--- date: 2023-01-27 17:54 title: Alessandra Sarti Topics on K3 surfaces aliases: - Alessandra Sarti Topics on K3 surfaces status: ✅ Started page_current: 0 page_total: 100000 flashcard: Research::K3s created: 2023-03-30T13:17 updated: 2024-01-05T21:48 --- # Progress - [x] Discuss the number of fixed points of a symplectic automorphism with finite order $n$. ✅ 2023-01-31 - [ ] Computed by Mukai using the Lefschetz fixed point formula: ![](attachments/2023-01-31-fixed.png) - [x] Discuss the classification of symplectic actions. ✅ 2023-01-31 - [ ] Xiao: 80 possible groups, maximal size of 960. - [x] Give $\dim \PP(V)$ in terms of $\dim V$. ✅ 2023-02-01 - [ ] $\dim \PP(V) = \dim V - 1$. - [x] Why does every K3 contain a rational curve? ✅ 2023-02-15 - [ ] Bogomolov-Mumford: degenerate to a Kummer which is a product of two special elliptic curves - [ ] Construct a configuration of rational curves there - [ ] Use irreducibility of $F_{2d}$ to deform to the general K3, possibly making it reducible, but has at least one irreducible. - [ ] Use one of the irreducible curves, then specialize to any K3 in $F_{2d}$, the curve can only break into rational curves. - [x] Discuss rational curves on K3s. ✅ 2023-02-15 - [ ] Always has at least one. - [ ] Any nonzero $D\in \Pic(X)^\eff$ satisfies $D\sim \sum R_i$ for some rational curves. - [ ] Over $k=\CC$ $X$ is never uniruled, i.e. no rational curves move. - [ ] Conjecture: $X/\bar k$ has infinitely many rational curves. - [ ] What is an elliptic K3? - [ ] What is a supersingular K3? - [ ] What is spreading out? - [ ] What is the Noether-Lefschetz locus? ## Lecture 1 - [x] What is a Kummer surface? ✅ 2023-02-01 - [ ] $\widetilde{T/\iota}$ where $T = \CC^2/\Lambda$ is a torus and $\iota$ is an involution with 16 ordinary double point singularities of type $A_1$. - [x] What are the dimensions of moduli of all and just algebraic K3s? ✅ 2023-02-01 - [ ] All K3s: 20-dimensional - [ ] Algebraic K3s: 19-dimensional - [x] What are the dimensions of moduli of all and just algebraic Kummer surfaces? ✅ 2023-02-01 - [ ] All: 4-dimensional - [ ] Algebraic: 3-dimensional. - [x] What is the dimension of moduli of Enriques surfaces? ✅ 2023-02-01 - [ ] 10-dimensional. - [x] What is the exceptional curve of a blowup of a point in a surface? ✅ 2023-01-27 - [ ] A rational $(-1)\dash$curve, so $E^2 = -1$ with $E\cong \PP^1$. - [x] What is a minimal surface? ✅ 2023-01-27 - [ ] No rational $(-1)\dash$curves - [x] What is a ruled surface? ✅ 2023-01-27 - [ ] $\exists S\rational C\times \PP^1$ with $g(C) \geq 0$. - [x] What is an abelian surface? ✅ 2023-01-27 - [ ] $\CC^2/\Lambda \embeds \PP^N$ with $\Lambda$ a rank 4 lattice. - [x] What is an Enriques surface? ✅ 2023-01-27 - [ ] Exactly quotients of K3s by fixed point free involutions. - [ ] Smooth projective $Y$ with $2K_Y \sim 0$ and $h^1(\OO_Y) = h^2(\OO_Y) = 0$; the canonical induces an unramified 2-to-1 cover $X\to Y$ for $X$ a K3. - [x] What are bielliptic surfaces? ✅ 2023-01-27 - [ ] Quotients of $E\times E'$ elliptic curves. - [x] What is a special fact about proper elliptic surfaces? ✅ 2023-01-27 - [ ] Always have an elliptic fibration. - [x] Classify surfaces by $\kappa(S)$. ✅ 2023-01-27 - [ ] $-\infty$: Rational or ruled. - [ ] $0$: Four types; abelian, K3s, Enriques, bielliptic. - [ ] $1:$ Proper elliptic. - [ ] $2$: General type. - [x] What's the etymology of K3? ✅ 2023-01-27 - [ ] Named after K2 mountains - [ ] Or Kummer, Kahler, Kodaira. - [x] Let $X_4$ be the Fermat quartic, what is $K_{X_4}$? ✅ 2023-01-27 - [ ] By adjunction, $K_{X_4} = (K_{\PP^3} + X_4)\mid_{X_4} = (-4H + 4H)\mid_{X_4} = 0$. - [x] What is the irregularity of $S$? ✅ 2023-01-27 - [ ] $q(S) \da h^1(\OO_S)$ or $h^0(\Omega^1)$, so $h^{1, 0}$ or $h^{0, 1}$. - [x] What is the geometric genus of $S$? ✅ 2023-01-27 - [ ] $h^{0}(\Omega^2)$, so $h^{2, 0}$. - [ ] What is the Lefschetz theorem of hyperplane sections? - [x] What is $\pi_1(X_4)$? ✅ 2023-01-27 - [ ] By Lefschetz, $\pi_1(X_4) = \pi_1(\PP^3) = 1$. - [x] Compute $q(X_4)$. ✅ 2023-01-27 - [ ] $H_1(X; \ZZ) = \pi_1(X)_\ab = 1$ so $1 = H^1(X;\CC) = H^{0,1 } \oplus H^{1, 0}$ so $q(X_4) = 0$. - [ ] Why does a K3 have a global 2-form with neither poles nor zeros? - [x] Give a birational definition of a K3. ✅ 2023-01-27 - [ ] A compact complex surface with $K_S \sim 0$ and $q(S) = 0$. - [x] How does one show a surface is K3? ✅ 2023-01-27 - [ ] Show $q(S) = 0$, e.g. using Lefschetz, and show $K_S \sim 0$ e.g. using adjunction. - [x] Give examples of K3s ✅ 2023-01-27 - [ ] Any smooth quartic in $\PP^3$. - [ ] Complete intersections of type (2,3) or (2,2,2) - [ ] $S\to \PP^2$ 2x ramified over a sextic curve. - [x] Show that the double plane is K3. ✅ 2023-01-27 - [ ] Check $K_S = \pi^* K_{\PP^2} + R$ where $\pi_*(R) = C_6$ and $\pi_*(C_6) = 2R$, use $K_{\PP^2} = -3H$ and $C_6 = 6H$ to get $$2K_S = \pi^*(2K_{\PP^2}) + 2R = \pi^*(2K_{\PP^2}) + \pi^*(C_6) = \pi^*(-6H) + \pi^*(6H) = 0$$ - [ ] If $K_S\neq 0$ but $2K_S = 0$, note $h^0(K_S) = 0$. - [ ] Check $$\chi(S) = 2(\chi(\PP^2) - \chi(C_6)) + \chi(R) = 2(3+18)-18 =24$$ using that $\chi(C_6) = 2\cdot 2g$. - [ ] Check $\chi(\OO_S) = 2$ and use Noether to conclude $1-h^1(\OO_S) = 2$, a contradiction. - [x] What is the Noether formula? ✅ 2023-01-27 - [ ] $\chi(\OO_X) = {1\over 12}(K_X^2 + \chi(X))$. - [x] Give an equation for the double plane. ✅ 2023-01-27 - [ ] $S: t^2 = f_6(x_0, x_1, x_2) \subseteq \PP(3,1,1,1)$ where $C_6 = V(f_6)$. ## Lecture 2: Kummer Surfaces - [x] What is a Kummer surface? ✅ 2023-01-28 - [ ] Example: the minimal resolution $K_m A$ of $A/\iota$ over 16 singularities, equivalently blowup 16 fixed points and extend $\iota$ to $\tilde\iota$ and take $( \Bl_{16} A)/\tilde\iota$. - [ ] This is a K3 since $K_X \sim 0$ and $q(X) = 0$. - [x] What is the involution on an abelian surface? What are its fixed points? ✅ 2023-01-28 - [ ] $A = \CC^2/\Lambda$ with $\rank_\ZZ \Lambda = 4$, take $(x,y)\mapsto (-x,y)$ on $\CC^2$ to get $2(x,y)\in \Lambda$, there are 16 such points and thus 16 $A_1$ singularities in $A/\iota$. - [x] What does an $A_1$ singularity look like? ✅ 2023-01-28 - [ ] Locally the vertex of a cone, rational double points. ![](attachments/2023-01-28-cone.png) - [x] If $A\selfmap_\iota$ is the involute on an abelian surface, what is the induced involution on the blowup at the 16 fixed points? ✅ 2023-01-28 - [ ] Write $\ts{\tv{ x_0: x_1}, \tv{x, y} \in \PP^1 \times \CC^2 \st x_0 y = x_1 x}$ and send $\tv{x_0, x_1}, \tv{x, y}\mapsto \tv{ x_0: x_1}, \tv{-x, -y}$. - [x] What type of curve is the resolution of an $A_1$ singularity? ✅ 2023-01-28 - [ ] $D\cong \PP^1$ with $D^2 = -2$. - [ ] What is a divisible divisor class? - [ ] How is $K_X\sim 0$ related to having a differential form with no zeros or poles? - [ ] For an abelian variety $A$, what is $A\dual$? - [x] What is $\chi(\K3)$? ✅ 2023-01-28 - [ ] $\chi(\K3) = 24$, all diffeomorphic to the Fermat quartic. - [x] Describe $\Lambda_{\K3}$. ✅ 2023-01-28 - [ ] Lattice structure on (all of) $H^2(\K3; \ZZ)$. - [ ] Rank $22$ - [ ] Signature $(3, 19)$ - [ ] Isometric to $U\sumpower 3 \oplus E_8(-1)\sumpower{2}$ where - [ ] $U = \qty{\ZZ^2, \left(\begin{array}{ll}0 & 1 \\1 & 0\end{array}\right)}$ - [ ] $E_8(-1) =$ unique even unimodular negative-definite lattice of rank 8. - [x] What is the rank of $\Lambda_{\K3}$? Proof? ✅ 2023-01-28 - [ ] $\chi(\K3) = 24$ and $p_{\K3}(t) = 1 + b_2 t^2 + t^4$ since $b_1 = b_3 = 0$ from $\pi_1(\K3) = 1$ and $b_0 = b_4 = 1$ from connectedness, thus $\beta_2 = 22$. - [x] Why is $H^2(\K3; \ZZ)$ torsionfree? ✅ 2023-01-28 - [ ] By UCT, $H^2(\K3;\ZZ)_\tors \cong H_2(\K3; \ZZ)_\tors$. If $m\tau = 0$ in the latter, construct an unramified cover $i: \tilde S\to S$ with $\chi(\tilde S) = m\chi(S) = 24m$. - [ ] Since $i$ is unramified, pulling back a top form yields $K_{\tilde S} \sim 0$. - [ ] Noether formula: $\chi(\OO_{\tilde S}) = {1\over 12} (K_{\tilde S}^2 + \chi(\tilde S)) = 2m$. - [ ] The LHS is $2-h^1(\OO_{\tilde S}) \leq 2$, the RHS is $\geq 2$, so $m=1$. ## Lecture 3: Basic properties of K3s - [x] What is the lattice associated to a K3? ✅ 2023-01-28 - [ ] The intersection pairing: $$\begin{align*}H^2(X; \ZZ)\tensorpower{\ZZ}{2} &\to \ZZ \\ \alpha\tensor \beta \mapsto \int_X \alpha\wedge \beta\end{align*}$$ - [ ] Nondegenerate and unimodular by Poincare duality. - [x] What is $\sgn \lkt$? ✅ 2023-01-28 - [ ] Have $\sgn\Lambda_{\K3} = (s_+, s_-)$ where $s_+ + s_- = 22$, can take topological trace to get $$\Trace(\beta) = s_+- s_- = {1\over 3}(c_1^2 -2c_2) = {1\over 3}(0-2\cdot24)= -16$$ - [ ] $c_1 = c_1(\T X) = -K_X$ - [ ] $c_2 = \chi(X) = 24$. - [ ] Combine $s_+ + s_- = 22$ and $s_+ - s_- = -16$ to solve for $\sgn \lkt = (3, 19)$, - [x] Why is $\lkt$ even? ✅ 2023-01-28 - [ ] Take an irreducible curve $C\subseteq X$ and use the genus formula: $$g(C) = 1 + {1\over 2}\qty{C.K_X + C^2} \implies C^2 = 2g-2 \in 2\ZZ$$ - [x] Discuss Milnor's classification. ✅ 2023-01-28 - [ ] Let $\Lambda$ be an indefinite lattice of signature $(s_+, s_-)$ where $s_+,\, s_- > 0$. - [ ] If $\Lambda$ is odd the $\Lambda \cong (1)\sumpower n \oplus (-1)\sumpower n$ where $n>0$. - [ ] If $\Lambda$ is even then $\Lambda \cong U\sumpower h \oplus E_8\sumpower{k}$ where $h > 0$ and $k\geq 0$. - [x] What is $\sgn U$? ✅ 2023-01-28 - [ ] $(1, 1)$ - [x] What is $\sgn E_8$? ✅ 2023-01-28 - [ ] $(8, 0)$ - [x] Discuss positivity of $\beta_{\K3}(\wait, \wait)$ on $\omega_X$. ✅ 2023-01-28 - [ ] $$\beta(\omega_X, \bar\omega_X)=\int_S \omega_X \wedge \bar\omega_X = \int_X \abs{f}\dz_1\wedge\dz_2 \wedge\bar{\dz_1} \wedge \bar{\dz_2} > 0$$ - [ ] Similarly $\beta(\omega_X, \omega_X) = 0$. - [ ] If $\beta(\omega_X, H^{1, 1}) = 0$ so $H^{1, 1} \perp_\beta (H^{2, 0} \oplus H^{0,2})$ - [x] Discuss the period domain. ✅ 2023-01-28 - [ ] $\CC\omega_X \in \PP H^2(X;\CC)$. - [ ] Take a marking $\Phi: H^2(X;\ZZ)\to \lkt$ and define $$\Omega = \ts{[\omega]\in \PP(\lkt\tensor\CC) \st \omega^2 = 0, \abs{\omega} > 0}, \quad \abs{\omega} \da \omega.\overline{\omega}$$ - [ ] Can write as $$\Omega \cong \ts{\tv{z_0:\cdots:z_{21}}\in \PP^{21}\st z_0^2+z_1^2 + z_2^2 - z_3^2 -\cdots - z_{21}^2 = 0,\, \sum_{i=0}^{21} \abs{z_i}^2 > 0}$$ which is an open subset of a quadric in $\PP^{21}$ and thus 20-dimensional. - [ ] One has $\Phi(\CC\omega_X) \subseteq \Omega$; define this as the period domain of a marked K3. - [ ] Torelli gives surjectivity. - [x] What is a marking? ✅ 2023-01-28 - [ ] $\Phi: H^2(X; \ZZ)\iso \lkt$ an isometry. - [x] What is $\dim \PP^n$. ✅ 2023-01-28 - [ ] $\dim \PP^n = n$, locally $n$ degrees of freedom: $p \sim \tv{1: x_1/x_0:\cdots :x_n/x_0}$. - [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. - [x] What is the dimension of quartics in $\PP^3$? Up to coordinate changes? ✅ 2023-01-28 - [ ] $h^0(\OO_{\PP^3}(4)) = {4+3\choose 4} = 35$ and its projective dimension is 34. - [ ] $\dim \PGL_4(\CC) = 16-1 = 15$ - [ ] So $34-15 = 19$ moduli. - [x] What is $h^0(\OO_{\PP^n}(m))$? ✅ 2023-01-28 - [ ] ${m+n\choose m}$. - [x] What is $\dim \PGL_n(\CC)$? ✅ 2023-01-31 - [ ] $n^2 - 1$. - [x] What are the dimensions of moduli for $S_{2,3}, S_{2,2,2}$, and the double plane $S\to \PP^2$? ✅ 2023-01-28 - [ ] All 19. ## Lecture 4: $\NS(X)$ and $\Aut(X)$. - [x] What does $\Omega/\Orth(\lkt)$ parameterize? ✅ 2023-01-29 - [ ] All K3s, including non-projective. - [x] What is the ample divisor associated to an embedding? ✅ 2023-01-29 - [ ] If $\phi: S\injects \PP^n$, take the hyperplane section $L = \phi(S) \intersect H$ for $H$ any hyperplane. - [x] What is the degree of a projective K3? ✅ 2023-01-29 - [ ] $L^2 = 2d$ for $L$ a hyperplane section of its projective embedding. - [ ] Even since $L = \phi(S) \intersect H \in H^2(S; \ZZ) \intersect H^{1, 1}(S)$ which is an even lattice. - [x] What is $L^\perp$? ✅ 2023-01-29 - [ ] $L^\perp \da \ts{v\in \lkt \st \inp{v}{x} = 0 \, \forall x\in L}$. - [x] Describe $[\omega_S]$. ✅ 2023-01-29 - [ ] $$[\omega_S]\in \ts{[\omega] \in \PP(L^\perp\tensor \CC) \st \abs{\omega} > 0,\, \omega^2 = 0 }$$ - [ ] Here $\dim L = 1$ so $\dim L^\perp = 20-1 = 19$. - [x] What is the degree of $F_4$? ✅ 2023-01-29 - [ ] Embed $F_4\injects \PP^3$ and slice by a hyperplane to get 4 points, so $L^2 = 4$ and $d=2$. - [x] Describe $c_1(L)$ for a K3 and its relation to $\Pic$ and $\NS$. ✅ 2023-01-29 - [ ] Take the exponential SES $\ZZ\injects \OO_S \surjects \OO_S\units$ to get $\delta^1: H^1(S; \OO_S\units) \to H^2(S; \ZZ)$, identify $H^1$ with $\Pic$ and $c_1 = \delta^1$. - [ ] Get $\Pic^0(S) = \ker c_1$ and $\Pic(S)/\Pic^0(S)\cong \im c_1 = \NS(X)$. - [x] Discuss $\NS(X)$ for $X$ a K3. ✅ 2023-01-29 - [ ] $\Pic^0(X) = 0$ so $\Pic(X) \cong \NS(X)$. - [ ] The sublattice of $H^{2}(X; \ZZ)$ generated by algebraic cycles. - [x] Describe $\NS(X)$ and $\Pic(X)$ in terms of divisors. ✅ 2023-01-29 - [ ] $\NS(X) = \CDiv(X)/\sim$ modulo algebraic equivalence. - [ ] $\Pic(X) = \CDiv(X)/\sim$ modulo linear equivalence. - [ ] $\CH_k(X) = Z_k(X)/\sim$ modulo rational equivalence - [ ] What are the following notions of equivalence: algebraic, linear, numerical? - [x] What is the lattice structure on $\NS(X)$? ✅ 2023-01-29 - [ ] $\Pic(X) = \NS(X)\injects H^2(X; \ZZ)$ so it is an even primitive sublattice of $\lkt$. - [ ] $T_X \da \NS(X)^\perp$ is also a lattice (transcendental) - [x] What is the Lefschetz 1-1 theorem? ✅ 2023-01-31 - [ ] There is a surjection $c_1: \Pic(X) \surjects H^{1,1} \intersect H^2(X; \ZZ)$. - [x] How is the Lefschetz 1-1 theorem used for K3s? ✅ 2023-01-29 - [ ] $\NS(X) = \Pic(X) = H^{1, 1}(X) \intersect H^2(X; \ZZ) = \omega_S^\perp \intersect H^2(X;\ZZ)$. - [x] What is $\rho(X)$? ✅ 2023-01-29 - [ ] $\rank_\ZZ \Pic(X) = \rank_\ZZ \NS(X)$. - [ ] Since $\NS(X) \leq H^{1, 1}(X)$, there is a bound $\rho(X) \leq 20$. - [x] What is $\sgn H^2(X; \ZZ)$? ✅ 2023-01-29 - [ ] $(3, 19)$ - [x] Discuss $\sgn \NS(X)$ and $\sgn T_X$. ✅ 2023-01-29 - [ ] $(a, b)$ where $a\leq 1$ since $\sgn H^2(X;\ZZ) = (3, 19)$ and if $H^{0, 1} \oplus H^{1,0} = \CC\omega_X \oplus \CC\bar{\omega_X}$ one can form a 2-dimensional space $V = \gens{\omega_X + \bar\omega_X, i(\omega_X - \bar\omega_X)}$ where both generators have positive square. and $V\subseteq (H^{1,1})^\perp$. - [ ] For $X$ projective, $L$ an ample class with $L^2 > 0$ yields $$\sgn \NS(X) = (1, \rho(X) - 1) \implies \sgn T_X = (2, 20-\rho(X))$$ since they must add up to $(3, 19)$. - [x] Discuss the transcendental lattice $T_X$. ✅ 2023-01-29 - [ ] $T_X \da \NS(X)^\perp \subseteq H^2(X; \ZZ)$. - [ ] $T_X \tensor \CC \contains H^{2, 0} \oplus H^{0, 2}$, so the periods move in $T_X$. - [ ] $[\omega_X] \in \ts{ \PP(T_X^L\tensor\CC) \st \omega^2 = 0, \abs{\omega} > 0}$ where $T_X^L = L^\perp$. - [x] Discuss $\Pic(X)$ for a generic projective K3. ✅ 2023-01-29 - [ ] Generically $\Pic(X) = \gens{L}_\ZZ$ for $L$ associated to the hyperplane section. - [ ] What is the base component of $\mcl\in \Pic(X)$? - [ ] What is a complete intersection in $\PP^n$? - [x] Discuss projective models of K3s. ✅ 2023-01-29 - [ ] Saint-Donat 74: if $\mcl \in \Pic(X)$ with $\mcl^2 > 0$ with no base components (so bpf), then the induced embedding $\phi_{\abs{\mcl}}:X\to \PP^N$ where $N = h^0(\OO_X(\mcl)) - 1 = {1\over 2}\mcl^2 -1$ is either - [ ] Hyperelliptic: 2-to-1 cover and $\deg \im \phi_{\abs \mcl} = {1\over 2}\mcl^2$, or - [ ] Non-hyperelliptic: birational with $\deg \im \phi_{\abs \mcl} = \mcl^2$. - [ ] $\mcl^2 = 2$ is the double plane. - [ ] $\mcl^2 = 4$ non-hyperelliptic is $X\to \PP^3$ a quartic - [ ] $\mcl^2 = 6$ non-hyperelliptic is $X\to \PP^4$ a complete intersection $(2, 3)$ - [ ] $\mcl^2 = 8$ non-hyperelliptic is $X\to \PP^5$ a complete intersection $(2, 2, 2)$ - [x] Discuss $\Aut(X)$. ✅ 2023-01-29 - [ ] Torelli induces $\Aut(X) \injects \Orth(H^2(X; \ZZ))$ - [ ] Image is effective Hodge isometries: preserves 2-forms and Kahler cones. - [ ] Discrete and finitely generated. - [ ] Generically trivial when $\Pic(X) = \gens{L}_\ZZ$ for $d\neq 1$. - [ ] If $\sigma$ preserves an ample class then it must have finite order. - [ ] What is a Kahler metric? - [ ] What is a hyperkahler variety? - [x] What is a symplectic automorphism? ✅ 2023-01-29 - [ ] $\sigma^* \omega_X = \omega_X$, i.e. preserves the symplectic form. - [ ] Possibly forces $\rank_\ZZ \Pic(X) \geq 8$. - [ ] Prove that if $\rho(X) = 1$ then $\Aut(X) = 1$. ## Lecture 5: Finite automorphism groups - [x] What is the SES involving the symplectic automorphisms of $G\leq \Aut(X)$ a finite subgroup? ✅ 2023-01-30 - [ ] Take $\alpha: G\to \CC\units$ where $f\mapsto \alpha(f)$ where $f^* \omega_X = \alpha(f)\omega_X$. - [ ] Restrict to its image $\mu_m$ to get a SES defining $G_0 \injects G\surjects \mu_m$ where $G_0 \da \ts{f\in G\st f^*\omega_X = \omega_X}$ . - [x] What are symplectic, non-symplectic, and purely non-symplectic automorphisms? ✅ 2023-01-30 - [ ] Symplectic: $g^*\omega_X = \omega_X$ - [ ] Non-symplectic: $g^*\omega_X = \alpha(g)\omega_X$ where $\alpha\neq 0,1$. - [ ] Purely non-symplectic: $g^* \omega_X = \alpha(g)\omega_X$ where $\alpha(g) = \zeta_m$ a *primitive* root of unity $m=\abs{g}$. - [x] Why are non-symplectic automorphisms important? (Proof sketch) ✅ 2023-01-30 - [ ] If $X$ admits a finite order non-symplectic automorphism, $X$ is automatically projective. - [ ] Proof idea: use Kodaira embedding, find a rational Kahler class. - [ ] Use that $H^2(X;\QQ)\injects H^2(X;\RR)$ is dense and $\mck$ is open in $H^{1, 1}\tensor \RR$. - [ ] Find $c\in H^2(X; \QQ)$, decompose into $c_{2,0} + c_{1,1} + c_{0, 2}$ in $H^2(X;\CC)$ where $c_{1,1}$ is a real Kahler form, and kill $c_{2,0}, c_{0, 2}$ using the automorphism: - [ ] Average to define $h \da \sum_{i=0}^{\abs{g} - 1}(g^*)^i c$ which is $g\dash$invariant. - [ ] Use that $g$ respects the Hodge decomposition, and $h_{1,1} = h\in H^2(X;\QQ)$ since $H^{2,0}, H^{0, 2}$ does not contain $g\dash$invariant elements. - [ ] By Kodaira embedding, some multiple of $h$ gives the embedding. - [x] Give examples of automorphisms of K3s. ✅ 2023-01-30 - [ ] $\iota: F_4\to F_4$ where $\tv{x_0:x_1:x_2:x_3}\mapsto \tv{x_0:x_1:-x_2:-x_3}$, symplectic, $\Fix(\iota)$ is 8 points. - [ ] $\iota: F_4\to F_4$ where $\tv{x_0:x_1:x_2:x_3}\mapsto \tv{x_0:x_1:\zeta_4 x_2: \zeta_4 x_3}$. - [ ] Set $\omega\da (\del_0 f_4)\inv \dx_2 \wedge \dx_3$, then $\omega\mapsto -\omega$, so non-symplectic but not purely non-symplectic. - [x] Describe symplectic automorphisms locally. ✅ 2023-01-30 - [ ] Can find local coordinates where $\sigma \in \GL_2(\CC)$ is of the form $\matt \lambda 0 0 {\bar\lambda}$ with $\lambda$ an $n$th root of unity where $n=\abs{\sigma}$. - [ ] Has only isolated fixed points, at most 8. - [ ] Nikulin counts $\size \Fix(\sigma)$ when $\abs{\sigma}$ has small prime order. - [ ] Quotient has only type $A$ singularities. - [x] What is an Enriques surface? ✅ 2023-01-30 - [ ] Quotient of a K3 by a fixed point free involution (non-symplectic!) - [x] Describe the Lefschetz number of a symplectic automorphism. ✅ 2023-01-30 - [ ] $L(\sigma) \da \sum (-1)^i \Tr(\sigma^* \actson H^i(\OO_S) )$. - [ ] $L(\sigma)\neq 0 \implies \Fix(\sigma)\neq\emptyset$ - [ ] $L(\sigma) = 2$ for a symplectic automorphism. - [x] Why are symplectic automorphisms important? ✅ 2023-01-30 - [ ] Quotients are again K3. - [ ] Not the case for non-symplectic automorphisms! - [x] For $\sigma$ symplectic on $X$ of order $p$, sketch how to compute $\size \Fix(\sigma)$. ✅ 2023-01-30 - [ ] Take $X\surjects X/\sigma$ and resolve to $Y$ to get $X\to Y$ a $p\dash$to-$1$ map; resolving $A_{p-1}$ singularities. - [ ] Write $X' \da X\sm \Fix(\sigma)$ and $Y' \da Y\sm \widetilde{\Fix(\sigma)}$ where $\widetilde{\Fix}(\sigma)$ are the corresponding rational curves to get $\chi(X') = p\chi(Y')$. - [ ] Check $\widetilde{\Fix}(\sigma)$ is a chain of $p-1$ copies of $\PP^1$ meeting at $p-2$ points, so $\chi\widetilde{\Fix}(\sigma) = 2(p-1) - (p-2) = p$. - [ ] Combine these to get $$\chi(X) - \size \Fix(\sigma) = p\cdot \chi(Y)- p\cdot \size \Fix(\sigma) \implies \size \Fix(\sigma) = {24\over p-1}$$ since $Y$ is also a K3 and thus $\chi(X) = \chi(Y) = 24$. - [x] What is the Kodaira embedding theorem? ✅ 2023-01-31 - [ ] If $X$ is compact Kahler and $\mcl\in \Pic(X)$ then $\mcl > 0 \iff \mcl\in \Pic(X)^\amp$. - [ ] $\impliedby$: if $\mcl\tensorpower{}{n} = \OO_{\PP^N}(1)\mid_X$, the hyperplane bundle is positive since its curvature is the Fubini-Study form. - [ ] Positive: $\mcl > 0 \iff \mcl$ admits a metric $h$ such that the associated curvature $\sqrt{-1}\Theta(\mcl, h)$ is a positive 1-1 form. - [ ] Comes from the Chern connection induced by $\delbar:\mcl \to \mcl \tensor H^{0, 1}(M)$, namely $\nabla^{0, 1} = \delbar$. - [ ] How does the Atiyah-Singer index theorem related to the Lefschetz number? - [ ] Draw local pictures of $A_p$ singularities. ## Lecture 6: Classification - [x] Give an example of how having too many rational curves can be used in a proof. ✅ 2023-01-31 - [ ] If $\abs{\sigma} = p = 11$ then $\size \Fix(\sigma) = 2$ yields 2 $A_{10}$ singularities, thus 20 independent $(-2)\dash$curves in $H^2(X; \ZZ)$ which generate a sublattice of signature $(0, 20)$. - [ ] This contradicts $\sgn H^2 = (3, 19)$. - [x] What is an elliptic fibration? ✅ 2023-01-31 - [ ] Proper morphism, connected fibers are smooth genus 1 algebraic curves. - [ ] Equivalently, by proper base change, the generic fiberis a smooth genus 1 curve. - [ ] E.g. any product of elliptic curves, surfaces with $\kappa(X) = 1$, Enriques surfaces - [x] What is the classification of $G\in\zmod^\fin \actson X$ symplectically? ✅ 2023-01-31 - [ ] By Nikulin, 14 possibilities: - [ ] $C_n$ where $n=2,\cdots, 8$. - [ ] $C_n^2$ where $n=2,3,4$. - [ ] $C_2\times C_n$ where $n=4,6$. - [ ] $C_2^3$ - [ ] $C_2^4$. - [x] What is the classification of $G\in\Fin\Grp \actson X$ symplectically? ✅ 2023-01-31 - [ ] Mukai: maximal order $\size G \leq 360$, with equality if $G = M_{20}$. - [x] Discuss $G\in \Fin\Grp\actson X$ not necessarily symplectically. ✅ 2023-01-31 - [ ] Kondo: $\size G \leq 3840$, proved using isometries of lattices. - [ ] Equality if $M_{20}\injects G\surjects C_4$. - [x] What is the Mathieu group? ✅ 2023-01-31 - [ ] $M_{20} = A_5\semidirect C_2^4$, order 960. - [x] What are the possible orders of CM on an elliptic curve? ✅ 2023-01-31 - [ ] $2,3,4,6$. - [x] Give an example of an elliptic curve with CM. ✅ 2023-01-31 - [ ] $y^2 = x^3 + x$ has CM by $\zeta_3$. - [x] Define the invariant lattice. ✅ 2023-01-31 - [ ] $$H^2(X; \ZZ)^\sigma \da \ts{x\in H^2(X;\ZZ) \st \sigma.x = x}$$ - [x] Give an example of a K3 with maximal Picard rank. ✅ 2023-01-31 - [ ] $X = \mathrm{Kummer}(E_3\cartpower{2})$ where $E_3$ is an elliptic curve with CM by $\zeta_3$. - [ ] Satisfies $\rho(X) = 20$. - [ ] Given by $X\injects \PP^3$ by $\sum x_i^4 - 12\prod x_i$. - [x] Discuss $(H^2(X; \ZZ)^\sigma)^\perp$. ✅ 2023-01-31 - [ ] $$(H^2(X; \ZZ)^\sigma)^\perp \subseteq \NS(X) \implies H^2(X;\ZZ)^\sigma \subseteq T_X$$ - [ ] Nikulin shows $(H^2(X;\ZZ)^\sigma)^\perp$ is a negative definite lattice. - [x] Compute $\rank_\ZZ (H^2(X; \ZZ)^\sigma)^\perp$. ✅ 2023-01-31 - [ ] Start with $\rank_\ZZ (H^2(X;\ZZ))^\perp = m(p-1)$. - [ ] $$\begin{align*}\size \Fix(\sigma) &= \chi(\Fix(\sigma)) \\ &= \sum_{i=0}^4 (-1)^i \Tr (\sigma \actson H^i(X;\ZZ)) \\ &= 2 + \Tr(\sigma \actson H^2(X;\ZZ)^\sigma) + \Tr(\sigma \actson ( H^2(X;\ZZ)^\sigma) ^\perp) \\ &= 2 + (22 - m(p-1)) - m \\&\implies m = {24\over p+1}\end{align*}$$ using that $\sum_{i=1}^{m-1}\zeta_m^i = -1$. - [x] Give lower bounds on $\rho(X)$ when $\zeta_p\actson X$. ✅ 2023-01-31 - [ ] ![](attachments/2023-01-30-maybetable.png) - [x] Why are K3s with automorphisms special? ✅ 2023-01-31 - [ ] Gives a lower bound on $\rho(X)$. - [x] Compute the dimension of moduli of K3s with an order 3 symplectic automorphism. ✅ 2023-01-31 - [ ] Dimension 7. How to show: ...? - [x] Discuss $\NS(X)$ when $X$ has an order 3 automorphism. ✅ 2023-01-31 - [ ] Always get the Coxeter-Todd lattice, whose points give highest density sphere packing in dimension 12.