathbb{R}} \mid\left(h+h^{-1}\right)(x)=\chi(h) x+\bar{\chi}(h) x \text { for all } h \in H\right\}$$ - Generic transcendental lattice: - $T(\chi)\leq L$ is the smallest primitive such that $T(\chi)_\CC \contains L_\CC^\chi$. - Generic Neron-Severi: - $\NS(\chi) \da T(\chi)^\perp$ - $L^{H_s}$ is the fixed lattice, $L_{H_s} \da (L^{H_s})^\perp$. - For us: - $1 \to \Aut_s(X) \to \Aut(X) \to \GL(H^{2, 0}(X))$ induces $1\to H_s\to H\to H/H_s\to 1$ with $H_s = C_2$ symplectic and $H/H_s = \mu_2$ non-symplectic. - Some consequences of $\chi$ being effective: - $\sgn L_{H_s} = (?, 0)$ is negative definite - $L_{H_s}$ and $\NS(\chi)_H$ contain no roots - $\sgn L^H = (1, ?)$ is hyberolic - $\sgn L_\RR^{\chi + \bar \chi} = (2, ?)$. - $H_S$ is independent of $\chi$. - Say $H$ is symplectic if $H = H_s$ and non-symplectic otherwise. - There are at most two characters $\chi: H\to \CC\units$, they are complex conjugate. - Corollary: $T(H) \da T(\chi), \NS(H)\da \NS(\chi)$ do not depend on the character. - $\mcm_H$ the fine space of $H\dash$marked surfaces $(X, \eta, G)$: non-Hausdorff. - The forgetful map $(X,\eta, G) \mapsto (X, \eta)$ yields a closed embedding $\mcm_H \embeds \mcm_L$ into $L\dash$lattice-polarized K3s - Period domain: $\DD^\chi \da \ts{\omega \in \PP L_\CC^\chi \st \omega^2 = 0, \norm\omega > 0}$. - Period map: $\mcp: \mcm_H^\chi \to \DD^\chi$. - Discriminant locus: $\Delta \da \Union \ts{\PP(\delta^\perp) \st \delta \in L_H,\,\, \delta^2 = -2}\subset \PP L_\CC$ the union of root hyperplanes. - Image: $\mcp(\mcm_H^\chi)\subseteq \DD^\chi\sm \Delta$. - Change of markings: given by the normalizer $$N(H) \da \ts{f\in \Orth(L) \st fH=Hf} \leq \Orth(L)$$ - Coarse space: $\mcf_H \da \mcm_H / N(H)$ is quasiprojective with only finite quotient singularities. - $\DD_H \da \DD^\chi \union \DD^{\bar\chi}$. - Bijective period map: $$\mcf_H \iso (\DD_H\sm \Delta)/N(H)$$ - They classify finite subgroups $G\leq \Aut(X)$ up to deformation by exhibiting biection with conjugacy classes of finite-order isometrices in a given lattice. - Show deformation classes biject with connected components of $\mcf_H$. - Specifically: deformation classes of $(X, G)$ nonsymplectic ($G\neq G_s$) biject with $\Union_{H\in \mcc} \pi_0(\mcf_H)$ where $\mcc$ is a transversal of the set of conjugacy classes of effective non-symplectic subgroups of $\Orth(L)$. - Enumerate all effective $H\leq \Orth(L)$ up to conjugacy. - $\size \pi_0 \mcf_H = {2\over [N(H): N(\chi)]}$ where $N(\chi) \da \Stab_{N(H)}(\chi)$ and $[H: H_s] > 2$ - If $[H: H_s] = 2$ then $\size \pi_0 \mcf_H = {2\over [N_T : N_T^+]}$ where $N_T \da \pi(N(H))$ where $\pi: N(H) \to \Orth(T) \da \Orth(T(\chi))$ is the restriction. Compute this number using Miranda-Morrison theory. - Small group ids: - $(4, 1) = C_4$, $k(C_4) = 5$ deformation classes - $(4, 2) = C_2^2$, 354 deformation classes. ## Questions - Does each deformation class correspond to a different degeneration? - Which classes are Types I/II/II? - Need to determine isotropic lines and planes in $T\da N(H) \leq \Orth(L)$ and their incidence relations. - Expect this to be $p\dash$elementary, so as to use classification to get invariants $(r,a,\delta)$ and apply [Lemma 5.4](https://arxiv.org/pdf/2208.10383.pdf#page=21&zoom=180,-45,270)? - Expect a diagram like [Figure 1](https://arxiv.org/pdf/2208.10383.pdf#page=5&zoom=180,-45,689)? - Still expect to define $\bar T \da e^\perp/e$ for $e\in T$ primitive isotropic? - Still expect to split $T = H\oplus \bar{T}$ with $e\in H$ and get a result similar to [Prop 5.5](https://arxiv.org/pdf/2208.10383.pdf#page=22&zoom=180,-45,712)? - How to construct $\overline{ \mcf_H}^{\KSBA}$? Need pairs $(X, \eps R)$ to replace $(X, \eps \bar{C_g})$ in [this description](https://arxiv.org/pdf/2208.10383.pdf#page=8&zoom=180,-45,300). - Here $\bar{C_g} \in \Fix(\iota)$ is a smooth genus $g$ curve in 50/75 cases. - Still source monodromy invariants $\lambda \in \bar{T} \intersect \mcc$? - Mirror: need to match $\lambda\in \bar{T}$ with nef line bundles (or forms) $\omega$ in $\hat{S}$? Get an analog of [theorem 6.16](https://arxiv.org/pdf/2208.10383.pdf#page=32&zoom=180,-45,393),