# Hermitian Symmetric Domains ## Calculus on Schemes :::{.remark} Much of the material for this section can be found in \cite{PMIHES_1970__39__175_0} or \cite{ramananGlobalCalculus2005}. In this section, let $X$ be a smooth scheme over $k$ with structure morphism $\pi: X\to k$. Typically we will take $S = \spec \CC$. Also let $E\in \QCoh(X)$ be a quasicoherent sheaf of $\OO_X\dash$modules on $X$, e.g. the sheaf of sections of a vector bundle $E\to X$. Similarly, $\T_X \da \T_{X/k}$ denotes the relative tangent sheaf, and $\Omega_X \da \Omega_{X/k} \da \T_{X/k}\dual \da \Hom_{\OO_X}(\T_{X/ k}, \OO_X)$ the sheaf of relative relative differentials. For an $\OO_X$-module $F$, we let $F_x$ denote its stalk at $x$, and $F(x) \da F\tensor_{\OO_{X, x}}\kappa(x)$ denote its fiber over $x$, where $\kappa(x) \da \OO_{X, x}/\mfm_x$ is the residue field at $x$. ::: :::{.definition title="Derivations"} \cite[\href{https://stacks.math.columbia.edu/tag/08RL}{Tag 08RL}]{stacks-project} Let $\pi: X\to k$ be a $k\dash$scheme and let $E$ be an $\OO_X\dash$module. A **derivation** into $E$ is a morphism of $\pi\inv\OO_K$-modules $D: \OO_X\to E$ which - is additive, - annihilates the image of $\phi$, and - is not $\OO_X$-linear, but satisfies the Leibniz rule \[ D(f\cdot g) = f\cdot D(g) + D(f)\cdot g \] for all sections $f,g\in \OO_X$. We write $\Der_{\pi\inv \OO_k}(\OO_X, F) \subseteq \Hom_{\pi\inv \OO_k}(\OO_X, F)$ for the sheaf of derivations into $F$. ::: :::{.remark} By definition, the functor $F\mapsto \Der_{\pi\inv \OO_k}(\OO_X, F)$ is representable by the $\OO_X\dash$module $\Omega_{X/k}$, and thus we have \[ \Der_{\pi\inv\OO_k}(\OO_X, F) \cong \Hom_{\OO_X}(\Omega_{X/k}, F) .\] ::: :::{.definition title="Vector fields on schemes"} A **vector field $v$ of $X$** (relative to $k$) is a section $v \in \T_{X/k}$. Note that by the universal property of $\Omega_{X/k}$, we can equivalently regard \[ v \in \T_{X/k} \da \Omega_{X/k}\dual \da \Hom_{\OO_{X}}(\Omega_{X/k}, \OO_X) \cong \Der_{\pi\inv\OO_k}(\OO_X, \OO_X) ,\] and we denote the associated derivation by $\partial_v$. Explicitly, it is given by the formula \begin{align*} \partial_v: \OO_X &\to \OO_X \\ f & \mapsto \del_v(f) \da df(v) \end{align*} where here we regard $df \in \Omega_{X/k} = \T_{X/k}\dual = \Hom_{\OO_X}(\T_{X/k}, \OO_X)$. ::: :::{.definition title="$E$-valued $k$-forms"} For an $\OO_X$-module $E$, define the **sheaf of $E\dash$valued $p\dash$forms** (relative to $k$) as \begin{align*} \Omega_{X/k}^p(E)\da \Omega^p_{X/k}\tensor_{\OO_X} E \end{align*} By convention, we define \[ \Omega_{X/k}^0(E) \da \Omega^0_{X/k} \tensor_{\OO_X} E \da \OO_X \tensor_{\OO_X} E \cong E .\] We also note that \[ \Omega_{X/k}^1(E) \da \Omega^1_{X/k} \tensor_{\OO_X} E \cong \Hom_{\OO_X}(\T_{X/k}, E) ,\] and in general \[ \Omega_{X/k}^p(E) \cong \wedge^p \T_{X/k}\dual \tensor_{\OO_X} E \cong \Hom_{\OO_X}(\wedge^p \T_{X/k}, E) .\] ::: :::{.definition title="Connections on quasicoherent sheaves"} Let $X/k$ be a scheme and let $E$ be an $\OO_X$-module. Let $d: \Omega^p_{X/k} \to \Omega^{p+1}_{X/k}$ be the exterior derivative, so in particular $d: \OO_X \to \Omega_{X/k}$. A **connection on $E$** (relative to $k$) is a morphism \[ \nabla \in \Hom_{\pi\inv \OO_k}( \Omega^0_{X/k}(E), \Omega^1_{X/k}(E)) \] which explicitly is a map of sheaves \[ \nabla: E \to \Omega_{X/k} \tensor_{\OO_X} E .\] which is not $\OO_X\dash$linear, but rather is $\pi\inv\OO_k$-linear and satisfies the Leibniz rule \[ \nabla(f.s) = (df\tensor s) + (f.\nabla(s)) \qquad \forall f\in \OO_X,\, s\in E .\] ::: :::{.definition title="Flat sections"} Let $X/k$ be a scheme and let $E$ be an $\OO_X$-module with connection $\nabla$. Any section $s\in E$ satisfying $\nabla(s) = 0$ is said to be **flat**, **covariantly constant**, **horizontal**, or **parallel**. Equivalently, this holds if $\nabla_v(s) = 0$ for all $v\in \T_{X/k}$. ::: :::{.definition title="Covariant derivatives on scheme"} Letting $v\in \T_{X/k}$ be a vector field on $X$, one can define a $\pi\inv\OO_K$-linear composite map $\nabla_v: E\to E$ by \begin{align*} \nabla_v: E \mapsvia{\nabla} \Omega_X \tensor_{\OO_X} E \iso \Hom_{\OO_X}(\T_{X/k}, E) \mapsvia{f \mapsto f(v)} E \end{align*} This produces a morphism of $\OO_X$-modules \begin{align*} \nabla \in \Hom_{\OO_X}( \T_{X/k}, &\, \Endo_{\pi\inv\OO_k}(E)) \\ v &\mapsto \nabla_v: E\to E \\ &\qquad\quad \,\,\, s \mapsto \nabla(s)(v) \end{align*} which satisfies 1. $\OO_X\dash$linearity of $\nabla$: \[ \nabla_{f.v + g.w} = f.\nabla_v + g.\nabla_w \qquad \forall f,g\in \OO_X,\, v,w\in \T_{X/k} ,\] and 2. The Leibniz rule for $\nabla_v$: \[ \nabla_v(f.s) = \del_v(f).s + f.\nabla_v(s) \qquad \forall f\in \OO_X,\, v\in \T_{X/k},\, s\in E ,\] where here we identify $v\in \T_{X/k}$ with $\partial_v \in \Der_{\pi\inv\OO_k}(\OO_X, \OO_X)$. We call $\nabla_v(s)$ the **covariant derivative of $s$ along $v$** with respect to $\nabla$. ::: :::{.definition title="Connections on schemes"} Let $X/k$ be a smooth scheme. A **connection** on $X$ is a connection on $\T_{X/k}$ relative to $k$. ::: :::{.remark} Here the action of vector fields on smooth functions is defined by the derivation \[ \partial_V: C^\infty(X, \RR) &\to C^\infty(X, \RR) \\ f &\mapsto \partial_V(f): X \to \RR \\ & \qquad\qquad \,\,\,\, p \mapsto (df(V))_p ,\] Here $df = \sum_{i=1}^n \dd{f}{x_i}\dx_i$, and we have the pairing \begin{align*} df(V) &= \qty{\sum_{i=1}^n \dd{f}{x_i} \dx_i}(V) \\ &= \sum_{i=1}^n \dd{f}{x_i} \dx_i(V) \\ &= \sum_{i=1}^n V(x_i)\cdot \dd{f}{x_i} \end{align*} where $x_i\in C^\infty(M, \RR)$ are coordinates functions on $M$, and thus if we set $V(x_i)_p \da V(x_i(p)) = (v_1, \cdots, v_n)$, we obtain \begin{align*} (df(V))_p &= \qty{ \sum_{i=1}^n V(x_i)\cdot \dd{f}{x_i} }_p \\ &= \sum_{i=1}^n V(x_i(p))\cdot \dd{f}{x_i}(p) \\ &= \sum_{i=1}^n v_i \cdot \dd{f}{x_i}(p) \end{align*} i.e. we recover the directional derivative of $f$ in the direction of $V(p) \in \T_{X, p}$. ::: :::{.definition title="Exterior covariant derivative on a scheme"} Let $\nabla: E\to E\tensor_{\OO_X} \Omega_{X/k}$ be a connection on $E$. Note that $\nabla$ can be extended to morphisms \begin{align*} \nabla^n: \Omega_{X/k}^n \tensor_{\OO_X} E &\to \Omega_{X/k}^{n+1} \tensor_{\OO_X} E \\ \alpha\tensor \beta &\mapsto (d\alpha \tensor \beta) + (-1)^n(\alpha \wedge \nabla(\beta)) ,\end{align*} the **exterior covariant derivative on $E$ relative to $\nabla$**, where $\alpha \wedge \nabla(\beta)$ is defined to be the image of $\alpha \tensor \nabla(\beta)$ under the map \begin{align*} \Omega^n_{X/k}\tensor_{\OO_X}\qty{\Omega_{X/k}\tensor_{\OO_X} E} &\to \Omega^{n+1}_{X/k} \tensor_{\OO_X} E \\ \alpha \tensor \eta \tensor \beta &\mapsto (\alpha\wedge \eta)\tensor \beta \end{align*} ::: :::{.definition title="Curvature of a connection on a scheme"} \label{def:curvature} The **curvature of $\nabla$** is the composite morphism \[ E \mapsvia{\nabla} \Omega_{X/k}\tensor_{\OO_X} E \mapsvia{\nabla^1} \Omega^2_{X/k} \tensor_{\OO_X} E ,\] This yields an $\OO_X$-linear map \[ R_{\nabla}: \Omega^0(E) \to \Omega^2(E) .\] ::: :::{.definition title="Flat connections"} A connection $\nabla$ on $E$ is **flat** or **integrable** if $R_\nabla = 0$. If $E$ admits a flat connection, we say $E$ is flat. ::: :::{.remark} In the analytic category, the following are equivalent: 1. $\nabla$ is flat, 2. $R_\nabla = 0$, 3. Parallel transport along small loops in $X$ is the identity, 4. Parallel transport along loops depends only on their homotopy class, 5. For each $x\in X$, parallel transport induces a representation $\rho: \pi_1(X, x)\to \GL(E(x))$. ::: :::{.remark} Note that the curvature is explicitly a morphism \[ R_\nabla: E \to \Omega^2_{X/k}\tensor_{\OO_X} E \] which, by tensoring on the left with $\qty{\Omega_{X/k}^2}\dual = \wedge^2 \T_{X/k}$, can be regarded as \[ R_\nabla: \wedge^2 \T_{X/k} \tensor_{\OO_X} E \to E .\] Tensoring on the right by $E\dual$, we can further regard the curvature as a map \[ R_\nabla: \wedge^2 \T_{X/k} \to E\tensor_{\OO_X} E\dual \cong \Endo_{\OO_X}(E) \] and thus $R_\nabla \in \Hom_{\OO_X}(\wedge^2 \T_{X/k}, \Endo_{\OO_X}(E)) \cong \Omega^2_{X/k}(\Endo_{\OO_X}(E))$ is an $\Endo_{\OO_X}(E)$-valued 2-form. ::: :::{.definition title="Brackets of vector fields on schemes"} Let $X$ be a smooth scheme over $k$ and recall that $\T_{X/k} \cong \Der_{\pi\inv\OO_k}(\OO_X, \OO_X)$. The **bracket** of two sections $v, w\in \T_{X, k}$ is defined as \[ [v, w] \da \partial_v \circ \partial_w - \partial_w \circ \partial_v .\] This endows $\T_{X/k}$ with the structure of a sheaf of Lie algebras. ::: :::{.definition title="Torsion of connections on schemes"} Let $X$ be a smooth scheme with connection $\nabla$. The **torsion $T_\nabla$ of $\nabla$** is defined on sections $v, w\in \T_{X/k}$ by \begin{align*} T_\nabla: \wedge^2 \T_{X/k} &\to \T_{X/k} \\ (v, w) &\mapsto T_\nabla(v,w) = \nabla_v(w) - \nabla_w(v) - [v, w] .\end{align*} where we regard $\nabla_v(w), \nabla_w(v), [v, w]$ as sections of $\T_{X/k}$. Note that $T_\nabla \in \Omega^2(\T_{X/k})$. ::: :::{.definition} \cite{hanselkaPositiveUlrichSheaves2024}. Let $X/\spec \RR$ be a scheme and let $E_1, E_2$ be $\OO_X$-modules. A pairing $g: E_1 \tensor_{\OO_X} E_2 \to \OO_X$ naturally induces two morphisms: \[ g_1: E_1 \to \Hom_{\OO_X}(E_2, \OO_X), \qquad g_2: E_2 \to \Hom_{\OO_X}(E_1, \OO_X) .\] Setting $E = E_1 = E_2$, we say a bilinear pairing \[ g: E\tensor_{\OO_X} E \to \OO_X \] is **symmetric** if $g_1 = g_2$. Letting $p: \spec \RR\to X$ be an $\RR$-point of $X$, we say $g$ is **positive (semi)definite at $p$** if the induced symmetric bilinear form on the finite-dimensional $\RR$-module $p^* E$ is positive (semi)definite. We say $g$ is **positive (semi)definite** if it is positive (semi)definite at every point $p$. We say that $g$ is **nondegenerate at $p$** if the morphism $p^* E \to \Hom_{\RR}(p^* E, p^* \OO_X)$ induced by $p^* g$ is an isomorphism, and **nondegenerate** if it is nondegenerate at every point $p$. ::: :::{.definition title="Metrics on $\OO_X$-modules"} Let $X$ be a smooth scheme over $k$ and $E$ an $\OO_X$-module on $X$ A **(Riemannian) metric on $E$** is a morphism of $\OO_X$-modules \[ g\in \Hom_{\OO_X}(E \tensor_{\OO_X} E, \OO_X) = (E\tensor_{\OO_X} E)\dual \cong E\dual \tensor_{\OO_X} E\dual \] which induces a nondegenerate bilinear form on fibers \[ g_x: E(x) \tensor_{\kappa(x)} E(x) \to \kappa(x) \] for all closed points $x\in X$. We refer to $(E, g)$ as a **metric $\OO_X$-module**. A **(Riemannian) metric** on a smooth scheme $X/k$ is a metric $g$ on $E\da \T_{X/k}$. We call $(X,g)$ a **Riemannian scheme**. ::: :::{.definition title="Metric connections on schemes"} Let $X$ be a smooth scheme and let $(E, g)$ be a metric $\OO_X$-module. We say $\nabla$ is **compatible with the metric $h$** on $E$ if \[ \del_v (g (s, t)) = g(\nabla_v(s), t) + g(s, \nabla_v(t)) \in \OO_X \qquad \forall v\in \T_{X/k}, \, \forall s,t\in E ,\] noting that $g(s, t)\in \OO_X$ and $\del_v: \OO_X\to \OO_X$ is the derivation associated to $v$. We also say that $\nabla$ is **orthogonal** with respect to $h$. This can be abbreviated \[ d(g(s, t)) = g(\nabla(s), t) + g(s, \nabla(t)) \in \Omega_{X} ,\] which explicitly acts on vectors as \[ d(g(s, t))(v) = g(\nabla_v(s), t) + g(s, \nabla_v(t)) \in \OO_X \qquad \forall v\in \T_{X/k} .\] This happens precisely if \[ \tilde \nabla(g) = 0 ,\] where $\tilde \nabla$ is the induced connection on $(E\tensor \overline{E})\dual$. ::: ## Riemannian manifolds and symmetric spaces :::{.definition title="Affine connections/covariant derivatives on manifolds"} Let $X$ be a smooth manifold, regarded as a scheme $(X, \OO_X) \da (X, C^\infty(X, \RR))$ over $k=\RR$. An **affine connection** on $X$ is a global section of a connection $\nabla$ on $\T_{X/\RR}$. Concretely, letting $\mfX_X \da H^0(\T_{X/\RR})$ be the $C^\infty(X, \RR)$-module of global vector fields on $X$, this is a map \[ \nabla: \mfX_X &\to \Endo_\RR(\mfX_X) \\ V &\mapsto \nabla_V .\] which assigns every vector field $V\in \mfX_X$ an $\RR$-linear map $\nabla_V$, the **covariant derivative along $V$**, satisfying 1. $C^{\infty}(X,\RR)$-linearity: $\nabla_{f\cdot V + g\cdot W} = f\cdot\nabla_V + g\cdot \nabla_W$, and 2. The Leibniz rule: $\nabla_V(f.\cdot W) = \partial_V(f)\cdot W + f\cdot \nabla_V(W)$ for all vector fields $V, W\in \mfX_X$ and all smooth functions $f,g\in C^\infty(X, \RR)$. ::: :::{.definition title="Curvature on manifolds"} Let $X$ be a smooth manifold with an affine connection $\nabla$. The **curvature** $R_\nabla$ of $\nabla$ is a global section of the curvature as defined in \cref{def:curvature} and is determined by the formula \begin{align*} R_\nabla: \wedge^2 \mfX_X &\to \Endo_{\RR}( \mfX_X ) \\ (V, W) &\mapsto R_\nabla(V,W) = \nabla_V \circ \nabla_W - \nabla_W \circ \nabla_V - \nabla_{[V, W]} \end{align*} where we regard $\nabla_V, \nabla_W, \nabla_{[V, W]} \in \Endo_{\RR}(\mfX_X)$. Note that $R_\nabla \in \Omega^2(\Endo_{C^\infty(X, \RR)}(\mfX_X))$. ::: :::{.definition title="Riemannian metrics on manifolds"} A **Riemannian metric** on a smooth manifold $M$ is a global section of a metric $g$ on $\T_M$ such that for every $x\in X$, \[ g_x: \T_{M}(x) \tensor_\RR \T_{M}(x)\to \RR \] is a symmetric positive definite bilinear form on the tangent space at $x$, so $g_x\in \Sym^2(\T_{M/k}(x))$ for every $x\in M$. A **Riemannian manifold** is a pair $(M, g)$. ::: :::{.remark} Every connected Riemannian manifold $(M, g)$, there exists a unique connection $\nabla$, the Levi-Cevita connection, which is 1. Torsion-free: \[ T_\nabla(X, Y) = 0 \qquad \forall X,Y\in \mfX_M, \text{ and } \] 2. Compatible with $g$: letting $\tilde\nabla$ is the connection on $\T_M\dual \tensor \T_M\dual$ induced by $\nabla$, \[ \tilde \nabla(g) = 0 .\] ::: :::{.definition title="Sections parallel along curves"} Let $(M, g, \nabla^M)$ be a Riemannian manifold and $\gamma: [a, b]\to M$ be a smooth curve. Let $E$ be a real vector bundle on $M$ with connection $\nabla$. Note that $\gamma^* E$ is a vector bundle over $[a,b]$ with an induced connection $\gamma^* \nabla$. A **section of $E$ along $\gamma$** is a section $s\in \gamma^* E$, and we say $s$ is **parallel along $\gamma$** if $(\gamma^* \nabla)(s) = 0$. ::: :::{.definition title="Parallel transports"} Let $(M, g, \nabla^M)$ be a Riemannian manifold, $\gamma: [a,b]\to M$ a smooth curve, and $E$ a real vector bundle on $M$ with connection $\nabla$. For any $v\in E(\gamma(a))$, there exists a unique parallel section $s_v$ of $E$ along $\gamma$ with $s_v(a) = v$. The element $s_v(b)\in E(\gamma(b))$ is called the **parallel translate of $v$ along $\gamma$**, which we denote $\tau_{\gamma, a, b}(v)$. This induces a family of isomorphisms \begin{align*} \tau_{\gamma, a, b}: E(\gamma(a)) &\iso E(\gamma(b)) \\ v &\mapsto s_v(b) \end{align*} which we call **parallel transports along $\gamma$**. ::: :::{.definition title="Geodesics"} Let $(M, g, \nabla)$ be a Riemannian manifold and $\gamma: I\to M$ a smooth curve. We say $\gamma$ is a **geodesic** if $\gamma' \da \gamma_*(\dd{}{t})$ is parallel along $\gamma$, i.e. \[ (\gamma^* \nabla)_{\dd{}{t}}(\gamma') = 0 \] where we regard $\gamma' \in H^0(\gamma^* \T_M)$ and $\dd{}{t}\in H^0(\T_I)$ as the standard vector field on $I$. ::: :::{.definition title="Sectional curvature"} Let $(M, g, \nabla)$ be a Riemannian manifold, let $p\in M$ be a point, and let $S_p \subseteq \T_M(p)$ be a 2-dimensional subspace of the tangent space at $p$. The **sectional curvature of $S_p$** is defined by \begin{align*} K_p: \Gr_2(\T_M(p)) &\to \RR \\ S_p & \mapsto g_p(R_{\nabla, p}(u, v)(u), v) \end{align*} where $u,v$ is an orthonormal basis of the plane $S_p \subseteq \T_M(p)$. ::: :::{.definition} A Riemannian manifold $(M, g, \nabla)$ is said to have **constant sectional curvature $\kappa$** if $K_p(S_p) = \kappa$ is a constant independent of $p\in M$ and $S_p \subseteq \T_M(p)$. ::: :::{.remark} The universal cover of any Riemannian manifold of constant curvature is isometric to one of the following three examples: - $\RR^n$ with $g$ the standard Euclidean metric, which satisfies $\kappa = 0$, - The sphere $\SS^n$ with the round metric, which satisfies $\kappa = 1$, or - The hyperbolic space $\HH^n$ with the hyperbolic metric, which satisfies $\kappa = -1$. ::: ### Homogeneous and Symmetric Spaces :::{.definition title="Homogeneous spaces"} Let $M$ be a topological space. We say that $M$ is a **homogeneous space** if there exists a topological group $G$ that with a faithful transitive action $G\to \Homeo(M)$. ::: :::{.example} Let $G$ be a topological group and $K\leq G$ a closed subgroup which contains no normal subgroups of $G$. There is a natural action of $G$ on $G/K$ given by \begin{align*} G &\to \Homeo(G/K) \\ g &\mapto \ell_g: G/K \to G/K \\ &\qquad\qquad aK \mapsto (ga)K \end{align*} This makes $M \da G/K$ a homogeneous space under $G$. ::: :::{.theorem} \cite[Thm. 3.2]{helgasonDifferentialGeometryLie2001} Let $G$ be a locally compact group with a countable base acting on a locally compact Hausdorff space $M$ giving $M$ the structure of a homogeneous space. Then $K \da \Stab_G(p)$ is closed for every $p\in M$, and there is a homeomorphism \begin{align*} \psi_p: \dcosetr{G}{K} &\to M \\ gK &\mapsto g.p \end{align*} Thus every such space is homeomorphic to the quotient of a topological group by a closed subgroup. ::: :::{.theorem} \cite[Prop. 4.3]{helgasonDifferentialGeometryLie2001} Let $M$ be a smooth connected real manifold admitting a transitive faithful action by a Lie group $G$. Then $K\da \Stab_G(p)$ is closed in $G$ for every $p\in M$, $G/K$ is a smooth manifold. If $\psi_p$ is a homeomorphism, then 1. $\psi_p$ is a diffeomorphism, and 2. $G^0$, the identity component of $G$, acts transitively on $M$. ::: :::{.definition title="Isometries"} Let $(M, g^M, \nabla^M)$ and $(N,g^N, \nabla^N)$ be Riemannian manifolds. An **isometry** between $M$ and $N$ is a diffeomorphism $f: M\to N$ such that $f^* g^N = g^M$, i.e. \[ g^N_{f(p)}(df_p(u), df_p(v)) = g^M_p(u, v) \qquad \forall p\in M, \, \forall u, v\in \T_M(p) .\] The group if isometries $f: M\to M$ is denoted $\Isom(M)$, which forms a locally compact topological group with compact stabilizers and acts faithfully and transitively on $M$. ::: :::{.definition title="Geodesic symmetry"} Let $(M, g, \nabla)$ be a connected Riemannian manifold and let $x\in M$. A local diffeomorphism $\iota_x\in \Diff(U)$ for some neighborhood $U\ni x$ is a **(local) geodesic symmetry** if $\iota_x(x) = x$ and for all geodesics $\gamma$ with $\gamma(0) = x$, $\iota_x$ reverses geodesics in the sense that $\iota_x(\gamma(t)) = \gamma(-t)$. ::: :::{.remark} Note that if $\iota_x$ is a local geodesic symmetry, then its derivative at $x$ satisfies $(d\iota_x)_x = -\id_{\T_{M}(x)}$. Also note that it is not necessarily the case that $\iota_x \in \Isom(U)$, i.e. it is generally a local diffeomorphism but not a local isometry. ::: :::{.definition title="Locally symmetric spaces"} A **locally symmetric space** is a Riemannian manifold $(M, g, \nabla)$ which satisfies any of the following equivalent conditions: 1. For all $x\in M$, there is a neighborhood $U$ of $x$ and a geodesic symmetry $\iota_x \in \Diff(U)$ such that $x$ is the unique fixed point of $\iota_x$ and $\iota_x\in \Isom(U)$ is a local isometry. 2. $\nabla(R_\nabla) = 0$, 3. The sectional curvature $K(S_p)$ is invariant under parallel transport, or ::: :::{.definition title="Symmetric spaces"} Let $(M, g,\nabla)$ be a locally symmetric space. We say $M$ is **(globally) symmetric** if $M$ is locally symmetric and each $x\in M$ is an isolated fixed point of a global involutive isometry $\iota_x\in \Isom(M)$. Equivalently, every geodesic symmetry $\iota_x \in \Isom(U)$ for all $x\in M$ extends to a global isometry $\tilde \iota_x\in \Isom(M)$ with $x$ as an isolated fixed point. We call $M$ a **symmetric space**. ::: :::{.theorem} \cite[Thm. 3.3]{helgasonDifferentialGeometryLie2001} Let $M$ be a Riemannian globally symmetric space and let $x\in M$ be a point. Then both $G\da \Isom(M)$ and $G^0 \da \Isom^0(M)$ are locally compact Lie groups that act faithfully and transitively on $M$. Letting $K\da \Stab_{G^0}(x)$, which is a compact subgroup of a connected group, there is a diffeomorphism \begin{align*} \psi_p: \dcosetr{G^0}{K} &\to M \\ gK &\mapsto g.x \end{align*} ::: :::{.remark} With notation as above, by \cite[Thm. 3.3]{helgasonDifferentialGeometryLie2001}, there is an involution \begin{align*} \sigma: G^0 &\to G^0 \\ g &\mapsto s_x g s_x \end{align*} where $s_x$ is the geodesic symmetry about $x\in M$. This exhibits $K$ as a subgroup of $G^0$ satisfying \[ (K^\sigma) \subseteq K \subseteq K^\sigma \] where $K^\sigma$ is the closed subgroup of fixed points of $\sigma$. This motivates the following: ::: :::{.definition} Let $G$ be a connected Lie group and $K$ a closed subgroup. We say the pair $(G, K)$ is a **symmetric pair** if there exists an involution $\sigma \in \Diff(G)$ such that \[ (K^\sigma)^0 \subseteq K \subseteq K^\sigma .\] If the group $\Ad_G(K)$ is compact, we say $(G, K)$ is a **Riemannian symmetric pair**. ::: :::{.remark} By \cite[Prop. 3.4]{helgasonDifferentialGeometryLie2001}, every Riemannian symmetric pair induces a Riemannian globally symmetric space $G/K$ under the action of $G$. Moreover, by \cite[Thm. 4.1]{helgasonDifferentialGeometryLie2001}, if $G$ is semisimple then one has - $G \cong \Isom^0(G/K)$ as Lie groups, and - setting $p \da \ts{K}$ in $G/K$, $\Stab_{p}(G) \cong K$. Thus letting $M \da G/K$ and $K \da \Stab_{\ts{K}}(G)$, there is a diffeomorphism \[ M \da \dcosetr{G}{K} \cong \dcosetr{\Isom^0(M)}{\Stab_{\Isom^0(M)}(p) } .\] Conversely, any globally symmetric space $M$ can be written in such a form. ::: :::{.remark} By \cite[Thm 5.6, Cor.5.7]{helgasonDifferentialGeometryLie2001}, if $M$ is a complete locally symmetric space, then its universal cover $\tilde{M}$ is a globally symmetric space under the lifted Riemannian metric. Letting $\Gamma \da \pi_1(M)$ be the fundamental group of $M$, there is an isometric and proper action $\Gamma\actson \tilde M$ and $M \cong \dcosetl{\Gamma}{\tilde M}$. Thus every locally symmetric space is diffeomorphic to a double coset space of the form \[ M \cong \dcosetl{\Gamma}{\tilde M} \cong \dcoset{\Gamma}{G}{K} \] where $G\da \Isom^0(M)$, $K \da \Stab_G(p)$ is a closed compact stabilizer, and $\Gamma \leq \Isom(\tilde M)$ is a discrete subgroup of isometries which acts properly discontinuously and fixed-point-free on $\tilde M$. Thus every locally symmetric space is determined by a triple $(\Gamma, G, K)$. ::: :::{.example} Let $(\Gamma, G, K) = (\SL_{2}(\ZZ), \SL_2(\RR), \SO_2(\RR))$. We first note that there is an isomorphism \[ \dcosetr{\SL_{2}(\RR)}{\SO_2(\RR)} &\iso \HH \\ g. \SO_2(\RR) &\mapsto g.x_0 ,\] where, for example, we can choose $x_0 \da i\in \CC$ as a basepoint. This follows from the fact that there is a transitive isometric action \begin{align*} \SL_2(\RR) \times \HH &\to \HH \\ \qty{ \matt abcd, z} &\mapsto {az+b\over cz+d} .\end{align*} We note that $\HH$ is a symmetric space under the geodesic isometry $\iota_i(z) = -1/\zbar$, which extends to a global isometry. One can then compute the stabilizer of $x_0 = i$ as \[ \Stab_{\SL_2(\RR)}(i) = \SO_2(\RR) .\] Thus the associated locally symmetric space is the modular curve \[ \dcoset{\SL_2(\ZZ)}{\SL_2(\RR)}{\SO_2(\RR)} \iso \dcosetl{\Gamma}{\HH} ,\] which is the moduli space of elliptic curves over $\CC$. Note that $\HH\cong \DD$ is biholomorphic to the open unit disc, which is bounded. This gives rise to a Hermitian symmetric space \[ \HH \cong \dcosetr{\SL_2(\RR)}{\SO_2(\RR)}\cong \dcosetr{\SU_{1, 1}(\CC)}{\U_1(\CC)} \] where \[ \SU_{1, 1}(\CC) \da \ts{\matt a b {\overline b}{\overline a} \st a,b\in \CC, \abs{a}^2 - \abs{b}^2 = 1} \] acts transitively on $\DD$ and one can take $x_0 = 0$ to show $\Stab_{\SU_{1, 1}(\CC)}(0) \cong \U_1(\CC)$. ::: :::{.definition} If $M$ satisfies - $K_p = 0$, and generalizes $\EE^n$, we say $M$ is **Euclidean type**, - $K_p \geq 0$, and generalizes $\SS^n$, we say $M$ is **compact type**, - $K_p\leq 0$ and generalizes $\HH^n$, $M$ is said to be **non-compact type**. ::: :::{.example} We give some examples of symmetric spaces: - $\EE^n$ with $\Isom(\EE^n) = \Orth_n(\RR)\semidirect \RR^n$ - $\SS^n$ with $\Isom(\SS^n) = \Orth_n(\RR)$ - $\HH^n$ with $\Isom(\HH^n) = ?$ ::: :::{.theorem} \cite[Prop 4.2]{helgasonDifferentialGeometryLie2001} If $M$ is a simply connected Riemannian globally symmetric space, it decomposes as a product \[ M \cong M_0 \times M_- \times M_+ \] where - $M_0 \cong \EE^n$ is a Euclidean space, - $M_-$ is a globally symmetric space of compact type, and - $M_+$ is a globally symmetric space of noncompact type. ::: :::{.example} Examples of symmetric spaces (noncompact/compact): - A1: $\SL_n(\RR)/\SO_n(\RR)$ and $\SU_n(\CC)/\SO_n(\RR)$ - A2: $\SU^*_{2n}(\CC)/\Sp_{2n}(\RR)$ and $\SU_{2n}(\CC)/\Sp_{2n}(\RR)$ - A3: $\SU_{p, q}(\CC)/S(\U_p(\CC)\times \U_q(\CC))$ and $\SU_{p+q}(\CC)/S(\U_p(\CC)\times \U_q(\CC))$ - All Hermitian symmetric. - Noncompact acts on $\ts{A\in \Mat_{p\times q}(\CC) \st A^t \overline{A} - I_q < 0}$ - BD1: $\SO^0_{p, q}(\RR)/\SO_p(\RR) \times \SO_q(\RR)$ and $\SO_{p+q}(\RR)/\SO_p(\RR) \times \SO_q(\RR)$ - $q=2$ Hermitian symmetric - Noncompact acts on $\ts{A\in \Mat_{2\times n}(\RR) \st A^tA - I_2 < 0}$ - D3: $\SO^*_{2n}(\RR)/\U_n(\CC)$ and $\SO_{2n}(\RR)/\U_n(\CC)$ - Hermitian symmetric - Noncompact acts on $\ts{A\in \Mat_{n\times n}(\CC) \st A^t = -A, A^t\overline{A} - I_n < 0}$. - C1: $\Sp_{2n}(\RR)/\U_n(\CC)$ and $\Sp_{0, n}(\CC)/\U_n(\CC)$. - Hermitian symmetric - Noncompact acts on $\ts{A\in\Mat_{n\times n}(\CC) \st A^t = A, A\overline{A} - I_n < 0}$ - C2: $\Sp_{p, q}(\CC)/\Sp_p(\CC)\times \Sp_q(\CC)$ and $\Sp_{p+q}(\CC)/\Sp_p(\CC)\times \Sp_q(\CC)$ The Hermitian symmetric spaces: - A3 - BD1 when $q=2$ - D3 - C1 - E3 - E7 The tube domains: - A3 for $p=q$, - D3 for $n$ even, - C1 - BD1 for $p=2$ - E7 ::: :::{.example} Some examples of diffeomorphisms of homogeneous spaces: - $\EE^n \cong \dcosetr{\RR^n\semidirect \Orth_n(\RR)}{\Orth_n(\RR)}$ - $\SS^n \cong \dcosetr{\SO_{n+1}(\RR)}{\SO_n(\RR) \times \SO_1(\RR)} \cong \Gr_1^{\or}(\RR^{n+1})$ - $\PP^n(\RR) = \dcosetr{\SO_{n+1}(\RR)}{S(\Orth_n(\RR) \times \Orth_1(\RR))} \cong \Gr_1(\RR^{n+1})$. - $\PP^n(\CC) \cong \dcosetr{\SU_{n+1}(\CC)}{S( \U_n(\CC) \times \U_1(\CC))} \cong \Gr_1(\CC^{n+1})$. - $\PP^n(H) \cong \dcosetr{\Sp_{n+1}(\CC)}{\Sp_n(\CC) \times \Sp_1(\CC)}$. - $\HH^2(\RR) \cong \Sym\mathrm{SPD}_2(\RR) \cong \dcosetr{\SL_2(\RR)}{\SO_2(\RR)} \cong \dcosetr{\SO^0_{2, 1}(\RR)}{\SO_2(\RR) \times \SO_1(\RR)} \cong \mch^1_\HH \cong \dcosetr{\Sp_2(\RR)}{\U_1(\CC)}$ - $\HH^2_\DD(\RR) \cong H^1_\DD(\CC) \cong \dcosetr{\SU_{1, 1}(\CC)}{\SU_1(\CC) \times \SU_1(\CC)}$. - $\HH^n(\RR) \cong \dcosetr{\SO^0_{n, 1}(\RR)}{\SO_n(\RR) \times \SO_1(\RR)} \cong \Gr_n^{+, \or}(\RR^{n, 1})$. - $\HH^n(\CC) \cong \dcosetr{\SU_{n, 1}(\CC)}{S(\U_n(\CC) \times \U_1(\CC)) } \cong \Gr_n^+(\CC^{n, 1})$ - $\HH^n(H) \cong \dcosetr{\Sp_{n, 1}(\CC)}{\Sp_n(\CC) \times \Sp_1(\CC)}$ - $\Gr_{p}(\RR^{p+q}) \cong \dcosetr{\Orth_{p+q}(\RR)}{\Orth_p(\RR) \times \Orth_{q}(\RR)} \cong \dcosetr{\SO_{p+q}(\RR)}{S(\Orth_p(\RR) \times \Orth_{q}(\RR))}$. - $\Gr^{\mathrm{or}}_p(\RR^{p+q}) \cong \dcosetr{\SO_{p+q}(\RR)}{\SO_p(\RR) \times \SO_{q}(\RR)}$ - $\Gr_k^+(\RR^{p, q}) \cong\dcosetr{\Orth_{p, q}(\RR)}{\Orth_k(\RR) \times \Orth_{p-k, q}(\RR)}$ - $\Gr_p^+(\RR^{p, q}) \cong\dcosetr{\Orth_{p,q}(\RR)}{\Orth_p(\RR) \times \Orth_{q}(\RR)}$ - $\Gr_k^{+, \mathrm{or}}(\RR^{p, q}) \cong\dcosetr{\Orth_{p,q}(\RR)}{\SO_k(\RR) \times \Orth_{p-k, q}(\RR)}$ - $\Gr_p^{+, \mathrm{or}}(\RR^{p, q}) \cong \dcosetr{\Orth_{p, q}(\RR)}{\SO_p(\RR) \times \Orth_q(\RR)}$ which has connected components of the form $\dcosetr{\Orth^+_{p, q}(\RR)}{\SO_p(\RR) \times \Orth_q(\RR)} \cong \dcosetr{\Orth^0_{p,q}(\RR)}{\SO_p(\RR) \times \SO_{q}(\RR)} \cong \dcosetr{\SO^0_{p,q}(\RR)}{\SO_p(\RR) \times \SO_{q}(\RR)}$ - $\Gr_{p}(\CC^{p+q}) \cong \dcosetr{\U_{p+q}(\CC)}{\U_p(\CC) \times \U_{q}(\CC)} \cong \dcosetr{\SU_{p+q}(\CC)}{S(\U_p(\CC) \times \U_{q}(\CC) )}$. - $\Gr_p^+(\CC^{p, q}) \cong \dcosetr{\U_{p, q}(\CC) }{\U_p(\CC) \times \U_q(\CC)} \cong \dcosetr{\SU_{p, q}(\CC)}{S(\U_p(\CC) \times \U_q(\CC))}$ - $\Gr_{p}(H^{p+q}) \cong \dcosetr{\Sp_{p+q}(\RR)}{\Sp_p(\RR) \times \Sp_q(\RR)}$ - $\mathcal{H}^n_{\HH} \cong \dcosetr{\Sp_{2n}(\RR)}{\U_n(\CC)}$ - $\mch^n_\DD \cong \dcosetr{\Sp_{2n}(\RR)}{\Orth_n(\RR)}$ - $\Sym\mathrm{PD}_n(\RR) \cong \dcosetr{\GL_n(\RR)}{\Orth_n(\RR)}$. - $\Sym\mathrm{SPD}_n(\RR) = \dcosetr{\SL_n(\RR)}{\SO_n(\RR)}$, - $\mathrm{SHerm}(\CC) \cong \dcosetr{\SL_n(\CC)}{\SU_n(\CC)}$. - $\OGr_k(\RR^{p, q}) \cong \dcosetr{\Orth_{p+q}(\RR)}{\U_k(\CC) \Orth_{p+q-2k}(\RR)}$ - $\SGr_k(\RR^{2n}) \cong \dcosetr{\Sp_{2n}(\RR)}{\U_k(\CC) \times \Sp_{2n-2k}}$ Others - $\dcosetr{\SO_{2n}(\RR)}{\U_n(\CC)}$: orthogonal complex structures on a $2n$-dimensional vector space - $\dcosetr{\U_n(\CC)}{\Orth_n(\RR)}$: Lagrangian subspaces of $\RR^{2n}$. - $\dcosetr{\U_n(\CC)}{\SO_n(\RR)}$: oriented Lagrangian subspaces of $\RR^{2n}$. - $\dcosetr{\Sp_{p+q}(\CC)}{\Sp_p(\CC) \times \Sp_{q}(\CC)}$, the Grassmannian of oriented $p$-planes in $H^n$ (quaternions) - $\dcosetr{\Sp_{2n}(\RR)}{\U_n(\CC)}$: complex Lagrangian subspaces of $\CC^{2n}$, where $U_n(\CC) \cong \Sp_{2n}(\RR) \intersect \Orth_{2n}(\RR)$. - $\dcosetr{\Sp_{p, q}(\CC)}{\Sp_p(\CC) \times \Sp_q(\CC) }$. Defining some spaces: - $\Sym_n(\RR) \da \ts{ A \in \Mat_{n\times n}(\RR) \st A^t = A}$, symmetric matrices - $\mathrm{PD}_n(\RR) \da \ts{A \in \Mat_{n\times n}(\RR) \st A > 0}$, the cone of positive definite matrices, i.e. inner products on $\RR^n$. - $\Sym\mathrm{PD}_n(\RR) \da \ts{A \in \Sym_n(\RR) \st A > 0}$, the cone of symmetric positive definite matrices, i.e. inner products on $\RR^n$. - $\Sym\mathrm{SPD}_n(\RR) \da \ts{A\in \mathrm{PD}_n(\RR) \st \det A = 1}$, the space of "special" symmetric positive definite matrices, i.e. positive definite quadratic forms of determinant 1. - $\mathrm{SHerm}(\CC)$: positive definite Hermitian quadratic forms of determinant 1. - $\mathcal{H}^n_{\HH} \da \ts{A + iB \st A\in \Sym_n(\RR), B\in \mathrm{PD}_n(\RR)}$, the Siegel upper half-plane - $\mathcal{H}^n_\DD \da \ts{A \in \Sym_n(\RR) \st I - \overline{A} A > 0} = \ts{A\in \Sym_n(\RR) \st \norm{A}_{\op} < 1}$, the Siegel disc, where $\norm{\wait}_{\op}$ is the operator norm. - $H^n_k \da \ts{v\in \PP(k^n) \st h(v, v) < 0}$ where $h$ is a Hermitian form on $k^{n+1}$ of signature $(n, 1)$, i.e. $h(x,y) = \sum_{i=1}^{n-1}\overline{x_i}y_i - \overline{x_n}y_n$. - $\Gr_k^+(\RR^{p, q}) \da \ts{V \subseteq \RR^{p, q} \st \dim_\RR V = k, \ro{q_{p, q}}{V} > 0}$ where $q_{p, q}$ is the bilinear form on $\RR^{p, q}$. - $\Gr_k^{+, \mathrm{or}}(\RR^{p, q}) \da \ts{V \subseteq \RR^{p, q} \st \dim_\RR V = k, \ro{q_{p, q}}{V} > 0, V \text{ is oriented}}$ - $\OGr_k(\RR^{p, q}) = \ts{ V \subseteq \RR^{p, q} \st \dim V = k, \ro{q_{p, q}}{V} = 0}$, the orthogonal Grassmannian. - $\SGr_k(\RR^{2n}) = \ts{ V \subseteq \RR^{2n} \st \dim V = k, \ro{\omega}{V} = 0}$, the symplectic Grassmannian - $\LGr(\RR^{2n}) = \SGr_n(\RR^{2n})$. ::: ## Hermitian Manifolds Much of the material for the following sections can be generalized from \cite[Ch. 8]{helgasonDifferentialGeometryLie2001}. :::{.remark} In this section, we define - $(X, \OO_X)$ is a scheme over $\spec(\RR)$ where $\OO_X \da C^\infty_X \da C^\infty(X, \RR)$ is the sheaf of smooth real-valued functions on $X$, - $X_\CC \da X\fiberprod{\spec \RR} \spec \CC$ - $\pi: X_\CC \to X$ the natural projection - $\tau: X_\CC \to X_\CC$ the automorphism induced by complex conjugation - $\OO_{X_\CC} \da \pi^* \OO_X \cong C^\infty(X, \CC)$ is the sheaf of smooth $\CC$-valued functions on $X$, - $\OO_{X_\CC}^h$ is the sheaf of holomorphic functions on $X$, - $E$ is a $\OO_X$-module, and $s\in E$ denotes a smooth (real) section of $E$, - $(E, J)$ is a complex $\OO_X$-module, and $s\in E_J$ denotes a smooth (complex) section of $E$, - If $E$ is an $\OO_{X_\CC}$-module, $s\in E$ denotes a smooth complex section of $E$. - $E^h$ denotes the sheaf of holomorphic sections of $E$, and $s\in E^h$ denotes a holomorphic section ::: ### (Almost) Complex Schemes :::{.definition title="Almost complex structures"} Let $(X, \OO_X)$ be a scheme over $\spec \RR$ and let $E$ an $\OO_X$-module. A **complex structure on $E$** is an endomorphism \[ J_E \in \Endo_{\OO_X}(E),\qquad J_E^2 = -\id_{E} .\] This induces a $\OO_{X_\CC}$-module structure defined by \[ (u + iv).s \da u.s + v.J_E(s) \qquad \forall s\in E, \forall u,v\in \OO_X .\] We call this a **complex $\OO_{X}$-module** and write it as a pair $(E, J_E)$. It has the same underlying sections as $E$, and when we wish to emphasize the $\OO_{X_\CC}$-module structure, we denote it $E_J$. An **almost complex scheme $X_\CC$** is is a scheme $(X, \OO_X)$ with a complex structure $J$ on its tangent sheaf $\T_{X}$ over $(X, \OO_{X})$, and we write this as a pair $(X, J)$. ::: :::{.remark} Remark on splitting of $\pi^* E$. ::: :::{.definition title="Integrable almost complex structures and complex schemes"} Let $(X, J)$ be an almost complex scheme. We say $J$ is **integrable** if the Lie bracket $[u, v]$ of two holomorphic sections $u, v\in \T_{X_\CC}^{0, 1}$ is again a holomorphic section, i.e. $\T_{X_\CC}^{0, 1}$ is closed under the bracketing operation on $\T_{X_\CC}$ induced from the bracket on $\T_X$, i.e. \[ [\T_{X_\CC}^{1, 0}, \T_{X_\CC}^{1, 0}]_{\T_{X_\CC}} \subseteq \T_{X_\CC}^{1, 0} .\] We say that $(X, J)$ is a **complex scheme** if $J$ is integrable, and write it as a pair $(X_\CC, \OO_{X_\CC}^h)$. ::: :::{.definition title="Conjugate modules"} The following comes from \cite{hanselkaPositiveUlrichSheaves2024}. Given an $\OO_{X_\CC}$-module $E$, the **conjugate $\OO_{X_\CC}$-module** is defined as $\overline{E} \da \tau^* E$ where $\tau: X_\CC\to X_\CC$ is the involution induced by complex conjugation on $X_\CC$. Moreover, conjugating is an involution in the sense that $\tau^* \overline{E} = E$ and there is an $\OO_X$-linear isomorphism $\tau_E: E \to \overline{E}$. Given a section $s\in E$, we write its image in $\overline{E}$ as $\overline{s} \da \tau_E(s)$. If $(E, J^E)$ is a complex $\OO_X$-module, we define the **conjugate complex $\OO_X$-module** as $(E, -J^E)$ and denote it by $\overline{E}$ as a complex $\OO_X$-module and $\overline{E_J}$ as an $\OO_{X_\CC}$-module. Note that there is again an $\OO_X$-linear isomorphism $\tau_E: E_J \to \overline{E_J}$. ::: :::{.remark} Concretely, letting $\pi: X_\CC\to X$, we have a "complexified" $\OO_{X_\CC}$-module $\pi^* E$ where $J$ extends to $\pi^* J \in \Endo_{\OO_{X_\CC}}(\pi^* E)$ which yields an eigensheaf decomposition $\pi^* E = E^{1, 0} \oplus E^{0, 1}$ where \[ E^{1, 0} = \ts{s\in \pi^* E \st \pi^*J(s) = i.s},\qquad E^{0, 1} = \ts{s\in \pi^* E\st \pi^*J(s) = -i.s} .\] We can thus decompose any section $s\in \pi^* E$ as $s = s^{1, 0} \bigoplus s^{0, 1}$. There are then isomorphisms of $\OO_{X_\CC}$-modules \begin{align*} \eta: (E, J) &\to (E^{1, 0}, i) \\ s & \mapsto {1\over 2}(s - i.J(s)) \end{align*} and \begin{align*} \overline{\eta}: (E, -J) &\to (E^{0, 1}, -i) \\ s & \mapsto {1\over 2}(s + i.J(s)) .\end{align*} Moreover, $\tau: X_\CC\to X_\CC$ induces an $\OO_X$-linear endomorphism $\tau^*: \pi^* E\to \pi^* E$, "conjugation", which restricts to an $\OO_X$-linear isomorphism $\tau^*: E^{1, 0} \to E^{0, 1}$. We can then write $\tau_E: E_J\to\overline{E_J}$ as the composition $\overline{\eta}\inv \circ \tau^* \circ \eta$, ::: :::{.remark} An almost complex structure $J$ on $X$ induces complexifies to an endomorphism $J_\CC \in \Endo_{\OO_{X_\CC}}(\T_{X_\CC})$. Since $J_\CC$ has eigenvalues $i, -i$, it induces eigensheaf decompositions on the complexifications of the tangent and cotangent sheaves: \[ \T_{X_\CC} \da \pi^* \T_X \cong \T^{1, 0}_{X_\CC} \oplus \T^{0, 1}_{X_\CC},\qquad \Omega_{X_\CC} \da \pi^* \Omega_{X} \cong \Omega^{1, 0}_{X_\CC} \oplus \Omega^{0, 1}_{X_\CC} .\] We refer to $\T^{1,0}_{X_\CC}$ (resp. $\Omega^{1, 0}_{X_\CC}$) as the **holomorphic tangent (resp. cotangent) sheaf of $X$**. Correspondingly, if we denote the complexification of $d: \OO_X \to \Omega^1_X$ as $d_\CC: \OO_{X_\CC} \to \Omega^1_{X_\CC}$, then $J_\CC$ induces a decomposition $d_\CC = \del \oplus \delbar$ where \[ \del: \OO_{X_\CC} \to \Omega^{1, 0}_{X_\CC}, \qquad \delbar: \OO_{X_\CC}\to \Omega^{0, 1}_{X_\CC} .\] We define a subsheaf $\OO_{X_\CC}^h \da \ker(\delbar)$, the sheaf of **holomorphic functions on $X$**, and refer to the $\CC$-scheme $(X_\CC, \OO_{X_\CC}^h)$ as an **almost complex manifold**. Later we will see that if $(E, \delbar_E)$ is a *holomorphic* $\OO_{X}$-module, the above splitting induces a splitting $\nabla = \nabla^{1, 0} + \nabla^{0, 1}$. ::: :::{.definition title="$J-holomorphic maps"} Let $(X_1, J_1)$ and $(X_2, J_2)$ be two almost complex schemes. A **$J$-holomorphic map $f: X_1\to X_2$** is a morphism $(f, f^\#): (X_1, \OO_{X_1}) \to (X_2, \OO_{X_2})$ of $\RR$-schemes whose differential satisfies \[ df \circ J_1 - f^* J_2 \circ df = 0 \in \Hom_{\OO_{X_1}}(\T_{X_1}, f^* \T_{X_2}) \] where $df \in \Hom_{\OO_{X_1}}( \T_{X_1/k}, f^* \T_{X_2})$ is the differential of $f$ and \[ f^* J_2 \in f^* \Endo_{\OO_{X_2}}(\T_{X_2})\cong \Endo_{f^* \OO_{X_2}}(f^* \T_{X_2}) \cong \Endo_{\OO_{X_1}}(f^* \T_{X_2}) ,\] is the induced endomorphism under pullback, where we've used that $f^* \OO_{X_2} \da f\inv \OO_{X_2}\tensor_{f\inv\OO_{X_2}}\OO_{X_1}\cong \OO_{X_1}$. If $J_1$ and $J_2$ are integrable, a **holomorphic map** is simply a morphism of $\CC$-schemes $(f, f^\#): (X_{1, \CC}, \OO_{X_{1, \CC}}^h ) \to (X_{2, \CC}, \OO_{X_{2, \CC}}^h)$. ::: :::{.definition title="Holomorphic functions"} Let $(X, J)$ be an almost complex scheme. Recall that a smooth $\CC$-valued function $f\in \OO_{X_\CC}$ can be regarded as a morphism $f: X_\CC \to \AA^1_\CC$, and since $\AA^1_\CC$ is smooth, this yields a morphism $df: \T_{X_\CC} \to f^*\T_{\AA^1_\CC}$. Since $\T_{\AA^1_\CC}$ is trivial, we can regard the differential as a map $df: \T_{X_\CC} \to \OO_{X_\CC}$ and thus an element of $\T_{X_\CC}\dual \cong \Omega_{X_\CC}$. Define a holomorphic structure on the trivial $\OO_X$-module $E = \OO_X$ by \begin{align*} \delbar_J: \OO_{X_\CC} &\to \Omega_{X_\CC}^{0, 1} \\ f &\mapsto df\circ J - j \circ df: \T_{X_\CC} \to \OO_{X_\CC} \end{align*} where $i$ is the natural complex structure on $\AA^1_\CC$. A function $f\in \OO_{X_\CC}$ is **$J$-holomorphic** if $\delbar_J(f) = 0$. The **sheaf of $J$-holomorphic (holomorphic) functions on $X$** is defined as \[ \OO_{X_\CC}^J \da \ker(\delbar_J) .\] ::: ### Holomorphic modules and connections :::{.definition title="Holomorphic modules and sections"} This can be found in \cite{narasimhanModuliVectorBundles1969}. Let $(X, J_X)$ be an almost complex scheme and let $(E, J_E)$ be a complex $\OO_{X}$-module. A **holomorphic structure on $E$** is a $\CC$-linear morphism of $\OO_{X_\CC}$-modules \[ \delbar_E: E_J \to \Omega^{0, 1}_{X_\CC}(E_J) \da \Omega^{0, 1}_{X_\CC} \tensor_{\OO_{X_\CC}} E_J \] satisfying the Leibniz rule \[ \delbar_E(f.s) = (\delbar(f) \tensor s) + f.\delbar_E(s) \qquad \forall f\in \OO_{X_\CC}, \, s\in E_J \] and $\delbar_E^2 = 0$. We refer to $(E, J_E, \delbar_E)$ as a **holomorphic $\OO_{X}$-module**. A smooth section $s\in E$ is said to be a **holomorphic section** if $\delbar_E(s) = 0$. We write $E^h \da \ker(\delbar_E)$ for the **sheaf of holomorphic sections of $E$**, noting that this is an $\OO_{X_\CC}^h$-module. ::: :::{.remark} We say a connection $\nabla$ is **compatible with the holomorphic structure on $E$** if \[ \nabla^{0, 1} = \delbar_E .\] ::: :::{.definition title="Holomorphic connections"} Let $(E, J_E, \delbar_E)$ be a holomorphic $\OO_{X}$-module over an almost complex scheme $(X, J_X)$. A **holomorphic connection** on $E$ is a $\CC$-linear morphism of $\OO_{X_\CC}^h$-modules \begin{align*} \del_E: E^h \to \Omega_{X_\CC}^{1, 0}(E^h) \da \Omega_{X_\CC}^{1, 0} \tensor_{\OO_{X_\CC}^h} E^h \end{align*} which satisfies the Leibniz rule \[ \del_E(f.s) = (\del(f)\tensor s) + f.\del_E(s)\qquad \forall f\in \OO_{X_\CC}^h, \, s\in E^h .\] We write this as a tuple $(E, J_E, \del_E, \delbar_E)$. ::: :::{.remark} Given a holomorphic $\OO_{X}$-module with connection $(E, J_E, \delbar_E, \del_E)$, one can construct a $C^\infty_X$ connection on $E$ by setting $\nabla \da \del_E \oplus \delbar_E$. Conversely, given a flat $C^\infty_X$ connection $\nabla$ on an $\OO_{X}$-module $E$ which admits a complex structure $J^E$, setting $\delbar_E \da \nabla^{0, 1}$ defines a holomorphic structure on $E$ and $\del_E \da \nabla^{1, 0}$ defines a holomorphic connection on $E$. ::: ### Hermitian schemes :::{.definition title="Hermitian modules"} Let $(X, J)$ be an almost complex scheme and $(E, J^E)$ a complex $\OO_X$-module. A pairing \[ h \in \Hom_{\OO_X}(E \tensor_{\OO_X}, \OO_X) .\] is **Hermitian** if $h$ is a positive-definite symmetric bilinear form and is compatible with $J^E$ in the following sense: \[ h(s, t) = h(J^E(s), J^E(t)) \qquad \forall s,t\in E .\] We call the tuple $(E, J^E, h)$ a **Hermitian $\OO_X$-module**. ::: :::{.definition title="Hermitian metrics"} Let $E$ be an $\OO_{X_\CC}$-module. A pairing \[ h \in \Hom_{\OO_{X_\CC}}(E \tensor_{\OO_{X_\CC}} \overline{E}, \OO_{X_\CC} ) \] is **Hermitian** if it is positive definite and $h_1: E\to \Hom_{\OO_{X_\CC}}(\overline{E}, \OO_{X_\CC})$ agrees with the pullback of $h_2: \overline{E} \to \Hom_{\OO_{X_\CC}}(E, \OO_{X_\CC})$ along $\tau$. We call $h$ a **Hermitian metric on $E$**, and call the tuple $(E, h)$ a **Hermitian $\OO_{X_\CC}$-module**. ::: :::{.remark} Let $(E, J^E, h)$ be a Hermitian $\OO_X$-module. It is then the case that the complexified form \begin{align*} \pi^* h: \pi^* E \tensor_{\OO_{X_\CC}} \pi^*E &\to \OO_{X_\CC}\\ s\tensor t &\mapsto \pi^* h(s^{1, 0}, t^{0, 1}) = {1\over 2}(h(s,t) + i.h(s, J(t))) \end{align*} defines a Hermitian $\OO_{X_\CC}$-module. ::: :::{.remark} The pullback of $h_2$ along $\tau$ is given by \begin{align*} \tau^*h_2 &\in \tau^* \Hom_{\OO_{X\CC}}(\overline{E}, \Hom_{\OO_{X_\CC}}(E, \OO_{X_\CC}) ) \\ &\cong \Hom_{\OO_{X_\CC}}(\tau^* \overline{E}, \tau^* \Hom_{\OO_{X_\CC}}(E, \OO_{X_\CC}) ) \\ &\cong \Hom_{\tau^* \OO_{X_\CC}}(E, \Hom_{\OO_{X_\CC}}(\tau^* E, \tau^* \OO_{X_\CC}) ) \\ &\cong \Hom_{\OO_{X_\CC}}(E, \Hom_{\OO_{X_\CC}}(\overline{E}, \OO_{X_\CC}) ) ,\end{align*} where we've used that $\tau^* \OO_{X_\CC} = \tau^* \pi^* \OO_X = \pi^* \OO_X = \OO_{X_\CC}$, since $\pi \circ \tau = \pi$, and for any $\OO_{X_\CC}$-modules $A, B$, we have $\tau^* \Hom_{\OO_{X_\CC}}(A, B) \cong \Hom_{\OO_{X_\CC}}(\tau^* A, \tau^* B)$. Concretely, the condition $h_1 = \tau^* h_2$ enforces \[ h(s\tensor t) = \tau_{\OO_{X_\CC}}(h( \tau_{E}\inv(t)\tensor \tau_E(s) )) \qquad \forall s\in E_J, t\in \overline{E_J} ,\] which on fibers reduces to the well-known identity for Hermitian forms \[ h(s, \overline{t}) = \overline{h(t, \overline{s})} \qquad \forall s\in E, \overline{t}\in \overline{E} .\] Similarly, the positive definite condition enforces $h(s, \bar{s}) > 0$ for all nonzero $s\in E_J$. Thus $h$ induces a Hermitian form on each fiber \[ h_x: E(x) \tensor_{\kappa(x)} \overline{E(x)} \to \kappa(x) \] for all closed points $x\in X_\CC$. ::: :::{.definition title="Hermitian connections"} Let $(E, J^E, h)$ be a Hermitian $\OO_X$-module over an almost complex scheme $(X, J)$. A connection $\nabla$ on $E$ is a **Hermitian connection** if it is compatible with $h$, i.e. if \[ \nabla(h) = 0 .\] ::: :::{.remark} Let $(E, \delbar_E)$ be a holomorphic vector bundle with Hermitian metric $h$. Then there is a unique connection $\nabla_{\Ch}$ on $E$, the **Chern connection**, which is a Hermitian connection, so $\nabla_{\Ch}(h) = 0$, and is compatible with the holomorphic structure on $E$, i.e. $\nabla_{\Ch}^{0, 1} = \delbar_E$. ::: :::{.definition title="Hermitian schemes"} Let $(X, J)$ be an almost complex scheme. A **Hermitian metric on $X$** is a Hermitian metric $h$ on the $\OO_{X_\CC}$-module $E = \T_{X_\CC}^{1, 0}$. The triple $(X, h, J)$ is an **almost Hermitian scheme**, and if $J$ is integrable, we say $X$ is a **Hermitian scheme**. ::: :::{.remark} Suppose $(X, g, J)$ is a Riemannian scheme with an almost complex structure. If $g$ is compatible with $J$, then one can define a Hermitian form on $\T_X$ \[ h(u, v) \da g(u, v) - ig(J(u), v) \] and a symplectic form, the associated Kahler form, \[ \omega(u, v) \da g(J(u), v) \] which are related by \[ h(u, v) = g(u, v) - i\omega(u, v) \] and are both compatible with $J$. Thus this determines an almost Hermitian scheme $(X, h, J)$. Conversely, given an almost Hermitian scheme $(X, h, J)$, one can define a compatible Riemannian metric $g = \Re(h)$ and a compatible nondegenerate $(1, 1)$-form $\omega = -{1\over 2}\Im(h)$. Thus $(X, h, J)$ gives rise to a Riemannian scheme $(X, g)$ with Levi-Cevita connection $\nabla_{\LC}$, as well as a symplectic scheme $(X, \omega)$. ::: ### Kahler schemes :::{.definition title="Kahler structures"} Let $(X, J, g)$ be an almost complex Riemannian scheme and let $(E, \nabla_E, J^E)$ an $\OO_X$-module with connection. We say $E$ admits a **Kahler structure** if \[ \nabla^E \circ J^E = 0 ,\] i.e. $J^E$ is invariant under parallel transport. ::: :::{.definition title="Kahler schemes"} An almost Hermitian scheme $(X, h, J)$ is an **almost Kahler scheme** if the associated Kahler form $\omega$ is closed, i.e. $d\omega = 0$, and thus $\omega$ is a symplectic form. An almost Hermitian scheme is a **Kahler scheme** if any of the following equivalent conditions hold: 1. $X$ is Hermitian and almost Kähler, i.e. $J$ is integrable and $d\omega = 0$, 2. $(\T_X, \nabla_{\LC})$ admits a Kahler structure, i.e. $\tilde \nabla_{\LC}(J) = 0$. 3. $\tilde\nabla_{\LC}(\omega) = 0$. 3. $\nabla_{\LC} \circ J = J \circ \nabla_{\LC}$ We write a Kahler scheme as a tuple $(X, h, J)$. ::: ### Hermitian Symmetric Spaces :::{.definition} Let $M$ be a connected Hermitian manifold. We define $\Hol(M)$ for the group of self-holomorphisms of $M$, and $\Isom(M)$ for the group of self-isometries of $M$ with respect to the hermitian metric $h$. We write $\Hol\Isom(M) \da \Hol(M) \intersect \Isom(M)$ for the group of holomorphic isometries of $M$. ::: :::{.definition} A **bounded domain** $\Omega$ is a bounded, open subset of $\CC^n$ for some $n$. ::: :::{.remark} Any bounded domain $\Omega$ admits a metric which is invariant under $\Hol\Isom(\Omega)$ called the *Bergman metric*, which is a Kahler metric. ::: :::{.remark} Idea: Hermitian symmetric manifolds are manifolds that are homogeneous spaces such that every point has an involution preserving the Hermitian structure. These were first studied by Cartan in the context of Riemannian symmetric manifolds. They show up often as orbifold covers of moduli spaces, e.g. polarized abelian varieties (with or without level structure), polarized K3 surfaces, polarized irreducible holomorphic symplectic manifolds, etc. There is a structure theorem: any Hermitian symmetric manifold $M$ decomposes as a product $M\cong \CC^n/\Lambda \times M_c \times M_{nc}$ where $\Lambda$ is some lattice, $M_c$ is an HSM of compact type, and $M_{nc}$ is an HSM of non-compact type. Every HSM of compact type is a flag manifold $G/P$ for $G$ a semisimple complex Lie group and $P$ is a parabolic subgroup. Every HSM of non-compact type admits a canonical so-called Harish-Chandra embedding whose image is a bounded symmetric domain $D\subseteq \CC^N$ for some $N$. Moreover, every HSM of non-compact type admits an associated Borel embedding into an associated HSM of compact type called its compact dual. Moreover, there is a Lie-theoretic classification of HSMs of compact and non-compact type -- they are all of the form $G/K$ for $G$ a simple compact (resp. non-compact) Lie group and $K\leq G$ is a maximal compact subgroup with center isomorphic to $S^1\cong \U_1(\CC)$. By the Harish-Chandra embedding, non-compact HSMs can be realized as bounded domains $D\subseteq \CC^N$ and admit a compactification by taking the closure $\bar{D} \supseteq D$ in $\CC^N$. There is a partition of $\bar{D}$ by an equivalence relation related to being connected through chains of holomorphic discs, and each equivalence class is called a boundary component of $D$. Boundary components are in bijection with their normalizer subgroups, which are precisely maximal parabolic subgroups of $G\da \Aut(D)$. ::: :::{.definition title="Hermitian symmetric space"} Let $M$ be a connected Hermitian manifold. We say $M$ is a **Hermitian symmetric space** if every $p\in M$ is an isolated fixed point of an involution $\iota_p \in \Hol\Isom(M)$. ::: :::{.remark} Such an $M$ is a Riemannian global symmetric space of even dimension. Both $G\da \Hol\Isom(M)$ and $G^0\da \Hol\Isom^0(M)$ act faithfully and transitively on $M$, and if one sets $K \da \Stab_{G^0}(p)$ for any $p\in M$, then $(G, K)$ is a Riemannian symmetric pair and $M$ is diffeomorphic to $G/K$. Conversely, by \cite[Prop 4.2]{helgasonDifferentialGeometryLie2001}, any Riemannian symmetric pair $(G, K)$ for which $M \da G/K$ admits a $G\dash$invariant Riemannian metric $g$ and a compatible complex structure $J$ for which $J_0$ is in the center of $K^*$, $g$ can be upgraded to a Hermitian structure making $M$ a Hermitian symmetric space. Also note that every Hermitian symmetric space is Kähler. Note that if $\Isom^0(M)$ is semisimple, then $\Isom^0(M) \cong \Hol\Isom^0(M)$. ::: :::{.definition} Let $M$ be a Hermitian symmetric space, and set $G^0 \da \Hol\Isom^0(M)$ and $K \da \Stab_{G^0}(p)$ for some $p\in M$. A Hermitian symmetric space is - **compact type** if $(G^0, K)$ is a compact type Riemannian symmetric space, - **noncompact type** if $(G^0, K)$ is a noncompact type Riemannian symmetric space. ::: :::{.proposition} If $M$ is a simply connected Hermitian symmetric space, there is a decomposition \[ M = M_0 \times M_- \times M_+ \] where - $M_0\cong \CC^n$ is Euclidean, - $M_-$ is noncompact type, and - $M_+$ is compact type. ::: :::{.definition title="Irreducible Hermitian symmetric spaces"} We say $M$ is **irreducible** if it is not the cartesian product of two symmetric Hermitian spaces; every irreducible such space is either $\RR^n$ for some $n$ or a homogeneous space $G/K$ for $G$ a real Lie group and $K$ a maximal subgroup. ::: :::{.proposition} \cite[Prop. 5.5, Thm.6.1]{helgasonDifferentialGeometryLie2001} If $M$ is a Riemannian globally symmetric space of either compact or noncompact type, then $M \cong \prod_{i=1}^n M_i$ is a product of irreducible symmetric spaces. If $M$ is a Hermitian symmetric space, then each $M_i$ is Hermitian as well. Moreover, in the compact (resp. noncompact) case, each $M_i$ is of the form $\dcosetr{G_i}{K_i}$ where $G_i$ is a connected compact (resp. noncompact) simple Lie group with trivial center where $K_i$ is a maximal compact (resp. connected) subgroup ::: :::{.definition title="Bounded symmetric domains"} A **bounded symmetric domain** is a bounded domain $\Omega \subseteq \CC^n$ such that for every $z\in \Omega$, there exists an involution $\iota_z\in \Hol(\Omega)$ such that $z$ is an isolated fixed point of $\iota_z$. If $M$ is an irreducible Hermitian symmetric domain of non-compact type, there is an open embedding $M\injects \cD_L \subseteq \CC^n$ onto a bounded subset $\cD$ of complex $n\dash$space, in which case we call $\cD$ a **bounded Hermitian symmetric domain**. ::: :::{.remark} If $\Omega$ is a bounded symmetric domain, then it has the structure of a Riemannian symmetric space under the Bergman metric. Moreover, $\Aut(\Omega)$ is semisimple and acts transitively on $\Omega$ with maximal compact stabilizers $K$. If $\Gamma \leq \Aut(\Omega)$ is a discrete subgroup, then $\Gamma$ acts on $\Omega$ properly discontinuously, making $\dcosetl{\Gamma}{\Omega}$ a normal complex analytic space. ::: :::{.remark} Every bounded symmetric domain $\Omega$ can be equipped with the Bergman metric, making it a Hermitian symmetric space of noncompact type. Moreover, if $M$ is any Hermitian symmetric space of noncompact type, there exists a bounded symmetric domain $\Omega$ and a holomorphism $M\to \Omega$. ::: :::{.remark} Example: $G = \Sp_{2n}(\RR)$ with $K = \Sp_{2n}(\RR) \intersect \SO_{2n}(\RR)\cong \U_n(\CC)$ under the isomorphism \begin{align*} \SO_{2n}(\RR) \intersect \Sp_{2n}(\RR) & \to \U_n(\CC) \\ g = \matt{A}{B}{-B}{A} &\mapsto A+ iB \end{align*} which follows from the relations $A^t B - B^t A = 0$ and $A^t A + B^t B = I_n$. Then $K$ is a maximal compact subgroup, and $G/K$ yields the Siegel upper half plane $\mch^n$. It is a Hermitian symmetric space of noncompact type. There is a map $Z\mapsto (I+iZ)(I-iZ)\inv$ which sends $\mch^n \to \mch_\DD^n = \ts{W\in \Mat_{n\times n}(\CC) \st W\in \Sym_n(\CC), I - \overline{W}W > 0}$ which is a bounded domain in $\CC^{n(n+1)\over 2}$. ::: :::{.example} The simplest example is the upper half plane $\HH \da G/K$ for $G= \SL_2(\RR)$ and $K = \SO_2(\RR)$, which is biholomorphic to the bounded domain $\Delta$ via the Cayley transformation, which is a homogeneous space for $(G, K) = (\SU_{1, 1}, B)$ where $B$ is the subgroup of diagonal matrices. Note that $\HH \cong \cH_1$ is the Siegel upper half space of genus 1. If $(V, q)$ is a real quadratic space where $V \da L_\RR$ for $L$ a lattice, we can define a corresponding domain \[ \cD^\pm_L \da \ts{\CC z \in \PP(V_\CC) \mid z^2 = 0,\, \abs{z} > 0} ,\] the set of lines spanned by isotropic vectors of positive Hermitian norm $\abs{z} \da z\bar{z}$ in $V_\CC$. If $\signature(L) = (2, n)$, so $L$ is hyperbolic, this has an irreducible component decomposition into two parts $\cD^\pm_L = \cD^+_L \coprod \cD^-_L$ interchanged by conjugation $z\mapsto \overline{z}$. Each component is an irreducible Hermitian symmetric domain of type \[ (G, K) = (\SO(V) \da \SO_{2, n}(\RR), \SO_2(\RR) \times \SO_n(\RR)). ,\] i.e. a Type IV domain for $\SO_{2, n}$. We let $\cD_L \da \cD_L^+$ be a choice of one component and write $\Orth^+(L) \leq \Orth(L)$ for the subgroup which preserves $\cD_L$ setwise. There is a distinguished divisor attached to $\cD_L$, the **discriminant divisor**: \[ \cH_L \da \Union_{v \in R_2(L)} H_v \intersect \cD_L ,\] the hyperplane configuration defined by mirrors of roots. When Global Torelli is satisfied, there is a period map $\phi$ whose image is typically the complement of some hyperplane arrangement $\cH$. In good cases, the relevant arrangement is precisely $\cH_L$. Note that $\cD_L$ is isomorphic to a flag variety $G_\CC/P$ for $P$ some parabolic subgroup, and thus the compact form $\tilde\cD_L$ is a projective algebraic variety containing $\cD_L$. We say $\cD_L$ as above is a **Hermitian symmetric domain of orthogonal type** or a **type IV Hermitian symmetric domain** in Cartan's classification. The period domains of K3 and Enriques surfaces are examples of such Type IV domains for $1\leq n \leq 19$. ::: :::{.definition title="Hermitian symmetric domain"} A **Hermitian symmetric domain** is a Hermitian symmetric space of non-compact type. ::: :::{.example} Some very basic examples of Hermitian symmetric manifolds: - Tori $\CC/\Lambda$ with Hermitian structure $g = dx dx + dy dy$ induced from $\RR^2$ (constant zero curvature). - The upper half space $\cH^1$ with Hermitian structure the hyperbolic metric $g = y^{-2}dx dy$ (constant negative curvature) - $\PP^1(\CC)$ with the Fubini-Study metric (constant positive curvature). More advanced examples of symmetric spaces: - $\EE^n$, Euclidean space $\RR^n$. - $\SS^n$, the spherical geometry, - $\HH^n$, hyperbolic space, - $\Sym_{n\times n}^{> 0}(\RR) \leq \SL_n(\RR)$ the Riemannian manifold of positive-definite symmetric matrices with real entries - $X$ defined in the following way: let $V$ be a Hermitian $\CC\dash$module with Hermitian form $h$ of signature $(p, q)$ and let $X \subseteq \Gr_p(V)$ be the Grassmannian of $p\dash$dimensional subspaces $W$ such that $\ro h W$ is positive definite. We first record their isometry groups: - $\Isom(\EE^n) = \RR^n\semidirect \Orth_n(\RR)$. - $\Isom(\SS^n) = \Orth_{n+1}(\RR)$ - $\Isom(\HH^n) = \Orth_{n+1}^+(\RR)$ the index 2 subgroup of $\Orth_{n+1}(\RR)$ which preserves the upper sheet $(\HH^n)^+$.\footnote{Note that for $n=1$, we can take the upper half-plane model which has isometry group $\PSL_2(\RR)$ or the disc model which has isometry group $\PSU_{1, 1}(\CC)$. These are actually isomorphic as Lie groups.} - $\Isom(\Sym_{n\times n}^{> 0}(\RR)) = \SL_n(\RR)$. - $\Isom(X) = \SU_{p, q}(\CC)$? Computing stabilizers of points, one can show \begin{align*} \RR^n &\cong {\RR^n\semidirect \Orth_n(\RR) \over \Orth_n(\RR)} \\ \SS^n &\cong {\Orth_{n+1}(\RR) \over \Orth_n(\RR)} \\ \HH^n &\cong {\Orth_{n+1}^+(\RR) \over \Orth_n(\RR) } \\ \Sym_{n\times n}^{> 0}(\RR) &\cong {\SL_n(\RR) \over \SO_n(\RR)}\\ X &\cong {\SU_{p, q}(\CC) \over \SU_p(\CC) \times \SU_q(\CC)} \end{align*} where $\Orth^+_{n, 1}(\RR) \leq \Orth_{n, 1}(\RR)$ is the index two subgroup which preserves the upper sheet of $\HH^n$. Note that taking $(p, q) = (1,1)$ yields $\HH^2$. ::: :::{.definition} > % Probably need to cite:https://dept.math.lsa.umich.edu/~idolga/EnriquesOne.pdf#page=556&zoom=160,-136,697 ::: :::{.definition title="Arithmetic subgroups"} Let $G$ be a simple linear algebraic group defined over $\QQ$. We define \[ G(\ZZ) \da \GL_n(\ZZ) \intersect G(\QQ) \] where we use the natural embedding of algebraic groups $G\injects \GL_n$ over $\QQ$. A subgroup $\Gamma \leq G(\QQ)$ is **arithmetic** if $\Gamma \intersect G(\ZZ)$ has finite index in both $\Gamma$ and $G(\ZZ)$. ::: :::{.definition title="Parabolic subgroups"} Let $G$ be a linear algebraic group over $\QQ$. We say $P\leq G$ is a parabolic subgroup if $G/P$ is a projective variety. ::: :::{.remark} As the notation suggests, there are other types of irreducible Hermitian symmetric domains. The following are some typical examples of the form $\Gamma\backslash \Omega$ for various definitions of $\Omega$: - Type $\rm III$: Siegel modular varieties corresponding to $\Gamma \leq \Sp(\Lambda)$, the isometry group of a symplectic lattice, of rank $n\geq 3$. - Type $\rm IV$: Orthogonal modular varieties corresponding to $\Gamma\leq \Orth^+(\Lambda)$, a connected component of the isometry group of a lattice of signature $(2, n)$ for $n \geq 3$, - Type ${\rm I}_{n, n}$: Hermitian modular varieties/Hermitian upper half spaces. These are attached to $\Gamma\leq \U(\Lambda)$ for $\Lambda$ a Hermitian form $q$ of signature $(n, n)$ with $n\geq 2$. The compact dual is the Grassmannian $\Gr_{n, 2n}$. - Type ${\rm II}_{2n}$: Quaternionic modular varieties/quaternionic upper half spaces. These are attached to $\Gamma\leq \Sp_{2n}(H)$ for $H$ Hamilton's quaternions, attached to a skew-Hermitian space of dimension $2n$ with $n\geq 2$. The compact dual is the orthogonal Grassmannian $\Orth \Gr_{2n, 4n}$ \dzg{Where do $\cH_g$ and $\Gamma\backslash \HH^n$ fit in?} ::: ## Period domains :::{.remark} For $\Lambda$ a lattice of signature $(2, n)$, the Hermitian symmetric domain attached to $\Lambda$ is the following: define $Q \subseteq \PP(\Lambda_\CC)$ be the quadric cut out by $(\omega, \omega) = 0$, then $\Omega_\Lambda$ is a choice of one of the two connected components of the open set $Q$ defined by $(\omega, \bar\omega) > 0$. Letting $\Orth^+(\Lambda) \leq \Orth(\Lambda)$ be the subgroup preserving the component $\Omega_\Lambda$ and $\Gamma \leq \Orth^+(\Lambda)$ be any finite index subgroup, we obtain \[ X_\Lambda(\Gamma) \da \Gamma \backslash \Omega_\Lambda .\] Embedding $\Omega_\Lambda$ in its compact dual, it has 0 and 1-dimensional boundary strata, corresponding to 1 and 2-dimensional isotropic subspaces of $\Lambda_\QQ$. The BB compactification $\bbcpt{X_\Lambda(\Gamma)}$ is the union of $\Omega_\Lambda$ and these rational boundary components, quotiented by the action of $\Gamma$, equipped with the Satake topology. A toroidal compactification $\torcpt{X_\Lambda(\Gamma)}$ is specificed by a finite collection of suitable fans $\ts{F_I}$, one for each 0-cusp (i.e. each $\Gamma\dash$orbit of isotropic lines $I$ in $\Lambda_\QQ$). For each $I$ there is a tube domain realization given by taking the linear projection from the boundary point, which defines an isomorphism \[ \Omega_\Lambda/U(I)_\ZZ \cong U \subseteq T_I \da U(I)_\CC/ U(I)_\ZZ \] The partial compactifications for the 1-cusps are completely canonical, so the overall compactification is defined by gluing onto the boundary of $X_\Lambda(\Gamma)$ certain natural quotients of all of these partial compactifications to obtain $\torcpt{X_\Lambda(\Gamma)}$. This yields a compact algebraic space which is proper over $\spec \CC$, and there is a natural morphism $\torcpt{X_\Lambda(\Gamma)}\to \bbcpt{X_\Lambda(\Gamma)}$. ::: :::{.example} Let $G \da \SL_2$ defined over $\QQ$ and let $\Gamma \leq \SL_2(\QQ)$ be an arithmetic subgroup. The (noncompact) modular curve attached to $\Gamma$ is \[ Y(\Gamma) \da \Gamma\backslash \HH^1 .\] In this case, rational boundary components are given by $\PP^1(\QQ) = \QQ\union \ts{\infty}\subseteq \PP^1(\CC)$, and a cusp of $Y(\Gamma)$ is a $\Gamma\dash$orbit in $\Gamma \backslash \PP^1(\QQ)$, of which there are finitely many. Adding them yields a compactification \[ X(\Gamma) \da \overline{Y(\Gamma)} \da Y(\Gamma) \union \ts{\text{cusps}} \] topologized appropriately, where e.g. $\ts{\infty}$ is one such cusp. Note that one typically takes the following groups for moduli of elliptic curves with level structure: - $Y(N) \da Y(\Gamma(N))$ where \[\Gamma(N) \da \ker\qty{\phi_N: \SL_2(\ZZ) \to \SL_2(\ZZ/N\ZZ)}.\] The level structure is a basis for $E[n]$. - $Y_0(N) \da Y(\Gamma_0(N))$ where $\Gamma_0(N) \supseteq \Gamma(N)$ is the pullback $\phi_N\inv\qty{\matt ab0d}$. The level structure is an identification $\mu_N \injects E_\tors$. - $Y_1(N) \da Y(\Gamma_1(N))$ where $\Gamma_1(N)$ is the pullback $\phi_N\inv\qty{1b01}$. The level structure is a point $p\in E$ of order $N$ in the group structure. How parabolic subgroups appear here: for $G \da \SL_2$, parabolic subgroups are all conjugate to the subgroup $P$ of upper-triangular matrices, and $G(\QQ)/P(\QQ) \cong \PP^1(\QQ)$ parameterizes all such parabolic subgroups. Why automorphic forms matter: consider $\Gamma \da \SL_2(\ZZ)$. The graded ring of modular forms $\bigoplus_k M_k$ is graded-isomorphic to $\CC[x,y]$ where $\abs x = 4, \abs y = 6$, and $\Proj \CC[x,y]\cong \PP^1(\CC)$. Letting $\ts{f_0, \cdots, f_N}$ be a basis of $M_k$, we can write down a map \begin{align*} \phi_k: Y(\SL_2(\ZZ)) &\to \PP^n(\CC) \\ z &\mapsto [f_0(z): \cdots : f_N(z)] \end{align*} For $k=12$ this separates points and tangent directions, giving a projective embedding. Explicitly, the morphism is \[ \phi_{12}(z) = [E_4(z): E_4(z)^3 - E_6(z)^2] \approx j(z) \] modulo some missing constants. In general, finding enough automorphic forms yields a projective embedding. \dzg{Would like to spell this out in terms of line bundles and linear systems too, in this easy case.} ::: ## Misc :::{.remark} Let $L$ be a lattice of signature $(2, n)$ and the associated period domain $\Omega_L^\pm = \Omega_L^+ \disjoint \Omega_L^-$. Let $\Orth(L)^+\leq \Orth(L)$ be the finite index subgroup fixing $\Omega_L^+$, equivalently the subgroup of elements of spinor norm one. A modular variety of orthogonal type is a homogeneous space of the form $F_L(\Gamma) \da \Gamma\backslash \Omega_L^+$ for an arithmetic subgroup $\Gamma \leq \Orth(L_\QQ)^+$. By general theory, such spaces admit BB compactifications $\bbcpt{F_L(\Gamma)}$ where rational maximal parabolic subgroups correspond to stabilizers of isotropic subspaces of $L_\QQ$; since $\signature(L) = (2, n)$ these are always isotropic lines or planes. For period spaces of K3 surfaces, one takes $\Gamma \da \Orth(L_{2d})\intersect \ker\qty{\Orth(L) \to \Orth(\discgroup L)}$. Boundary strata correspond to central fibers of KPP models of Type II and Type III. ::: :::{.remark} Let $L$ be a symplectic lattice of rank $2g$, ie.e. a free $\ZZ\dash$module with a nondegenerate alternating form $(\wait, \wait)$. Define the associated period space \[ D_L \da \ts{V\in \Gr_g(L_\CC) \mid (V, V) = 0, \, i(V, \bar V) > 0} \cong \Sp_{2g}(\RR)/ \U_g(\CC) \cong \cH^g \] which is a Hermitian symmetric domain of type III that can be identified with the Siegel upper half-space of dimension $g$. We can form the moduli space of PPAV as \[ \cA_g \da \Sp_{2g}(\ZZ) \backslash \cH^g \cong \Sp_{2g}(\ZZ) \backslash \Sp_{2g}(\RR) / \U_g(\CC) \] Rational boundary components of $\bbcpt{\cA_g}$ correspond to $\Gamma \da \Sp_{2g}(\ZZ)$ orbits of totally isotropic subspaces in $L_\QQ$. Since $\Gamma$ acts transitively, such spaces are indexed by their dimension $i = 0,1,\cdots, g$ and there is a stratification \[ \bbcpt{\cA_g} = \disjoint_{k=0}^g \cA_{k} \implies \partial \bbcpt{\cA_g} = \disjoint_{k=0}^{g-1} \cA_k \] ::: :::{.remark} The BB compactification of a locally symmetric domain $D$: write $D = H/K$ as a homogeneous space where $H \da \mathrm{Hol}(D)^+$ and $K\leq H$ is a maximal compact subgroup. Then cusps in $\partial \bbcpt{\Gamma\backslash D}$ correspond to rational maximal parabolic subgroups of $H$. To get boundary components: apply the Harish-Chandra embedding to $D$ to embed $HC: D\injects D^{cd}$ and let $F_P \in \overline{HC(D)}$ be a boundary component. Its normalizer $N(F_P)\da\ts{g\in H \mid g(F_P) = F_P} \leq H$ is a maximal parabolic in $H$. We say $F_P$ is **rational** if $N(F_P)$ can be defined over $\QQ$. Since $\Gamma$ preserves such rational $F_P$, we can set $\bd D \da$ the disjoint union of all rational $F_P$ and set $\bbcpt{\Gamma\backslash D} = {D \disjoint\bd D \over \Gamma}$. ::: ### Explicit realizations of symmetric spaces :::{.remark} The symmetric space associated with a Lie group $G$ is in some sense the most natural space $G$ acts on. For $G=\Orth_{p, q}(\RR)$, the symmetric space is $\Gr^+(\RR^{p, q})$, the Grassmannian of maximal positive-definite subspaces of $\RR^{p, q}$. The right choice of maximal compact subgroup here is $K\da \Orth_p(\RR) \times \Orth_q(\RR)$, the subgroup fixing $\RR^{m, 0}$.\dzg{Typo maybe.} When $(p, q) = (2, n)$, these symmetric spaces admit special descriptions. Note that $\Orth_{n+1}(\RR)$ is the group of isometries of $S^n$, so its projectivization $\PP\Orth_{n+1}(\RR)$ is the isometry group of an elliptic geometry. One can similarly obtain isometries of hyperbolic geometry: - Start with $\RR^{1, n}$ - Take the norm 1 vectors $H^\pm \da \ts{v\in \RR^{1, n} \mid v^2 = 1} = H^+ \disjoint H^-$ to get a 2-sheeted hyperboloid; the pseudo-Riemannian metric on $\RR^{1, n}$ restricts to a Riemannian metric on $H$. - Take one sheet $H^+$; this is a model of $\HH^{n}$\dzg{Indexing might be off here} The group of isometries of $H^+$ is now $\PP\Orth_{1, n}(\RR)$. Note that in $\Orth_{1, n}(\RR)$ there is an index 2 subgroup\footnote{ Apparently, these are elements whose spinor norm equals their determinant. } \[ \Orth_{1, n}(\RR)^\pm = \ts{\gamma \in \Orth_{1, n}(\RR) \mid \gamma(H^+) = H^+, \gamma(H^-) = H^-} .\] ::: :::{.remark} Forming the symmetric spaces for $\Orth_{2, n}(\RR)$: the maximal compact is $K = \Orth_2(\RR) \times \Orth_n(\RR)$ and $\Orth_2(\RR)$ is similar enough to $\U_1(\CC)$ that we should expect the associated symmetric space to be Hermitian. It will be an open subset of a certain quadric: - Start with $\PP(\CC^{2, n})$. - Take the quadric of isotropic vectors $Y = \ts{z\in \PP(\CC^{2, n}) \mid z^2 = 0}$. - Take the open subset $U \da \ts{z\in Y \mid (z, \bar z) > 0}$. Why this matches the previous description: write $z = x+iy$, then $x^2 = y^2 > 0$ and $(x, y) = 0$, so $V\da \RR x\oplus \RR y$ are an orthogonal basis for a positive definite subspace of $\RR^{2, n}$. Multiplying by a scalar only changes basis, so we essentially get a map $\PP(U) \to \Gr^+(\RR^{2, n})$ naturally. This symmetric space can also be identified with points $z\in \CC^{1, n-1}$ with $\Im(z) \in C^+$, one of two cones of $\RR^{1, n-1}$, realizing this as a tube domain generalizing $\HH$. ::: :::{.definition title="Endomorphisms compatible with metrics"} Let $X$ be a scheme and $F\in \Endo_{\OO_X}(E)$ for $(E, g)$ a metric $\OO_X$-module. We say $F$ is **compatible with $g$** if \[ g(F(s), F(t)) = g(s, t)\,\qquad \forall s,t\in E ,\] i.e. $F$ induces isometries on all fibers. :::