--- created: 2023-02-27T23:05 updated: 2024-05-15T17:51 aliases: - Hermitian symmetric domains - Siegel upper half space - Siegel - Hermitian - Hermitian metric - Hermitian space - Hermitian manifold - symmetric domain - bounded symmetric domain - Hermitian symmetric space - Hermitian symmetric spaces - tube domain tags: - dissertation --- --- - Refs: - - [Paper by Looijenga on BB](https://webspace.science.uu.nl/~looij101/kyoto2013.pdf) - [Vancover lectures](https://www.math.stonybrook.edu/~rlaza/vancouver_lect_final.pdf) - Links: - [Siegel modular forms](Unsorted/Siegel%20modular%20forms.md) - [Hermitian form](Hermitian%20form) - [flag variety](Unsorted/flag%20variety.md) --- ![](../../attachments/2023-04-09-3.png) # Hermitian symmetric domains - **Hermitian manifolds**: $(M, g, \nabla, J)$ a Riemannian manifold with $J\in \T^{1, 1}X$ where $J^2 = -\id$ and $g(JX, JY) = g(X, Y)$. - **Hermitian locally symmetric** if $\nabla R = \nabla J = 0$ for $R$ the curvature tensor or the symmetric $s_x: y\mapsto \exp_x(\log_x y)$ leaves $\nabla, J$ invariant. - Always Kahler under $\omega(X, Y) \da g(X, JY)$. - **Hermitian symmetric** if $\forall p\in M$ there exists an involution $s_p\in \Herm(M)$ with $s_p(p) = p$ an isolated fixed point. - Idea: $(M, g)$ is homogeneous, i.e. $\Riem(M, g)\actson M$ transitively. - Alternative definition of **Hermitian symmetric spaces**: - Define a **Riemannian symmetric space**: $\forall p\in M$ there is a global isometry $F_p$ such that $F_p(p) = p$ and $(dF_{p})_p = -\id\actson \T_p M$. - Take a complex Hermitian manifold $(M, h)$, then $\Riem(M) \da (\Re(M), \Re(h))$ is a Riemannian manifold. - If $\Riem(M)$ is a Riemannian symmetric space and each isometry $F_p$ is a Hermitian isometry, then $M$ is a **Hermitian symmetric space**. - Equivalently, the [covariant derivative](Unsorted/covariant%20derivative.md) of the [Riemann curvature tensor](Riemann%20curvature%20tensor) vanishes. - **Bounded symmetric domains**: - A **domain** is a connected open set in $\mathbb{C}^n$. - A domain $D$ is called a **bounded symmetric domain**, if $D$ is bounded and for every $z \in D$ there exists a biholomorphic map $s_z: D \rightarrow D$ which is involutive and has $z$ as an isolated fixed point. - Any bounded symetric domain is a HSS under the [[Bergman metric]]. - **Euclidean type**: can be written in the form $\CC^n/\Gamma$. - Irreducible: not of Euclidean type and not a product of two lower dimensional Hermitian symmetric spaces. - **Compact type**: product of compact irreducible HSSs. - **Hermitian symmetric domains**: HSSs not of compact type. - Noncompact types embed into their compact duals under the Borel embedding. - **Tube domains**: $D\iso V + i\Omega$ where $V\in \mods{\RR}$ and $\Omega\subseteq V$ is a properly convex (contains no lines) open cone. - E.g. $\HH = \RR + i\RR_{\gt 0}$. ![](../../attachments/2024-04-13-5.png) Actual definition in part 3 ![](../../attachments/2023-02-27hermsymdom.png) ![](../../attachments/2023-02-27hermdomterm.png) As a quotient: ![](../../attachments/2023-02-27quotient.png) Relation to [Shimura varieties](Shimura%20variety.md): ![](../../attachments/2024-04-13-6.png) ![](../../attachments/2024-04-13-7.png) # Tube domains ![](../../attachments/2023-03-10tube.png) ![](../../attachments/2023-03-14tube.png) # Siegel upper half space ![](../../attachments/2024-04-13-16.png) Toward [moduli space of abelian varieties](abelian%20variety.md): ![](../../attachments/2024-04-13-17.png) - Uniformize to write $X\in \Ab\Var$ as $\CC^{g}/\Gamma$ where $\Gamma = \gens{v_1,\cdots, v_{2g}}_\ZZ$ are generators. - Choose $H^0(\Omega^1_X) = \gens{\omega_i}^{1\leq i \leq g}_\CC$ and $H^1(X; \ZZ) = \gens{\gamma_k}^{1\leq k \leq 2g}_\ZZ \cong \Lambda$ to get a change of basis $v_j = \tv{\int_{\gamma_1} \omega_j, \cdots, \int_{\gamma_{2g}} \omega_j }$ and thus a transition matrix $$V = \tv{v_1^t, \cdots, v_{2g}^t}\in \Mat_{2g \times 2g}(\CC) \quad V\sim \tv{T \mid \id_{g\times g}}$$ - If $X$ is projective then the Riemann-Frobenius conditions hold: $$T = T^t, \qquad \Im(T) > 0 \implies T\in \Sieg_{g\times g}(\RR) \da \Sym_{g\times g}(\RR) + \sqrt{-1}\Sym_{g\times g}^{\gt 0}(\RR) \subseteq \Sym_{g\times g}(\CC)$$ - Here $\Sym^{\gt 0}_{g\times g}$ is the cone of positive definite symmetric matrices. - The assignment $X\mapsto T \in \Sieg_{g\times g}(\RR)$ is the period map and the latter is the [period domain](Unsorted/period.md). - **Markings** of $(A, L)$: choose an isomorphism $\phi: H^1(A; \ZZ) \iso\Lambda\dual$ such that $c_1(L) \in H^2(X; \ZZ)\cong \Extpower^2 H^1(A;\ZZ) \cong \Extpower^2 \Lambda\dual$ is expressed as a matrix $J_{D_g}:=\left(\begin{array}{cc}0_g & D_g \\-D_g & 0_g\end{array}\right),$ where $D_g = \diag(d_1,\cdots, d_g)$ with $d_1\mid d_2\mid\cdots\mid d_g$. - Building the moduli space: define $$\mca_{d_1,\cdots, d_g} \da \Sieg_{g\times g}(\RR)/\Gamma, \qquad \Gamma \da \Sp(J_{D_g})(\ZZ)$$ where $\Gamma$ is the change of markings group $\Sp(J_{D_g})(\ZZ)$, since any two markings corresponding to $J_{D_g}$ and $J'_{D_g}$ differ by a change of basis $M$ where $MJ_{D_g}M^t = J'_{D_g}$. - This is a **Type III** Hermitian symmetric space. - **Principal polarization**: corresponds to the choice $D_g \da \id_g$. ![](../../attachments/2023-03-14siegel.png) ![](../../attachments/2023-03-14-8.png) ![](../../attachments/2023-03-14-9.png) ![](../../attachments/2023-03-14-10.png) # Examples - Examples of Hermtian symmetric spaces: - $\HH^1 = \SU_{1, 1}/\U_1$. - Why: Hermitian metric $h(x, y) = y^{-2}\abs{\dz}^2 = {1\over 2}y^{-2}(dz\tensor \dzbar +\dzbar \tensor \dz)$, natural $\SL_2(\RR)$ action, yielding $\Isom(\HH) = \PSL_2(\RR)\da \SL_2(\RR)/\gens{\pm 1}$. - Involution: $z\mapsto -z\inv$ is an involution at $\sqrt{-1}$, and $\HH$ is connected and thus Hermitian symmetric. - $\PP^1_\CC = ?$ - $\CC/\Lambda$ with $\Lambda \leq \GG_a(\CC)$ a discrete subgroup (Euclidean type) - $\DD$ - Why: natural Hermitian metric, $z\mapsto -z$ with $z=0$ as an isolated fixed point and there is a transitive action $$\SU_{1,1}\actson \DD \text{ by } \matt{\bar a}b{\bar b}a \cdot z = {az+b \over \bar bz + \bar a}$$ - $\Sieg_{2n} = \Sp_{2n}(\RR)/\U_n(\CC)$. - $X_{m, n}$ corresponding to Hermitian $h$ with $\sgn h = (m, n)$: $$X_{m, n} = \ts{W\in \Gr_{m}(\CC^m \oplus \CC^n) \st \ro h W > 0} = {\SU_{m, n} \over \SU_m \times \SU_n}$$ - Note a tube domain. ![](../../attachments/2023-03-10.png) - **Theorem**: if $M$ is a symmetric space, then $M \cong \Isom(M)/K$ where $K = \Stab_{\Isom(M)}(p)$ for any point $p\in M$. - $\Isom(\RR^n) = \Orth_n(\RR) \semidirect \RR^n$ - $\Isom(\SS^n) = \Orth_{n+1}(\RR)$. - $\Isom(\HH^n) = \Orth_{n, 1}^+$. - $\Isom(P_n(\RR)) = \SL_n(\RR)$ for $P_n(\RR) \leq \SL_n(\RR)$ the subspace of symmetric positive definite matrices - Hermitian symmetric space iff $n=2$. - Thus $$\begin{aligned}& \mathbf{R}^n=\mathbf{R}^n \ltimes \Orth_n(\RR) / \Orth_n(\RR) \\& \SS^n=\Orth_{n+1}(\RR) / \Orth_n(\RR) \\& \mathbf{H}^n= \Orth_{n, 1}^{+}(\RR) / \Orth_n(\RR) \\& P(n, \mathbf{R})=\SL_n(\mathbf{R}) / \SO_n(\RR)\end{aligned}$$ - See [a vector space example here](https://www.aimath.org/WWN/surfacegroups/strubel.pdf#page=27)describing $\SU_{p, q} \over \SU_p\times \SU_q$, where $p=q=1$ yields $U$. - Example: $$M\da {\SO_{2, n} \over \SO_2 \times \SO_n}\cong \ts{V\in \Gr_2(\RR^2 \oplus \RR^n) \st \ro{ \beta_{2, n}}V >0 \text{ is positive definite}}$$ where $\beta_{p, q}$ is the bilinear form of signature $(p, q)$. - Can write its complex structure as $$M\cong \ts{z\in \CC^n\st 1 - 2z^2 + \abs{zz^t} > 0,\,\, \abs{z} < 1}$$. # Types For $\beta$ a bilinear form, set $\beta_\CC$ its complexification and $h(v,w)\da \beta_\CC(v, \bar w)$. - Type $\mathrm{I}_{p, q}$: e.g. $\HH$. - Take $(V, h)$ a Hermitian space $V\in \mods{\CC}$ with $\sgn h = (p, q)$ - Take $G\da \SU(V)$ and $X\leq \Gr_p(V)$ the open subset of positive-definite subspaces, then $G\actson X$ is transitive with $\Stab_G(F) \subseteq K \da \U(F) \times \U(F^\perp) \leq \SU(V)$ a maximal compact. - $X \subseteq \T_F \Gr_p(V) = \Hom(F, V/F) = \Hom(F, F^\perp)$ is a bounded domain. - The automorphic bundle is $\mcl_V\da \Extpower^p \qty{\ro{\eta}{V}}$ where $\eta\to \Gr_p(V)$ is the tautological, and $\mcl_V = \sqrt[n]{\eta'}$ where $\eta' \to X$ is its canonical. - $p=1 \implies \Gr_1(V) = \PP(V) \contains X$ a complex ball. - $p=q=1$ yields $\HH$. - Type $\mathrm{II}_g$: - $(V, \beta)$ with $\sgn \beta = (g, g)$ and $V\in \mods{\RR}$. - Let $\DD_V$ is the set of $\beta_\CC$ isotropic subspaces $F\in \Gr_g(V_\CC)$ where are $h\dash$positive for $h$ a Hermitian extension of $\beta$. - $\Orth(V)\actson \DD_V$ transitively with $\Stab_{\Orth(V)}(F) \cong \U(F)$ a maximal compact. - Type $\mathrm{III}_g$: e.g. $\Sieg_g$ as a tube domain. See - $(V, \omega)$ with $V\in \mods{\RR}$ and $\omega$ a symplectic form, $\dim_\RR V = 2g$. - Extend to a Hermitian form $h(a,b) = \sqrt{-1}\omega_\CC(a, \bar b)$ with $\sgn h = (g,g)$. - $G = \Sp(V)$ and $X\da \HH_V \subseteq \Gr_g(V_\CC)$ the totally isotropic subspaces $F$ for $\omega_\CC$ which are PD wrt $h$. - $G\actson X$ transitively, $\Stab_G(F) \cong \U(F)$. - Any $F\in \HH_V$ defines a weight 1 Hodge structure $V$ polarized by $\omega$ with $V^{1, 0} \da F, V^{0, 1}\da \bar F$. - $\ro\eta{\HH_V}$ is the [hodge bundle](Unsorted/hodge%20bundle.md) where $\eta\to \Gr_g(\VV_\CC)$ is the tautological. - $\mcl_V \da \Extpower^g (\ro\eta{\HH_V})$ the $g$th power of the Hodge bundle. - $\TT_F \HH_V$ is the space of quadratic forms on $F$, and $\HH_V$ is a bounded subset of such forms. - Type $\mathrm{IV}_n$: period domains - $(V, \beta)$ with $V\in \mods{\RR}, \dim_\RR V = n+2$ and $\sgn \beta = (2, n)$. - $V^+ \da \ts{v\in V_\CC \st \beta_\CC(v,v) = 0,\,\, h(v,v) > 0}$, then $\PP(V^+) \subset \Gr_{2}(V)$ is the set of positive-definite oriented planes. - $\Orth(V) \actson \PP(V^+)$ with $\Stab_{[v]} = \SO(P_{[v]}) \times \Orth(P_{[v]}^\perp )$. ## Type IV Affine model: take $V\in \mods{\ZZ}$ and $\sgn \beta = (1, n-1)$. Let $C^\pm$ be the light cone $\ts{v\in V \st v^2 > 0}$ which has 2 components. Then $$\Omega_n \da V_\RR + iC^+ \subset \CC^n$$ noting that $V_\RR \cong \RR^{1, n-1}$. Projective model: for $\Lambda \da V \oplus U$, so $\sgn \beta = (2, n)$, can map $\Omega_n \to \PP(\Lambda \tensor \CC)\subset \PP^{n+1}$ whose image is $\ts{\omega \st \omega^2 = 0, \, \norm\omega > 0 }$. ![](../../attachments/2023-03-12proj.png) Example for $n=1$: $V = \RR^{1, 0}$ and $C^+ = \RR_{\gt 0}$ with $\Lambda_\RR \cong \RR^{2, 1}$ and $Q$ is the conic $V(z_1^2 +z_2 z_3)\subset \PP(\RR^{2,1}) \cong \PP^2$, the image is $Q^0 = \ts{\abs{z_1}^2 + z_2 \bar z_3 + \bar z_2 z_3 > 0,\,\, \Re(z_2) > 0}$. Project from $g\da\tv{0:0:1}$ to $\PP^1$ to get $\ts{\tv{z_1: z_2} \st z_2\neq 0,\,\, \Im(z_1/z_2) > 0} \cong \HH$. Grassmannian model: take $w\in Q^0$ to $\RR\Re(w) \oplus \RR\Im(w)$ to get $P \in \Gr_2(L)$; then $P$ is naturally oriented and $\norm w > 0 \implies P$ is positive definie, so $P\in \Gr_2(L)^+$ the positive definite oriented planes. # Riemannian symmetric spaces ![](../../attachments/2023-03-14riem.png) # Diagrammatic classification of irreducible Hermitian symmetric domains ![](../../attachments/2023-03-14herm-1.png) ![](../../attachments/2023-03-14-13.png) ![](../../attachments/2023-03-14-14.png) # Parabolic subgroups Example: ![](../../attachments/2023-04-06-4.png) ![](../../attachments/2023-04-06-5.png) # Siegel Modular Varieties ![](../../attachments/2023-05-16.png) The [Baily-Borel](Baily-Borel.md) compactification: ![](../../attachments/2023-05-16-1.png) ![](../../attachments/2024-04-13-19.png) oo # Hilbert modular varieties ![](../../attachments/2023-05-16-2.png)