## ICTP Day 8

Today was a half day, and we finished Carlson’s series. I think this afternoon I’ll clean some things up and maybe post notes from the finished series on my website, along with links from here. In any case, here are my notes:

1. Murre 4

We have ${\mathcal{X}_d\subset \mathbb{P}^N}$ an irreducible smooth variety. Look at the complex torus ${J^p(X)}$, the intermediate Jacobian of Griffiths. It won’t in general be an abelian variety.

We define ${J^p_{alg}(X)\subset J^p(X)}$ to be the largest abelian subvariety of ${J^p(X)}$.

1.1. Abel-Jacobi Map

Let ${Z\in Z^p_{hom}(X)}$

There exists a homomorphism ${\mathcal{AJ}^p:Z^p_{hom}(X)\rightarrow J^p(X)}$ which factors through ${CH^p_{hom}(X)}$.

an outlike of the construction, we note that ${H^{2p-1}(X)}$ and ${H^{2d-2p+1}(X)}$ are dual, and we take ${V}$, the universal cover of ${J^p(X)}$ and ${F^{d-p+1}H^{2d-2p+1}}$, which are also dual. Now, for ${Z\in Z_{hom}^p(X)}$, we have ${Z=\partial C}$ where ${C}$ is a topological chain of dimension ${2d-2p+1}$.

So take ${\alpha\in F^{d-p+1}H^{2d-2p+1}(X)}$, and take the cohomology class of ${\alpha}$ representing this ${\alpha}$, ${v_C:\alpha\rightarrow \int_C\alpha}$ for ${v_C\in V}$ does not depend on the choice of cohomology class, ${\alpha=\alpha+d\beta}$, and we can choose ${\beta}$ such that it contains at least ${d-p+1}$ ${dz}$‘s. Then ${\int_Cd\beta=\int_Z\beta=0}$.

Now, take ${C'}$, and we have ${\int_{C'}\alpha\in H^{2p-1}(X,\mathbb{Z})}$ and ${\partial(C'-C)=0}$, and ${\int_{C'}-\int_C=\int_Z\alpha}$. So we have a well defined element ${\mathcal{AJ}^p(Z)\in J^p(X)}$.

Let ${X=C}$ be a curve, then ${J(C)=H^1(X,\mathbb{C})/H^{1,0}+H^1(X,\mathbb{Z})}$, and if we take ${D=\sum(p_i-p_i')\in \mathrm{Div}(X)}$, ${\alpha_1,\ldots,\alpha_g}$ a basis for ${H^0(X,\Omega)=H^{1,0}}$, then ${\mathcal{AJ}^1(D)=\left(\int_{T_i}\alpha_1,\ldots,\int_{T_i}\alpha_g\right)\in \mathbb{C}^g/\Lambda=J(X)}$.

Remark: Suppose that ${Z\in Z^p_{alg}(X)}$. We claim that ${\mathcal{AJ}^p(Z)\in J^p_{alg}(X)}$.

1.2. Algebraic Equivalence vs. Homological Equivalence

We look at ${Z^i_{alg}(X)\subseteq Z^i_{hom}(X)}$. For divisors ${i=1}$, we have equality for every algebraically closed ${k}$ by a theorem of Matsusaka in 1956.

There exists ${\mathcal{X}_d\subset\mathbb{P}^N}$ smooth over ${\mathbb{C}}$ such that for certain ${i>1}$ the ${Z^i_{alg}(X)\neq Z^i_{hom}(X)}$.

We define ${Gr^i(X)=Z^i_{hom}(X)/Z^i_{alg}(X)}$ to be the Griffiths group, which is discrete.

There exist varieties such that for ${i>1}$, ${Gr^i(X)\neq 0}$ and, in fact with ${Gr^i(X)\otimes \mathbb{Q}\neq 0}$.

There exist Chow varieties, given ${X_d\subset\mathbb{P}^N}$ fixing the dimension ${q}$ and degree ${m}$, then ${Z^p(X)_{\deg m}=Ch(X,r,m)}$ is the Chow variety.

Preparation:

Lefschetz hyperplane Theorem: ${W=V\cap H\subset V\subset \mathbb{P}^N}$ with ${H}$ a hyperplane, ${i^*:H^j(V,\mathbb{Z})\rightarrow H^j(W,\mathbb{Z})}$ is an isom for ${j<\dim W}$ and is injective for ${j=\dim W}$.

In particular, ${W\subset\mathbb{P}^N}$ a hypersurface section then ${H^j(W)=0}$ for ${j}$ odd and ${j\neq \dim W}$ and ${H^{2j}(W,\mathbb{Z})=\mathbb{Z} h}$ where ${h=d(WH)}$ for ${2j<\dim W}$ and ${2j>\dim W}$.

We need the notion of a Lefschetz pencil: Let ${V\subset \mathbb{P}^N}$. Take ${H_0,H_1}$ hyperplanes in general position with respect to ${V}$, then ${H_t=H_0+tH_1}$ for ${t\in \mathbb{P}^1}$ is a pencil, and we call ${W_t=V\cap H_t}$ a Lefschetz pencil.

1. For all except finitely many ${t}$, ${W_t}$ is smooth
2. ${W_t}$ for ${t\in S}$ only are ordinary double points, where ${S}$ is the locus of singular sections.

The action of ${\Gamma}$ (the image of the monodromy representation) on ${H^j(W,\mathbb{Q})}$ is completely reducible.

Fact: ${H^j(W)_{van}=0}$ if ${j\neq d=\dim W}$ where the vanishing means it is in the kernel of ${i_*:H^j(W)\rightarrow H^{j+2}(V)}$.

Let ${V_{d+1}\subset\mathbb{P}^N}$ and ${d+1=2m}$. Consider a Lefschetz pencil ${\{W_t\}}$ on ${V}$ with ${\dim W=2m-1=d}$, and assume that ${H^{2m-1}(V)=0}$, but ${H^{2m-1}(W)\neq H^{m,m-1}+H^{m-1,m}}$. Let ${Z\in Z^m(V)}$ and set ${Z_t=Z\cdot W_t\in Z^m(W_t)}$. Assume that for very general ${t}$, the ${Z_t}$ is algebraically equivalent to zero. Then ${Z}$ is homologically equivalent to 0 on ${V}$.

Indication of proof: Let ${Z\in Z_{hom}(W_t)}$, then step 1 is to look at ${\mathcal{AJ}(Z_t)\in J_{alg}^{2m-1}(W_t)}$, then to take a normal function ${\nu:U\rightarrow \cup_{t\in U} J^{2m-1}(W_t)}$, and step 2 is to show that ${\nu=0}$ is homologically zero on ${V}$.

2. Carlson 5

Recall that ${H^{n+1}(\mathbb{P}^{n+1}\setminus X)=\{A\Omega/Q^\ell\}}$ the set of meromorphic forms with a pole at ${X}$. By the adjoint of the tube map, we have a residue map to ${H^n(X)}$.

This map is holomorphic, because if we take ${\frac{\partial}{\partial\bar{t}}\left(\mathrm{res} \frac{A\Omega}{Q^\ell}\right)=0}$, and for Horizontality, we take ${\frac{\partial}{\partial t}\left(\mathrm{res}\frac{A\Omega}{(Q+tR)^\ell}\right)=-\ell \mathrm{res} \frac{RA\Omega}{(Q+tR)^{\ell+1}}}$, and this is essentially Griffiths transversality for hypersurfaces.

Now, let ${D}$ be a period domain of filtrations. It’s an open subset of a closed subset of a product of Grassmannians ${\prod_{p} Gr(F^p,\mathbb{C}^n)}$. What are the tangent spaces of a Grassmannian?

Look at ${Gr(r,\mathbb{C}^N)}$. For ${S\in Gr(r,\mathbb{C}^N)}$, we have that ${T_S Gr(r,\mathbb{C}^N)=\hom(S,\mathbb{C}^N/S)}$.

So in general we have that ${\frac{d}{dt} F^p_t\in \hom(F^p,\mathbb{C}^N/F^p)}$, and this lies in ${\hom(F^p,F^{p-1}/F^p)}$.

Now take ${G=G/V}$ where ${G}$ is ${\mathrm{Sp}(n,\mathbb{R})}$ or ${\mathrm{SO}(Q,\mathbb{R})}$, and ${\mathfrak{g}=Lie(G)}$, then ${\mathfrak{g}_\mathbb{C}\subset \hom(H_\mathbb{C},H_\mathbb{C})}$ which is a HS of weight 0, inducing one on ${\mathfrak{g}_\mathbb{C}}$.

Let ${S}$ be a ${V}$-module, these correspond to homogeneous vector bundles on ${G/V}$ by taking ${G\times_V S}$, ${\mathfrak{g}^{-}}$ corresponds to the Holomorphic tangent bundle of ${D=T_{hol}D}$ and ${\mathfrak{g}^{-1,1}}$ is the horizontal part, and a family ${\{X_t\}}$ gives a family of Hodge structures, ${H^n(X_t)\in \mathfrak{g}^{-1,1}}$.

So then we have ${D=G/V\rightarrow G/K}$ with fiber ${K/V}$ compact complex, and ${G/K}$ a symmetric space. Then we set ${\mathfrak{k}_\mathbb{C}=\oplus_{p\mbox{ odd}} \mathfrak{g}^{p,-p}}$ and ${\mathfrak{p}_\mathbb{C}=\oplus_{p\mbox{ even}}\mathfrak{g}^{p,-p}}$.

2.1. Curvature of ${D}$

.

It is fairly well known that ${\mathbb{H}_g}$ has holomorphic sectional curvature negative and bounded above by a negative constant. On ${D}$ more general, the sectional curvatures can be positive or negative: the positive ones correspond to ${\mathfrak{g}^{p,-p}}$ for ${p}$ even, and for ${p}$ odd we get negative directions, and especially important is ${\mathfrak{g}^{-1,1}}$.

Distance Decreasing Principal: Let ${f:M\rightarrow D}$ a holomorphic, horizontal, negatively curved and bounded away from zero manifold, then this map is distance decreasing. Specifically, ${f^*ds^2_D\leq ds^2_M}$.

If ${M}$ is a disk of radius ${R}$, then ${ds_R^2=\frac{R^2dzd\bar{z}}{(R^2-|z|^2)}}$.

A nontrivial variation of Hodge structures has at least 3 singularities.

For elliptic curves, assume not. then we have ${\mathbb{C}^*\rightarrow \mathbb{H}/\Gamma}$, this lifts to the universal cover ${\mathbb{C}\rightarrow \mathbb{H}}$ the unit disc, and this gives an entire holomorphic function, which is a contradiction.

In generality, we look at ${\mathbb{C}^*\stackrel{f}{\rightarrow} D/\Gamma}$ and ${\mathbb{C}\stackrel{\tilde{f}}{\rightarrow}D}$. Now, we have ${\tilde{f}^*ds^2_D\leq ds^2_R}$ This is then ${Cdzd\bar{z}\leq \frac{dzd\bar{z}}{R^2}}$ for ${C\neq 0}$ and in fact ${C>0}$, and so ${C\leq \frac{1}{R^2}}$, which is a contradiction.

Application 2: The eigenvalues of a monodromy transformation ${T}$ are roots of unity. Let ${f}$ map the unit disc to ${D/\Gamma}$ and ${\tilde{f}:\mathbb{H}\rightarrow D}$. We’ll look at ${in}$. Then ${d_D(\tilde{f}(in),\tilde{f}(\gamma\cdot in))\leq d_\mathbb{H}(in,in+1)}$. Now, ${ds^2_\mathbb{H}=\frac{dx^2+dy^2}{y^2}}$, so the RHS is ${\frac{1}{n}}$.

We have ${\tilde{f}(\gamma x)=\rho(\gamma)\tilde{f}(x)}$, which rewrites LHS as ${d_D(\tilde{f}(\gamma in),\rho(\gamma)\tilde{f}(in))}$, and this is just ${gV}$ when we write it as ${G/V}$, and so we have ${d_D(g_n V, \rho(\gamma)g_n V)\leq \frac{1}{n}}$, but we can use the homogeneous nature of our space to write ${d_D(V,g_n^{-1}\rho(\gamma) g_n V)\leq \frac{1}{n}}$.

So, the conjugacy class of ${\rho(\gamma)}$ has a limit point in ${V}$ (compact), os the eigenvalues of ${\rho(\gamma)}$ have absolute value ${1}$, and are algebraic integers, so then a theorem of Kronecker that implies that they are roots of unity.

3. Griffiths 1 – Mumford-Tate Groups

Mumford-Tate groups are a rich class of groups and are the basic symmetry groups of Hodge structures. They capture both the ${\mathbb{Q}}$ and ${\mathbb{C}}$ structures.

They sit at the intersection of algebraic geometry, representation theory and arithmetic geometry.

We’re going to focus on the higher weight case, because the classical case is much more well understood.

Some notation: ${V}$ a ${\mathbb{Q}}$-vector space, ${V_\mathbb{R}}$ its tensor product with ${\mathbb{R}}$, etc, ${\hat{V}}$ the dual, ${Q}$ a nondegenerate bilinear form which is alternating or symmetric, and ${G=\mathrm{Aut}(V,Q)}$ as a rational algebraic group.

We’ll denote by ${\mathbb{S}=Res_{\mathbb{C}/\mathbb{R}}\mathbb{C}^*}$ the restriction of scalars.

This talk was given from the lecture notes on an overhead, and went too fast to take notes.