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Section 5.6 Moduli of linearized \(\CC\)-structures (RICKY)

Subsection 5.6.1 Motivation: Period morphisms

Recall for \(A\) a polarized AV we get a lattice \(H_1(A,\ZZ)\) with some structure. To keep track of the \(\CC\)-structure we record the Hodge structure induced on \(H_1(A,\RR)\) via the Hodge decomposition theorem. If we want to say construct a moduli space of Elliptic Curves we might try to create a moduli space of \(\CC\)-structures on a fixed torus \(T\text{.}\)

The linearized version of this is to fix \(H^1(T, \RR)\) and consider possible Hodge structures on it.

Example 5.6.1.
\begin{equation*} E_\lambda \colon y^2 = x(x-1) (x-\lambda) \end{equation*}
\begin{equation*} \mathcal E \xrightarrow f S = \PP^1 \smallsetminus \{0,1,\infty\} \end{equation*}

then we can identify

\begin{equation*} V_\lambda = H^1_\sing(E_\lambda, \RR) \end{equation*}

for nearby \(\lambda \in S\text{.}\) Then the Hodge structure looks like:

\begin{equation*} F^1 V_{\lambda, \CC} = \langle \frac{\diff x}{y} \rangle \hookrightarrow V_{\lambda, \CC} \end{equation*}

this induces a period map

\begin{equation*} S\supseteq U \to \PP^1 \end{equation*}

sending \(s \mapsto F^1V_{s, \CC}\text{.}\)

Today generalise the role of \(\PP^1\) in this.

Subsection 5.6.2 Moduli of Hodge structures

Recall: a Hodge structure on a real vector space \(V\) is equivalent to a morphism \(h \colon \mathbf S \to \GL(V)\) where \(\mathbf S = \Res_\RR^\CC \mathbf G_m\) Given \(h\text{,}\) let

\begin{equation*} V^{p,q} = \{ v \in V_\CC : h(z) v= z^{-p} \bar z ^{-q} v\} \end{equation*}

(the characters of \(\mathbf S\) are of the form \(\chi_{p,q} = z^{-p} \bar z ^{-q}\) for \((p,q)\in \ZZ^2\text{.}\) So a general Hodge structure on a Lie group \(G\) is defined to be a map \(\mathbf S \to G\text{.}\)

If \(h\) and \(h'\) are conjugate by \(g\) then conjugation by \(g\) takes \(V^{p,q}\) of one into the other (b/c it preserves the character spaces of \(\mathbf S\)). Conversely if \(\{V_1^{p,q}, V_2^{p,q}\}\) are two HS with the same combinatorial data then we can take \(g \colon V_{\CC} \to V_{\CC}\text{.}\) Taking \(V_1^{p,q} \cong V_2^{p,q}\) and satisfying \(g(\bar v) = \overline{g(v)}\) (using Hodge symmetry) since \(g\) commutes with \(\bar \cdot\text{,}\) it descends to a map on \(V\text{.}\)

Let \(X\) be a conjugacy class of morphisms \(h\colon \mathbf S \to G\text{.}\)

Impose the condition that:

\begin{equation} h(\RR^\times) \text{ lies in the center of }G(\RR) \forall h\tag{5.6.1} \end{equation}

(If the HS on \(V\) is of weight \(k\) then \(h(t) = t^k I\text{,}\) the converse is also true.)

\(G\) acts transitively on \(X\) (via conjugation). So

\begin{equation*} X= G/K \end{equation*}

for \(K = \Stab(h) \) for some \(h\) in \(X\text{.}\) This gives \(X\) the structure of a \(C^\infty\)-manifold.

The \(\CC\)-structure on \(X\).

We give \(T_hX = \Lie G / \Lie K\) a \(\CC\)-v.s. structure let \(\psi_g (x) = g x g \inv\) gives

\begin{equation*} G\to \Aut(G) \end{equation*}

and its derivative is the adjoint map \(\ad\text{.}\) If we compose with \(h \colon \mathbf S\to G\) we get a hodge structure on \(L= \Lie G\text{.}\)

As \(h(\RR^\times)\) is in the center of \(G(\RR)\text{,}\) have \(\ad h(\RR^\times)\) is the identity on \(L\text{.}\) Hence the hodge structure on \(L\) is of weight 0. By above remark.

Let \(L^{0,0} = L_\CC^{0,0} \cap L\) be the real \((0,0)\) part of the HS on \(L\text{.}\)

By the definition of \(K\text{,}\) \(\psi_h(k) = k\) for all \(k \in K\text{.}\) Differentiating gives

\begin{equation*} (\ad h) (v) = v \end{equation*}

for all \(v \in \Lie K\) So \(\Lie K \subseteq L^{0,0}\text{.}\) Conversely if \(v \in L^{0,0}\) then \((\ad h) (v)=v\) implies

\begin{equation*} (\ad h)(\exp v) = \exp v \end{equation*}

so \(\exp v \in K\) i.e. \(v \in \Lie K\text{.}\)

see notes.

These lemmas combined give \(T_h X\) a \(\CC\)-structure.

To get a \(\CC\)-manifold structure on \(X\) we embed \(X\) into a \(\CC\) manifold in a way that respects the \(\CC\)-structures on the tangent spaces.

Pick a faithful representation \(G \hookrightarrow \GL(V)\text{.}\) Then \(h \in X\) we get a Hodge structure on \(V\) via

\begin{equation*} \mathbf S \xrightarrow h G \xrightarrow \rho \GL(V) \end{equation*}

all other \(h' \in X\) have the same combinatorial data.

Let \(\mathbf F\) be the flag variety parameterises filtrations of the type associated to \(h \in X\text{.}\)

To be safe assume \(V\) of weight \(k\text{.}\)

We have an injective map

\begin{equation*} X \hookrightarrow \phi \mathcal F \end{equation*}

this induces a complex structure on \(X\text{,}\) see notes for deets.

Subsection 5.6.3 Geometric conditions and chill (on VHS)

Recall that a VHS parameterised by a space \(S\) must satisfy “Griffiths transversality”, this translates to the condition

Background on Cartan involutions.

Let \(G\) be a real algebraic group with involution \(\sigma\text{.}\) Then a real form of \(G\) associated to \(\sigma\) is

\begin{equation*} G^\sigma (A) = \{ g \in G(A \otimes \CC) : \sigma(g) = \bar g\} \end{equation*}

for all \(\RR\)-algebras \(A\text{.}\)

Example 5.6.6.

\(G = \GL_n\text{,}\) \(\sigma(g) = (g^\perp)\inv\) then

\begin{equation*} G^\sigma = U(n) \end{equation*}

observe that this is compact!

Definition 5.6.7. Cartan involutions.

\(\sigma\) is called a Cartan involution if \(G^\sigma\) is compact, i.e. \(G^\sigma(\RR)\) is compact and meets every connected component of \(G^\sigma(\CC)\text{.}\)

\(K\) compact, take any \(H_0(u,v)\) a pos. def. herm. form on \(V\text{.}\) Then

\begin{equation*} H(u,v) = \int_K H_0(Ku, Kv) \diff K \end{equation*}

is \(K\)-invariant with some properties. For the converse statement the conditions imply \(K \hookrightarrow U(K)\) hence \(K\) is compact.

Remark 5.6.10.

One source of involutions on \(G\) come from \(C\in G\smallsetminus Z\) s.t. \(C^2 \in Z\) then

\begin{equation*} g \mapsto C g C\inv \end{equation*}

is such an involution. e.g. \(J\text{!!}\)