--- title: Notes Hodge Theory Looijenga aliases: - Notes Hodge Theory Looijenga annotation-target: https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=7c5703ab981992ae3f9fb07d825d12f8f11c865e status: Done created: 2023-02-11T23:11 updated: 2024-04-19T16:21 flashcard: Reading::Hodge --- # Notes Hodge Theory Looijenga ## Misc - [x] What is a QHS? ✅ 2023-02-12 - [ ] For $V\in\qmod^\fd$, a decomposition $V = \bigoplus_{p, q} V^{p, q}$ with $\bar{V^{p, q}} = V^{q, p}$ where $V_m \da \bigoplus_{p+q=m} V^{p, q}$ is defined over $\QQ$. - [x] What is a ZHS? ✅ 2023-02-12 - [ ] A QHS $V$ with a lattice $V(\ZZ) \subset V(\QQ)$. - [x] What is the Weil element of a QHS? ✅ 2023-02-12 - [ ] $C\in \GL(V)$ which acts as $i^{p-q}$ in $V^{p, q}$. - [x] What is a pure QHS of weight $m$? ✅ 2023-02-12 - [ ] $p+q\neq m\implies V^{p, q} = 0$ - [x] What is the colevel of a HS? ✅ 2023-02-12 - [ ] $\min\ts{p\st h^{p, m-p} \neq 0}$. - [x] What are the filtrations on a HS? ✅ 2023-02-12 - [ ] Increasing Hodge filtration: $\Fil^n V \da \sum_{p > n, q} V^{p, q}$ - [ ] Decreasing weight filtration: $\Fil_n V \da \sum_{p+q \leq n} V^{p, q}$. - [x] What are the Weil and Griffiths intermediate Jacobians? ✅ 2023-02-12 - [ ] Define $V_W$ the Weil complex structure $V_W^{1, 0}$ and Griffiths complex structure $V_G^{1 ,0} = F^{k+1}V$ - [ ] Then $$J_W(V)=V /\left(V(\mathbb{Z})+V_W^{1,0}\right) \text { and } J_G(V)=V /\left(V(\mathbb{Z})+F^{k+1} V\right)$$ - [x] What is the Serre-Deligne torus $S$? Why is it important? ✅ 2023-02-12 - [ ] $S(\RR) =\CC\units$, so $S = \Restriction{\CC}{\RR} \GG_m$. - [ ] Alternatively $\matt a {-b} b a \in \GL_2(\RR)$. - [ ] There is a cocharacter $w: \GG_m\to S$ which on real points is $\RR\units\injects \CC\units$, a QHS on $V$ is the same as an action $S\actson V$ defined over $\RR$ with $w$ defined over $\QQ$, so $V^{p, q}$ are eigenspaces with characters $z^p\bar{z}^q$. - [x] What is the Tate HS? Where does it come from? ✅ 2023-02-12 - [ ] $\ZZ(1)$ the ZHS on $\CC$ of bidegree $(-1, -1)$ with lattice $2\pi i \ZZ$. - [ ] Natural HS on $H^1(\GG_m(k))$ for $k=\kbar \subseteq \CC$, regard as $H^2(\CP^1; \ZZ)$. - [x] What is $\ZZ(m)$? $V(m)$? ✅ 2023-02-12 - [ ] $\ZZ(m) = \ZZ(1)\tensorpower{}{m}$ with lattice $(2\pi i)^m \ZZ$ - [ ] $V(m) = V\tensor_\ZZ \ZZ(m)$. - [x] What is a polarization of a QHS? ✅ 2023-02-12 - [ ] For a pure HS of weight $m$, $Q:V\tensorpower{}{2}\to \ZZ(-m)$ defined over $\QQ$ such that the form $(v,w)\mapsto (2\pi i)^m Q(Cv, w)$ is Hermitian and positive-definite, where $C$ is the Weil element. - [ ] Alternatively a morphism $V\to V\dual(-m)$, motivated by Poincare duality. - [x] What is the Hodge spectral sequence? ✅ 2023-02-12 - [ ] $E_1^{p, q} = H^q(M; \Omega_M^p) \abuts H^{p+q}(M; \CC)$. - [x] What is the primitive part of cohomology? ✅ 2023-02-12 - [ ] $H_\prim^{n-k}(M) \da \ker\qty{H^{n-k} \mapsvia{L^{k+1}} H^{n+k+2} }$. - [x] Discuss the Hodge spectral sequence for $M$ connected and projective (or more generally compact Kahler). ✅ 2023-02-12 - [ ] Degenerates at $E_1$, yielding a decreasing filtration $\Fil^* H^d(M;\CC)$ with $\gr^p H^d(M; \CC) \cong H^{d-p}(M; \Omega_M^p)$. - [ ] Puts a HS of pure weight $d$ on $H^d(M; \CC)$ with Hodge filtration $\Fil^*$. - [ ] $H^{2n}(M; \CC)= \ZZ(-n)$. - [ ] For $\eta\in H^2(M; \QQ)$ a hyperplane class, $\eta$ is type $(1,1)$ and defines a Lefschetz operator $L(\wait) \da \eta\cup(\wait)$. - [ ] Hard Lefschetz holds: $L^k: H^{n-k}(M; \QQ)\iso H^{n+k}(M; \QQ)$ for all $n$. - [ ] $$\bigoplus_{k=0}^n H_\prim^{n-k}(M; \QQ)\tensor_\RR {\CC[L] \over \gens{L^{k+1}}}\iso H^*(M; \CC)$$ - [ ] There is a polarization $$\begin{align}I: H_\prim^{n-k}(M; \QQ)\tensorpower{}{2} &\to \QQ \\ a\tensor b &\mapsto(-1)^{(n-k)(n-k-1)\over 2} \int_M L^k(a\wedge b)\end{align}$$ - [x] Discuss relative differentials? ✅ 2023-02-12 - [ ] For $f: M\to S$, $\Omega^*_M / \gens{f^* \Omega^1_S}$, a quotient of a sheaf of dg $\OO_M\dash$algebras. - [ ] Fiber over $s\in S$ is the de Rham sheaf complex of the fiber $M_s$. - [ ] Resolves $f\inv \OO_S$ by coherent $\OO_M$ modules - [x] What is a VHS? ✅ 2023-02-12 - [ ] Over a smooth variety $S$, a local system $V$ on $S$ of $\cmod^\fd$ defined over $\QQ$ - [ ] With an increasing weight filtration by sub-local systems - [ ] With a decreasing Hodge filtration on the underlying vector bundle $\mcv \da \OO_S \tensor{\CC_S} V$ by sub-bundles satisfying - [ ] Having a HS on every fiber - [ ] The covariant derivative maps $\Fil^p \mcv\to \Fil^{p-1}\mcv$. - [x] What is a PVHS over $S$? ✅ 2023-02-12 - [ ] For a VHS of pure weight $m$, a polarization $V\tensor_{\CC_S}V\to \ZZ_S(-m)$. - [x] What is the Gysin sequence? ✅ 2023-02-12 - [ ] $$\cdots \rightarrow \mathrm{H}_{\mathrm{Y}}^{\mathrm{d}}(\mathrm{X}) \rightarrow \mathrm{H}^{\mathrm{d}}(\mathrm{X}) \rightarrow \mathrm{H}^{\mathrm{d}}(\mathrm{X}-\mathrm{Y}) \rightarrow \mathrm{H}_{\mathrm{Y}}^{\mathrm{d}+1}(\mathrm{X}) \rightarrow \cdots$$ - [x] What is an MHS? ✅ 2023-02-12 - [ ] A decreasing Hodge filtration $\Fil_H V$ and an increasing weight filtration $\Fil^W V$ such that the filtration on $\gr_\ell^W$ induced by $\Fil_H$ gives a HS of pure weight $\ell$. - [x] Discuss MHSs. ✅ 2023-02-12 - [ ] For $X\in \Sch\slice \CC^\ft$ a variety, $H^d(X)$ carries a MHS with weights in $\ts{0,1,\cdots, 2d}$. - [ ] If $X$ is smooth, the weights are $\geq d$. - [ ] If $X$ is complete, the weights are $\leq d$. - [ ] This extends the functor $\Mfd_\CC\to \mathsf{HS}$ to $\Var\slice \CC^\ft\to \mathsf{MHS}$. - [ ] What are the semisimple and unipotent parts of an operator? - [x] What is a primitively polarized K3 of genus $g$? ✅ 2023-02-12 - [ ] $(X, \eta)$ where $\eta\in H^{1,1}(X; \RR)$ is a Kahler class. - [ ] Torelli: iso type of $(X, \eta)$ completely determined by $(H^2(X;\ZZ), \cap)$, $\eta$, and the HS on $H^2(X;\CC)$. - [ ] Primitive: not a proper multiple of an integral class. - [ ] $g\da 1+{1\over 2}(\eta.\eta)$; the linear system $\eta$ defines contains curves of arithmetic genus $g$. - [x] What is the polarization on $H^1(X)$ for $X$ smooth Kahler? ✅ 2023-02-12 - [ ] $(\alpha, \beta)\mapsto \int_M \alpha\wedge\beta\wedge \omega^{\dim M - 1} \in \RR$. - [x] Where do MHSs appear? ✅ 2023-02-12 - [ ] Cohomology of noncompact $\CC\dash$varieties. - [ ] Limits of PVHS - [ ] $\pi_1(X) \tensor \RR$ for $X\in\Alg\Var\slice\CC$. - [ ] $\pi_*(X)\tensor \RR$ for $X\in \Alg\Var\slice \CC$ with $\pi_1(X) = 1$. - [x] What is a polarized Hodge structure on $V\in\mods{\CC}$? ✅ 2023-02-05 - [ ] $V = \bigoplus V^{p, q}$ with $Q:V\tensor_\CC \bar V \to \CC$ such that - [ ] the decomposition is orthogonal with respect to $Q$ and - [ ] $(-1)^p Q$ is positive definite on $V^{p, q}$, so there is a positive definite Hermitian product $\inp{v}{w} \da \sum_{p, q} (-1)^p Q(v^{p, q}, w^{p, q})$. - [x] What is a polarized VHS for $(E, d: A^0(E) \to A^1(E))$ a flat bundle on $X$? ✅ 2023-02-05 - [ ] Puts a Hermitian metric on a vector bundle! - [ ] $E = \bigoplus_{p+q=k} E^{p, q}$ as sub-bundles such that $$d(A^0\left(E^{p, q}\right)) \subseteq A^{1,0}\left(E^{p, q}\right) \oplus A^{1,0}\left(E^{p-1, q+1}\right) \oplus A^{0,1}\left(E^{p, q}\right) \oplus A^{0,1}\left(E^{p+1, q-1}\right)$$ - [ ] Each fiber $E_x$ is a Hodge structure of weight $k$. - [ ] Polarization: $Q: A^0(E) \tensor _{A^0}\bar{A^0(E)} \to A^0$ which satisfies the Leibniz rule wrt $d$ and induces polarizations on every fiber. - [ ] Yields a positive definite Hermitian metric $h(u, w)\da \sum_{p+q=k} (-1)^p Q(v^{p, q}, w^{p, q})$. - [x] What is Griffiths transversaility? ✅ 2023-02-12 - [ ] $$\nabla(\mcf^p)\subseteq \mcf^{p-1} \tensor \Omega^1$$ - [ ] Needed since the Hodge filtration on the fibers are not necessarily preserved by differentiation. - [x] What is the Leray spectral sequence? ✅ 2023-02-12 - [ ] For $f:X\to Y$, $$H^p\left(Y, \RR^q f_* \mathbb{Q}\right) \Rightarrow H^{p+q}(X, \mathbb{Q})$$ - [x] Give a sufficient condition for the Leray spectral sequence to degenerate. ✅ 2023-02-12 - [ ] $f:X\to Y$ smooth projective, by Deligne, as a conseuence of Hard Lefschetz. - [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 $\nabla(fv) = df\tensor v + f\nabla(v)$. - [x] When is a connection integrable? ✅ 2023-02-12 - [ ] E.g. if $\nabla^2 = 0$; equivalent to having $\rank V$ solutions to $\nabla v = 0$. - [x] What is the Gauss-Manin connection? ✅ 2023-02-12 - [ ] For $f:X\to Y$, the integrable connection on $V \da \OO_Y \tensor_\CC \RR^i f_* \ul{\CC}$, the vector bundle associated to the local system $L\da \RR^1 f_* \ul{\CC}$. - [ ] Sections $L$ are precisely solutions to $\nabla v = 0$. - [x] Motivate perverse sheaves. ✅ 2023-02-12 - [ ] The Leray spectral sequence doesn't degenerate for arbitrary morphisms $f:X\to Y$, nor is there a decomposition $\RR f_* \QQ = \bigoplus \RR^q f_* \QQ [-q]$. - [ ] Instead $\RR f_* \QQ$ decomposes (in a weaker sense) into perverse sheaves. - [ ] Decomposition theorem: Let $f: X \rightarrow Y$ be a projective map of complex algebraic varieties, with $X$ smooth, then $\mathbb{R} f_* \mathbb{Q}$ decomposes into a sum of translates of simple perverse sheaves. - [ ] What is a limit MHS? - [x] Motivate $\dmod$. ✅ 2023-02-12 - [ ] Generalize integrable connections. - [ ] Coherent such correspond to a system of PDEs. - [ ] A PVHS gives a regular holonomic $\dmod$. - [x] Define $\dmod$. ✅ 2023-02-12 - [ ] For $X$, let $D_X$ be the sheaf of algebras of holomorphic differential operators. - [ ] For $X = \AA^n$, $H^0(D_X) = W^n$ the Weyl algebra: $\gens{x_1,\cdots, x_n, \del_1,\cdots, \del_n}/\gens{\left[x_i, x_j\right]=\left[\partial_i, \partial_j\right]=0,\left[\partial_i, x_j\right]=\delta_{i j}}$. - [ ] $M\in \mods{\mathcal D_X} \subset\oxmods$ if $M\in \Sh(X; \Ab\Grp)$ where $D_X\actson M$ from the left. - [x] Give an example of $M\in \dmod$. ✅ 2023-02-12 - [ ] $(V, \nabla)$ an integrable vector bundle, with $\del_i .v \da \nabla_{\del_i}(v)$. - [ ] Need integrability to get $[\del_i \del_j] = 0$. - [ ] What is a holonomic $\dmod$? - [x] What is a perverse sheaf? ✅ 2023-02-12 - [ ] An object of $D_{\const}^b(X, \cmod)$ (complexes of sheaves of vector spaces with bounded constructible cohomology) isomorphic to $\mathrm{RH}(M)$ for some $M\in \dmod^{\hol}$ and $\mathrm{RH}(\wait) \da \RHom(\wait, \OO_X)$ is the functor inducing the Riemann-Hilbert equivalence. - [x] Motivate the sheaves $\mathrm{IC}(L)$ for a local system $L$. ✅ 2023-02-12 - [ ] For singular spaces, only allow chains which meet the singular strata with appropriate codimensions. - [ ] Always a perverse sheaf. - [x] What is Saito's theorem? ✅ 2023-02-12 - [ ] For $f:X\to Y$ projective, $\RR f_* K$ decomposes into a sum of IC complexes associated to PVHS when $K$ is in a certain subcategory of regular holonomic $\dmod$. - [x] What is the significant of the weight filtration? ✅ 2023-02-12 - [ ] $\gr_\ell W_*$ carries a pure HS of weight $\ell$, so looks like $H^\ell$ of a smooth projective variety. # Main Results - [x] Discuss VHSs. ✅ 2023-02-12 - [ ] For $f:X\to Y$, suppose $H^k(X_y; \CC)$ carries a filtration - [ ] Organize $\ts{H^k(X_y; \CC)}_{y\in Y}$ into a local system. - [ ] Tensor with functions on the base to get a flat bundle $(V, \nabla)$ - [ ] Filtrations organize into sub-bundles. - [x] What is the monodromy theorem for VHSs? ✅ 2023-02-12 - [ ] For $V$ a PVHS over $\DD^\circ$, lift via $\pi: \HH\to \DD$ to get a monodromy morphism $\rho: \pi_1(\DD^\circ) \to \Orth(H^0(\HH; \pi^* V))$ with the form $I$. - [ ] Theorem: the eigenvalues of $T\da \rho(1)$ are of the form $\zeta_n$ for some $n$, and in particular $\abs{\lambda} = 1$. - [ ] $T$ has Jordan blocks of size at most $d\da \max\ts{\abs{p-q}\st h^{p, q}\neq 0}$. - [ ] What is the nilpotent orbit theorem?