## ICTP Day 7

Today we kept going with the advanced topics, and started Shimura varieties, though that one went VERY fast. As always, comments and corrections are welcome!

1. Carlson 3

1.1. Monodromy

Let ${X\rightarrow S}$, ${\Delta}$ the discriminant and ${X_s}$ the fiber over ${s\in S\setminus \Delta}$ where ${X_s}$ is nonsingular.

Let’s look at the example ${y^2=x(x-1)(x-t)}$ then ${\Delta=\{0,1,\infty\}}$. Set ${\phi_t(x)}$ to be the flow where ${\phi_t:X_0\rightarrow X_t}$ is a diffeomorphism. Then set ${\rho(\gamma)={\phi_1}_*:H_n(X_0,\mathbb{Z})\rightarrow H_n(X_0,\mathbb{Z})}$, and so we get ${\rho:\pi_1(S\setminus \Delta,0)\rightarrow \mathrm{Aut}(H_n(X_0,\mathbb{Z}))}$ the monodromy representation, and ${\rho(\gamma)\in \mathrm{GL}(r,\mathbb{Z})}$.

For ${n}$ odd it preserves a skew-symmetric form, for ${n}$ even a symmetric form, so we have ${\mathrm{Sp}(r,\mathbb{Z})}$ or ${\mathrm{SO}(Q,\mathbb{Z})}$.

And this gives ${\tilde{f}:\widetilde{S\setminus \Delta}\rightarrow D}$, and we can descend this to the quotient ${S\setminus \Delta\rightarrow \Gamma\backslash D}$ where ${\Gamma}$ is at least as big as the image of the monodromy, at at most ${\mathrm{Aut}(H_n(X,\mathbb{Z}))}$.

To get a specific example we can compute, let’s look at the family ${y^2=x^2-t}$. Let ${t=\epsilon e^{i\theta}}$. Following around this loop, we see that its image is ${T=\left(\begin{array}{cc}1 & 1\\ 0 & 1\end{array}\right)}$, and it’s a Dehn twist.

For the family of cubics, we have a map ${\rho:\pi_1(\mathbb{P}^1\setminus\{0,1,\infty\})\rightarrow \mathrm{SL}(2,\mathbb{Z})}$, in general, the image of ${\rho}$ is large.

Beauville showed that for hypersurfaces, ${\rho}$ is surjective or has finite index.

1.2. Asymptotics of the Period Map

Let ${y^2=x(x-1)(x-t)}$. Recall that ${\omega=\frac{dx}{y}=\frac{dx}{\sqrt{x(x-1)(x-t)}}}$, and ${A(t)}$ and ${B(t)}$ as yesterday. We want to do the asymptotics of ${A(t)}$ as ${t\rightarrow 0}$, we get ${\int_\delta \frac{dx}{x}\frac{1}{\sqrt{-t}}=\frac{2\pi}{\sqrt{t}}}$, and for ${B(t)}$ we take ${t>>2}$ and get ${-2\int_1^t \frac{dx}{x\sqrt{x-t}}=\frac{4}{\sqrt{t}}\arctan\frac{\sqrt{1-t}}{\sqrt{t}}}$, and this simplifies to ${\frac{2i}{\sqrt{t}}\log t}$ by using trig identities.

So the ${A}$-period is ${2\pi i t^{-1/2}}$ and the ${B}$-period is ${2it^{-1/2}\log t}$, and the period map is then ${Z(t)=\frac{i}{\pi}\log t}$.

${Z_{ij}(t)=t^a(\log t)^b}$. For algebraic surfaces, ${b\leq 2}$. In general, ${b\leq}$ weight of the Hodge structure.

1.3. Period Domains and Hodge structures of weight 1

First, note that the number of moduli is ${3g-3}$ unless ${g=1}$, where it is 1.

An abelian variety is a complex torus ${\mathbb{C}^g/\Lambda}$ thatis a projective variety.

The moduli of abelian varieties of dimension ${g}$ is ${g(g+1)/2}$.

Now, given a weight 1 Hodge structure, ${J(H)=H_\mathbb{C}/H^{1,0}+H_\mathbb{Z}}$ is a projective variety, by the Riemann Bilinear Relations and the Kodaira Embedding Theorem.

The Torelli Theorem says that the map ${M\mapsto J(H^1(M))}$ is injective, so we get a map ${\mathcal{M}_g\rightarrow \mathcal{A}_g}$, and finding equation for this image is called the Schottky problem.

What is ${D}$? In the general case of weight 1. We write down a matrix ${(A,B)}$ of ${A}$-periods and ${B}$-periods, with the basis ${\omega_1,\ldots,\omega_g}$ a basis for ${H^{1,0}}$ and ${\delta_1,\ldots,\delta_g,\gamma_1,\ldots,\gamma_g}$ a symplectic basis for ${H_1(\mathbb{Z})}$.

We define ${A_{ij}=\int_{\delta_j}\omega_i}$ and ${B_{ij}=\int_{\gamma_j}\omega_i}$. The Riemann Bilinear Relations tell us that ${i\int_M \omega\wedge\bar{\omega}>0}$ for ${\omega}$ a nonzero abelian differential, ${\omega=\sum v_m \omega_m}$ and ${\omega_m=\sum A_{mi}\delta^i+\sum B_{nj}\gamma^j}$.

Putting this all together, we get that we get that ${i(v_m A_{mi}\bar{v}_n\bar{B}_{ni}-v_m B_{mj}\bar{v}_n\bar{A}{nj})>0}$, which is just ${iv(AB^*-BA^*)\bar{V}>0}$, and ${A,B}$ nonsingular matrices. So we can find a basis such that the period matrix is ${(1,Z)}$. The first RB relation gives that ${Z=Z^t}$.

So, for weight 1, ${D=\mathbb{H}_g=\{Z|Z^t=Z, \mathrm{Im}(Z)>0\}}$.

2. Murre 3

2.1. Comparison

Let ${X}$ be a smooth projective irreducible variety over ${\mathbb{C}}$, there are several topologies, in particular the Zariski topology and the étale topology, as well as teh underlying analytic space ${X_{an}}$ which is a complex analytic manifold which is compact and connected in the classical topology.

We have a comparison theorem saying that ${H^i(X_{an},\mathbb{Q})\otimes\mathbb{Q}_\ell\cong H_{et}^i(X,\mathbb{Q}_\ell)}$ for ${\ell\neq}$ the characteristic.

Similarly, we have a theorem of Serre in 1956, GAGA, that ${\mathscr{F}\mapsto \mathscr{F}_{\mathscr{O}_X}\mathscr{O}_{X_{an}}}$ is an equivalence of categories between algebraic and analytic coherent sheaves, and the cohomologies are the same.

In particular, we find that the Picard groups are isomorphic, and so ${CH^1(X)}$ doesn’t depend on whether we look analytically or algebraically.

2.2. Cycle Map

Let ${Z\in Z^p(X)}$ with ${q=d-p}$.

There exists a homomorphism ${\gamma_\mathbb{Z}:Z^p(X)\rightarrow H^{2p}(X_{an},\mathbb{Z})}$ given by taking the inclusion ${Z\rightarrow X}$ and using the exact sequence of a pair for the analytic spaces.

In particular, we have maps ${T:H^0(Z,\mathbb{Z})\rightarrow H^{2p}(X,Z,\mathbb{Z})}$ and ${\rho:H^{2p}(X,Z,\mathbb{Z})\rightarrow H^{2p}(X,\mathbb{Z})}$, and ${H_{2p}(Z,\mathbb{Z})\rightarrow H^0(X,\mathbb{Z})}$. Taking ${1\in H_{2p}(Z,\mathbb{Z})}$ to an element of ${H^0(Z,\mathbb{Z})}$, we define ${\gamma(Z)=\rho T(1)\in H^{2p}(X,\mathbb{Z})}$.

What is the position of ${\gamma(Z)\in H^{2p}(X,\mathbb{C})}$?

Lemma 1 Let ${j:H^{2p}(X,\mathbb{Z})\rightarrow H^{2p}(X,\mathbb{C})=\oplus_{r+s=2p}H^{r,s}(X)}$ then ${j\gamma_\mathbb{Z}(Z)\in H^{p,p}(X)}$, and hence ${\gamma_\mathbb{Z}(Z)=Hdg^p(X)=j^{-1}(H^{p,p})}$.

Proof: We have a nondegenerate pairing between ${H^{r,s}}$ and ${H^{d-r,d-s}}$. Set ${\alpha=j\gamma_\mathbb{Z}(Z)}$, and take a ${\beta}$. Look at ${\langle j\gamma_\mathbb{Z}(Z),\beta\rangle=\int_Z \beta|_Z}$, and this is zeor unless ${\beta\in H^{q,q}(X)}$. $\Box$

Reference for details: Griffiths and Harris, or Voisin.

2.3. Hodge Classes, Cycles and Conjecture

We have the theorem that the image of ${\gamma_\mathbb{Z}}$ is contained in ${Hdg^p(X)}$.

Is ${\gamma_\mathbb{Z}}$ surjective?

This is true for ${i=1}$, by the Lefschetz (1,1) Theorem. But for ${i>1}$, this is no longer true! See Hirzebruch-Atiyah in 1962, Kollar in 1992 and Totaro 1998.

However, it is still wide open if it is true after tensoring with ${\mathbb{Q}}$.

But we do have:

${CH^1(X)\rightarrow Hdg^1(X)}$ is onto.

Sketch of proof, following Kodaira and Spencer:

Here we have that ${CH^1(X)=\mathrm{Pic}(X)}$, and we have the exponential sequence ${0\rightarrow \mathbb{Z}\rightarrow \mathscr{O}_{X_{an}}\rightarrow \mathscr{O}^*_{X_{an}}\rightarrow 1}$ exact, so we get an exact sequence on cohomology. This gives us an exact sequence ${H^1(X_{an},\mathscr{O}^*_{X_{an}})\stackrel{\gamma_\mathbb{Q}}{\rightarrow} H^2(X_{an},\mathbb{Z})\rightarrow H^2(X_{an},\mathscr{O}_{X_{an}})}$ and so we get that ${D\mapsto \gamma_\mathbb{Z}(D)}$, and every (1,1) class has image zero in ${H^{0,2}(X)}$, so the map is surjective onto them.

2.4. Intermediate Jacobians

Let ${J^p(X)}$ be the ${p}$th intermediate Jacobian of ${X}$ over ${\mathbb{C}}$, then ${J^p(X)=H^{2p-1}(X_{an},\mathbb{C})/F^pH^{2p-1}+H^{2p-1}(X,\mathbb{Z})}$. This is the Griffiths intermediate Jacobian.

Lemma 2 ${J^p(X)}$ is a complex torus of dimension ${\frac{1}{2}\dim H^{2p-1}(X_{an},\mathbb{C})}$.

Proof: To see if ${\alpha_1,\ldots,\alpha_m}$ is a ${\mathbb{Q}}$ basis for ${H^{2p-1}(X,\mathbb{Q})}$, we need ${v=\sum R_i\alpha_i\in F^p\Rightarrow v=0}$. We have that the ${r_i\in \mathbb{Q}}$, so ${\bar{v}=v}$, so ${\bar{v}\in V}$, and ${F^p\cap V=-}$. $\Box$

This is in general not an abelian variety. However, if ${V\subset H^{p-1,p}}$, then it is.

This will always happen for ${i=1}$ and ${i=d}$ where ${d=\dim X}$, and these are the Picard variety of ${X}$ and the Albanese variety of ${X}$, and, in the case of curves, these are the same and are the Jacobian of of ${X}$.

3. Brosnan 3

Last time, we looked at ${\mathrm{Ext}^1_{MHS} (\mathbb{Z},H)=J(H)=H_\mathbb{C}/F^0H+H_\mathbb{Z}}$ is the Griffiths intermediate Jacobian for ${H}$ pure of negative weight, by Carlson’s formula.

For ${X}$ a smooth projective variety, ${J^p(X)=J(H^{2p-1}(X,\mathbb{Z})(-p))}$ which has weight -1.

Recall from Cattani that a variation of Hodge structure of weight ${k}$ on a complex manifold ${S}$ consists of a pair ${(H,F)}$ where ${H}$ is a local system on ${S}$, ${F^*}$ is a decreasing filtration on ${H\otimes_\mathbb{Z} \mathscr{O}_S}$ and at every point ${s\in S}$, we have ${(H,F^p)_s}$ defines a Hodge structure, along with Griffiths Transversality, that ${\nabla F^p\subset F^{p-1}\otimes \Omega_S}$.

Now, Griffiths notices that if ${f:X\rightarrow S}$ is a smooth projective morphism, then if you set ${\mathscr{H}=R^kf_*\mathbb{Z}}$ so that ${\mathscr{H}_s=H^k(X_s,\mathbb{Z})}$, then we have a variation of Hodge structures.

A variation of MHS is a triple ${(V,F,W)}$, with ${V}$ a local system, ${F}$ and ${W}$ appropriate filtrations, see Cattani.

And, just to be clear:

A local system of ${A}$ modules on ${S}$ for ${A}$ a ring is a locally constant sheaf of ${A}$-modules. If ${S}$ is connected, then local systems of ${A}$ modules are in bijection with ${\pi_1(S,s)-A}$ modules.

Let ${S}$ be a complex manifold, ${\mathscr{H}}$ a variation of pure Hodge structures on ${S}$. The group of normal functions on ${S}$ is the group ${NF(S,\mathscr{H})=\mathrm{Ext}^1_{VMHS(S)}(\mathbb{Z},\mathscr{H})}$.

In this, ${VMHS(S)}$ denotes teh abelian category of variations of MHS on ${S}$ and ${\mathbb{Z}}$ denotes the constant pure Hodge structure on ${S}$ which is ${\mathbb{Z}(0)}$ at every point, and the weight of ${H}$ is negative.

Given ${\mathscr{H}}$, we have a family ${J(\mathscr{H})\rightarrow S}$ of intermediate Jacobians and the fiber at ${s\in S}$ is ${J(H_s)}$, so by restriction, an element ${\nu\in NF(S,\mathscr{H})}$ determines a section ${\sigma_\nu}$ of ${J(\mathscr{H})/S}$.

Fact: ${\sigma_\nu}$ determines ${\gamma}$. In other words, there is an injection ${NF(S,\mathscr{H})}$ to the group of section ${\sigma:S\rightarrow J(\mathscr{H})}$ of a family ${J(\mathscr{H})\rightarrow S}$.

The idea of hte proof is to try to use the section ${\sigma:S\rightarrow J(\mathscr{H})}$ to construct an extension of mixed Hodge structures. What would be nice would be to have a tautological extension of ${\mathbb{Z}}$ by ${\mathscr{H}}$ sitting over ${J(\mathscr{H})}$. Then you can pull it back by ${\sigma}$ and can ignore Griffiths transversality and find such a tautological extension to get injectivity.

Look at ${S=\mathbb{P}^1\setminus \{0,1,\infty\}}$ and ${\mathscr{H}=\mathbb{Z}(1)}$, then ${\mathrm{Ext}^1_{VMHS(S)}(\mathbb{Z},\mathbb{Z}(1))=\mathscr{O}_S^*(S)}$, and we recall from last time that we constructed an element corresponding to ${\lambda\in \mathscr{O}_S^*(S)}$, and we gave it as an extension ${0\rightarrow\mathbb{Z}\rightarrow H^1(\mathbb{P}^1\setminus\{0,\infty\},\{1,\lambda\})\rightarrow \mathbb{Z}(-1)\rightarrow 0}$, and this is in ${\mathrm{Ext}(\mathbb{Z}(-1),\mathbb{Z})}$, which is isomorphic to ${\mathrm{Ext}(\mathbb{Z},\mathbb{Z}(1))}$.

3.1. Singularities

Suppose that ${j:S\rightarrow \bar{S}}$ open immersion of complex manifolds, and ${\mathscr{H}}$ a VHS on ${S}$. Typical situation: ${f:X\rightarrow \bar{S}}$ with ${X,\bar{S}}$ smooth projective varieties, and by generic smoothness, there is a dense oepn ${S\subset\bar{S}}$ where ${f}$ is smooth. Therefore, if we set ${\mathscr{H}=R^kf_*\mathbb{Z}}$, we get that ${\mathscr{H}}$ is a variation of Hodge structure on ${S}$, but won’t in general extend to ${\bar{S}}$.

Take ${\nu\in NF(S,\mathscr{H})}$ then ${cl(\nu)\in \mathrm{Ext}_{Sheaves(S)}(\mathbb{Z},\mathscr{H}_\mathbb{Z})=H^1(S,\mathscr{H}_\mathbb{Z})}$ is a topological invariation of ${\nu}$, the singularity of ${\nu}$ at a point ${s\in \bar{S}\setminus S}$. We define it to be ${sing_s(\nu)}$ which captures the topology of ${\nu}$ near ${s}$ as follows: take a small ball around ${s}$ in ${\bar{S}}$. Defien ${sing_s(\nu)}$ to be the image of ${cl(\nu)}$ under the map ${H^1(S,\mathscr{H}_\mathbb{Z})\rightarrow H^1(B\cap S,\mathscr{H}_\mathbb{Z})}$. As long as ${B}$ is small enough, this is independent of the ball.

4. Carlson 4

Just a few more things about weight 1 before going on to weight 2.

${D=\mathbb{H}_g=\{Z|Z^t=Z, \mathrm{Z}>0\}}$. Another way to view it is that ${\mathbb{H}_1=\mathrm{SL}(2,\mathbb{R})/U(1)}$. We can see this by noting that ${\mathrm{SL}(2,\mathbb{R})}$ acts transitively on ${\mathbb{H}_1}$, and the subgroup fixing ${i}$ is ${U(1)}$.

More generally, ${\mathbb{H}_g}$ has a transitive action by ${\mathrm{Sp}(2g,\mathbb{R})}$. Let ${K}$ be the isotropy group for ${H}$, then ${\mathbb{H}_g=G/K=\mathrm{Sp}(2g,\mathbb{R})/U(g)}$, and this is an example of a hermitian symmetric space, though higher weight period domains generally aren’t.

4.1. Higher weight

Let ${H}$ be a Hodge structure of weight ${k}$, ${D}$ the period domain, then ${G}$ is ${\mathrm{Sp}(n,\mathbb{R})}$ or ${\mathrm{SO}(Q,\mathbb{R})}$, and acts transitively on ${D}$, and let ${V}$ be the isotropy group of ${H}$

In the weight 2 case, if ${p=\dim H^{2,0}}$ and ${q=\dim H^{1,1}}$, then ${G=\mathrm{SO}(2p,q)}$ and ${V=U(p)\times \mathrm{SO}(q)}$ with ${K}$, the maximal compact, ${\mathrm{SO}(2p)\times \mathrm{SO}(q)}$.

${D}$ is not in general a Hermitian symmetric space, but it is when ${K=V}$. For instance, if ${p.

For instance, K3 surfaces.

Fact: ${K/V}$ is compact complex subvariety of ${D}$.

Period map for hypersurfaces ${X\subset\mathbb{P}^{n+1}}$ has a holomorphic part and a horizontal part, ${F^p}$ and ${F^{p-1}}$.

Poincaré Residue: Look at the cohomology of ${\mathbb{P}^{n+1}\setminus X}$. Grothendieck looked at this cohomology and its meromorphic forms with pole along ${X}$.

Let ${z_0,\ldots,z_{n+1}}$ be homogeneous coordinates on ${\mathbb{P}^{n+1}}$, let ${U_0}$ be ${z_0\neq 0}$, then we have affine coordinates by dividing by ${z_0}$, and so ${d(z_1/z_0)\wedge\ldots\wedge d(z_{n+1}/z_0)=\frac{\sum (-1)^iz_i dz_0\wedge\ldots\wedge\hat{dz}_i\wedge\ldots\wedge dz_{n+1}}{z_0^{n+2}}}$. Set the numerator as ${\Omega}$.

Define ${\Omega_A=A\Omega/Q^2}$ where ${Q}$ is the equation of ${X}$ with ${\deg A+n+2=2\deg Q}$, so then ${[\Omega]}$ is acohomology class in ${H^{n+1}(\mathbb{P}^{n+1}\setminus X)}$.

We have a sequence ${H^{n+1}(\mathbb{P}^n)\rightarrow H^{n+1}(\mathbb{P}^{n+1}\setminus X)\rightarrow H^n(X)\rightarrow H^{n+2}(\mathbb{P}^{n+1}}$ so we define the local residue map in the usual way.

Claim: The period map is both holomorphic and horizontal.

To see that it’s holomorphic, let ${X_t}$ be given by ${Q+tR=0}$. Then ${\frac{\partial}{\partial \bar{t}} \left(\frac{A\Omega}{(Q+tR)^2}\right)=0}$, so ${\frac{\partial}{\partial\bar{t}}\frac{1}{2\pi i}\int_{\gamma} \frac{A\Omega}{(Q+tR)^2}=0}$ and by the residue formula, the map is holomorphic. Horizontality then follows from Griffiths Transversality.

5. Kerr 1 – Shimura Varieties

The plan of the course is:

1. Hermitian Symmetric Domains, ${D}$
2. Locally Symmetric Varieties, ${\Gamma\backslash D}$
3. Theory of CM
4. Shimura Varieties
5. Fields of Definition

5.1. Algebraic Groups

An algebraic group ${G}$ defined over ${k}$ is an algebraic variety with morphisms ${G\times G\rightarrow G}$ and ${G\rightarrow G}$ along with ${e\in G(k)}$ subject to the rules making ${G(L)}$ a group for all ${L/k}$.

${\mathbb{G}_m^*=\{xy=1\}\subset \mathbb{A}^2}$ then ${\mathbb{G}_m(k)=k^*}$.

${G}$ is connected iff ${G_{\bar{k}}}$ is irreducible and ${G}$ is simple iff ${G}$ nonabelian with no normal connected subgroups other than the trivial ones.

If ${k=\mathbb{C}}$, then examples are the usual algebraic groups, ${\mathrm{SL}_n,\mathrm{SO}_n,\mathrm{Sp}_n,E_6,E_7,E_8,F_4,G_2}$, over ${k=\mathbb{R}}$, we have the real forms, and over ${\mathbb{Q}}$ “all hell breaks loose”

We call ${G}$ a torus if ${G_{\bar{k}}}$ is a product of ${\mathbb{G}_m}$‘s.

Inside ${\mathrm{GL}_2}$, there is ${\mathbb{S}\supset\mathbb{U}}$, ${\mathbb{G}_m}$ being the matrices ${\left(\begin{array}{cc}a&b\\-b&a\end{array}\right)}$ with ${a^2+b^2\neq 0}$, the next being ${a^2+b^2=1}$ and the last being ${b=0}$.

${\mathbb{S}\cong \mathbb{C}^*\times \mathbb{C}^*}$ and ${\mathbb{U}=\mathbb{C}^*}$.

Let ${G=\mathrm{Res}_{E/\mathbb{Q}}\mathbb{G}_m}$ be Weil restriction. Then ${\dim_\mathbb{Q} G=[E:\mathbb{Q}]}$, ${G(\mathbb{Q})=E^*}$ and ${G(k)=E^*\otimes_\mathbb{Q} k\cong (k^*)^{[E:\mathbb{Q}]}}$.

We call ${G}$ semisimple iff it is (almost) the direct product of simple groups, and reductive if linear reps are completely reducible.

One representation is ${G\stackrel{\mathrm{ad}}{\rightarrow} \mathrm{GL}(\mathfrak{g})}$ where ${\mathfrak{g}=T_eG}$ is the Lie algebra, by ${g\mapsto \{X\mapsto gXg^{-1}\}}$

For semisimple ${G}$, we say that ${G}$ is adjoint if ${\mathrm{ad}}$ is injective, and simply connected if any isogeny ${G'\rightarrow G}$ with ${G'}$ connected is an isomorphism.

Let ${G}$ be a reductive real algebraic group and ${\theta:G\rightarrow G}$ an involution.

${\theta}$ Cartan if adn only if ${\{g\in G(\mathbb{C})|g=\theta(\bar{g})\}=G^{(\theta)}(\mathbb{R})}$

This holds iff ${\theta=\mathrm{Ad}(C)}$ for some ${C\in G(\mathbb{R})}$ with ${C^2\in Z(\mathbb{R})}$, ${G\subset \mathrm{Aut}(V,Q)}$ such that ${Q( \cdot, C\cdot)>0}$ on ${V}$.\

${\theta=1}$ if adn only if ${G(\mathbb{R})}$ compact.

5.2. Three Characterizations of Hermitian Symmetric Domains

Let ${X}$ be a connected open subset of ${\mathbb{C}^n}$ with compact closure, such that ${Hol(X)}$ acts transitively and contains the symmetries ${s_p}$. We call it a bounded symmetric domain, and these are analytic and extrinsic objects.

Hermitian symmetric spaces of noncompact type (analytic, intrinsic) are ${(X,g)}$ a connected complex manifold with Hermitian metric such that ${Is(X,g)}$ acts transitively, contains symmetries ${s_p}$ for all ${p\in X}$, and ${Is(X,g)^+}$ is semisimple adjoint and noncompact.

The third type, called circle conjugacy class, is ${X=\mathbb{G}(\mathbb{R})}$ a conjugacy class of a homomorphism ${\phi:\mathbb{U}\rightarrow G}$ in an algebraic group over ${\mathbb{R}}$ where ${G}$ is a real adjoin algebraic group and only ${z,1,z^{-1}}$ appear as eigenvalues in the red og ${\phi}$ on ${Lie(G)_\mathbb{C}}$, ${\theta}$ is ${ad(\phi(-1))}$ is Cartan, and ${\phi(-1)}$ doesn’t project to 1 in any simple factor of ${G}$.

These are equivalent notions.

Under the equivalence, ${Is(X,g)^+\cong Hol(X)^+\cong G(\mathbb{R})^+}$ and if ${K_p=stab(p)}$ for some ${p\in X}$, then ${X}$ can be written ${G(\mathbb{R})^+/K_p}$.

3 to 2: Let ${p=\phi}$, ${K=Z_{G(\mathbb{R})^+}(\phi)\subset G^{(\theta)}(\mathbb{R})}$ then ${K_c}$ is a 1-eigenspace of ${\phi(-1)}$ and ${K}$ is compact. We have ${\mathfrak{g}_\mathbb{C}=1\oplus P^+\oplus P^-}$ with eigenvalues ${1,z,z^{-1}}$.

Using ${G(\mathbb{R})^+}$ to translate ${J=d(\phi(i))}$ to all of ${TX}$ yields and almost complex structure.

${X\rightarrow \check{X}=G(\mathbb{C})/P^+\cong G^{(\theta)})\mathbb{R})/K}$ makes ${X}$ a complex manifold. Then there exists a ${K}$-invariant symmetric and definite bilinear form on ${T_\phi X}$, so there exists ${G(\mathbb{R})^+}$-invariant riemannian metric ${g}$ which commtues with ${J}$, and so is Hermitian. ${s_\phi}$ is ${\mathrm{Ad}\phi(-1)}$, and so ${G}$ is noncompact.

(2) to (3): ${Is(X,g)^+}$ is adjoint and semisimple, so is ${G(\mathbb{R})^+}$ for some ${G\subset \mathrm{GL}(Lie(Is^+))}$. For ${p\in X}$, we get ${s_p\in \mathrm{Aut}(X)}$ and ${s_p^2=id}$, so ${ds_p}$ is multiplication by ${-1}$ on ${T_pX}$, because ${p}$ is an isolated fixed-point.

In fact, for any ${|z|=1}$, there exists a unique isometry ${u_k(z)}$ of ${(X,g)}$ such that on ${T_pX}$ ${du_p(z)}$ is mult by ${z}$.

The uniqueness means that ${u_p:U_1\rightarrow Is(X,g)^+}$ is a homomorphism, it algebraizes to ${\phi_p:\mathbb{U}\rightarrow G}$ over ${\mathbb{R}}$.

${\phi_p(z)=\mathrm{Ad}(g)\phi_p(z)}$. Now, ${\mathfrak{g}_\mathbb{C}=k_c\oplus T^{1,0}_pX\oplus T^{0,1}_pX}$ have eigenvalues ${1,z,z^{-1}}$.

5.3. Cartan’s Classification of irreducible HSD’s

Let ${X}$ be an irreducible HSD, ${G}$ a connected simple ${\mathbb{R}}$-alg group, ${T\subset G_\mathbb{C}}$ a maximal algebraic torus over ${\mathbb{C}}$. ${\mathfrak{g}_\mathbb{C}=\mathfrak{t}\oplus\bigoplus_{\alpha\in K} \mathfrak{g}_\alpha}$ where ${R\subset \hom(T,\mathbb{G}_m)\cong \mathbb{Z}^n}$, ${R=R^+\coprod R^-}$. Over ${\mathbb{C}}$, ${\phi}$ defines a cocharacter which we may conjugate with any factor from ${T}$ and have ${\langle \mu,\alpha\rangle\geq 0}$ for all ${\alpha\in R^+}$.

Thus, ${\mu}$ must act through ${z,1,z^{-1}}$, and so ${\langle\mu,\alpha\rangle=0}$ or 1 for each ${\alpha\in R^+}$. Then, ${\mu}$ is a miniscule coweight.

Therefore, ${\langle \mu,\alpha_i\rangle=1}$ for a unique simple root ${\alpha_i}$ and for this ${i}$, ${\hat{m}_i=1}$.

So we have a 1-1 correspondence between irreducible HSD and special nodes on a connected Dynkin diagram, and we can even say how many there are:

$\displaystyle \begin{array}{ccccccccc}A_n&B_n&C_n&D_n&E_6&E_7&E_8&F_4&G_2\\n&1&1&3&2&1&0&0&0\end{array}$

For ${A_n}$, we get ${X\cong SU(p,q)/S(U_p\times U_q)}$ for ${p+q=n}$, for ${B_n}$, ${X\cong SO(n,2)^+/SO(n)\times SO(2)}$ corresponds to K3 surfaces, for ${C_n}$ we get ${X\cong \mathrm{Sp}_n(\mathbb{R})/U(n)}$, which are the Siegel upper half planes.

6. Brosnan 4

6.1. Hodge Classes and Normal Functions

Lefschetz realized that you can start with a Hodge class ${\alpha\in H^2(X)}$ where ${X\rightarrow\mathbb{P}^1}$ is a smooth projective algebraic surface, and if you assume that ${\alpha}$ is primitive, meaning that ${\alpha|_{X_s}=0}$ for ${s\in \mathbb{P}^1}$ such that ${X_s}$ is smooth, then there is a normal function associated to ${\alpha}$ in ${NF(U,\mathscr{H}}$ where ${U}$ is teh smooth locus of ${\mathbb{P}^1}$.

Let ${X}$ be a smooth projective complex variety. Then Deligne sets ${\mathbb{Z}(p)_D=\mathbb{Z}(p)\rightarrow \mathscr{O}_X\rightarrow\Omega^1_X\rightarrow\ldots\rightarrow\Omega_X^{p-1}}$ in degrees ${0}$ to ${p}$.

Note that ${\mathbb{Z}(1)_D}$ is just ${\mathbb{Z}(1)\rightarrow \mathscr{O}_X}$, and so gives the exponential sequence.

Thus, ${\mathbb{Z}(1)_D\cong \mathscr{O}_X^*[-1]}$, and ${H^2(X,\mathbb{Z}(1)_D)=H^0(X,\mathbb{Z}(1)_D[2])=H^0(X,\mathscr{O}_X^*[1])=H^1(X,\mathscr{O}_X^*)\cong \mathrm{Pic} X}$.

For ${p\in \mathbb{Z}}$, ${Hodge^{2p}(X)=H^{2p}(X,\mathbb{Z}(p))\cap H^{p,p}}$

If ${X}$ is a smooth projective variety, then is an exact sequence ${0\rightarrow J(H^{2p-1}(X)(p))\rightarrow H^{2p}(X,\mathbb{Z}(p)_D)\stackrel{cl}{\rightarrow} Hodge^{2p}(X)\rightarrow 0}$.

Strictly, we’ve been working with hypercohomology, but we’re not going to worry about distinguishing it.

Suppose that ${f:X\rightarrow \bar{S}}$ is a morphism with ${X, \bar{S}}$ smooth projective and irreducible, and ${S}$ the smooth locus. Write ${Prim(X/S)=\{\alpha\in Hodge^{2p}(X):\alpha|_{X_s}=0\forall s\in s\}}$ to be the primary classes.

Given ${\alpha\in Prim^p(X/S)}$, we can find a lift ${\tilde{\alpha}}$ of ${\alpha}$ ot ${H^{2p}(X,\mathbb{Z}(p)_D)}$. Then for every ${s\in S}$ we have ${cl(\tilde{\alpha}_s)=\alpha|_s=0\in H^{2p},X_s,\mathbb{Z}(p)_D)}$.

So for every ${\tilde{\alpha}}$, we get a map ${s\mapsto \tilde{\alpha}_s\in J(H^{2p-1}(X_0)(-p))}$ and so we write ${\mathscr{H}=R^{2p-1}f_*\mathbb{Z}(p)|_S}$.

The association ${s\mapsto \tilde{\alpha}_s}$ defines a normal function ${\gamma(\tilde{\alpha})\in NF(S,\mathscr{H})}$.

The dependence on ${\tilde{\alpha}}$ is pretty simple: any other lift of ${\alpha}$ will differ from ${\tilde{\alpha}}$ by an element of ${J(H^{2p-1}(X)(p))}$.

We have a map ${Prim^p(X/S)\rightarrow NF(S,\mathscr{H})/J(H^{2p-1}(X)(-p))}$.

${0\rightarrow J(H^{2p-1}(X)(-p))\rightarrow H^{2p}(X,\mathbb{Z}(p)_D)\stackrel{cl}{\rightarrow} Hodge^{2p}(X)\rightarrow 0}$.

So we have a triangle ${\Omega_X^{\leq p-1}[-1]\rightarrow \mathbb{Z}(p)_D\rightarrow \mathbb{Z}(p)\rightarrow \Omega_X^{\leq p-1}}$ and so we jsut need to know ${\mathbb{H}(X,\Omega^{\leq p-1}_X)}$. In fact, we have ${\Omega^{\geq p}\rightarrow \Omega^*\rightarrow |\Omega^{\leq p-1}\rightarrow \Omega^{\geq p}[1]}$, and by degeneration of Hodge to deRham spectral sequence, we know that ${H^n(X,\Omega^{\geq p}_X)=F^pH^n(X,\mathbb{C})}$.

So ${H^n(X,\Omega_X^*)=H^n(X,\mathbb{C})}$, and therefor ${H^n(X,\Omega^{\leq p-1})=H^n(X,\mathbb{C})/F^pH^n(X/\mathbb{C})}$.

So we have ${H^{2p-1}(X,\mathbb{Z}(p))\rightarrow H^{2p-1}(X,\mathbb{C})/F^p\rightarrow H^{2p}(X,\mathbb{Z}(p)_D)\rightarrow H^{2p}(X,\mathbb{Z}(p))\rightarrow H^{2p}(X,\mathbb{C})/F^p}$.

6.2. A construction of Green-Griffiths

If ${X}$ is a smooth projective variety with ${\dim X=2n}$ and ${\mathscr{O}(1)}$ fixed very ample line bundle, from this let’s construct a sequence of normal functions on a sequence of spaces.

First, we construct the spaces. For each ${d\in \mathbb{Z}_{>0}}$, set ${\mathbb{P}_d=\mathbb{P}(H^0(X,\mathscr{O}_X(d))=|\mathscr{O}_X(d)|}$ is the complete linear system on ${\mathscr{O}_X(d)}$.

Now, write ${Prim(X)=\{\alpha\in Hodge^{2n}X:\alpha|_D=0\mbox{ for }D\mbox{ a smooth divisor in }\mathscr{O}_X(d)\}}$. Now for each ${d}$ we have an incidence variety ${\mathcal{X}_d=\{(x,f)\in X\times \mathbb{P}_d:f(x)=0\}}$ and this is smooth over some dense open ${U\subset\mathbb{P}_d}$. Set ${\mathscr{H}_d=R^{2n-1}{\pi_d}_*\mathbb{Z}(p)}$.

By definition, if ${\alpha\in Prim(X)}$, and ${\pi:\mathcal{X}_d\rightarrow X}$ the projection, then ${\pi^*\alpha\in Prim^{2n}(\mathcal{X}_d/\mathbb{P}_d)}$, and so we get a map ${Prim(X)\rightarrow NF(U_d,\mathscr{H})/J(H^{2n-1}\mathcal{X}(n))}$.

Now let ${s\in \mathbb{P}_d\setminus U_d}$, the set of singular hyperplane sections. Then Green-Griffiths noticed that teh singular ${sing_s\nu}$ of the normal function ${\nu}$ associated to a primitive class ${\alpha\in Prim(X)}$ is related to the restriction of ${\alpha}$ to the divisor ${X_s\subset \mathcal{X}_d}$