## ICTP Day 13 – The Conference

Sorry these are going up a bit later than usual, had some difficulties last night, and was working on my talk. I’m just not taking notes whenever there are slides or lecture notes on the overhead, because it’s impossible for me to keep up while taking notes on those talks, so that will limit these a bit, but here are the other talks:

1. Arapura – Beilinson-Hodge cycles on semiabelian varieties

Joint work with Manish Kumar.

Reminder: If ${X}$ is a smooth projective variety over ${\mathbb{C}}$, then there’s a cycle map ${CH^p(X)\otimes \mathbb{Q}\rightarrow H^{2p}(X,\mathbb{C})\cap H^{p,p}}$.

This map is surjective.

We’ll abbeviate the Hodge Conjecture as HC.

Now, assume ${X}$ smooth but not necessarily projective, so open (quasiprojective). Then the cohomology carries a MHS.

We define ${BH^{ab}(X)=\hom_{MHS}(\mathbb{Q}(-b),H^a(X))}$ the Beilinson-Hodge cycles, and then the Hodge cycles correspond, in the projective case, to ${BH^{2p,p}}$.

Do these cycles have a geometric origin?

We should make this rigorous, so we look at Bloch’s higher Chow groups. Try the singular homology.

Look at ${\Delta^n\rightarrow X}$, and we can look at ${\Delta^n}$ as ${\mathbb{A}^n\subset \mathbb{A}^{n+1}}$ embedded to contain the hyperplane ${\sum x_i=1}$.

So we replace ${\Delta\rightarrow X}$ with cycles on ${X\times \mathbb{A}^q}$ satisfying that the ?? is in the expected dimension.

So we get a complex ${\rightarrow Z^p(X\times \mathbb{A}^q)\stackrel{\partial}{\rightarrow} Z^p(X\times \mathbb{A}^{q-1})\rightarrow\ldots}$ and we define ${CH^p(X,q)}$ to be the ${q}$th homology of this complex.

The usual Chow groups ${CH^p(X)=CH^p(X,0)}$.

${CH^1(X,1)\cong \mathscr{O}(X)^*}$, the group of units in the ring of regular functions on ${X}$, because ${\Gamma_f\in Z^1(X\times \mathbb{A}^1)}$.

${CH^*(X,*)}$ is the motivic cohomology, though we’re going to focus on the concrete cycles.

Given a cycle ${Z\in Z^p(X\times\mathbb{A}^q)}$, the fundamental class of ${(Z,\partial Z)}$ gives a class ${[Z]\in H^{2p}(X\times \mathbb{A}^q,X\times \partial\mathbb{A}^q)}$, and if we pass to topology, we can explain this as ${\tilde{H}^{2p}(X\times S^q)}$ and then we integrate out the sphere componenet to get ${H^{2p-q}(X)}$. And it turns out, as MHS, we have ${H^{2p-q}(X)\cong H^{2p}(X\times \mathbb{A}^q,X\times\partial \mathbb{A}^q)}$.

So we have ${CH^p(X,q)\otimes \mathbb{Q})\rightarrow BH^{2p-q,p}(X,\mathbb{Q})}$ a cycle map.

This map is surjective for all ${p,q}$.

We’ll call this ${\mathrm{BHC}_{p,q}}$.

If ${X}$ is smooth and projective and ${q=0}$, then this is just HC.

BHC is false.

This can be seen in the case ${p=3}$ and ${q=2}$, and is related to Mumford’s example of a surface where ${CH_0(X)}$ is different from ${\mathrm{Alb}(X)}$.

However, the conjecture if believed to be ok if ${X}$ is defined over ${\bar{\mathbb{Q}}}$, and the work of Asakure-S. Saito suggests that it seems ok so far for ${p=q}$.

Suppose that ${X}$ is a product of smooth affine curves or semiabelian over ${\mathbb{C}}$. Then ${\mathrm{BHC}_{p,p}}$ holds for ${X}$ unconditionally, and if HC holds for a smooth compactification ${\bar{X}}$, then ${\mathrm{BHC}_{p,q}}$ is true for all ${p,q}$.

Suppose that ${X}$ is as before, but defined on a finitely generated field ${k/\mathbb{Q}}$. The cycle map ${CH^p(X,p)\otimes \mathbb{Q}_\ell\rightarrow H^{2p-p}(\bar{X}_{et},\mathbb{Q}_\ell(p))^G}$ is surjective where ${G=Gal(\bar{k}/k)}$ and if the Tate conjecture holds, then a similar statement holds for all ${p,q}$.

The above can be considered as a Beilinson-Tate conjecture.

Idea of the Proof of Theorem 1

Just as the HC has the easy case of the (1,1)-theorem, there’s an easy case of the BHC.

Recall that ${CH^1(X,1)=\mathscr{O}(X)^*}$, then the cycle map ${c:\mathscr{O}(X)^*\rightarrow H^1(X,\mathbb{Z})}$ is given via the exponential map.

Lemma 1 (Abel) The map ${c}$ is surjective onto the Hodge cycles in ${H^1(X,\mathbb{Z})}$. That is, ${\mathrm{BHC}_{1,1}}$ is true

In fact, ${\oplus CH^*(X,*)}$ is a ring, and likewise, ${BH^{*,*}(X)}$ is. The Mumford-Tate group is isomorphic to the Galois group.

Now, given a ${\mathbb{Q}}$-MHS ${H}$, then there exists a ${\mathbb{Q}}$-algebraic group ${MT(H)}$ such that ${Rep(MT(H))}$ is equivalent to the tensor category generated by ${H}$.

We’ll take a subgroup the Special Mumford-Tate group ${SMT(H)=\cap \ker (MT(H)\rightarrow characters)}$.

${SMT(H)}$ leaved Beilinson-Hodge cycles on tensor powers of ${H}$ invariant, and conversely.

Note that ${SMT(H)}$ is generally not a reductive group, but we need to do invariant theory for it, and this reduces the theorem to ${\mathrm{BHC}_{1,1}}$.

2. Illusie – Semistable reduction and vanishing theorems revisited, after J. Suh

This talk is dedicated to the memory of Eckart Viewhwig

This talk was on slides, and I couldn’t keep up.

3. Griffiths – Hodge domains and automorphic cohomology

Talk based in part on joint work with Mark Green and Matt Kerr

3.1. Introduction

From the early days, there was a mystery about the relation, if any, between automorphic cohomology ${H^d_{(a)}(\Gamma\backslash D,\mathscr{L}_{\rho_\lambda})}$ and variations of Hodge structure ${\Phi:S\rightarrow \Gamma\backslash D}$. Recent work of EGW and C and on Mumford-Tate groups and domains have shed some light on this issue and we will discuss this and related manners.

Here, ${D=M(\mathbb{R})/H}$ is a Hodge domain (period domains are special cases), ${\lambda}$ is a weight in it with Harish-Chandra character ${\Theta_\lambda}$ for the corresponding discrete series representation in ${L^2(M(\mathbb{R}))}$. ${\mathscr{L}_{\rho_\lambda}}$ a line bundle on ${D}$ in Schmid’s construction of the automorphic cohomology ralizing the discrete series representation with character ${\Theta_\lambda}$ where ${d=\dim_\mathbb{C} K/H}$, and ${S=\bar{S}\setminus Z}$ is a smooth quasiprojective variety where ${Z}$ is a NCD with unipotent monodromies ${T_i\neq I}$ and ${\Phi}$ is a locally liftable holomorphic mapping with ${\Phi_*(TS)\subset W}$ where ${W}$ si ${\{dF^p\subset F^{p-s}\otimes \Omega^s_S\}}$ is ???

(here, the slide changed faster than I could type)

Classicaly, ${D}$ is a hermititan symmetry domain and automorphic forms give a map between representation and algebraic geometry. ${H^0_{(a)}(\Gamma\backslash D,\mathscr{L}_{\rho_\lambda})\rightarrow H^0(S,\Phi^*\mathscr{L}_{\rho_\lambda})}$

The question is, what is the relation between these in the nonclassical case?

In the very classical case, that is, the weight ${n=1}$, and ${D\subset \mathbb{H}_g}$ parameterize the complex points of a component of a Shimura variety, there is an additional rich an extensive arithmetic aspect and one has relateions between algebraic geometry, arithmetic and representation theory via variations of Hodge structure, discrete series and cuspidal automorphic representations, and L-functions and Galois representation.

Recently, it has been found that the ${\mathbb{Q}}$-algebraic semisimple ${M}$ whose ${M(\mathbb{R})}$ have discrete series are exactly the Mumford-Tate groups that arise in Hodge theroy. And these are the ${M}$ that are hoped to have cuspidal automorphic representations in ${L^2(M(\mathbb{Q})\backslash M(\mathbb{A}))}$.

The use of cycle spaces and their enlargements has given a way to map ${H^d}$ to ${H^0}$ the (slide changed too fast)

3.2. Hodge Groups/Domains

Let ${V}$ be a ${\mathbb{Q}}$-vector space, ${Q:V\otimes V\rightarrow \mathbb{Q}}$ a polarization ${G=\mathrm{Aut}(V,Q)}$ and ${\mathbb{S}=Res_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m,\mathbb{C}}}$ and ${S^1}$ the maximal compact subgroup of ${\mathbb{S}}$.

A Hodge structure is ${\tilde{\phi}:\mathbb{S}\rightarrow GL(V_\mathbb{R})}$ and a polarized Hodge structure is a map ${\phi:S^1\rightarrow G(\mathbb{R})}$ (slide change)

(The slides changed too quickly for me to keep up after this)

3.3. Cycle Spaces and their enlargements – Examples

4. Green – Vanishing of Chern Polynomials for Hodge Domains

This is joint work with Carlson and Griffiths. It has appeared in the online journal SIGMA.

Let ${D}$ be a period domain, we have ${p:S\setminus \Sigma\rightarrow \Gamma\backslash D}$ a period map. Griffiths discovered the infinitesimal period relations, which to a differential geometer is a colletion of 1-forms ${\theta_1,\ldots,\theta_m}$ which is not an integrable exerior differential sysem.

But, for any VHS, we have the integrability conditions.

On the period domain, we have the bundles from the Hodge filtration ${F^p\rightarrow D}$, and the quotient ${F^p/F^{p+1}}$ is ${H^{p,n-p}}$, as we’re only looking at polarized Hodge structures of weight ${n}$, with given Hodge numbers.

So we have ${0\rightarrow F^p\rightarrow V\rightarrow V/F^p\rightarrow 0}$ where ${V}$ is a flat bundle. Now, because we have ${Q}$, ${V/F^p\cong (F^{n-p+1})^*}$ naturally. This, of course, gives relations on the Chern classes, because ${c(F^p)c^((F^{n-p+1})^*)=1}$, but this isn’t what we’ll be talking about.

We also have that ${H^{n,n-p}}$ is conjugate to ${H^{n-p,p}}$.

Now, let ${T_S\rightarrow \hom(F^p, V/F^p)}$ the natural map, then the IPR is that ${T_S\rightarrow \hom(F^p, F^{p-1}/F^p)}$ and ${dp:T_S\rightarrow \oplus \hom(F^p/F^{p+1},F^{p-1}/F^p)}$.

So we get ${B^p:F^p/F^{p+1}\rightarrow T_S^*\otimes F^{p-1}/F^p}$, but this is just ${H^{n,n-p}\rightarrow T_S^*\otimes H^{p-1,n-p+1}}$.

So we look at ${B^{n+1-p}}$ and dualize, and we het ${(H^{n-p,p})^*\stackrel{(B^{n+1-p})^*}{\rightarrow} T_S^*\otimes (H^{n+1-p,p-1})^*}$, and this is just ${H^{p,n-p}\rightarrow T_S^*\otimes H^{p-1,n+1-p}}$, and the IPR says that for all ${p}$, ${(B^{n+1-p})^*=B^p}$.

For the integrability conditions, we map ${H^{p,n-p}\stackrel{B^p}{\rightarrow} T_S^*\otimes H^{p-1,n+1-p}\stackrel{B^{p-1}}{\rightarrow} \bigwedge^2 T_S^*\otimes H^{p-2,n+2-p}}$, and the integrability condition is that ${B^{p-1}\circ B^p=0}$. (Here we mean that we compose and wedge)

This always holds for VHS.

The integrable tangent space is then ${T\subseteq \oplus_p \hom (H^{p,n-p},H^{p-1,n+1-p})}$.

We have a map ${F^p\rightarrow T_D^*\otimes V/F^p}$ and the curvature of ${F^p}$ is ${\Theta_{F^p}=\left(\begin{array}{cc}(\bar{B}^p)^*& 0 \\ 0 & 0\end{array}\right)}$ with columns ${H^{p,n-p}}$ and ${F^{p-1}}$ and similar rows, and the block is an ${h^{p,n-p}\times h^{p,n-p}}$ matrix of ${(1,1)}$-forms.

To compute the Chern forms, say, by Chern-Weil, we look at ${\det \left(\frac{1}{2\pi i}\Theta_{F^p}-\lambda I\right)=\sum (-1)^i c_i(F^p)\lambda^{\dim F^p-i}}$, with ${c_0=1}$ by convention.

Restricted to any integrable ${T}$, ${c_i(F^p)c_j(F^{n-p})=0}$ if ${i+j>h^{p,n-p}}$.

So we have ${\Theta_{F^p}=\bar{B}_p^* B_p}$ (well, in the nonzero block), then ${\det\left(\frac{1}{2\pi i} \bar{B}^*_p B_p-\lambda I\right)=\sum (-1)^i c_i(F^p)\lambda^{h^{p,n-p}-i}}$.

Now, ${\Theta_{F^{n-p}}}$ is in the nonzero block ${\bar{B}^*_{n-p} B_{n-p}}$, and ${\Theta^*_{F^{n-p}}=B_{p+1}\bar{B}^*_{p+1}}$ in the nonzero block.

Now, I want to multiply these together: ${\Theta_{F^p}\Theta^*_{F^{n-p}}}$, we get (in the nonzero block, of course) ${\bar{B}_p^* B_p B_{p+1} \bar{B}^*_{p+1}}$, but ${B_p B_{p+1}=0}$, and so the product is zero when restricted to ${T}$!

So how can we make use of this?

Notation: If ${M}$ is a ${d\times d}$ matrix, set ${\chi_M(\lambda)=\det(M-\lambda I)=\sum (-1)^i c_i(M) \lambda^{d-i}}$.

If ${A,B}$ are ${d\times d}$ matrices such that ${AB=0}$, then ${c_i(A)c_j(B)=0}$ for ${i+j>d}$.

Note: We’re not working over a field, if we were, then this would all be obvious.

Over a field, ${AB=0}$ implies ${\mathrm{rank}(A)+\mathrm{rank}(B)\leq d}$, and ${c_i(A)=0}$ if ${i>\mathrm{rank}(A)}$.

Proof: We want ot look at ${\det(A-\lambda I)\det(B-\mu I)}$. This is also ${\det(A-\lambda I)(B-\mu I)=\det(AB-\lambda B-\mu A+\lambda\mu I)}$, and as ${AB=0}$, every term has degree ${\geq d}$, because everything has a ${\lambda}$ or ${\mu}$. Thus, ${c_i(A)c_j(B)=0}$ if ${\lambda^{d-i}\lambda^{d-j}}$ of degree less than ${d}$, which is the same as ${i+j>d}$. $\Box$

5. Schnell – Neron models and Poincare bundles

Let ${H}$ be a polarized HS over ${\mathbb{Z}}$ of weight ${-1}$ and ${J(H)=H_\mathbb{C}/F^0H_\mathbb{C}+H_\mathbb{Z}}$ which the polarization ${Q}$ makes isomorphic to ${(F^0H_\mathbb{C})^*/H_\mathbb{Z}}$.

The points of ${J(H)}$ correspond to extensions of MHS, ${0\rightarrow H\rightarrow V\rightarrow \mathbb{Z}(0)\rightarrow 0}$. this gives ${0\rightarrow H_\mathbb{Z}\rightarrow V_\mathbb{Z}\rightarrow \mathbb{Z}\rightarrow 0}$ with a splitting, which we call ${v_\mathbb{Z}}$, and by dualizing we get ${0\rightarrow \mathbb{Z}(0)\rightarrow V^*\rightarrow H(-1)\rightarrow 0}$ which gives us that ${F^1V_\mathbb{C}^*\cong F^0H_\mathbb{C}}$, and ${[v_\mathbb{Z}]\in J(H)}$ classifies the extension.

Problem: Given a family of intermediate Jacobians that degenerates, extend it in a good way (Green-Griffiths).

Let ${X\subset \bar{X}}$ open and ${\mathscr{H}}$ a ${\mathbb{Z}}$-VHS, polarized of weight ${-1}$, then we have ${J(\mathscr{H})\rightarrow X}$.

Why do we want to do this? Because we want to study the behavior of normal functions on ${\bar{X}\setminus X}$.

Let ${W^{2n}\subset \mathbb{P}^N}$ be smooth and projective, ${\mathscr{O}_W(1)}$. Then we have a family ${\pi:\mathcal{Y}\rightarrow B}$ open in ${|\mathscr{O}_W(1)|}$ which is a VHS, and ${R^{2n-1}\pi_* \mathbb{Z}(n)}$ the variable part, and we have ${J^n(\mathcal{Y}/B)\rightarrow B}$.

History:

Green-Griffiths-Kerr deals with the case where ${\dim X=1}$. This was generalized by Brosnan-Pearlstein-Saito who gave a more topological construction, and there’s also wrok by Katz-Nakayama-Usui.

So now, ${J(\mathscr{H})=(F^0\mathscr{H}_\mathscr{O})^*/\mathscr{H}_\mathbb{Z}}$ and it is natural to extend ${F^0\mathscr{H}_\mathscr{O}}$ and then take the quotient by ${\mathscr{H}_\mathbb{Z}}$.

${X=\Delta^*}$ and ${\bar{X}=\Delta}$, then a VHS is the same as a nilpotent orbit.

We can ${\mathbb{R}}$-split a polarized MHS, and use the Deligne decomposition with ${I^{p,q}}$ conjugate to ${I^{q,p}}$ and ${W_\ell=\oplus_{p+q\leq \ell} I^{p,q}}$ and ${F^k=\oplus_{p\geq k} I^{p,q}}$.

This allows us to construct a VHS on ${\Delta^*}$ with monodromy ${T=e^N}$ and the period map is ${e^{zN}F}$ on ${\mathbb{H}\rightarrow \Delta^*}$, it will take ${z}$ to ${t=e^{2\pi i z}}$.

One way: ${\mathscr{H}_{\mathscr{O}}}$ has a canonical extension ${\tilde{\mathscr{H}}}$ with ${\nabla:\tilde{\mathscr{H}}_\mathscr{O}\rightarrow \Omega^1_\Delta(\log 0)\otimes \tilde{\mathscr{H}}_\mathscr{O}}$, and the sections of ${\tilde{\mathscr{H}}_\mathscr{O}}$ come from ${e^{zN}v}$ for ${v\in H_\mathbb{C}}$, and ${F^p\tilde{\mathscr{H}}_\mathscr{O}}$ is generated by ${e^{zN}v}$ for ${v\in F^p}$.

So what happens to the integral classes?

We take a sequence ${h_n\in H_\mathbb{Z}}$ over points ${t_n}$ which go to ${0}$ such that ${Q(h_n, e^{z_n N} v)}$ converges for all ${v\in F^0}$. Now, we can rewrite this as ${Q(e^{-z_n N} h_n,v)}$, and ${e^{-z_n N}h_n}$ is just ${h_n-z_n N h_n}$.

This imples that ${h_n^{-1,1}}$ converges and that ${h_n^{-2,0}-z_n N h_n^{-1,1}}$ converges.

This is not enough to control ${h^{p,q}_n}$ individually, though.

Fix: Suppose we also allow derivatives ${\nabla_{\partial/\partial t} (e^{zN}v)}$. This is just ${e^{zN}Nvdz}$, and in fact we get ${\frac{1}{2\pi i} \frac{1}{t} e^{zN}Nv}$ for ${v\in F^1}$. Test ${h_n}$ on this, and ${\frac{1}{t} N h_n^{-1,1}}$ converges. Call our three sequence ${a_n,b_n,c_n}$.

Solve: ${h_n=h_n^{-1,1}+\bar{h}_n^{-1,1}+h_n^{-2,0}+\bar{h}_n^{-2,0}}$, and this is just ${b_n+t_n z_n c_n +a_n}$ plus conjugates, and ${t_n z_n\rightarrow 0}$.

Let ${j:\Delta^*\rightarrow \Delta}$ the inclusion, then we have that ${j_*^{reg}\mathscr{H}_\mathscr{O}}$ along with ${\nabla_{\partial/\partial t}}$ is a D-module, and we can take a sub-D-module generated by ${\tilde{\mathscr{H}}_\mathscr{O}}$, and call it ${\mathscr{M}}$.

We set ${F^1\mathscr{M}=F^1\tilde{\mathscr{H}}_\mathscr{O}}$, then ${F^0\mathscr{M}=F^0\tilde{\mathscr{H}}_{\mathscr{O}}+\frac{\partial}{\partial t} F^1\tilde{\mathscr{H}}_\mathscr{O}}$, etc, and this leads to the “minimal extension”

General: consistent use of Saito’s theorem.

Let ${MHM(X)}$ be the category fo mixed Hodge modules on ${X}$. An object here is a perverse sheaf ${rat(M)}$ and a filtered ${D}$-module ${(\mathscr{M},F)}$.

An example is a VMHS, with ${\mathscr{H}_\mathbb{Q}[\dim X]}$ and ${(\mathscr{H}_\mathscr{O},\nabla,F_*\mathscr{H}_\mathscr{O}=f^{-*}\mathscr{H}_\mathscr{O})}$.

Construction: ${\mathscr{H}}$ a polarized ${\mathbb{Z}}$-VHS on ${X}$ of weight -1, ${j:X\rightarrow \bar{X}}$, then take an integral extension ${M}$, which gives ${(\mathscr{M},F)}$ with ${F_0\mathscr{M}|_X\cong F^0\mathscr{H}_\mathscr{O}}$ a coherent sheaf on ${X}$.

${T(F_0\mathscr{M})=\mathrm{Spec}(\mathrm{Sym} F_0\mathscr{M})}$ and has section ${(F_0\mathscr{M})^\vee}$. Then ${T_\mathbb{Z}}$ is the etale space of ${j_*\mathscr{H}_\mathbb{Z}}$, and finally, there exists ${T_\mathbb{Z}\rightarrow T(F_0\mathscr{M})}$ taking ${h}$ to ${Q(h,-)}$.

${\mathscr{E}}$ is closed analytic embedding, then ${\bar{J}(\mathscr{H})=T(F_0\mathscr{M})/T_\mathbb{Z}}$ is a Hausdorff-analytic space, the “Neron model”

\mbox{}

1. Any admissible normal function without singularities extends to a holomorphic section of ${\bar{J}(\mathscr{H})\rightarrow \bar{X}}$ which is horizontal.
2. For all ANF, ${\Gamma(\nu)}$ has analytic closure on ${\bar{J}(\mathscr{H})}$.

${Z(v)}$ is algebraic if ${X}$ is algebraic.

Interpretation for singularities: ANF has a singularity at ${x\in \bar{X}\setminus X}$ if and only if it does not extend to a section.

Now, we look at Poincaré bundles.

Green-Griffiths: try to construct a Poincaré bundle on ${\bar{J}(\mathscr{H})\times \bar{J}(dual)}$ and use it to get invariants of ANF.

What can we do? Start from an ANF for ${\mathscr{H}^*(1)}$ without singularities, then construct a line bundle ${\mathscr{L}\rightarrow \bar{J}(\mathscr{H})}$, and given ${v,v'}$, we construct ${\mathscr{L}(v,v')}$ on ${\bar{X}}$.

6. Pearlstein – The locus of the Hodge classes in admissible variations of mixed Hodge structure

This is joint work with Brosnan and Schnell

Let ${V}$ be a MHS with lattice ${V_\mathbb{Z}}$, then a Hodge class of ${V}$ is an element of ${V_\mathbb{Z}\cap F^0V_\mathbb{Z}\cap W_0 V_\mathbb{Z}}$.

Let ${\mathscr{V}\rightarrow S}$ be a VMHS with lattice ${\mathscr{V}_\mathbb{Z}}$, let ${T(\mathscr{V}_\mathbb{Z})\subset E(\mathscr{V}_\mathscr{O})}$ the etale space of ${\mathscr{V}_\mathbb{Z}}$.

${\mathrm{Hdg}(\mathscr{V})=T(\mathscr{V}_\mathbb{Z})\cap E(F^0\mathscr{V}_\mathscr{O})\cap E(W_0\mathscr{V}_\mathscr{O})}$. A choice of graded polarization ${\mathrm{Hdg}(\mathscr{V})=\coprod_k \mathrm{Hdg}_k(\mathscr{V})}$, where ${\mathrm{Hdg}_k(\mathscr{V})}$ is the locus of elements ${\alpha\in \mathrm{Hdg}(\mathscr{V})}$ with ${Q_0(\alpha+w_{-1},\alpha+w_{-1})=K}$.

Let ${S}$ be a Zariski open subset of a complex manifold ${\bar{S}}$ and ${\mathscr{V}\rightarrow S}$ be an admissible VMHS wrt ${\bar{S}}$. Then ${\mathrm{Hdg}_k(\mathscr{V})}$ extends to an analytic space, finite and proper over ${\bar{S}}$.

If ${\mathscr{V}}$ pure and weight 0, this is due to CDK.

If ${S}$ is a quasi-projective variety, then ${\mathrm{Hdg}_k(\mathscr{V})}$ is also quasiprojective.

Let ${\nu}$ be an admissible higher normal function on ${S\subset\bar{S}}$, ie, ${\nu}$ comes from ${0\rightarrow \mathscr{G}\rightarrow \mathscr{V}\rightarrow \mathbb{Z}(0)\rightarrow 0}$ where ${\mathscr{H}}$ is pure of weight ${k<0}$. Then the closure of the zero locus of ${\nu}$ in ${\bar{S}}$ is complex analytic.

If ${S}$ is quasi-projective then ${Z(nu)}$ is algebraic.

Let ${\mathscr{V}}$ be an admissible VMHS on a quasiprojective variety ${S}$, then the locus of points where ${\mathscr{V}}$ is split over ${\mathbb{Z}}$ is algebraic.

WLOG, assume that each ${W_k}$ is defined over ${\mathbb{Z}}$, and ${Gr^W_k}$ is torsion free, ${S}$ connected. Replace ${\mathscr{V}}$ by ${W_0\mathscr{V}}$.

A generalized normal function on ${S\subset \bar{S}}$ is an admissible extension ${0\rightarrow \mathscr{H}\rightarrow \mathscr{V}\rightarrow \mathbb{Z}(0)\rightarrow 0}$ where ${\mathscr{H}}$ is a ${\mathbb{Z}}$-VMHS with weight ${<0}$.

Let ${\nu}$ be an admissible generalized normal function on ${S\subset\bar{S}}$ then the closue of ${Z(\nu)}$ in ${\bar{S}}$ is complex analytic.

Proof: Let ${\mathscr{V}}$ be the corresponding VMHS. If ${\mathscr{H}}$ is pure, we’re done by theorem 3. Otherwise, let ${m\leq -1}$ be the smallest integer such that ${W_m\neq 0}$.

We define ${\mathscr{V}^1=\mathscr{V}/W_m\mathscr{V}}$ and ${\nu_0}$ the associated generalized normal function. ${Z(\nu)\subset Z(\nu_0)}$, and by induction, we can assume ${\bar{Z}(\nu_0)\subset \bar{S}}$ is complex analytic.

Let ${S_0}$ denote the regular locus of an irreducible component of ${Z(\nu_0)}$, then ${\pi:\bar{S}_0\rightarrow \bar{S}}$ which is an isomorphism over ${S_0}$. ${\pi}$ is proper implies that we can replace ${\bar{S}}$ by ${\bar{S}_0}$ and ${\nu}$ by the pullback to ${S_0}$, so we can assume ${\nu_0=0}$.

${0\rightarrow NF(S,W_n\mathscr{H})\rightarrow NF(S,\mathscr{H})\rightarrow NF(S,\mathscr{H}/W_n\mathscr{H})}$, get ${\nu'\in NF(S,W_n\mathscr{H})}$ and ${W_m\mathscr{H}}$ pure.

Theorem 3 implies that ${Z(\nu')}$ has analytic closure in ${\bar{S}}$, so ${Z(\nu)=Z(\nu')}$. $\Box$

Now, we prove Theorem 1:

Let ${\mathscr{V}\rightarrow S}$ be an admissible VMHS, ${\mathscr{V}=W_0\mathscr{V}}$. For any point ${s\in S}$, we have ${0\rightarrow \mathrm{Hdg}(\mathscr{V}_s)\rightarrow \mathrm{Hdg}(Gr^W_0 \mathscr{V}_s)\rightarrow \mathrm{Ext}^1_{MHS}(Z(0),W_0\mathscr{V}_s)}$ implies that ${\mathrm{Hdg}(\mathscr{V})}$ extends into ${\mathrm{Hdg}(Gr_0^W\mathscr{V})}$. Let ${Z\in \mathrm{Hdg}_k(\mathscr{V})}$ an irreduciblecomponent and ${Y\in \mathrm{Hdg}_k(Gr^W_0\mathscr{V})}$ irreducible and containing ${Z}$.

${\bar{Y}}$ extends to an analytic space which is finite and proper over ${\bar{S}}$. We have ${\mu:\bar{Y}'\rightarrow \bar{Y}}$ with ${\mu^{-1}(Y)\cong Y}$. Pull back to ${Y}$, and call it ${\mathscr{V}'}$.

By construction, we have a section ${\mathbb{Z}(0)\rightarrow Gr^W_0\mathscr{V}'}$, and so we get a generalized normal function ${\nu'\in NF(Y,\mathscr{H}')}$ where ${\mathscr{H}'=W_{-1}\mathscr{V}'}$.

So ${Z(\nu')=Z}$ and then by the proposition, we have ${\bar{Z}(\nu')\subseteq \bar{S}}$ which is analytic.

7. Movasati – Automorphic functions for moduli of polarized Hodge structures

No notes, due to fast moving slides