--- title: COMPACT MODULI OF K3 SURFACES WITH A NONSYMPLECTIC AUTOMORPHISM aliases: - COMPACT MODULI OF K3 SURFACES WITH A NONSYMPLECTIC AUTOMORPHISM annotation-target: https://arxiv.org/pdf/2110.13834.pdf flashcard: Research::Papers created: 2023-01-07T21:15 updated: 2024-07-31T13:14 --- #in_progress # Intro ## Priority - [x] What is a stable pair $(S, \eps D)$? Give an example. ✅ 2023-01-17 - [ ] slc singularities - [ ] $\K_S + \eps D \in \QQ\dash\CDiv(X)^\amp$. - [x] What is a Kulikov model? ✅ 2023-03-30 - [ ] $X\to \DD$ an extension of $X^\circ \to \DD^\circ$ where - [ ] $X$ is a smooth algebraic space, - [ ] $K_X\sim 0$, - [ ] $X_0$ is RNC - [x] What are Types I, II, II of Kulikov models? ✅ 2023-01-17 - [ ] I: $X_0$ smooth, - [ ] II: $X_0$ has double curves but no triple points, - [ ] III: $X_0$ has triple points - [ ] What is a KSBA compactification? - [ ] What is $\mcm_\rho$? - [x] What is $F_{2d}$? ✅ 2023-03-30 - [ ] Moduli of polarized K3s $(X, L)$, - [ ] ADE singularities, - [ ] $L\in \Pic(X)^{\amp, \prim}$ with $L^2 = 2d$ - [ ] What is $\mcm^{\slc}$? - [x] What is $F_\rho$? ✅ 2023-03-30 - [ ] Moduli of $\rho\dash$markable K3s. - [ ] What is $\bar{F_\rho}^\slc$? - [x] What is a nonsymplectic automorphism? ✅ 2023-01-07 - [ ] $\sigma: X\to X$ with finite order $n > 1$ where $\sigma^*(\omega_X) = \zeta_n \omega_X$ for some primitive root of unity $\zeta_n$, where $\omega_X \in H^{2, 0}(X)$ is a nonzero 2-form. - [ ] Describe the moduli of smooth quasipolarized K3s. - [ ] Describe the moduli of smooth K3s with a nonsymplectic automorphism. - [x] What is $F_\rho^{\mathrm{ade}}$? ✅ 2023-02-15 - [ ] $(\DD_\rho \sm \Delta_\rho)/\Gamma_\rho$ - [ ] Obtained from smooth K3s by contracting $(-2)\dash$curves orthogonal to $C_1 \in \Fix(\sigma)$ with $g\geq 2$ - [ ] Has a separated stack. - [x] What are the lattice signatures for $S_X, T_X$? ✅ 2023-02-15 - [ ] $S_X: (1, \rho - 1)$ - [ ] $T_X: (2, 20-\rho)$. - [x] What is $\Delta_\rho$? ✅ 2023-02-14 - [ ] Discriminant locus: $\DD_\rho \intersect \Union_{\text{roots }\delta \in (L^G)^\perp} \delta^\perp$. - [ ] $L^G \da \Fix(\rho) \subset S_\rho$ the fixed classes for $G = \gens{\rho}$ - [ ] $L \da \lkt$. - [ ] $S_\rho \da T_\rho^\perp$, $T_\rho \da L^\prim_\CC \intersect L$ generic Picard/transcendental lattices. - [ ] $L^\prim_\CC$ the sum of primitive eigenspaces of $\rho$. ## Background - [x] How does an involution act on $H^2$? ✅ 2023-02-15 - [ ] Fixes $S_X$ and acts by $\cdot(-1)$ on $T_X$. - [x] Classify $(X, \sigma)$ with $\sigma$ an involution, up to deformation. ✅ 2023-02-15 - [ ] Primitive 2-elementary sublattices $S \leq \lkt$, classified by Nikulin using $(g, k, \delta)$, 75 cases. - [x] What is a K3 surface? ✅ 2023-01-17 - [ ] Complete, smooth, $\dim_\CC X = 2$, $\omega_{X/k}\cong \OO_X$, and $H^1(\OO_X) = 0$. - [ ] Interpret $q(X) \da h^{0, 1}(X)$ as the irregularity. - [x] What is the Picard number $\rho(X)$? ✅ 2023-02-07 - [ ] For $X$ compact complex, $$\rho(X) \da \rank_\ZZ \im(\Pic(X) \to H^{1, 1}(X))$$ - [ ] If $X$ is Kahler, $\rho(X) = \rank H^{1,1}(X) = \rank \NS(X)$. - [x] What is the Jacobian of a complex manifold? ✅ 2023-02-07 - [ ] $$\Jac(X) \da H^1(\OO_X)/H^1(X; \ZZ)\cong \ker(\Pic(X) \to H^2(X; \ZZ)) \cong \Pic^0(X)$$ - [ ] The continuous part of $\Pic(X)$. - [x] What is primitive cohomology? ✅ 2023-02-07 - [ ] Let $(X, g)$ be a compact Kähler manifold. Then the primitive cohomology is defined as $$H^k(X, \mathbb{R})_{\prim}:=\operatorname{Ker}\left(\Lambda: H^k(X, \mathbb{R}) \longrightarrow H^{k-2}(X, \mathbb{R})\right)$$ and $$H^{p, q}(X)_{\prim}:=\operatorname{Ker}\left(\Lambda: H^{p, q}(X) \longrightarrow H^{p-1, q-1}(X)\right)$$s - [ ] I.e. any Kahler class induces a representation of $\liesl_2(\CC)$ on $H^*(X; \RR)$. - [ ] What is a Type I domain? - [ ] What is a Type IV domain? - [ ] What is a polarizing divisor? - [ ] What are the invariants $g, k, \delta$? - [ ] What are canonical singularities? - [ ] What are slc singularities? - [ ] What is a log center? - [ ] What is an algebraic space? - [ ] What is a unipotent operator $T$? - [ ] What is a semifan? - [ ] What are Atiyah flops of types 0, I, II? ## Section 2 - [x] What is the Neron-Severi lattice? ✅ 2023-01-17 - [ ] Generally $S_X \da \NS(X) = \coker(\Pic^0(X) \injects \Pic(X))$. - [x] Discuss $H^{2, 0}(X)$ for $X$ a K3. ✅ 2023-01-17 - [ ] $H^{2, 0}(X) = \CC\omega_X$ for some nowhere vanishing holomorphic 2-form $\omega_X$ - [x] Discuss $\NS(X)$ for $X$ a projective K3. ✅ 2023-01-17 - [ ] $S_X$ is nodegenerate of signature $(1, r_X - 1)$ where $r_X \da \rank \NS(X)$ is the Picard number. - [ ] For a K3, By Lefschetz $(1, 1)$, $$\Pic(X) = \NS(X) = H^{2, 0}(X)^\perp \intersect H^2(X; \ZZ) \subseteq H^2(X; \CC)$$ - [x] What is the transcendental lattice? Its signature? ✅ 2023-01-17 - [ ] $T_X = (S_X)^\perp \subseteq H^2(X; \ZZ)$ the orthgonal complement of Neron-Severi. - [ ] Signature $(2, 20-r_X)$ - [x] What is the Kahler cone $\mck_X$? ✅ 2023-01-07 - [ ] $\mck_X \subseteq H^{1, 1}(X; \RR)$ the set of Kahler forms, forms an open convex cone. - [x] What is a marking of a K3? ✅ 2023-01-07 - [ ] Let $X$ be a K3 surface. A marking is an isometry $\phi: H^2(X, \mathbb{Z}) \rightarrow L$. - [x] What is the period domain $\DD$? ✅ 2023-01-17 - [ ] $$\mathbb{D}=\mathbb{P}\left\{x \in L_{\mathbb{C}} \mid x \cdot x=0, x \cdot \bar{x}>0\right\}, \quad \operatorname{dim} \mathbb{D}=20$$ - [x] Discuss the moduli space of marked K3s. ✅ 2023-01-07 - [ ] Admits a fine moduli space $\mcm$ - [ ] Has a surjective etale period map $$\begin{align}\pi: \mcm &\to \DD \\ (X, \phi) &\mapsto \phi(H^{2, 0}(X)) \in \PP(L\tensor \CC)\end{align}$$ - [ ] Non-Hausdorff 20-dimensional $\CC\dash$manifold. - [ ] $\pi_0 \mcm = \ts{\mcm_-, \mcm_+}$ two isomorphic connected components where are exchanged under $(X, \phi)\mapstofrom (X, -\phi)$. - [x] For $\rho \in \Orth(L)$, what is a $\rho\dash$marking of $(X, \sigma)$? What is $\rho\dash$markable? ✅ 2023-01-07 - [ ] An isometry $\phi: H^2(X, \mathbb{Z}) \rightarrow \lkt$ such that $\sigma^*=\phi^{-1} \circ \rho \circ \phi$. - [ ] We say that $(X, \sigma)$ is $\rho$-markable if it admits a $\rho$-marking. - [ ] ![](attachments/2023-02-12-marking.png) - [x] What is a family of $\rho\dash$marked surfaces? ✅ 2023-02-12 - [ ] A smooth morphism $f:(\mcx, \sigma_B) \to B$ - [ ] An automorphism $\sigma_B \in \Aut(X)$ over $B$ - [ ] An isomorphism f local systems $\phi_S: \RR^2f_* \ul{\ZZ} \to L\tensor \ul{\ZZ}_B$ where every fiber is a K3 with a $\rho\dash$marking. - [x] What are the moduli stacks $\mcm_\rho, F_\rho$? ✅ 2023-02-12 - [ ] Moduli of $\rho\dash$marked (resp. $\rho\dash$markable) K3 surfaces - [ ] $\mcm_\rho(B)$ is the groupoid of $\rho\dash$marked familes over $B$. - [x] How are deformation classes of K3 surfaces with nonsymplectic automorphisms $(X, \sigma)$ classified? ✅ 2023-01-07 - [ ] Primitive 2-elementary hyperbolic sublattices $S \subseteq L_{\K3}$. - [ ] What is the discriminant locus $\Delta_\rho \subseteq \DD_\rho$? ## Section 3: Stable Pair Compactifications - [x] Give an example of a stable slc pair. ✅ 2023-02-14 - [ ] Any K3 $X$ with ADE singularities and any effective ample divisor $R$, - [ ] CY: $\omega_X\cong \OO_X$ - [ ] Has canonical singularities, better than slc - [ ] No log centers. - [x] What is a stable slc surface pair? ✅ 2023-01-17 - [ ] A pair $(S, \epsilon D)$, where - [ ] $S$ is a connected, reduced, projective Gorenstein surface $S$ with $\omega_S \simeq \mathcal{O}_S$ which has semi log canonical singularities. - [ ] CRPG CY SLC - [ ] $D$ is an effective ample Cartier divisor on $S$ that does not contain any log canonical centers of $S$. - [ ] Thus $(S, \eps D)$ is stable pair for small $\eps \QQ_{\gt 0}$. - [ ] What is $\mcm^{\slc}$? - [x] What is a canonical choice of polarizing divisor? ✅ 2023-01-17 - [ ] An algebraically varying big and nef $R \in \Div(U)$ for $U$ a Zariski dense subset of $F_\rho$. - [x] What is a Kulikov surface? ✅ 2023-01-17 - [ ] $X_0$ the central fiber of a Kulikov model. - [x] What is Kulikov's key result on degenerations of K3s? ✅ 2023-01-17 - [ ] For $Y^*\to C^*$ a family, there is a finite base change $(C', 0) \to (C, 0)$ and a birational modifcation of its pullback $Y'\rational X$ such that $X$ has smooth total space, $K_X \sim_{C'} 0$, and $X_0$ is reduced normal crossings. - [x] What are the log monodromies for Kulikov degenerations. ✅ 2023-01-17 - [ ] Define $N \da \log T = (T-I) - {1\over 2}(T-I)^2 + \cdots$ - [ ] Type I $\implies N = 0$. - [ ] Type II $\implies N\neq 0, N^2 = 0$. - [ ] Type III $\implies N \neq 0, N^2 \neq 0, N^3=0$. - [ ] $N$ is always integral and of the form $$Nx = (x\cdot \lambda)\delta - (x\cdot \delta)\lambda,\qquad \delta\in H^2(X_t; \ZZ), \lambda \in \delta^\perp/\delta$$ - [x] What is a **recognizable divisor**? ✅ 2023-01-17 - [ ] A canonical choice of polarizing divisor $R$ for $U \subset F_\rho$ is **recognizable** if - [ ] For every Kulikov surface $X_0$ of Type I, II, or III which smooths to some $\rho$-markable $\mathrm{K} 3$ surface, - [ ] There is a divisor $R_0 \subset X_0$ such that on any smoothing into $\rho$-markable K3 surfaces $X \rightarrow(C, 0)$ with $C^* \subset U$, - [ ] The divisor $R_0$ is, up to the action of$\Aut^0\left(X_0\right)$, the flat limit of $R_t$ for $t \neq 0 \in C^*$. - [x] What is the importance of a regognizable divisor? ✅ 2024-01-29 - [ ] If $R$ is recognizable, then $\bar{F_\rho}^{\slc}$ is a semitoroidal compactification of $F_\rho$ for a unique semifan $\mcf_R$. - [x] What is the Baily-Borel compactification associated to a lattice $N$ with isometry $\rho$? ✅ 2023-02-14 - [ ] $\DD_N \da \ts{x\in \PP(N\tensor \CC)\st x^2=0, \abs{x} > 0}$ a type IV domain - [ ] $\DD_\rho = \ts{x\in \PP(N\tensor \CC)^{\zeta_n} \st \abs{x} > 0}$ a Type I subdomain - [ ] $\tilde \Gamma_\rho \da Z_{\Orth(N)}(\rho)\actson \DD_\rho$, and also for any finite index subgroup $\Gamma$. - [ ] Set $\bar{\DD_\rho}$ the closure of the compact dual (dropping $\abs{x} > 0$) - [ ] Define $\bar{\DD_\rho}^\mathrm{BB} = \DD_\rho \union \Union_{J} B_J$ by adding *rational boundary components*, usually $\HH \disjoint(-\HH)$ or points.s - [ ] BB compactification is $\overline{\DD_\rho/\Gamma}\da \bar{\DD_\rho}^{\mathrm{BB}}/\Gamma$. ## Section 4: Moduli of quotient surfaces ## Section 5: Extensions - [x] When is $G\subseteq \Aut(X)$ nonsymplectic? ✅ 2023-01-17 - [ ] $G\actson H^{2, 0}(X) = \CC\omega_X$ gives a SES $$G_0 \injects G \surjects \mu_n,\qquad \mu_n\in \cstar$$ and $G$ is non symplectic if $G\ne G_0$. - [x] What is a $(\rho, \chi)\dash$marking of $(X, \sigma)$ ✅ 2023-01-17 - [ ] Let $\rho: G\injects \Orth(L)$ be finite subgroup, $\chi: G\to \cstar$ a character, and $\sigma: G\to \Aut(X)$. - [ ] Marking: an isometry $H^2(X; \ZZ) \to L$ where $\phi \circ \sigma(g)^* = \rho(g) \circ \phi$ such that the induced character $\alpha: G\to \cstar$ equals $\chi$. - [x] What is a family of $(\rho, \chi)\dash$marked K3s? ✅ 2023-01-17 - [ ] A smooth model $(\chi, \sigma_B, \phi_B) \mapsvia{f} B$ where $\sigma_B: G\to \Aut(\mcx/B)$ and $\phi_B: \RR^2 f_* \ul{\ZZ} \to L\tensor \ul{\ZZ}_B$ where every fiber is a $(\rho, \chi)\dash$marked surface. - [x] What is the change-of-markings group? ✅ 2023-01-17 - [ ] The centralizer: $$\Gamma_\rho \da Z_{\Orth(\lkt)}(\rho) = \ts{\gamma\in \Orth(L) \st \gamma\circ \rho = \rho \circ \gamma}$$ # COMPACT MODULI OF K3 SURFACES WITH A NONSYMPLECTIC AUTOMORPHISM >%% >```annotation-json >{"created":"2023-01-08T02:33:55.910Z","updated":"2023-01-08T02:33:55.910Z","document":{"title":"2110.13834.pdf","link":[{"href":"urn:x-pdf:bd64dc45e0ebf59b8d61be818dd13cc3"},{"href":"https://arxiv.org/pdf/2110.13834.pdf"}],"documentFingerprint":"bd64dc45e0ebf59b8d61be818dd13cc3"},"uri":"https://arxiv.org/pdf/2110.13834.pdf","target":[{"source":"https://arxiv.org/pdf/2110.13834.pdf","selector":[{"type":"TextPositionSelector","start":649,"end":822},{"type":"TextQuoteSelector","exact":"An automorphismσ of X is called non-symplectic if it has finite order n > 1 and σ∗(ωX) = ζnωX,where ωX ∈H2,0(X) is a nonzero 2-form and ζn is a primitive nth root of identit","prefix":"rface over the complex numbers. ","suffix":"y.By changing the generator of t"}]}]} >``` >%% >*%%PREFIX%%rface over the complex numbers.%%HIGHLIGHT%% ==An automorphismσ of X is called non-symplectic if it has finite order n > 1 and σ∗(ωX) = ζnωX,where ωX ∈H2,0(X) is a nonzero 2-form and ζn is a primitive nth root of identit== %%POSTFIX%%y.By changing the generator of t* >%%LINK%%[[#^welf45uf8b|show annotation]] >%%COMMENT%% > >%%TAGS%% > ^welf45uf8b >%% >```annotation-json >{"created":"2023-01-08T02:34:12.596Z","updated":"2023-01-08T02:34:12.596Z","document":{"title":"2110.13834.pdf","link":[{"href":"urn:x-pdf:bd64dc45e0ebf59b8d61be818dd13cc3"},{"href":"https://arxiv.org/pdf/2110.13834.pdf"}],"documentFingerprint":"bd64dc45e0ebf59b8d61be818dd13cc3"},"uri":"https://arxiv.org/pdf/2110.13834.pdf","target":[{"source":"https://arxiv.org/pdf/2110.13834.pdf","selector":[{"type":"TextPositionSelector","start":1231,"end":1325},{"type":"TextQuoteSelector","exact":"the automorphism group Aut(X,σ), i.e. those automorphisms of Xcommuting with σ, may be infinit","prefix":" pairs (X,σ). Butto begin with, ","suffix":"e. To fix this, we will usually "}]}]} >``` >%% >*%%PREFIX%%pairs (X,σ). Butto begin with,%%HIGHLIGHT%% ==the automorphism group Aut(X,σ), i.e. those automorphisms of Xcommuting with σ, may be infinit== %%POSTFIX%%e. To fix this, we will usually* >%%LINK%%[[#^xjwe97qhs28|show annotation]] >%%COMMENT%% > >%%TAGS%% > ^xjwe97qhs28 >%% >```annotation-json >{"created":"2023-01-08T02:34:19.000Z","updated":"2023-01-08T02:34:19.000Z","document":{"title":"2110.13834.pdf","link":[{"href":"urn:x-pdf:bd64dc45e0ebf59b8d61be818dd13cc3"},{"href":"https://arxiv.org/pdf/2110.13834.pdf"}],"documentFingerprint":"bd64dc45e0ebf59b8d61be818dd13cc3"},"uri":"https://arxiv.org/pdf/2110.13834.pdf","target":[{"source":"https://arxiv.org/pdf/2110.13834.pdf","selector":[{"type":"TextPositionSelector","start":1328,"end":1441},{"type":"TextQuoteSelector","exact":"To fix this, we will usually additionally assume:(∃g ≥2) The fixed locus Fix(σ) contains a curve C1 of genus g ≥2","prefix":"muting with σ, may be infinite. ","suffix":".By looking at the μn-action on "}]}]} >``` >%% >*%%PREFIX%%muting with σ, may be infinite.%%HIGHLIGHT%% ==To fix this, we will usually additionally assume:(∃g ≥2) The fixed locus Fix(σ) contains a curve C1 of genus g ≥2== %%POSTFIX%%.By looking at the μn-action on* >%%LINK%%[[#^za6uv1xeut|show annotation]] >%%COMMENT%% > >%%TAGS%% > ^za6uv1xeut >%% >```annotation-json >{"created":"2023-01-08T02:47:20.859Z","updated":"2023-01-08T02:47:20.859Z","document":{"title":"2110.13834.pdf","link":[{"href":"urn:x-pdf:bd64dc45e0ebf59b8d61be818dd13cc3"},{"href":"https://arxiv.org/pdf/2110.13834.pdf"}],"documentFingerprint":"bd64dc45e0ebf59b8d61be818dd13cc3"},"uri":"https://arxiv.org/pdf/2110.13834.pdf","target":[{"source":"https://arxiv.org/pdf/2110.13834.pdf","selector":[{"type":"TextPositionSelector","start":7428,"end":7472},{"type":"TextQuoteSelector","exact":"Theorem 2.1 (Torelli Theorem for K3 surfaces","prefix":"on X; it is an open convex cone.","suffix":", [PSS71]). The isomorphismsσ: X"}]}]} >``` >%% >*%%PREFIX%%on X; it is an open convex cone.%%HIGHLIGHT%% ==Theorem 2.1 (Torelli Theorem for K3 surfaces== %%POSTFIX%%, [PSS71]). The isomorphismsσ: X* >%%LINK%%[[#^54sks7ze8ua|show annotation]] >%%COMMENT%% > >%%TAGS%% > ^54sks7ze8ua >%% >```annotation-json >{"created":"2023-01-08T03:27:04.682Z","updated":"2023-01-08T03:27:04.682Z","document":{"title":"2110.13834.pdf","link":[{"href":"urn:x-pdf:bd64dc45e0ebf59b8d61be818dd13cc3"},{"href":"https://arxiv.org/pdf/2110.13834.pdf"}],"documentFingerprint":"bd64dc45e0ebf59b8d61be818dd13cc3"},"uri":"https://arxiv.org/pdf/2110.13834.pdf","target":[{"source":"https://arxiv.org/pdf/2110.13834.pdf","selector":[{"type":"TextPositionSelector","start":7966,"end":8005},{"type":"TextQuoteSelector","exact":"A marking is an isometry φ: H2(X,Z) →L.","prefix":" [ast85].Let X be a K3 surface. ","suffix":" LetD = P{x ∈LC |x ·x = 0, x · ̄"}]}]} >``` >%% >*%%PREFIX%%[ast85].Let X be a K3 surface.%%HIGHLIGHT%% ==A marking is an isometry φ: H2(X,Z) →L.== %%POSTFIX%%LetD = P{x ∈LC |x ·x = 0, x · ̄* >%%LINK%%[[#^afmpo0hu6g|show annotation]] >%%COMMENT%% > >%%TAGS%% > ^afmpo0hu6g >%% >```annotation-json >{"created":"2023-01-08T03:27:17.710Z","updated":"2023-01-08T03:27:17.710Z","document":{"title":"2110.13834.pdf","link":[{"href":"urn:x-pdf:bd64dc45e0ebf59b8d61be818dd13cc3"},{"href":"https://arxiv.org/pdf/2110.13834.pdf"}],"documentFingerprint":"bd64dc45e0ebf59b8d61be818dd13cc3"},"uri":"https://arxiv.org/pdf/2110.13834.pdf","target":[{"source":"https://arxiv.org/pdf/2110.13834.pdf","selector":[{"type":"TextPositionSelector","start":8056,"end":8111},{"type":"TextQuoteSelector","exact":"There exists a fine moduli space M of marked K3 surface","prefix":"·x = 0, x · ̄x > 0}, dim D = 20.","suffix":"s and a period mapπ: M → D, (X,φ"}]}]} >``` >%% >*%%PREFIX%%·x = 0, x · ̄x > 0}, dim D = 20.%%HIGHLIGHT%% ==There exists a fine moduli space M of marked K3 surface== %%POSTFIX%%s and a period mapπ: M → D, (X,φ* >%%LINK%%[[#^0jr8fgcrr60s|show annotation]] >%%COMMENT%% > >%%TAGS%% > ^0jr8fgcrr60s >%% >```annotation-json >{"created":"2023-01-08T03:27:27.098Z","updated":"2023-01-08T03:27:27.098Z","document":{"title":"2110.13834.pdf","link":[{"href":"urn:x-pdf:bd64dc45e0ebf59b8d61be818dd13cc3"},{"href":"https://arxiv.org/pdf/2110.13834.pdf"}],"documentFingerprint":"bd64dc45e0ebf59b8d61be818dd13cc3"},"uri":"https://arxiv.org/pdf/2110.13834.pdf","target":[{"source":"https://arxiv.org/pdf/2110.13834.pdf","selector":[{"type":"TextPositionSelector","start":7325,"end":7342},{"type":"TextQuoteSelector","exact":"he K ̈ahler cone ","prefix":"ttice, of signature(2,20 −rX). T","suffix":"KX ⊂ H1,1(X,R) is the set of cla"}]}]} >``` >%% >*%%PREFIX%%ttice, of signature(2,20 −rX). T%%HIGHLIGHT%% ==he K ̈ahler cone== %%POSTFIX%%KX ⊂ H1,1(X,R) is the set of cla* >%%LINK%%[[#^tdlr901wy|show annotation]] >%%COMMENT%% > >%%TAGS%% > ^tdlr901wy >%% >```annotation-json >{"created":"2023-01-08T03:27:34.638Z","updated":"2023-01-08T03:27:34.638Z","document":{"title":"2110.13834.pdf","link":[{"href":"urn:x-pdf:bd64dc45e0ebf59b8d61be818dd13cc3"},{"href":"https://arxiv.org/pdf/2110.13834.pdf"}],"documentFingerprint":"bd64dc45e0ebf59b8d61be818dd13cc3"},"uri":"https://arxiv.org/pdf/2110.13834.pdf","target":[{"source":"https://arxiv.org/pdf/2110.13834.pdf","selector":[{"type":"TextPositionSelector","start":6938,"end":6979},{"type":"TextQuoteSelector","exact":"Denote by S = SX the Neron-Severi lattice","prefix":"ttice H2(X,Z) is isometric to L.","suffix":" Pic(X) = NS(X). By the Lefschet"}]}]} >``` >%% >*%%PREFIX%%ttice H2(X,Z) is isometric to L.%%HIGHLIGHT%% ==Denote by S = SX the Neron-Severi lattice== %%POSTFIX%%Pic(X) = NS(X). By the Lefschet* >%%LINK%%[[#^qnofvjqtqyc|show annotation]] >%%COMMENT%% > >%%TAGS%% > ^qnofvjqtqyc >%% >```annotation-json >{"created":"2023-01-08T03:27:41.958Z","updated":"2023-01-08T03:27:41.958Z","document":{"title":"2110.13834.pdf","link":[{"href":"urn:x-pdf:bd64dc45e0ebf59b8d61be818dd13cc3"},{"href":"https://arxiv.org/pdf/2110.13834.pdf"}],"documentFingerprint":"bd64dc45e0ebf59b8d61be818dd13cc3"},"uri":"https://arxiv.org/pdf/2110.13834.pdf","target":[{"source":"https://arxiv.org/pdf/2110.13834.pdf","selector":[{"type":"TextPositionSelector","start":6577,"end":6937},{"type":"TextQuoteSelector","exact":"ce is a free abelian group with an integral-valued symmetricbilinear form. Let L = H⊕3 ⊕E⊕28 be a fixed copy of the even unimodular latticeof signature (3,19), where H = II1,1 corresponds to the bilinear form b(x,y) = xyand E8 is the standard negative definite even lattice of rank 8. For any smooth K3surface X the cohomology lattice H2(X,Z) is isometric to L","prefix":"tomorphism2A. Notations. A latti","suffix":".Denote by S = SX the Neron-Seve"}]}]} >``` >%% >*%%PREFIX%%tomorphism2A. Notations. A latti%%HIGHLIGHT%% ==ce is a free abelian group with an integral-valued symmetricbilinear form. Let L = H⊕3 ⊕E⊕28 be a fixed copy of the even unimodular latticeof signature (3,19), where H = II1,1 corresponds to the bilinear form b(x,y) = xyand E8 is the standard negative definite even lattice of rank 8. For any smooth K3surface X the cohomology lattice H2(X,Z) is isometric to L== %%POSTFIX%%.Denote by S = SX the Neron-Seve* >%%LINK%%[[#^9c4jk8dw94h|show annotation]] >%%COMMENT%% > >%%TAGS%% > ^9c4jk8dw94h >%% >```annotation-json >{"created":"2023-01-08T03:27:52.010Z","updated":"2023-01-08T03:27:52.010Z","document":{"title":"2110.13834.pdf","link":[{"href":"urn:x-pdf:bd64dc45e0ebf59b8d61be818dd13cc3"},{"href":"https://arxiv.org/pdf/2110.13834.pdf"}],"documentFingerprint":"bd64dc45e0ebf59b8d61be818dd13cc3"},"uri":"https://arxiv.org/pdf/2110.13834.pdf","target":[{"source":"https://arxiv.org/pdf/2110.13834.pdf","selector":[{"type":"TextPositionSelector","start":7637,"end":7869},{"type":"TextQuoteSelector","exact":"For any lattice H, a root is a vector δ ∈ H with δ2 = −2. The set of all rootsis denoted by H−2. The Weyl group W(H) is the group generated by reflectionsv 7→v + (v,δ)δ for δ ∈H−2. It is a normal subgroup of the isometry group O(H).","prefix":"X)) = H2,0(X′) and σ∗(KX) = KX′.","suffix":"2B. Moduli of marked unpolarized"}]}]} >``` >%% >*%%PREFIX%%X)) = H2,0(X′) and σ∗(KX) = KX′.%%HIGHLIGHT%% ==For any lattice H, a root is a vector δ ∈ H with δ2 = −2. The set of all rootsis denoted by H−2. The Weyl group W(H) is the group generated by reflectionsv 7→v + (v,δ)δ for δ ∈H−2. It is a normal subgroup of the isometry group O(H).== %%POSTFIX%%2B. Moduli of marked unpolarized* >%%LINK%%[[#^82y51j5qp93|show annotation]] >%%COMMENT%% > >%%TAGS%% > ^82y51j5qp93 >%% >```annotation-json >{"created":"2023-01-08T03:27:58.378Z","updated":"2023-01-08T03:27:58.378Z","document":{"title":"2110.13834.pdf","link":[{"href":"urn:x-pdf:bd64dc45e0ebf59b8d61be818dd13cc3"},{"href":"https://arxiv.org/pdf/2110.13834.pdf"}],"documentFingerprint":"bd64dc45e0ebf59b8d61be818dd13cc3"},"uri":"https://arxiv.org/pdf/2110.13834.pdf","target":[{"source":"https://arxiv.org/pdf/2110.13834.pdf","selector":[{"type":"TextPositionSelector","start":7209,"end":7299},{"type":"TextQuoteSelector","exact":"In this case, its orthogonalcomplement TX = (SX)⊥ ⊂ H2(X,Z) is the transcendental lattice,","prefix":"nerate of signature (1,rX − 1). ","suffix":" of signature(2,20 −rX). The K ̈"}]}]} >``` >%% >*%%PREFIX%%nerate of signature (1,rX − 1).%%HIGHLIGHT%% ==In this case, its orthogonalcomplement TX = (SX)⊥ ⊂ H2(X,Z) is the transcendental lattice,== %%POSTFIX%%of signature(2,20 −rX). The K ̈* >%%LINK%%[[#^lzdyrwblm8l|show annotation]] >%%COMMENT%% > >%%TAGS%% > ^lzdyrwblm8l >%% >```annotation-json >{"created":"2023-01-08T03:28:06.319Z","updated":"2023-01-08T03:28:06.319Z","document":{"title":"2110.13834.pdf","link":[{"href":"urn:x-pdf:bd64dc45e0ebf59b8d61be818dd13cc3"},{"href":"https://arxiv.org/pdf/2110.13834.pdf"}],"documentFingerprint":"bd64dc45e0ebf59b8d61be818dd13cc3"},"uri":"https://arxiv.org/pdf/2110.13834.pdf","target":[{"source":"https://arxiv.org/pdf/2110.13834.pdf","selector":[{"type":"TextPositionSelector","start":6147,"end":6291},{"type":"TextQuoteSelector","exact":"n Section 5 we extend the results in two different ways: to K3 surfaces with afinite group of symmetries G ⊂Aut X which is not purely symplectic","prefix":" del Pezzo pairs (Y, n−1+\u000fn B).I","suffix":", and to moregeneral polarizing "}]}]} >``` >%% >*%%PREFIX%%del Pezzo pairs (Y, n−1+n B).I%%HIGHLIGHT%% ==n Section 5 we extend the results in two different ways: to K3 surfaces with afinite group of symmetries G ⊂Aut X which is not purely symplectic== %%POSTFIX%%, and to moregeneral polarizing* >%%LINK%%[[#^fgnxulknfu|show annotation]] >%%COMMENT%% > >%%TAGS%% > ^fgnxulknfu >%% >```annotation-json >{"created":"2023-01-08T03:28:18.824Z","updated":"2023-01-08T03:28:18.824Z","document":{"title":"2110.13834.pdf","link":[{"href":"urn:x-pdf:bd64dc45e0ebf59b8d61be818dd13cc3"},{"href":"https://arxiv.org/pdf/2110.13834.pdf"}],"documentFingerprint":"bd64dc45e0ebf59b8d61be818dd13cc3"},"uri":"https://arxiv.org/pdf/2110.13834.pdf","target":[{"source":"https://arxiv.org/pdf/2110.13834.pdf","selector":[{"type":"TextPositionSelector","start":8113,"end":8328},{"type":"TextQuoteSelector","exact":"and a period mapπ: M → D, (X,φ) 7→ φ(H2,0(X)) ∈ P(LC). M is a non-Hausdorff 20-dimensionalcomplex manifold with two isomorphic connected components interchanged bynegating φ. The period map is ́etale and surjective","prefix":"i space M of marked K3 surfaces ","suffix":".For a period point x ∈ D, the v"}]}]} >``` >%% >*%%PREFIX%%i space M of marked K3 surfaces%%HIGHLIGHT%% ==and a period mapπ: M → D, (X,φ) 7→ φ(H2,0(X)) ∈ P(LC). M is a non-Hausdorff 20-dimensionalcomplex manifold with two isomorphic connected components interchanged bynegating φ. The period map is ́etale and surjective== %%POSTFIX%%.For a period point x ∈ D, the v* >%%LINK%%[[#^61934vfh3so|show annotation]] >%%COMMENT%% > >%%TAGS%% > ^61934vfh3so