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Section 5.4 Ricky Show

Subsection 5.4.1 Moduli of PPAVs

Recall if \(A/ \CC\) is an abelian variety, then

\begin{equation*} A = A(\CC) = \CC^g/ \Lambda, \,g = \dim (A) \end{equation*}
\begin{equation*} \Lambda \cong H_1(A,\ZZ) \end{equation*}

Also a polarization \(\lambda \colon A \to A^\vee\) is equivalent to choosing a Riemann form

\begin{equation*} E \colon \Lambda \times \Lambda \to \ZZ \end{equation*}

s.t.

  1. \(E\) is bilinear alternating
  2. \(E_\RR \colon V\times V \to \RR\) has \(E_\RR(iv,iw) = E_\RR( v,w)\text{.}\)
  3. \begin{equation*} H(v,w) = E_\RR(iv, w) + i E_\RR(v,w) \end{equation*}
    is a positive definite Hermitian form on \(V\text{.}\)

A principal polarization corresponds to \(E\) being a perfect pairing.

Definition 5.4.1.

A PPAV (principally polarized abelian variety) is a pair \((A,\lambda)\text{.}\)

If \((\ZZ^{2g}, \Psi)\) is the standard \(2g\)-dim symplectic form \(\Psi\) then by linear algebra there is a symplectic isomorphism

\begin{equation*} \alpha \colon \ZZ^{2g} \xrightarrow\sim \Lambda \end{equation*}

with \(\Psi(v,w) = E(\alpha(v), \alpha(w))\text{.}\)

Recall the standard \(\Psi\) is

\begin{equation*} \Psi (v,w) = v^\transpose J w, J = \begin{pmatrix} 0 \amp I_g \\ -I_g \amp 0\end{pmatrix} \end{equation*}
Definition 5.4.2.

The Siegel upper half space is

\begin{equation*} \mathscr H_g = \{ Z = X+iY \in M_g(\CC) : Z^\transpose = Z ; \, X,Y \in M_g(\RR); \, Y \gt 0\} \end{equation*}

i.e. \(Y\) is pos. def.

Check: \(\mathscr H_1\) is the usual upper half plane.

(Sketch) First one can show that \(\Sp_{2g}(\RR)\) acts transitively on \(\mathscr H_g\) via linear fractional transformations:

\begin{equation*} M = \begin{pmatrix} A\amp B\\ C\amp D\end{pmatrix} \in \Sp_{2g}(\RR),\, Z\in \mathscr H_g \end{equation*}
\begin{equation*} M\cdot Z = (AZ+B)(CZ+D)\inv \in \mathscr H_g \end{equation*}

second one computes \(\Stab J = U(g)\)

For \(g= 1\text{,}\) \(\Sp_2(\RR) = \SL_2(\RR)\) acts transitively on \(\mathscr H_1\text{,}\) \(\Stab(i) = \specialorthogonal (2) = U(1)\text{.}\)

\begin{equation*} \begin{pmatrix} \cos \theta\amp \sin \theta \\ - \sin \theta \amp \cos \theta \end{pmatrix} i = i \end{equation*}

and if

\begin{equation*} \begin{pmatrix} a \amp b \\ c \amp d\end{pmatrix} i = i \end{equation*}

then \(ai + b = -c + di\) so \(M= \begin{pmatrix} a\amp b\\ -b \amp a \end{pmatrix}\in \specialorthogonal(2)\text{.}\)

We will construct a map

\begin{equation*} \{(A,\lambda ,\alpha)\} \xrightarrow\sim\Sp_{2g} (\RR)/U(g) \end{equation*}

first we construct a bijection between

\begin{equation*} \{(A,\lambda, \alpha)\} \end{equation*}

and some linear data on a fixed space , so given \((A, \lambda, \alpha)\) use \(\alpha\) to identify

\begin{equation*} \alpha\colon \ZZ^{2g} \xrightarrow \sim \Lambda = H_1(A,\ZZ) \end{equation*}

then tensor with \(\RR\) to get

\begin{equation*} \alpha_\RR \colon \RR^{2g} \xrightarrow\sim \Lambda\otimes \RR \cong \Lie(A) (= \CC^g) \end{equation*}

the action of \(i\) on the right induces \(J\) on the left with \(J^2 = -I\text{.}\)

From \(E_\RR(iv,iw) = E_\RR(v,w)\) we get \(J \) symplectic

\begin{equation*} \Psi_\RR(Jv, Jw) = \Psi_\RR( v,w) \end{equation*}

from \(E_\RR(iv,v) \gt 0\) we get \(J\) is positive

\begin{equation*} \Psi_\RR(Jv,v) \gt 0 \end{equation*}

conversely given \(J\) symplectic positive \(J^2 = -I\) on \(\RR^{2g}\) we can construct \((A,\lambda) = (V/\ZZ^{2g}, E)\) This comes with an \(\alpha\) for free since \(H_1(A, \ZZ) \cong \ZZ^{2g}\text{.}\)

Suppose \(J\) and \(J_0\) are two complex structures, symplectic positive matrices on \(\RR^{2g}\text{.}\) Then a lemma from linear algebra tells us that there exists a \(S \in \Sp_{2g}(\RR)\text{.}\) s.t. \(J_0 = S J S \inv\text{.}\) We see that this \(S\) is well defined up to an element of \(G = Z(J) \cap \Sp_{2g}(\RR)\text{.}\) But if \(\gamma \in G \) then \(\gamma\) preserves the associated \(\CC\)-str. on \(\RR^{2g}\text{.}\) then since \(\gamma\) is symplectic, it preserves

\begin{equation*} H(v,w) = E_\RR(iv,w) + iE_\RR(v,w) \end{equation*}

implies

\begin{equation*} \gamma \in U(g)\text{.} \end{equation*}

Subsection 5.4.2 Hodge structures

Let \(M\) be a \(\cinf\) compact \(\RR\)-manifold. Then \(H^i_{\mathrm{sing}}(M, \RR) \cong H^i_\dR(M)\text{.}\) What about for compact \(\CC\)-manifolds \(X\text{?}\) For \(M\) have \(H^i_\dR(M) = H^i(\Omega^\bullet (M))\text{.}\) This won't give de Rham isomorphism for \(X\text{:}\)

\begin{equation*} H^i_{\mathrm{sing}}(X) \end{equation*}

supported up to \(i = 2d\) with \(d = \dim_\CC(X)\text{.}\) but \(H^i(\Omega^\bullet_{\mathrm{hol}}(C))\) is supported up to \(i = d\text{.}\)

For \(M\)

\begin{equation*} 0 \to \underline \RR \to \Omega^0 \to \Omega^1 \to \cdots \to \Omega^d \to 0 \end{equation*}

is a resolution of \(\underline \RR\) by acyclic sheaves, by the existence of \(\cinf\) bump functions.

\begin{equation*} H^i_\dR(M) \cong H^i(M, \underline \RR) \cong H^i_{\mathrm{sing}}(M, \RR)\text{.} \end{equation*}

For \(X\) this doesn't work with \(\Omega_{\mathrm{hol}}^\bullet\) as there are no holomorphic bump functions.

\begin{equation*} 0 \to \underline \CC \to \Omega^0_{\mathrm{hol}} \to \Omega^1_{\mathrm{hol}} \to \cdots \to \Omega_{\mathrm{hol}}^\bullet \to 0 \end{equation*}

is still a resolution but not acyclic. Instead we use hypercohomology which takes as input any resolution and outputs a cohomology group. This has the property that

\begin{equation*} H^i(X, \underline \CC) \cong \mathbf H ^i(\Omega_X^\bullet) \end{equation*}

so we define \(H^i_\dR(X) = \mathbf H^i(\Omega_X^\bullet)\text{.}\) so that

\begin{equation*} H^i_\dR(X) \cong H^i(X, \underline \CC) \cong H^i_{\mathrm{sing}}(X, \CC) \end{equation*}

On \(X\) we have the sheaf of \((p,q)\) forms \(\Omega^{p,q}\) These are locally given by

\begin{equation*} \sum_{|I| = p, |J| = q} f_{I,J} \diff z_I \diff \bar z_J\text{.} \end{equation*}

We have

\begin{equation*} \bar \partial \colon \Omega^{p,q} \to \Omega^{p,q+1} \end{equation*}

satisfying \(\bar \partial ^2 = 0\text{.}\) So we can define \(H^{p,q}(X) = \ker \bar \partial/ \im \bar \partial\) (Dolbeaut cohomology).

Remark 5.4.6.
\begin{equation*} H^{p,q} (X) \cong H^q( X, \Omega^p) \end{equation*}

using \(\bar \partial\) Poincaré lemma

Example 5.4.7.

\(E/\CC\) elliptic curve.

\begin{equation*} H^0_\dR = H^{0,0} \end{equation*}
\begin{equation*} H^1_\dR = H^{1,0} \oplus H^{0,1} \end{equation*}
\begin{equation*} H^2_\dR = H^{2,0} \oplus H^{1,1} \oplus H^{0,2} \end{equation*}

outer terms 0, diamond is 1 , 1 , 1, 1.

Definition 5.4.8. Hodge structures.

A Hodge structure on \(V/\RR\) is a \(\ZZ\)-bigrading on \(V_\CC = V \otimes \CC\) such that

\begin{equation*} \overline V^{p,q} = V^{q,p} \end{equation*}

its of Hodge type \(S\subseteq \ZZ^2\) if \(V^{pq} \ne 0 \) iff \((p,q) \in S\text{,}\)

Example 5.4.9.

the Hodge decomposition gives a hodge structure on \(H^n_{\mathrm{sing}}(X, \RR)\text{.}\)

If \(V\) has a hodge structure of weight \(n\) (i.e. \(V^{pq} \ne 0\) iff \(p,q = n\)). Then we can recover the hodge structure from the associated hodge filtration

\begin{equation*} \Fil^p V_\CC = \bigoplus_{p' \ge p} V^{p' q} \end{equation*}
Example 5.4.10.
\begin{equation*} \Fil^0(H^1(E)) = H^{1,0} \oplus H^{0,1} \end{equation*}
\begin{equation*} \Fil^1(H^1(E)) = H^{1,0} \end{equation*}
\begin{equation*} \Fil^2(H^1(E)) = 0 \end{equation*}
\begin{equation*} V^{p,q} = \Fil^p V \cap \overline {\Fil^q V} \end{equation*}

in weight \(n\text{.}\)

Alternative definition.
\begin{equation*} \mathbf S = \Res_\RR^\CC \mathbf G_m \end{equation*}
\begin{equation*} \mathbf S(A) = \{ (a,b) \in A^2 : a^2+ b^2 \ne 0\} \end{equation*}
\begin{equation*} \mathbf S(\RR) = \CC^\times \end{equation*}

Hence for any lie group \(G\) we can define a hodge structure on \(G\) as a morphism of algebraic groups

\begin{equation*} \mathbf S \to G \end{equation*}

If \(G \to \GL(V)\) is a faithful rep this induces a hodge structure on \(V\text{.}\)

Definition 5.4.13.

A polarization of a HS \(h\colon \mathbf S \to \GL(v)\) is an alternating bilinear form

\begin{equation*} \Psi \colon V\times V \to \RR \end{equation*}

with

  1. \begin{equation*} \Psi (Jv,Jw) = \Psi(v,w) \text{ for } J = h(i) \end{equation*}
  2. \begin{equation*} \Psi(v,Jw) \text{ is pos. def.} \end{equation*}