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<h^{2,0}=1}.

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}


About Charles Siegel

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
This entry was posted in Conferences, Hodge Theory, ICTP Summer School. Bookmark the permalink.

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