--- created: 2023-03-26T11:58 updated: 2024-06-08T11:35 flashcard: Dissertation::April_2024 --- # 000 Enriques talk draft 1 # From [Notes Kodaira Dimension](Notes%20Kodaira%20Dimension.md) - [x] What is a weak CY? ✅ 2024-01-31 - [ ] $X$ smooth projective with $\omega_X \cong \OO_X$. - [x] What is a (strong) CY? ✅ 2023-01-14 - [ ] A weak CY with $H^i(\OO_X) = 0$ for $0 \lt i \lt \dim(X)$ and $\pi_1(X(\CC)) = 1$. - [x] How is $C.D$ defined for curves on a surface? ✅ 2023-01-25 - [ ] $$C.D = \sum_{p\in C\intersect D}(C.D)_p, \quad (C.D)_p \da \dim_k \OO_{X, p}/\gens{f, g}$$ where $C \transverse D \iff \mfm_x = \gens{f, g}$ for $f,g$ their local equations. - [x] What is RR for surfaces? ✅ 2023-01-25 - [ ] For $D\in \Div(X)$, $$\chi(\OO_X(D)) - \chi(\OO_X) = {D\cdot (D-K_X) \over 2}$$ # From [Alessandra Sarti Topics on K3 surfaces](Alessandra%20Sarti%20Topics%20on%20K3%20surfaces.md) - [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] 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] 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 $\pi_1(X_4)$? ✅ 2023-01-27 - [ ] By Lefschetz, $\pi_1(X_4) = \pi_1(\PP^3) = 1$. - [x] What is the Noether formula? ✅ 2023-01-27 - [ ] $\chi(\OO_X) = {1\over 12}(K_X^2 + \chi(X))$. - [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] 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 $m=1$. - [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] 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 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] 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 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. - [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] 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 $\nabla^{0, 1} = \delbar$. - [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] 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] Why are K3s with automorphisms special? ✅ 2023-01-31 - [ ] Gives a lower bound on $\rho(X)$. # From [Beauville Complex Surfaces](Books%20Beauville%20Complex%20Surfaces.md) - [x] How is $L.L'$ defined for $L, L'\in \Pic(X)$? ✅ 2023-01-25 - [ ] $L.L' = \chi(\OO_X) - \chi(L\inv) - \chi((L')\inv) + \chi(L\inv \tensor (L')\inv)$. - [x] For $C_1, C_2$ curves, what is $\OO_X(C_1) . \OO_X(C_2)$? ✅ 2023-01-25 - [ ] $\OO_X(C_1) . \OO_X(C_2) = C_1.C_2$ - [x] For $C$ a smooth irreducible curve and $L\in \Pic(X)$, what is $\OO_X(C) . L$? ✅ 2023-01-25 - [ ] $\OO_X(C) . L = \deg\ro{L}{C}$ # From [Books Lectures on K3 Surfaces (Huybrechts)](Books%20Lectures%20on%20K3%20Surfaces%20(Huybrechts).md) - [x] Why are K3s important in string theory? ✅ 2023-02-07 - [ ] 2nd simplest example of a compact Ricci-flat manifold after the torus. - [ ] Tends to show up wherever the heterotic string does - [x] What is the importance of $(-2)\dash$curves for K3s? ✅ 2023-02-07 - [ ] $C.C = 2g-2$, so rational curves are always -2. - [ ] So in fact all self-intersections are even. - [x] Why is Ricci-flatness important? ✅ 2023-02-07 - [ ] Vacuum solutions in string theory (expansions of Einstein's field equations) - [x] What is a geometric point? ✅ 2023-01-10 - [ ] A morphism $\spec k \to X$ where $k=\kbar$. - [x] What is a geometric fiber? ✅ 2023-01-10 - [ ] The fiber over a geometric point, i.e. for $f:X\to Y$, the fiber product with $p_i: \spec k \to Y$ given by $X\fiberprod{Y}\spec k$. - [x] What is a geometrically integral scheme? ✅ 2023-01-10 - [ ] Idea: prevents becoming reduced or reducible after a field extension. - [ ] The structure morphism $f: X\to \spec k$ is geometrically integral, i.e.for all geometric points $p_i: \spec L \to \spec k$ the fiber $X\fiberprod{\spec k} \spec L$ is integral. - [ ] Thm: geometrically integral iff geometrically reduced and geometrically irreducible. - [x] Discuss the cotangent sheaf of a K3. ✅ 2023-01-10 - [ ] Locally free of rank 2, $\Omega_X^2 = \det \Omega_X = \omega_X \cong \OO_X$. - [ ] Carries an algebraic symplectic structure $(\Omega_X)\cartpower{2} \to \OO_X$ inducing an isomorphism $\T_X\cong \Omega_X\dual \da \sheafhom(\Omega_X, \OO_X) \cong \Omega_X$. - [x] Why is a smooth quartic a K3? ✅ 2023-01-10 - [ ] If $X \subseteq \PP^3$ then take the LES for $\OO_{\PP^3}(-4) \injects\OO_{\PP^3} \surjects\OO_X$ and use that $H^1(\OO_{\PP^3}) = H^2(\OO_{\PP^3}(-4)) = 0$ to conclude $H^1(\OO_X) = 0$. - [ ] The adjunction formula yields $\omega_X = \omega_{\PP^3}\tensor \OO(4)\mid_X \cong \OO_X$. - [x] Give several examples of K3s. ✅ 2023-01-10 - [ ] A smooth quartic, e.g. $x_0^4 + \cdots + x_3^4\subset \PP^3$ when $\characteristic k \neq 2$. - [ ] A smooth complete intersection of type $(d_1,\cdots, d_n) \subset \PP^{n+2}$ when $\sum d_i = n+3$. - [ ] Kummer surfaces: the quotient of an abelian surface by the involution $x\mapsto -x$ after resolving 16 singular points. - [ ] A branched double cover $X\to \PP^2$ branched along a smooth curve $C$ of degree 6, using $\pi_* \OO_X \cong \OO_{\PP^2} \oplus \OO(-3) \implies H^1(\OO_X) = 0$ and $\omega_X \cong \pi^*(\omega_{\PP^2} \tensor \OO(3)) \cong \OO_X$. - [x] For $L_1, L_2\in \Pic X$, define $L_1.L_2$. ✅ 2023-01-10 - [ ] The coefficient of $ab$ in $\chi(L_1^1\tensor L_2^b)$. - [x] What is Riemann-Roch for line bundles on surfaces? ✅ 2023-01-10 - [ ] $\chi(L) - \chi(\OO_X) = {1\over 2} (L.L \tensor \omega_X\dual)$. - [x] What is the Picard number? ✅ 2023-01-10 - [ ] $p(X) = \rank_\ZZ \NS(X)$ where $\NS(X) \da \coker(\Pic^0(X) \injects \Pic(X))$. - [x] What is the signature of the intersection form on $\Num(X)$? ✅ 2023-01-10 - [ ] $(1, p(X) - 1)$ for $p(X)$ the Picard number, so its real form diagonalizes to $(1, -1,\cdots, -1)$. - [x] What is $\chi(\OO_X)$ for a K3? ✅ 2023-01-10 - [ ] 2: $h^0(\OO_X) = h^1(\OO_X) = 0$ by definition, and by Serre duality $h^2(\OO_X) = h^0(\omega_X) = 1$. - [x] What is $\pi_1^\et(X)$ for a K3? ✅ 2023-01-10 - [ ] Trivial: if $Y\to X$ is an irreducible finite etale cover of degree $d$ then $Y$ is a smooth complete surface, $\omega_Y \cong\OO_Y$, and $\chi(\OO_Y) = d\chi(\OO_X) =2d$ with $h^0(\OO_Y) = h^2(\OO_Y) = 1$. - [ ] Thus $2-h^1(\OO_Y) = 2d \implies d=1$. - [x] What is the statement of RR for $L\in \Pic(X)$ for $X$ a K3? How does this change if $L$ is ample? ✅ 2023-01-10 - [ ] $\chi(L) = {1\over 2}L^2 + 2$. - [ ] If $L$ is ample then $H^1(L) = 0$ so the LHS is $h^0(L)$. - [x] Give an equivalent characterization of when a line bundle is trivial. ✅ 2023-01-10 - [ ] Both $H^0(L)$ and $H^0(L\dual)$ are nontrivial. - [x] What is the Hirzebruch-Riemann-Roch theorem? ✅ 2023-01-10 - [ ] $\chi(\mcf) = \int \chern(\mcf)\Todd(X) = \chern_2(\mcf) + 2\rank(\mcf)$. - [x] What is the Noether formula for K3s? ✅ 2023-01-10 - [ ] $\chi(\OO_X) = {1\over 12}(c_1^2(X) + c_2(X)) = {c_2(X)\over 12}$. - [x] What is $c_2(X)$ for a K3? ✅ 2023-01-10 - [ ] 24 by the Noether formula. - [x] Discuss $h^0(\T_X)$ for $X$ a K3. ✅ 2023-01-10 - [ ] $h^0(\T_X) = h^0(\Omega_X)= 0$ using the isomorphism $\T_X\cong \Omega_X$. So no global vector fields. - [x] Discuss GAGA ✅ 2023-01-10 - [ ] To any $X\in\Sch^\ft\slice \CC$ there exist $X^\an \to X$ a morphism of ringed spaces from an analytic space where the points of $X^\an$ correspond to the closed points of $X$ and there is an induced equivalence $\Coh(X)\iso \Coh(X^\an)$. - [x] Discuss $H^*_\sing(X)$ for a K3. ✅ 2023-01-10 - [ ] Take the LES $H^1(X;\ZZ) \to H^1(X;\OO)\to H^1(X;\OO\units)$ to show $H^1(X;\ZZ) = H^3(X;\ZZ) = 0$ mod torsion and $H^0(X;\ZZ) = H^4(X;\ZZ) = \ZZ$. The only other nontrivial group is $H^2(X; \ZZ)$. - [x] Discuss $H^2(X; \ZZ)$ ✅ 2023-01-10 - [ ] Torsionfree, since $0\to \Pic(X) \to H^2(X;\ZZ) \to H^2(X; \OO)$ using $\Pic(X)\cong H^1(X; \OO\units)$; since $H^2(X;\OO)\cong \CC$ and $\Pic(X)$ are torsionfree, so is $H^2$. - [ ] The intersection form on $\Pic(X)$ corresponds to the intersection form on $H^2$ under this embedding. - [ ] $\rank_\ZZ H^2 = 22$ since $e(X) = c_2(X) = 24$ and $b_2(X) = 22$, and this defines a unimodular lattice $\E_8(-1)\sumpower{2} \oplus U\sumpower{3}$. - [x] What is the Hodge-Frölicher spectral sequence? ✅ 2023-02-01 - [ ] $$H^q(X; \Omega^p_X) \abuts H^{p+q}(X; \CC)$$ - [x] What is the signature of the intersection form on $H^2$? ✅ 2023-01-10 - [ ] By the Thom-Hirzebruch index theorem, ${1\over 3}p_1(X) = {1\over 3}(c_1^2(X) - 2c_2(X)) = -16$, use that $b_2(X) = 22$ to get $(3, 19)$. - [x] How is $\rho(X)\leq 22$ proved for K3s over positive characteristic fields? ✅ 2023-01-10 - [ ] Take the LES in $H^*_\et$ for the Kummer sequence $\mu_n\injects \GG_m\surjects \GG_m$ for $n$ prime to $p$, use $\Pic(X) \cong H^1(X;\GG_m)$ to show $H^1_\et(X; \mu_n) = k\units/(k\units)^n$. - [ ] Use $k$ separably closed and duality to get $H^1_\et(X; \ZZladic) = H^3(X;\ZZladic) = 0$. - [ ] Use $c_2(X) = 24$ to conclude $\rank_\ZZ H^2_\et(X; \ZZladic) = 22$. - [x] How is $c_1$ realized as a morphism: ✅ 2023-01-10 - [ ] $c_1: \Pic(X) \to H^2_\et(X; \ZZladic(1))$. - [x] What is an elliptic surface? ✅ 2023-01-10 - [ ] $\pi:X\to \PP^1$ surjective with generic fiber a smooth elliptic curve. Admit Weierstrass normal forms. - [x] Define the base locus $\mathrm{Bs}\abs{L}$. ✅ 2023-01-13 - [ ] The maximal closed subscheme of $X$ contained in every $D\in \abs{L}$. - [ ] Equivalently $$\mathrm{Bs}\abs{L} = \Intersect_{s\in H^0(L)} Z(s)$$ - [x] When does $L$ induce a rational map? ✅ 2023-01-13 - [ ] If $h^0(L) > 1$, inducing $\phi_L: X\rational \PP H^0(L)\dual$ regular on $X\sm \mathrm{Bs}\abs{L}$. - [x] What is the geometric genus of a reduced curve? ✅ 2023-01-13 - [ ] $g(\tilde C)$ where $\nu:\tilde C\to C$ is the normalization. - [x] How are geometric and arithmetic genus related for a reduced curve? ✅ 2023-01-13 - [ ] $p_a(C) = g(C) + h^0(\nu_* \OO_{\tilde C}/\OO_C)$ where the latter sheaf is concentrated on $C^\sing$. - [x] Discuss curves on a K3. ✅ 2023-01-12 - [ ] By adjunction, $C^2\geq -2$. Automatically smooth, since $g(C) \leq p_a(G)$ with equality iff $C$ is smooth, and here $p_a(C) = p_g(C) = 0$. Any $(-2)\dash$curve is integral. - [x] When is a line bundle on a K3 ample? ✅ 2023-01-12 - [ ] $L$ is ample iff $L\in \mcc_X \subset \NS(X)_\RR$ (the positive cone) and $(L.C) > 0$ for every smooth rational curve $C\cong \PP^1 \subseteq X$. - [x] What is a nef line bundle $L$ for a complete variety $X$? ✅ 2023-01-12 - [ ] $(L.C) > 0$ for all closed curves. - [x] What does it mean for a line bundle on a surface to be big and nef? Why care? ✅ 2023-01-12 - [ ] $L^2 > 0$ and $L$ is nef. - [ ] Weaker than ample, but results for ample often still hold for just big and nef. Note that $L^2 > 0$ is not quite the right definition of "big" alone. - [ ] Big and nef on a K3 implies effective. - [x] What is the generalization of Kodaira vanishing for projective surfaces (Kodaira–Ramanujam vanishing theorem)? ✅ 2023-01-12 - [ ] For $X$ a smooth projective surface over $\characteristic k = 0$, if $L$ is big and nef then $$H^{> 0}(L\tensor \omega_X) = 0$$ Since $H^1(\omega_X\tensor L)\dual \cong H^1(L\dual)$ by Serre duality, this is a vanishing theorem for $L\dual$ with sufficient "positivity". - [x] Does Kodaira vanishing hold in positive characteristic? ✅ 2023-01-13 - [ ] Generally no, see Mumford's normal projective surface as a counterexample, or Raynaud finds for any smooth projective surface $X$ over $k=\kbar$ an $L\in \Pic(X)$ with $H^1(L\tensor \omega_X) \neq 0$. - [ ] However it *does* hold for K3s, using $H^1(\OO_X) = 0$ and any big and nef $L$ on a K3 is effective. This is explained by Deligne-Illusie for surfaces that lift to characteristic zero (which K3s do). - [x] Discuss $L^2$ and $h^0(L)$ for $L = \OO_X(C)$, $C$ a curve on a K3 $X$. ✅ 2023-01-13 - [ ] Let $L= \OO_X(C)$ for $C\subseteq X$ a smooth irreducible curve of genus $g$ on a K3 $X$, then $$L^2 = 2g-2,\qquad h^0(L) = g+1$$ - [ ] Use adjunction to prove the first, and RR + Serre duality + $H^0(\ro L C) \cong H^0(\omega_C)$ and the SES $H^0(X; \OO_X) \injects H^0(X; L)\surjects H^0(C; \ro L C)$. - [x] Let $L\in \Pic(X)$ be big and nef. Show that $\abs{L}$ is bpf. ✅ 2023-01-17 - [ ] Decompose $L = M + F$. - [ ] Mobile $\implies$ effective $\implies$ nef, so $M^2 \geq 0$. - [ ] If $M^2 > 0$, then $M$ is big and nef so $L$ has at most isolated base points. - [ ] Check $H^1(X; M) = 0 = H^1(X; L)$ so $\chi(M) = h^0(M) = h^0(L) = \chi(L)$. - [ ] By RR, $M^2 = L^2$, so $2(M.F) + F^2 = 0$ - [ ] $L$ nef $\implies (M.F) + F^2 = L.F\geq 0 \implies M.F = F^2 = 0$ - [ ] By RR on $F\neq 0$, reach a contradiction $1 = h^0(F) \geq \chi(F)$, so $F=0$.| - [x] What is the Base Point Free theorem? ✅ 2023-01-17 - [ ] If $L\in \Pic(X)$ is big and nef, then $L^n$ is globally generated for some $n$. - [ ] Holds for smooth projective varieties over $\CC$ when $L\tensor \omega_X\dual$ is big and nef. - [x] Discuss when $L\in \Pic(X)$ for $X$ a K3 is bpf. ✅ 2023-01-17 - [ ] Sufficient condition: $L^2 > 0$ and $\abs{L}$ contains an irreducible curve. - [x] Discuss irreducible curves on a K3 in $\characteristic k \neq 2$. ✅ 2023-01-17 - [ ] Always have $C^2 > 0$, $\abs{\OO_X(C)}$ is bpf, its generic member $C'$ is smooth. - [ ] Either $C'$ is hyperelliptic (so $\deg \phi = 2$) or not (then $\deg \phi = 1$ is birational) - [x] When is a K3 surface elliptic? ✅ 2023-01-17 - [ ] $X$ is elliptic $\iff \exists L\in \Pic(X)$ with $L^2 =0$ away from characteristic 2,3. - [ ] Idea: pass from $L$ with $L^2 = 0$ to a nef $L'$ with $(L')^2 = 0$ through a series of reflections $L\mapsto L + (L.C)C$ for $C$ a $(-2)\dash$curve with $L.C < 0$. - [x] State Bertini's theorem. ✅ 2023-02-20 - [ ] For smooth projective varieties $X\subseteq \PP^N$, a generic hyperplane section $H\intersect X$ is smooth. - [x] What is a polarized K3 of degree $2d$? ✅ 2023-01-17 - [ ] A projective K3 $X$ with $L\in \Pic(X)^\amp$ which is primitive and satisfies $L^2 = 2d$. - [x] What is the genus of a polarized K3? ✅ 2023-01-17 - [ ] Write $2d = 2g-2$, then $g = g(C)$ for any $C\in \abs{L}$, we say $(X, L)$ is a polarized K3 of genus $g$. - [x] Do there exist K3s of arbitrary degree over $k=\kbar$? ✅ 2023-01-17 - [ ] Yes, for $g\geq 3$ over $k=\kbar$ there exists K3s of degree $2g-2$ in $\PP^g$. - [ ] Idea: for $g=3k$, take a generic quartic $X$ in $\PP^3$ containing a line $\ell$. - [ ] Let $H$ be the hyperplane section and consider $\abs{H-\ell}$ - [ ] This defines an elliptic pencil $\abs{E}$. - [ ] Consider $L_k \da H + (k-1)E \in \Pic(X)^\vamp$, for generic $X$ it is primitive. - [ ] Yields a polarized K3 of degree $L_k^2 = 6k-2$. - [x] Do there exist K3s of arbitrary degree over $k\neq \kbar$? ✅ 2023-01-17 - [ ] No: for $k=\FF_q$, the degree of a polarized K3 defined over $k$ is bounded. - [ ] Related to questions about rational points on the moduli of K3s. - [x] Discuss $\Pic(X)^\amp \subseteq \Pic(X)$. ✅ 2023-01-17 - [ ] An open subspace, i.e. ampleness is an open property. - [x] What is a Hodge structure of weight $n$? ✅ 2023-01-19 - [ ] For $V\in \mods{\CC}$, a Hodge structure of weight $n \in \mathbf{Z}$ on $V$ is given by a direct sum decomposition of the complex vector space $V_{\mathbf{C}}$ $$V_{\mathbf{C}}=\bigoplus_{p+q=n} V^{p, q} \quad\text{such that} \quad \overline{V^{p, q}}=V^{q, p}$$ - [x] What are isogenous Hodge structures? ✅ 2023-01-19 - [ ] $V, W$ with integral Hodge structures are isogenous iff $V_\QQ\cong W_\QQ$. - [x] What are Hodge classes? ✅ 2023-01-19 - [ ] For $V$ a Hodge structure of even weight $2k$, the intersection $V\intersect V^{k, k}$. - [x] What are $\ZZ(1), \QQ(1)$? ✅ 2023-01-19 - [ ] The Tate Hodge structure $\mathbf{Z}(1)$ is the Hodge structure of weight $-2$ given by the free $\mathbf{Z}$-module of rank one $(2 \pi i) \mathbf{Z}$ (as a submodule of $\mathbf{C}$ ) such that $\mathbf{Z}(1)^{-1,-1}$ is one-dimensional. Similarly, one defines the rational Tate Hodge structure $\mathbf{Q}(1)$. - [x] What is $\ZZ(k)$ for $k\in \ZZ$? ✅ 2023-01-19 - [ ] $\ZZ(-1) = \ZZ(1)\dual \da \Hom_{\cmod}(V_\CC, \CC)$ and $\ZZ(k) \da \ZZ(1)\tensorpower{\CC}{k}$ whose underlying module is $(2\pi i )^k \ZZ$. - [ ] $\ZZ(0)$ is the trivial Hodge structure of weight zero and rank one. - [x] What is the Tate twist $V(k)$ for $V$ a Hodge structure of weight $n$? ✅ 2023-01-19 - [ ] $V(k) \da V\tensor_\CC \ZZ(k)$, so $V(k)^{p, q} = V^{p+k, q+k}$ which is weight $n-2k$. - [x] What is the Hodge filtration? ✅ 2023-01-19 - [ ] $\Fil^i V_\CC \da \bigoplus_{p\geq i} V^{p, q}_\CC$, satisfies $\Fil^p V_\CC \oplus\bar{\Fil^q V_\CC} = V_\CC$ for all $p+q=n+1$. - [x] What is the Hodge structure associated to an arbitrary Hodge filtration? ✅ 2023-01-19 - [ ] For $\Fil^* V$ a Hodge filtration, define $V^{p, n-p} \da \Fil^p V \intersect \bar{\Fil^{n-p} V}$. - [x] Discuss the status of the Hodge conjecture. ✅ 2023-01-19 - [ ] Can fail for nonprojective hyperkählers, integral version known not to hold. - [ ] Known for $(1, 1)\dash$classes and $(\dim X-1, \dim X - 1)\dash$classes. - [ ] Known for K3s, but not for products of K3s. - [x] What is a polarization and a polarized Hodge structure? ✅ 2023-01-19 - [ ] A polarization of a rational Hodge structure $V$ of weight $n$ is a morphism of Hodge structures $\psi: V \otimes V \rightarrow \mathbf{Q}(-n)$ such that its $\mathbf{R}$-linear extension yields a positive definite symmetric form $(v, w) \longmapsto \psi(v, C w)$ on the real part of $V^{p, q} \oplus V^{q, p}$. Then $(V, \psi)$ is called a polarized Hodge structure. - [x] What is the transcendental lattice of a K3 surface? ✅ 2023-01-19 - [ ] If $V$ is the Hodge structure $H^2(X, \mathbf{Z})$ of a K3 surface $X$, then $$\begin{align}V^{1,1} \cap V=H^{1,1}(X) \cap H^2(X, \mathbf{Z}) \simeq \mathrm{NS}(X) \simeq \operatorname{Pic}(X),\end{align}$$see Section 1.3.3, and $T$ is called the transcendental lattice $$\begin{align} T(X) \subset H^2(X, \mathbf{Z})\end{align}$$of the K3 surface $X$. It is usually considered as an integral Hodge structure. - [x] Discuss the transcendental o of a K3 surface. ✅ 2023-01-19 - [ ] $T(X) = \NS(X)^\perp$, and if $X$ is projective then $T(X)$ is a polarizable irreducible Hodge structure. - [ ] For nonprojective K3s, $T(X) + \NS(X) \subseteq H^2(X; \ZZ)$ need not be direct and $T(X)$ need not be irreducible nor polarizable. - [x] What is the moduli functor $\mcm_d$? ✅ 2023-02-02 - [ ] $$\begin{align}\opcat{ \Sch\slice S} &\to \Set \\T &\mapsto \ts{(f: X\to T, L)}\modiso\end{align}$$ where $f$ is smooth proper and $L \in \Pic_{X/T}(T)$ where all geometric fibers $X_k$ are polarized K3s: a K3 $X_k$ with primitive ample $L_k \in \Pic(X_k)$ with $L_k^2 = 2d$. - [x] What is a fine moduli space? ✅ 2023-02-02 - [ ] For a functor $\mcm_d$, a scheme $M_d\in \Sch\slice S$ and an isomorphism of functors $\mcm_d \cong h_{M_d}$ (the functor of points) - [x] What is a coarse moduli space? ✅ 2023-02-02 - [ ] For $\mcm_d$ a functor, a scheme $M_d\in \Sch\slice C$ and a *morphism* of functors $\Psi:\mcm_d \to M_d$ where - [ ] If $k=\kbar$, there is an induced iso $\mcm_d(\spec k)\iso M_d(\spec k) \in \Set$. - [ ] Universality: $\forall N\in \Sch\slice S$ and $\forall \Phi: \mcm_d \to h_N$ there is a unique $\pi: M\to N$ factoring it. - [x] Discuss scheme-theoretic properties of $\mcm_d$. ✅ 2023-02-02 - [ ] Separated algebraic space, locally of finite type over $S$ - [ ] For $S = \spec \CC$, coarsely represented by a quasiprojective variety. - [ ] Its coarse space $M_d$ is etale-locally the quotient of a smooth scheme by a finite group, using Luna's etale slice theorem, thus is not smooth. - [x] What does it mean for $\mcm_d$ to be a stack? ✅ 2024-01-29 - [ ] A functor $\Sch\slice S\op\to \Set$ or equivalently - [ ] A category $G$ over $\Sch\slice S$: objects are pairs $(f: X\to T, L)$ with a projection$G\to \Sch\slice S$ where $(X\to T, L)\mapsto T$, whose fibers are groupoids. - [x] Discuss the stack-theoretic properties of $\mcm_d$. ✅ 2024-01-29 - [ ] Separated, DM, finite-type. - [ ] Keel-Mori show any such stack has a coarse moduli space in algebraic spaces. - [ ] Resolve singularities to get a smooth stack over $\spec \ZZ\adjoin{1\over 2d}$. - [x] What is a quasi-polarized K3? ✅ 2024-01-29 - [ ] $(X, L)$ with $L$ big and nef of square $2d$. - [x] Describe the moduli stack of quasipolarized K3s. ✅ 2024-01-29 - [ ] Smooth DM of finite type $\mcf_g$. - [ ] Coarse space is quasiprojective - [ ] Open substack of polarized K3s is separated. - [x] What is the Noether-Lefschetz locus $\mathrm{NL}^1(\mcf_d)$? ✅ 2024-01-29 - [ ] K3s $X$ with $\rho(X) \geq 1$. - [x] What is the positive cone? ✅ 2023-02-19 - [ ] $C_X\in \pi_0\ts{v\in \NS(X) \st v^2 > 0}\subseteq \NS(X)_\RR$ the component containing one (equivalently all) ample classes. - [x] What is the ample cone? ✅ 2023-02-19 - [ ] $\Amp(X) \da\ts{\sum a_i L_i \st L_i\in \NS(X)^\amp, a_i\in \RR_{\gt 0}} \subseteq \NS(X)_\RR$. - [ ] The cone spanned by ample line bundles. - [x] What is the nef cone? ✅ 2023-02-19 - [ ] $\Nef(X) \da \ts{v\in \NS(X) \st v.C \geq 0 \text{ for all curves } C}$. - [ ] Not spanned by nef bundles. - [x] How are $\Amp(X)$ and $\Nef(X)$ related? ✅ 2023-02-19 - [ ] Nakai-Moishezon-Kleiman: For $X$ smooth projective over any field and $L\in \Pic(X)$, $$L\in \Pic(X)^\amp \iff L^2>0 \text{ and } L.C \geq 0 \,\text{ for all curves } C$$ - [x] What is the effective cone $\NE(X)$? ✅ 2023-02-19 - [ ] The cone of curves, $$\NE(X) \da \ts{\sum a_i [C_i] \st C_i\text{ are irreducible curves}, a_i\in \RR_{\gt 0}}\subseteq \NS(X)_\RR$$ - [x] What is the Mori cone? ✅ 2023-02-19 - [ ] $\cl_{\NS(X)_\RR} \NE(X)$. # From [Flashcards Algebraic Geometry](Flashcards%20Algebraic%20Geometry.md) - [x] What is $\torsors{G}\slice X$? ✅ 2023-02-05 - [ ] Schemes $Y\to X$ with a fiberwise $G\dash$action which etale-locally looks like $G\times X$ with $G\actson G$ by right-translation, and morphisms are $G\dash$equivariant morphisms over $X$. - [x] What is an etale morphism in $\calg\slice\ZZ$? ✅ 2023-02-12 - [ ] $\pi: R\to S\in \calg\slice \ZZ$ is etale iff - [ ] $\pi$ is flat - [ ] $\del_2: S\tensorpower{R}{2}\to S$ is flat.s - [ ] $S$ is finitely presented over $R$ - [x] Discuss how etaleness is related to smoothness. ✅ 2023-02-12 - [ ] Etale $\implies$ smooth. - [ ] Smooth and $\Omega_{S/R} = 0 \implies$ etale. - [x] What is a big bundle? ✅ 2023-02-14 - [ ] Sections grow maximally: $h^0(L^k)\geq \bigo( k^{\dim X} )$. - [x] What is a nef bundle? ✅ 2023-02-14 - [ ] $L$ has non-negative degree on every irreducible curve $C$ - [ ] Degree: $\deg \div(s)$ for $s$ any rational section of $L$. - [x] What is a big and nef bundle? ✅ 2023-02-14 - [ ] $L^2 > 0$ and $L$ is nef. - [x] What is a large complex structure limit? ✅ 2023-02-12 - [ ] Idea: line bundle $L$ over the total space of a family which specifies $c_1(L)$ and thus a symplectic class. - [ ] A polarized algebraic family of $n$-dim CY manifolds $X \rightarrow S \backslash\{0\}$ over a punctured algebraic curve, whose 'essential skeleton' (simplicial subcomplex of dual complex) has dimension $n$. - [ ] Also called "maximal degeneration" - [ ] Semistable SNC model. - [x] What is a stable curve? ✅ 2023-02-08 - [ ] Nodal with $\size \Aut(X) < \infty$ (nodes: $xy=0$). - [x] What is the scheme-theoretic definition of an elliptic curve over $R$? ✅ 2023-02-12 - [ ] A smooth proper morphism $\pi: E\to \spec R$ with a section $0: \spec R\to E$ where the geometric fibers of $\pi$ are connected genus 1 curves. - [ ] Implies $h^0(E; \Omega_{E/R}) =1$. - [x] What is a hyperplane section? Give an example. ✅ 2023-02-12 - [ ] $Y \subseteq X$ of the form $Y = X\intersect H$ for $X\embeds \PP^N$ and $H\subseteq \PP^N$ a hyperplane. - [ ] Any hypersurface is an example. - [x] What is the Lefschetz decomposition? ✅ 2023-02-12 - [ ] $$H^k(X; \CC) = \bigoplus_{2r\leq k}L^r H^{k-2r}_\prim(X)$$ - [x] What is the hard Lefschetz theorem? ✅ 2023-02-07 - [ ] For $X$ compact Kahler, for $k\leq n\da \dim X$, - [ ] $L^{n-k}: H^k(X; \RR)\iso H^{2n-k}(X; \RR)$ - [ ] $H^k(X; \RR) = \bigoplus_{i\geq 0} L^i H^{k-2i}(X; \RR)_\prim$ - [ ] $H^k(X; \RR)_\prim \tensor_\RR \CC =\bigoplus_{p+q=k} H^{p, q}(X)_\prim$. - [x] What is the Hodge index theorem? ✅ 2023-02-07 - [ ] For $X$ a compact Kahler surface, the intersection form satisfies $$\sgn H^2(X; \ZZ) = (2h^{2, 0} + 1,\, h^{1,1} -1), \qquad \sgn H^{1,1}(X) = (1,\, h^{1,1}-1)$$ - [ ] Equivalently, if $L\in \NS(X)^\amp$ (or even $L^2 \gt 0$) then the intersection form is negative definite on $L^\perp$. - [x] What is the Lefschetz $(1, 1)\dash$theorem? ✅ 2023-02-07 - [ ] For $X$ compact Kahler, $\Pic(X) \surjects H^{1, 1}(X)$. - [x] What is Batyrev-Borisov's mirror symmetry? ✅ 2023-02-12 - [ ] For e.g. complete intersections in Fanos, $H^1(X; \T\dual X) \cong H^1(X\dual; \T X\dual)$. - [ ] LHS: variations of $\omega_X$ a symplectic form (A-model) - [ ] RHS: variations of complex structure (B-model) # From [Lattice Theory](Lattice%20Theory.md) - [x] What is the signature of a lattice? ✅ 2023-01-17 - [ ] The signature of the associated bilinear form $\beta$. - [ ] Change variables to write $\beta(x) = \sum \lambda_i x_i^2$ where $\lambda_i \in \ts{0, \pm 1}$, then the signature is $(n_+, n_-)$, the counts of how many times $+1$ and $-1$ appear respectively. - [x] What is a positive/negative definite form? A nondegenerate form? ✅ 2023-01-17 - [ ] Positive: $\lambda_i = +1$ for all $i$. - [ ] Negative: $\lambda_i = -1$ for all $i$. - [ ] Nondegenerate: $\lambda_i \neq 0$ for all $i$. - [x] What is an integral lattice? ✅ 2023-01-17 - [ ] $\inp{x}{y}\in \ZZ$ for all $x,y\in L$. - [x] What is the $E_8$ lattice? ✅ 2023-01-17 - [ ] The unique positive-definite, even, unimodular lattice of rank 8. - [ ] Points in $\ZZ^8$ whose coordinates are either all integers or all half-integers and the sum of coordinates is even. - [ ] Root lattice of $E_8$ ![](../attachments/2023-01-17-E8root.png) $$\left(\begin{array}{cccccccc} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 2 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2\end{array}\right)$$ - [ ] - [x] What is a lattice? ✅ 2023-01-07 - [ ] A lattice is a free abelian group with an $\ZZ\dash$valued symmetric bilinear form. - [x] What is a root in a lattice? ✅ 2023-01-07 - [ ] A vector $v\in \Lambda$ with $v^2 = -2$. - [x] What is the Weyl group of a lattice? ✅ 2023-01-07 - [ ] $W(\Lambda)\normal \Orth(\Lambda)$ generated by reflections $v\mapsto v + (v, \delta)\delta$ for $\delta$ the roots of $\Lambda$. - [x] What is the hyperbolic lattice $H$? What is its bilinear form? ✅ 2023-02-07 - [ ] The matrix of the quadratic form is $\matt 0 1 1 0$; so a basis with two isotropic elements. - [ ] Nondegenerate since $\disc H = -1 \neq 0$. - [ ] $Q(x,y)= xy$, equivalently to $Q(x, y) = x^2-y^2$. - [x] What is the dual of a lattice? ✅ 2023-02-07 - [ ] $L\dual \da \zmod(L\to \ZZ) \cong \ts{q\in L_\QQ \st (q, \Lambda) \subseteq \ZZ}$. - [x] What is $\len \Lambda$? ✅ 2023-02-07 - [ ] The minimal number of generators of $D_\Lambda$. - [x] What is a $p\dash$elementary lattice? ✅ 2023-02-07 - [ ] $D_\Lambda \cong C_p\sumpower{k}$ for some $k$; $\implies \len \Lambda = k$. - [x] What is a nondegenerate pairing on $\Lambda$? ✅ 2023-01-28 - [ ] The linear extension of $\beta$ to $V = \Lambda\tensor_\ZZ \RR$ induces an isomorphism $V\iso V\dual$. - [ ] Equivalently $\disc \Lambda \neq 0$. - [x] What is $\disc(\Lambda)$ for a lattice $(\Lambda, \beta)$? ✅ 2023-01-28 - [ ] $\disc(\Lambda) \da \det G(\beta)$ where $G(\beta)$ is the Gram matrix. - [ ] Interpret as $\disc(\Lambda) = \covol(\Lambda)^2 \da \vol(V/\Lambda)^2$. - [ ] When integral, $\disc(\Lambda ) = \size(\Lambda\dual/\Lambda)$. - [ ] Can compute by taking $M$ a generator matrix for $\Lambda$, then $\disc \Lambda = M^tM = (\det M)^2$. - [x] What is a unimodular lattice? ✅ 2023-01-28 - [ ] Integral with $\disc \Lambda = +1$. - [ ] Equivalently, $\covol(\Lambda) \da \vol(V/\Lambda) = 1$. - [ ] Equivalently, $\Lambda = \Lambda\dual$. - [ ] Equivalently $D_\Lambda = \ts{0}$. - [ ] More generally, $I\dash$modular if $\Lambda = I\Lambda\dual$. - [x] Give a consequence of having a unimodular lattice. ✅ 2023-02-07 - [ ] If $M\leq \Lambda$ is unimodular then there is an isometry $\Lambda \iso M\oplus M^\perp$. - [x] What is an overlattice? Give an example. ✅ 2023-02-07 - [ ] $\Lambda$ is an overlattice of $M$ iff $M\leq \Lambda$ is a sublattice and $[\Lambda: M] < \infty$, so $\rank_\ZZ \Lambda = \rank_\ZZ M$. - [ ] $\Lambda\dual$ is always an overlattice of $\Lambda$, - [ ] $\Lambda$ is always an overlattice of $M\oplus M^\perp$ for any $M\leq \Lambda$. - [x] What is an even unimodular lattices? ✅ 2023-02-07 - [ ] $\Lambda = \Lambda\dual$ and $Q(\Lambda) \subseteq R$, or equivalently $\beta(x,x)\in 2R$ for all $x\in \Lambda$. - [ ] Equivalently the Gram matrix is in $\GL_n(\ZZ)$ with even diagonal entries. - [x] What is the twist $\Lambda(n)$ of a lattice $\Lambda$? ✅ 2023-01-28 - [ ] $\beta_{\Lambda(n)}(x,y) = n\beta_\Lambda(x, y)$. - [x] What is the signature of a discriminant form $q_\Lambda$? ✅ 2023-01-28 - [ ] If $\sgn \Lambda = (\ell_+, \ell_-)$ then $\sgn q_\Lambda = \ell_+ - \ell_-$. - [x] What is the theta function of a lattice? ✅ 2023-01-28 - [ ] $$\theta_{\Lambda}(z)=\sum_{x \in \Lambda} q^{x^2/2}, \qquad q = \exp(2\pi i z), \,\,z\in \HH$$ - [ ] Coefficients count the number of vectors of a fixed norm. - [x] What is a 2-elementary lattice? ✅ 2023-01-28 - [ ] $\Lambda$ with discriminant group $D_\Lambda\cong C_2^r$ for some $r$. - [x] What is $\ell(D_\Lambda)$? ✅ 2023-02-01 - [ ] The minimal number of generators. - [x] What is a primitive sublattice? ✅ 2023-01-28 - [ ] $R \leq \Lambda$ a co-torsionfree sublattice, ie $\Lambda/R$ is torsionfree. - [ ] Yields a split SES $R\injects \Lambda \surjects \Lambda/R$ so $\Lambda \cong R \oplus \Lambda/R$, not necessarily orthogonal wrt $\beta$. - [ ] Examples: any $R \leq \Lambda$ of the form $R = S^\perp$ for some $S\leq \Lambda$ is always primitive. - [ ] What is the importance of primitive sublattices? - [ ] If $H \leq \Lambda$ is primitive, then $H$ admits a complementary sublattice so $H \oplus H^\perp = \Lambda$. - [x] What is a primitive element of a lattice? ✅ 2023-02-07 - [ ] $x\in \Lambda$ where $\gens{x}_\ZZ$ is a primitive sublattice. - [ ] Corresponds to $x$ being the closest element in $\gens{x}_\QQ \intersect \Lambda$ to the origin. - [ ] Equivalently, $v$ is primitive iff ${1\over n} v\not \in \Lambda$ for any $n$. - [x] What is a root lattice? ✅ 2023-02-07 - [ ] A sublattice $\Lambda' \leq \Lambda$ generated by all $-2$ vectors. - [x] What is a lattice isometry? ✅ 2023-01-30 - [ ] A map $f: (\Lambda_1, \beta_1) \to (\Lambda_2, \beta_2)$ where $\beta_2(f(x), f(y)) = \beta_1(x,y)$. - [ ] Forms a group $\Orth(\Lambda)$. - [x] What is the discriminant form? ✅ 2023-02-01 - [ ] For an even lattice, $$\begin{align*}q_\Lambda: D_\Lambda &\to \QQ/2\ZZ \\ x + \Lambda &\mapsto \beta(x,x) +2\ZZ\end{align*}$$ - [ ] Obtained by extending $\beta$ to $\Lambda\dual$ and then mapping to the quotient. - [x] What is the bilinear form associated to the discriminant form $q_\Lambda$? ✅ 2023-02-07 - [ ] $$\begin{align*}\beta_\Lambda: \Sym^2 D_\Lambda &\to \QQ/\ZZ \\x\tensor y&\mapsto {1\over 2}\qty{ q_{\Lambda}(x+y,x+y) - q_{\Lambda}(x,x)-q_{\Lambda}(y,y)} \\ \\ \implies q_\Lambda(x) &= \beta_\Lambda(x,x)\end{align*}$$ - [x] What is an isotropic element of a lattice $V$? An isotropic subspace? ✅ 2023-02-07 - [ ] $x\in V$ with $Q(x) = 0$. - [ ] Subspaces $U$: $U \subseteq U^\perp \iff\ro{Q}{U} = 0$. - [x] What is an integral lattice? What is a common mitake here? ✅ 2023-02-07 - [ ] $\im \beta \subseteq \ZZ$. - [ ] Equivalently $\Lambda \subseteq \Lambda\dual$. - [ ] Not sufficient for $\im Q \subseteq \ZZ$, e.g. $Q = \matt{2}{-1}{-1}{2}$ has $Q(\vector x) = x_1^2 -x_1 x_2 + x_2^2$ so $Q(\ZZ^2) \subseteq \ZZ$ but $\beta((1,0), (0, 1)) = -1/2\not\in \ZZ$. - [x] When are two lattices in the same genus? ✅ 2023-02-07 - [ ] $\sgn L_1 = \sgn L_2$ and their $p\dash$adic completions $(L_i)_p \da L_i \tensor_\ZZ \ZZpadic$ are isomorphic for all primes as $\ZZpadic\dash$bilinear forms. - [ ] Genus determines $L_\QQ$ uniquely, and each genus has only finitely many iso classes. - [x] What is the index of a lattice? ✅ 2023-02-07 - [ ] $\ind \Lambda = n_+ - n_- \in C_8$. - [x] What is the theta series of an even lattice of dimension $n=2k$ and level $\ell$? ✅ 2023-03-06 - [ ] $$\theta_\Lambda(q)\da \sum_{m\in \ZZ_{\geq 0}} f_\Lambda(m) q^m,\qquad f_\Lambda(m) \da \size\ts{x\in \Lambda \st \beta(x,x) = 2m}$$ - [ ] Modular form of weight $k$ for $\Gamma_0(\ell)$ and a quadratic character $\varepsilon: \Gamma_0(\ell) \to \mu_2$ - [ ] Alternatively, $\theta_\Lambda(q) = \sum_{x\in \Lambda} 1\cdot q^{{1\over 2}\beta(x,x)}$. - [x] What is the genus of a lattice? ✅ 2023-03-27 - [ ] All lattices with the same signature and discriminant form up to isomorphism. - [ ] Notation: $g(M) \da \rm{II}_{p, q}(D_M)$. - [x] What is an overlattice? Why care? ✅ 2024-01-29 - [ ] $M$ is an overlattice of $L$ iff $M\contains L$ and $[M: L] < \infty \in \zmod$. - [ ] $\sgn M = \sgn L$ for any $M$ over $L$ - [ ] Overlattices of $L$ biject with subgroups of $D_L$. - [x] What is a modular lattice of level $\ell$? Why care? ✅ 2024-01-29 - [ ] $\rank_\ZZ L = n$ with $\sigma: L\to L\dual$ satisfying $\sigma(v^2) = \ell v^2$. - [ ] $\theta_L(q)$ is a modular form of weight $n$ wrt a Fricke group of level $\ell$: $H(\ell)\leq \SL_2(\RR)$ with $[H(\ell): \Gamma_0(\ell)] = 2$. - [x] Discuss classification of lattices. ✅ 2023-03-27 - [ ] Over algebraic number fields: classified by Hasse-Minkowski - [ ] Unimodular, indefinite, integral: classified by Milnor. - [ ] Odd: $\Lambda \cong \gens{1}^{n_+} \oplus \gens{-1}^{n_-}$. - [ ] Even: Exist $\iff \sgn \Lambda \equiv 0 \mod 8$; if indefinite, classified by $\sgn \Lambda$. - [ ] Indefinite, integral: Nikulin, using discriminant forms. - [ ] Even: - [ ] Indefinite, unimodular, nondegenerate: determined by $\sgn \Lambda = (n_+, n_-)$ - [ ] (Negative) definite ($n_+ = 0$), unimodular, nondegenerate: Only in ranks $r\equiv 0 \mod 8$, not unique for $r\neq 8$. - [ ] Indefinite, $\len \Lambda \leq\rank \Lambda - 2$: determined by $(\sgn \Lambda, q_\Lambda)$. - [x] What is Nikulin's theorem on lattices? ✅ 2023-01-28 - [ ] For $\Lambda$ even, negative definite, $\rank_\ZZ(\Lambda) \geq \ell(D_\Lambda) + 2$: - [ ] $\Lambda$ is determined up to isometry by its rank, signature, and discriminant form. - [x] Discuss classification of 2-modular lattices? ✅ 2023-01-28 - [ ] If 2-modular and indefinite, uniquely determined by rank, signature, $\ell(A_\Lambda)$, and $\Delta$ where $\Delta = 0 \iff q_\Lambda(\Lambda) \subseteq \ts{0, 1}$ and 1 otherwise. - [x] Give a classification of indefinite unimodular integral lattices. ✅ 2023-01-30 - [ ] ![](../attachments/2023-01-30-lattice.png) - [x] Discuss classification of **even**, **unimodular** , **indefinite** integral lattices: ✅ 2023-01-31 - [ ] For signature $(p, q)$, exist if $p\equiv q \mod 8$, uniquely determined by rank and signature. - [ ] Given by $$\Pi_{p, q} \da U\sumpower{p} \oplus E_8 \sumpower{m}, \qquad p-q=-8m$$ or $$\Pi_{p, q} = U\sumpower{q}\oplus E_8(-1)\sumpower{m}, \qquad p-q = +8m$$ - [x] Discuss classification of **even**, **unimodular**, **definite** integral lattices ✅ 2023-02-01 - [ ] Definite means losing uniqueness, compared to indefinite. - [ ] $\dim = 8: E_8$ - [ ] $\dim = 16: E_8\sumpower{2}, \Lambda_{16}$. - [ ] $\dim = 24$: 24 Neimeier lattices, - [ ] Including the Leech lattice: uniquely has no roots. - [ ] $\dim = 32$: billions. - [ ] $\dim = 40: > 10^{80}$?? - [x] What is Rohklin's theorem? ✅ 2023-01-31 - [ ] If $X$ is a smooth closed spin 4-manifold, $H^2(X)$ is an even lattice and $\sgn H^2(X) \equiv 0 \mod 16$. - [ ] Converse: if $H^2(X)$ is even and $\pi_1 X = 1$, then $X$ is spin. - [ ] Counterexample if $\pi_1 X\neq 1$: the Enriques surface has $\pi_1 X = C_2$ and $H^2 = E_8 \oplus H$ which has signature 8. # From [Notes Geometry and Moduli of K3s](Notes%20Geometry%20and%20Moduli%20of%20K3s.md) - [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 the Hodge diamond of a K3? ✅ 2023-01-17 - [ ] ![](../attachments/2023-01-17-hodgeK3.png) . - [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] 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] 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 $\phi = f^*$. # From [Notes Hodge Theory Looijenga](Notes%20Hodge%20Theory%20Looijenga.md) - [x] What is a connection? ✅ 2023-02-12 - [ ] For $L$ a local system on $Y$,parallel transport induces $\nabla_\eta: V_y\to V_y$ for $\eta\in T_y Y$ and $V\da \OO_Y\tensor_\CC L$ the associated vector bundle. - [ ] Satisfies $\nabla_\eta(fv) = \eta(f)v + f\nabla_\eta(v)$. - [ ] View as an operator $\nabla: V\to \Omega^1_Y \tensor V$ satisfying $\nabla(fv) = df\tensor v + f\nabla(v)$.