# Normal functions over locally symmetric varieties

Ryan Keast and Matt Kerr
Department of Mathematics, Campus Box 1146
Washington University in St. Louis
St. Louis, MO63130, USA
_e-mail_<EMAIL_ADDRESS>
Department of Mathematics, Campus Box 1146
Washington University in St. Louis
St. Louis, MO63130, USA __
_e-mail_<EMAIL_ADDRESS>
###### Abstract.

We classify the irreducible Hermitian real variations of Hodge structure
admitting an infinitesimal normal function, and draw conclusions for cycle-
class maps on families of abelian varieties with a given Mumford-Tate group.

###### 2000 Mathematics Subject Classification:

14D07, 14C25, 14M17, 17B45, 32M15, 32G20

##  1\. Introduction

Normal functions are holomorphic horizontal sections of the intermediate
Jacobian bundle $J(\mathcal{V})$ associated to a variation of Hodge structure
$\mathcal{V}$ (of odd weight) over a complex analytic manifold. Given a family
of smooth projective varieties $\mathscr{X}\overset{\pi}{\to}\mathscr{S}$ and
a cycle $\mathfrak{Z}\in CH^{p}(\mathscr{X})$ whose restriction
$Z_{s}:=\mathfrak{Z}\cdot X_{s}$ to each fiber is homologous to zero, the
Abel-Jacobi images $AJ_{X_{s}}^{p}(Z_{s})$ yield such a section
$\nu_{\mathfrak{Z}}$ of $J(\mathcal{V}_{\pi}^{2p-1})$. These _geometric_
normal functions and their singularities are at the heart of the reformulation
of the Hodge Conjecture by Green and Griffiths [GG], which has brought about a
renewal of interest (cf. [KP] and references therein).

The most famous example arises from the image $C^{+}$ of an algebraic curve
$C$ in its Jacobian. Writing $\imath:J(C)\to J(C)$ for the map sending
$u\mapsto-u$, and $C^{-}$ for $\imath(C^{+})$, the _Ceresa cycle_
$Z_{C}:=C^{+}-C^{-}\in CH_{1}(J(C))$ is homologous to zero, with
$AJ_{C}^{g-1}(Z_{C})\in J(H_{prim}^{2g-3}(J(C)))$ of infinite order for very
general $C$ of genus at least 3 [Ce]. By studying cohomology of the mapping
class group, Hain has proved (up to a factor of 2) that the resulting section
essentially generates all normal functions over the moduli space
$\mathcal{M}_{g}(\ell)$ of curves with level $\ell$ structure (for $\ell\geq
3$) [Ha].111Here the underlying VHS $\mathcal{V}\to\mathcal{M}_{g}(\ell)$ is
assumed to lie in the Tannakian category generated by $H^{1}(C)$. By
“essentially” we mean up to sections of Jacobians of level-1 variations, which
are families of abelian varieties. In this paper we shall mostly ignore these,
which in the geometric case corresponds to considering only the normal
functions that factor through the Griffiths group of cycles modulo algebraic
equivalence. We shall also work rationally, i.e. consider torsion normal
functions to be zero.

A natural question is what happens over $\mathcal{A}_{g}(\ell)$, or more
generally over quotients $\mathcal{A}_{g}(\Gamma)$ of the Siegel domain
$D=\mathrm{III}_{g}:=Sp_{2g}(\mathbb{R})/U(g)$ by a torsion-free congruence
subgroup $\Gamma\leq Sp_{2g}(\mathbb{Q})$.222$\mathcal{A}_{g}(\ell)$ is the
case $\Gamma=\ker\left(Sp_{2g}(\mathbb{Z})\to
Sp_{2g}(\mathbb{Z}/\ell\mathbb{Z})\right)$, which is torsion-free for
$\ell\geq 3$. However, Ceresa’s normal function does not even survive the
(generically 2:1) map from $\mathcal{M}_{g}(\ell)$ to $\mathcal{A}_{g}(\ell)$,
let alone extend to the latter. In fact, a result of Raghunathan [Ra] implies
a much more general vanishing phenomenon: if $\mathcal{V}$ is a homogeneous
VHS (cf. $\S 2.3$) over a locally symmetric variety
$\mathrm{X}:=\Gamma\backslash D:=\Gamma\backslash G(\mathbb{R})/K$, associated
to a nontrivial $\mathbb{Q}$-irrep $V$ of $G$, and $G$ has $\mathbb{Q}$-rank
$>1$, then $J(\mathcal{V})$ admits no admissible normal functions over any
Zariski open subset of $\mathrm{X}$. Moreover, when the $\mathbb{Q}$-rank is
one, the boundary components in the smooth toroidal compactification of
$\mathrm{X}$ are smooth, and so any admissible normal function would have no
singularities to study.

This situation improves somewhat when one considers étale neighborhoods of
$\mathrm{X}$ instead of Zariski ones, motivated by the example of
$\mathcal{M}_{3}(\ell)$ minus its hyperelliptic locus (over
$\mathrm{X}=\mathcal{A}_{g}(\ell)$). To simplify matters, assume that the
group $G$ (of any $\mathbb{Q}$-rank), the Hermitian symmetric domain $D$, and
the representation $V$ are as above, but also simple resp. irreducible over
$\mathbb{R}$, with $V_{\mathbb{R}}$ of highest weight $\lambda$. (We also
consider some cases where $G_{\mathbb{R}}=U(1)\cdot\text{simple}.)$ One would
like to classify the pairs $(D,\lambda)$ for which $\mathcal{V}$ has odd
weight and the resulting $J(\mathcal{V})\to\mathrm{X}$ admits a normal
function after base change to some étale neighborhood. For this to happen, a
certain cohomology sheaf $\oplus_{j\geq 0}\mathcal{H}^{1}(j)$ of
_infinitesimal normal functions_ on $D$ (cf. $\S 2.1$) must not vanish, and it
is the pairs with this property that we shall classify in this paper using a
result of Kostant [Ko] (cf. $\S\S 2.2,2.4$). We should note that this approach
has already been carried out by Nori [No1] in the Siegel domain case, so the
results below involving $D=\mathrm{III}_{g}$ are not new.

When $D$ is of tube type, and ignoring variations of weight one, we obtain a
complete classification: the pairs include all of the weight-3 Calabi-Yau
variations

$\left(\mathrm{I}_{3,3},\omega_{3}\right),\;\left(\mathrm{II}_{6},\omega_{6}\right),\;\left(\mathrm{III}_{3},\omega_{3}\right),\;\left(\mathrm{EVIII},\omega_{7}\right)$

described by Gross [Gro] and Friedman-Laza [FL], as they obviously must (cf.
$\S 2.1$). The remainder of the list is rather short: namely, we have

$\left(\mathrm{I}_{2,2},\omega_{1}+a\omega_{2}\text{ or
}\omega_{3}+a\omega_{2}\right),\;\left(\mathrm{II}_{4},\omega_{1}+a\omega_{3}\text{
or }\omega_{1}+a\omega_{4}\right),$
$\left(\mathrm{III}_{1},(2a+1)\omega_{1}\right),\;\left(\mathrm{III}_{2},\omega_{1}+a\omega_{2}\right),\;\left(\mathrm{IV}_{2n-1},a\omega_{1}+\omega_{n}\right),$
$\text{and }\left(\mathrm{IV}_{2n-2},a\omega_{1}+\omega_{n-1}\text{ or
}a\omega_{1}+\omega_{n}\right),$

where $a$ is any positive integer. Our results in the non-tube cases are more
subtle and partial; while we enlarge $G$ by a 1-torus and thus allow a variant
of the half-twist construction [vG1] (cf. $\S 4$), we mainly restrict to
variations of Calabi-Yau type or appearing in the cohomology of abelian
varieties of generalized Weil type.

Returning to $\mathcal{A}_{g}(\ell)$ (or more generally
$\mathcal{A}_{g}(\ell)$), we can ask once more about the loci “supporting”
higher-weight normal functions.333The question is independent of $\ell$ (or
$\Gamma$), but we need $\Gamma$ torsion-free to be able to speak of a
homogeneous VHS over _all_ of $\mathcal{A}_{g}(\Gamma)$. By this we mean that
there is an irreducible _homogeneous_ variation $\mathcal{V}$ of odd level
$>1$ over $\mathcal{A}_{g}(\Gamma)$, a subvariety
$\mathscr{S}\overset{\imath}{\hookrightarrow}\mathcal{A}_{g}(\Gamma)$ with
étale neighborhood $\mathscr{T}\overset{\jmath}{\to}\mathscr{S}$, and a
nontorsion horizontal section of $J(\jmath^{*}\imath^{*}\mathcal{V})$ over
$\mathscr{T}$. Alternatively, one could consider only those $\mathcal{V}$
appearing in the cohomology of the universal abelian variety. In either case,
the loci $\overline{\imath(\mathscr{S})}$ are proper subvarieties for $g>3$
[No1], and are expected to be more like $\mathcal{M}_{g}(\Gamma)$ than locally
symmetric subvarieties. Indeed, $\mathcal{M}_{g}(\Gamma)$ supports in this
sense the Ceresa normal function for any $g$; another example is the Fano
normal function supported by the locus in $\mathcal{A}_{5}(\Gamma)$ of
intermediate Jacobians of cubic threefolds [CNP]. The approach in this article
is consequently analogous to the search for special subvarieties of the
Torelli locus $\overline{\mathcal{M}_{g}}\subset\mathcal{A}_{g}$ [MO], which
are expected not to exist for large enough $g$. So the following vanishing
result, which follows from our classification, should not be surprising:

######  Theorem.

The loci of Weil resp. quaternionic abelian varieties in
$\mathcal{A}_{g}(\Gamma)$ do not support higher weight normal functions for
$g>6$ resp. $8$. In particular, the image of the reduced Abel-Jacobi map

$\overline{AJ}^{r}:\,\mathit{Griff}^{r}(A)_{\mathbb{Q}}\to\overline{J}^{r}(A)_{\mathbb{Q}}$

is zero for a very general Weil resp. quaternionic abelian variety $A$ of
dimension at least $8$ resp. $10$.

(See Thms. 3.3 and 3.5 in $\S 3$.) There are related results for cycles in
certain codimensions for the generalized Weil abelian varieties over
$\mathrm{I}_{p,q}$ ($p+q=n+1$, $p<q$) studied in $\S 4.1$, which get weaker as
$|q-p|$ grows (cf. Theorem 4.3). However, both here and for the domains
$\mathrm{IV}_{m}$ parametrizing “spin abelian varieties” (for instance, those
arising from the Kuga-Satake construction), one has infinitesimal normal
functions for arbitrarily large $n$ and $m$.

For each of the “infinitesimal normal functions” described in this paper, the
next immediate problem is to determine whether it comes from an actual
(admissible) normal function, and if so, to geometrically realize it and
determine (outside the exceptional cases) its locus of support in
$\mathcal{A}_{g}(\Gamma)$. As an added incentive, whenever the geometric
realization is as a cycle on some finite pullback of the universal abelian
family over a locally symmetric subvariety
$D(\Gamma^{\prime})\subseteq\mathcal{A}_{g}(\Gamma)$, one obtains as a
corollary the infinite generation of the relevant Griffiths group of a very
general member of this family. This was observed by Nori [No2] for type
$\mathrm{III}_{3}$ (i.e. $\mathcal{A}_{3}(\Gamma)$ itself) using the Ceresa
cycle; the general argument is practically identical to that in [op. cit.],
but with [Ra] replacing the use of the congruence subgroup theorem.

We briefly carry out the geometric realization for
$(D,\lambda)=\left(\mathrm{I}_{3,3},\omega_{3}\right)$ in $\S 3.5$, using the
fact that for certain families of Weil abelian $6$-folds, the general member
is a Prym variety associated to an unramified 3:1 curve covering with base
curve of genus $g=4$ [F, Sc]. See Theorem 3.6 for the application to Griffiths
groups. The locus of support of the resulting normal function is (at least)
the $9$-dimensional locally symmetric quotient of $D$ in
$\mathcal{A}_{6}(\Gamma)$. The construction works for $g>4$ as well, but a
dimension count shows that the locus of support is a _proper_ subvariety of
the relevant $\Gamma\backslash\mathrm{I}_{p,p}$.

With an admissible normal function in hand, one can also try to determine its
zero locus, and its singularities along intersections of toroidal boundary
components. But even in the absence of this, one can probably still use
Kostant’s result, replacing $\mathfrak{g}$ by subalgebras of the form
$\mathfrak{sl}_{2}^{\oplus a}$, to compute spaces of singularity classes for
normal functions over homogeneous variations.

An equally intriguing prospect is to try to extend the computation of
infinitesimal normal function spaces (or singularity classes) to the
nonclassical case, which is more relevant to the Green-Griffiths program. Here
the homogeneous families of Hodge structures only become variations upon
restriction to the image of a period map. However, the computation only
requires an infinitesimal variation as input, so it may already be interesting
to start with tangent spaces to the Schubert VHS of [Ro].

#### Acknowledgments:

The authors thank P. Brosnan and G. Pearlstein for helpful discussions, and
gratefully acknowledge support from NSF Grant DMS-1361147. This paper was
written while MK was a member at the Institute for Advanced Study, and he
thanks the IAS for excellent working conditions and the Fund for Mathematics
for financial support.

##  2\. Infinitesimal normal functions and Kostant’s theorem

This section contains the abstract underpinnings of the calculations in
$\S\S$3-4. Apart from $\S$2.4, it is mostly expository and contains the
material on infinitesimal invariants, Lie algebra cohomology, and Hermitian
VHS that will be used subsequently.

###  2.1. Normal functions and infinitesimal invariants

Let $\mathcal{V}\to\mathscr{S}$ be a polarized $\mathbb{Q}$-VHS over a complex
manifold, pure of weight $-1$. By abuse of notation, $\mathcal{V}$ will denote
the underlying holomorphic vector bundle and its sheaf of sections, with Hodge
filtration $\mathcal{F}^{\bullet}$; $\mathbb{V}$ is the underlying
$\mathbb{Q}$-local system,
$Q:\mathbb{V}\times\mathbb{V}\to\mathbb{Q}_{\mathscr{S}}$ the polarization,
and $\nabla$ the Gauss-Manin connection.

Define a filtered complex of sheaves

(2.1)
$C^{\bullet}:=\left(\mathcal{V}\otimes\Omega^{\bullet}_{\mathscr{S}},\nabla\right)\;,\;\;F^{p}C^{\bullet}:=\left(\mathcal{F}^{p-\bullet}\mathcal{V}\otimes\Omega_{\mathscr{S}},\nabla\right)$

on $\mathscr{S}$, so that the differential of each

$Gr_{F}^{p}C^{\bullet}=\left(Gr_{\mathcal{F}}^{p-\bullet}\mathcal{V}\otimes\Omega_{\mathscr{S}}^{\bullet},\bar{\nabla}\right)$

is $\mathcal{O}_{\mathscr{S}}$-linear. Consider the exact sequence

(2.2) $0\to F^{0}C^{\bullet}\to C^{\bullet}/\mathbb{V}\to
C^{\bullet}/(F^{0}C^{\bullet}+\mathbb{V})\to 0\,;$

the sheaf of (quasi-)horizontal sections of the Jacobian bundle
$J(\mathcal{V})$ is

$\mathcal{J}_{hor}^{\mathbb{Q}}=\mathcal{H}^{0}\left(C^{\bullet}/(F^{0}C^{\bullet}+\mathbb{V})\right).$

The space of normal functions associated to $\mathcal{V}$ is

$\mathrm{NF}_{\mathscr{S}}(\mathcal{V}):=\Gamma\left(\mathscr{S},\mathcal{J}_{hor}^{\mathbb{Q}}\right),$

with infinitesimal invariant

$\delta:\mathrm{NF}_{\mathscr{S}}(\mathcal{V})=\mathbb{H}^{0}\left(\mathscr{S},\tfrac{C^{\bullet}}{F^{0}C^{\bullet}+\mathbb{V}}\right)\to\mathbb{H}^{1}\left(\mathscr{S},F^{0}C^{\bullet}\right)\to\Gamma\left(\mathscr{S},\mathcal{H}_{\nabla}^{1}(F^{0}C^{\bullet})\right)$

arising from (2.2).

######  Lemma 2.1.

If $H_{\nabla}^{0}(F^{0}C^{\bullet})=0$, then $\delta$ is injective.

###### Proof.

The assumption implies that we only need to check injectivity into
$\mathbb{H}^{1}(\mathscr{S},F^{0}C^{\bullet})$. By the Theorem of the Fixed
Part [S], the assumption also implies that
$H^{0}(\mathscr{S},\mathbb{V})=\\{0\\}$. Since
$\mathbb{H}^{0}(\mathscr{S},C^{\bullet}/\mathbb{V})=H^{0}(\mathscr{S},\mathbb{V}_{\mathbb{C}}/\mathbb{V})=H^{0}(\mathscr{S},\mathbb{V})\otimes\tfrac{\mathbb{C}}{\mathbb{Q}}$,
the result now follows from the long-exact sequence of (2.2).∎

######  Lemma 2.2.

If $\mathcal{H}_{\nabla}^{1}(F^{0}C^{\bullet})$ and
$\mathcal{H}_{\nabla}^{0}(F^{0}C^{\bullet})$ vanish, then
$\mathrm{NF}_{\mathscr{T}}(\jmath^{*}\mathcal{V})=\\{0\\}$ for any étale
neighborhood $\jmath:\mathscr{T}\to\mathscr{S}$.

###### Proof.

Vanishing of
$\mathcal{H}^{0}(F^{0}C^{\bullet})=\mathbb{V}_{\mathbb{C}}\cap\mathcal{F}^{0}$
implies (again by [S]) that $H^{0}(\mathscr{T},\jmath^{*}\mathbb{V})=\\{0\\}$;
now apply the previous Lemma. ∎

Now writing
$\mathcal{H}^{k}(j):=\mathcal{H}_{\bar{\nabla}}^{k}(Gr_{F}^{j}C^{\bullet})$,
consider the spectral sequence

(2.3)
$\mathcal{E}_{1}^{q,p}:=\left\\{\begin{array}[]{cc}\mathcal{H}^{p+q}(p),&p\geq
0\\\ 0,&p<0\end{array}\right.$

converging to $\mathcal{H}_{\nabla}^{*}(F^{0}C^{\bullet})$. We recover at once
the result of Green and Voisin [Gre], in the form stated by Nori [No1]:

######  Proposition 2.3.

If all the $\mathcal{H}^{0}(j)$ and $\mathcal{H}^{1}(j)$ vanish for $j\geq 0$,
then $\mathrm{NF}_{\mathscr{T}}(\jmath^{*}\mathcal{V})=\\{0\\}$ for any étale
neighborhood $\jmath:\mathscr{T}\to\mathscr{S}$.

Two obvious situations in which the vanishing conditions _fail_ are those of
VHS of level one, or level three and “Calabi-Yau type”. Recall that if

$\mathcal{V}=\mathcal{F}^{p_{min}}\mathcal{V}\supsetneq\mathcal{F}^{p_{min}+1}\mathcal{V}\supsetneq\cdots\supsetneq\mathcal{F}^{p_{max}}\mathcal{V}\supsetneq\mathcal{F}^{p_{max}+1}\mathcal{V}=\\{0\\}$

then the level $\ell(\mathcal{V})=p_{max}-p_{min}$; for weight $-1$ we have
$\ell(\mathcal{V})=2p_{max}+1=-2p_{min}-1$. Write
$h^{p}:=\dim(Gr_{\mathcal{F}}^{p}\mathcal{V})$ for the Hodge numbers, and
$\underline{h}=(h^{p_{min}},\ldots,h^{p_{max}})$; put
$d:=\dim_{\mathbb{C}}\mathscr{S}$. By naively computing ranks we have

######  Example 2.4.

(level 1) If $\underline{h}=(n,n)$ and $d>1$, then
$\mathcal{H}^{1}(0)\neq\\{0\\}$. If $d=1,$ then
$\mathcal{H}^{1}(1)\neq\\{0\\}$.

######  Example 2.5.

(level 3 CY) If $\underline{h}=(1,n,n,1)$ and $1<d<2n$, then
$\mathcal{H}^{1}(0)\neq\\{0\\}$. If $d=1$, then
$\mathcal{H}^{1}(2)\neq\\{0\\}$. (All CY VHS we consider have $d\leq n$.)

This makes the broad vanishing results we obtain in this paper somewhat
surprising. It also explains why we have to ignore the level-one cases below.

Finally, given a smooth family of varieties $\pi:\mathscr{X}\to\mathscr{S}$,
let $Z\in Z^{r}(X_{s_{0}})_{\mathbb{Q},hom}$ be a homologically trivial cycle
on a very general fiber. Suppose its class in the Griffiths group
$\mathit{Griff}^{r}(X_{s_{0}})=\tfrac{Z^{r}(X_{s_{0}})_{\mathbb{Q},hom}}{Z^{r}(X_{s_{0}})_{\mathbb{Q},alg}}$
has nonzero image under

(2.4)
$\overline{AJ}^{r}_{X_{s_{0}}}:\,\mathit{Griff}^{r}(X_{s_{0}})\to\overline{J}^{r}(X_{s_{0}}):=Ext^{1}_{\text{MHS}}\left(\mathbb{Q}(0),\overline{H}^{2r-1}(X_{s_{0}},\mathbb{Q}(r))\right),$

where $\overline{H}^{2r-1}(X_{s_{0}})$ is the quotient of
$H^{2r-1}(X_{s_{0}})$ by its maximal level-one sub-HS. Spreading out yields a
cycle $\mathfrak{Z}\in Z^{r}(\mathscr{X}_{\mathscr{T}})_{\mathbb{Q}}$ on an
étale pullback
$\mathscr{X}_{\mathscr{T}}\overset{\pi_{\mathscr{T}}}{\to}\mathscr{T}$ meeting
fibers properly (with each $\mathfrak{Z}\cdot X_{t}\underset{hom}{\equiv}0$),
and a normal function

$\nu_{\mathfrak{Z}}\in\mathrm{NF}_{\mathscr{T}}\left(\mathcal{V}\right),\;\;\mathcal{V}=\overline{R}^{2r-1}(\pi_{\mathscr{T}})_{*}\mathbb{Q}_{\mathscr{X}_{\mathscr{T}}}(r)\otimes\mathcal{O}_{\mathscr{T}},$

where $\overline{R}$ denotes the quotient by the maximal level-one sub-VHS. So
we have the

######  Corollary 2.6.

If all the $\mathcal{H}^{0}(j)$ and $\mathcal{H}^{1}(j)$ vanish for $j\geq 0$,
then $\overline{AJ}_{X_{s_{0}}}^{r}(Z)=0$ in (2.4).

While there is no converse result, nonvanishing of $\mathcal{H}^{1}(0)$ in
particular seems to be a good predictor of the existence of interesting
cycles. Note that its nonvanishing for level-one VHS $\mathcal{V}$ has a
geometric “origin”. Namely, these VHS correspond to the $H^{1}$ (or
$H^{2D-1}$) of families of abelian $D$-folds
$\mathscr{A}\overset{\pi}{\to}\mathscr{S}$. The existence of nontorsion points
on the geometric generic fiber over $\overline{\mathbb{C}(\mathscr{S})}$
yields nontrivial geometric normal functions in
$\mathrm{NF}_{\mathscr{T}}(\jmath^{*}\mathcal{V}$) over some étale
neighborhood $\mathscr{T}$. This explains why we only consider the Griffiths
group.

###  2.2. $\mathfrak{n}$-cohomology of finite-dimensional representations

Let $\mathfrak{g}$ be a complex semisimple Lie algebra of rank $n$,
$\mathfrak{b}\supset\mathfrak{t}$ Borel and Cartan subalgebras,
$\Delta=\Delta(\mathfrak{g},\mathfrak{t})\subset\mathfrak{t}^{*}$ the
corresponding roots. Denote by $\Delta^{+}=\Delta(\mathfrak{b})$ the positive
roots, $\Sigma=\\{\sigma_{1},\ldots,\sigma_{n}\\}\subset\Delta^{+}$ the simple
roots, $\Omega=\\{\omega_{1},\ldots,\omega_{n}\\}\subset\mathfrak{t}^{*}$ the
fundamental weights, and $\Lambda$ the (weight) lattice they generate. The
Killing form $B(X,Y)=\mathit{Tr}\left(\text{ad}X\circ\text{ad}Y\right)$ on
$\mathfrak{g}$ induces a symmetric bilinear form $\langle\,,\,\rangle$ on
$\Lambda$, a particular orthonormal basis of which (as in [Kn, App. C]) will
be denoted by $\\{e_{i}\\}$. We have in particular
$\langle\omega_{i},\sigma_{j}\rangle=\frac{1}{2}\langle\sigma_{j},\sigma_{j}\rangle\delta_{ij}$.

By the Theorem of the Highest Weight, the irreducible representations
$\\{V^{\lambda}\\}$ of $\mathfrak{g}$ (of finite dimension) are parametrized
by their highest weight $\lambda$; there is a 1-to-1 correspondence between
irreps and weights of the form $\lambda=\sum_{i=1}^{n}m_{i}\omega_{i}$ with
$m_{i}\geq 0$. Fix an element ${\tt E}\in\Lambda$ with all $\frac{1}{2}{\tt
E}(\sigma_{i})$ non-positive and integral, and let
$\mathfrak{g}=\oplus_{j\in\mathbb{Z}}\mathfrak{g}^{j,-j}$ be the decomposition
into $\text{ad}({\tt E})$-eigenspaces with eigenvalue $2j$. For any
representation $(V,\rho)$, there is a corresponding decomposition into
$\rho({\tt E)}$-eigenspaces, which can have odd or fractional eigenvalues.
Write $\mathfrak{p}=\oplus_{j\geq 0}\mathfrak{g}^{j,-j}$,
$\mathfrak{n}=\oplus_{j<0}\mathfrak{g}^{j,-j}$ (so that
$\Delta(\mathfrak{n})\subset\Delta^{+}$), and
$\mathfrak{g}^{0}=\mathfrak{g}^{0,0}$. We also denote
$\Delta_{0}=\Delta(\mathfrak{g}^{0},\mathfrak{t})$,
$\Delta_{0}^{+}=\Delta_{0}\cap\Delta^{+}$, and $\\{V_{0}^{\xi}\\}$ for the
highest-weight irreps of $\mathfrak{g}^{0}$.

Our main computational tool will be a result of Kostant (cf. [Ko, Thm. 5.14]).
It computes the decomposition of the cohomologies of the natural complex

$0\to V^{\lambda}\to\mathfrak{n}^{\vee}\otimes
V^{\lambda}\to\wedge^{2}\mathfrak{n}^{\vee}\otimes V^{\lambda}\to\cdots$

under the action of $\mathfrak{g}^{0}$. To state a version of it (Prop. 2.7
below), let $W=W(\mathfrak{g},\mathfrak{t})$ and
$W_{0}=W(\mathfrak{g}^{0},\mathfrak{t})$ be the Weyl groups, and consider the
set

$W^{0}:=\left\\{w\in
W\left|w(\Delta^{+})\supseteq\Delta_{0}^{+}\right.\right\\}$

of minimal-length representatives of cosets $W_{0}\backslash W$ (with length
$|w(\Delta^{+})\cap\Delta^{-}|$), which we partition into

(2.5) $W^{0}(j)=\left\\{w\in W^{0}\left|\text{length}(w)=j\right.\right\\}.$

Finally, write
$\rho:=\tfrac{1}{2}\sum_{\delta\in\Delta^{+}}\delta=\sum_{i=1}^{n}\omega_{i}$,
and

(2.6) $w\cdot\lambda:=w(\lambda+\rho)-\rho.$

######  Proposition 2.7.

__As a $\mathfrak{g}^{0}$-module,
$H^{k}(\mathfrak{n},V^{\lambda})\cong\oplus_{w\in
W^{0}(k)}V_{0}^{w\cdot\lambda}$.

Now suppose $\mathfrak{g}\supset\mathfrak{t}$ is the complexification of
$\mathfrak{g}_{\mathbb{R}}\supset\mathfrak{t}_{\mathbb{R}}$, with
$\mathfrak{t}_{\mathbb{R}}$ _compact_ (and $\mathfrak{g}_{\mathbb{R}}$ not).
Writing $\Delta=\Delta_{c}\amalg\Delta_{n}$ for the decomposition into compact
and noncompact roots, we assume $\tfrac{1}{2}{\tt E}(\Delta_{c})\subset
2\mathbb{Z}$ and $\tfrac{1}{2}{\tt E}(\Delta_{n})\subset 2\mathbb{Z}+1$. Let
$\tilde{V}$ be an irreducible representation (of finite dimension). We then
have either $\tilde{V}_{\mathbb{C}}\cong V^{\lambda}$ (_real_ case) or
$\tilde{V}_{\mathbb{C}}\cong V^{\lambda}\oplus\overline{V^{\lambda}}$ as
$\mathfrak{g}$-modules. Let $w_{0}\in W$ denote the unique element with
$w_{0}(\Delta^{+})=\Delta^{-}$, and write $-\tau$ for the induced involution
on $\Lambda$; for $\mathfrak{g}$ simple of Hermitian type other than
$A_{n},$$D_{2m+1}$, or $E_{6}$, $\tau=\text{id}_{\Lambda}$. In the non-real
case, we have $\overline{V^{\lambda}}\cong V^{\tau(\lambda)}$, and we say
$V^{\lambda}$ is _complex_ resp. _quaternionic_ when
$\lambda\neq\tau(\lambda)$ resp. $=\tau(\lambda)$. To distinguish the real and
quaternionic cases, we use the following

######  Proposition 2.8.

[GGK] For $\lambda=\sum M_{i}\sigma_{i}$ with $\tau(\lambda)=\lambda$,
$V^{\lambda}$ is real iff $\sum_{\sigma_{i}\text{
compact}}M_{i}\in\mathbb{Z}$.

Given $\lambda=\sum m_{i}\omega_{i}$ ($m_{i}\geq 0$), we shall write

(2.7) $\tilde{V}^{\lambda}:=\left\\{\begin{array}[]{cc}V^{\lambda},&\text{real
case}\\\ V^{\lambda}\oplus
V^{\tau(\lambda)},&\text{cx./quat.}\end{array}\right..$

It is to these representations that we shall apply Prop. 2.7. Note that since
$\tilde{V}^{\lambda}$ has an underlying $\mathbb{R}$-vector space
$\tilde{V}_{\mathbb{R}}^{\lambda}$ (which is irreducible as a representation
of $\mathfrak{g}_{\mathbb{R}}$), it can be viewed (up to Tate twist) as an
$\mathbb{R}$-Hodge structure via the action of ${\tt E}$, whose eigenvalues
are regarded as “$p-q$” (on Hodge type $(p,q)$).

Since differences of weights of $V^{\lambda}$ lie in the root lattice (which
${\tt E}$ sends into $2\mathbb{Z}$) and $-\tau(\lambda)=w_{0}(\lambda)$ is a
weight of $V^{\lambda}$, $\tilde{V}^{\lambda}$ _can be placed in weight $-1$
iff ${\tt E}(\lambda)$ is odd_. In this case

(2.8)
$\left(\tilde{V}^{\lambda}\right)^{p,-p-1}:=\left\\{(2p+1)\text{-eigenspace of
}\tt{E}\text{ on }\tilde{V}^{\lambda}\right\\},$

and (using $\overline{V^{\lambda}}\cong V^{\tau(\lambda)}$) the level is

(2.9)
$\ell\left(\tilde{V}^{\lambda}\right)=\text{max}\left\\{-\tt{E}(\lambda),-\tt{E}(\tau(\lambda))\right\\}.$

Moreover, when ${\tt E}(\lambda)$ is odd there exists (up to scale) a unique
$\mathfrak{g}$-invariant alternating bilinear form $Q$ on
$\tilde{V}_{\mathbb{R}}^{\lambda}$, which polarizes this Hodge structure. Note
that in the non-real cases, $Q$ pairs $V^{\lambda}$ and $V^{\tau(\lambda)}$.

######  Remark 2.9.

The level of the complex summands (as complex Hodge structures) is

(2.10)
$\ell(V^{\lambda})\left(=\ell(V^{\tau(\lambda)})\right)=-\tfrac{1}{2}\left(\tt{E}(\lambda)+\tt{E}(\tau(\lambda))\right).$

This is the minimal level that can be achieved using half-twists, which are
discussed in $\S 4$.

###  2.3. Homogeneous VHS over locally symmetric varieties

Let $G$ be a semisimple $\mathbb{Q}$-algebraic group of Hermitian type, such
that $G_{\mathbb{R}}$ has a compact maximal torus $T_{\mathbb{R}}$; and write
$\mathfrak{g}_{\mathbb{R}}\supset\mathfrak{t}_{\mathbb{R}}$ for the Lie
algebras (with complexifications $\mathfrak{g}\supseteq\mathfrak{t}$). Choose
a cocharacter $\chi_{0}:\mathbb{G}_{m}\to T_{\mathbb{C}}$ such that ${\tt
E}:=\chi_{0}^{\prime}(1)$ satisfies444One should also assume ${\tt E}$ doesn’t
project to zero in any simple factor of $\mathfrak{g}$, but in the
calculations below $\mathfrak{g}$ will be simple.

(2.11) ${\tt{E}}(\Delta_{c})=0,\;\;{\tt{E}}(\Delta_{n})=\\{\pm 2\\}.$

Denoting the composition
$\mathbb{G}_{m}\overset{\chi_{0}}{\to}T_{\mathbb{C}}\hookrightarrow
G_{\mathbb{C}}$ by $\varphi_{0}$, the corresponding _Hermitian symmetric
domain_ is the orbit under conjugation by the group of real points
$G(\mathbb{R})$:

$D:=G(\mathbb{R}).\varphi_{0}\cong G(\mathbb{R})/G^{0}(\mathbb{R}),$

with $G_{\mathbb{R}}^{0}\leq G_{\mathbb{R}}$ a maximal compact subgroup.
Taking $\Gamma\leq G(\mathbb{Q})$ a torsion-free congruence subgroup, the
_locally symmetric variety_ $\mathrm{X}:=\Gamma\backslash D$ is in fact a
quasi-projective variety by the Baily-Borel theorem.

The $G(\mathbb{R})$-orbit of
$\text{Ad}\circ\varphi_{0}:\mathbb{G}_{m}\to\mathit{Aut}(\mathfrak{g}_{\mathbb{C}},-B)$
gives a $(-B)$-polarized $\mathbb{Q}$-VHS of weight zero and level two over
$D$, with Hodge decompositions
$\mathfrak{g}=\oplus_{j=-1,0,1}\mathfrak{g}_{\text{Ad}(g){\tt E}}^{j,-j}$ at
$g\varphi_{0}g^{-1}\in D$; this descends to $\mathrm{X}$. More generally (with
$\mathbb{A}=\mathbb{Q}$ or $\mathbb{R}$), given an $\mathbb{A}$-representation
$\rho:G_{\mathbb{A}}\to Aut(V_{\mathbb{A}},Q)$ ($Q$ a $(-1)^{k}$-symmetric
bilinear form) such that $\rho\circ\varphi$ is an $\mathbb{A}$-PHS, we get a
homogeneous $\mathbb{A}$-PVHS $\mathcal{V}_{\mathbb{A}}$ (with weight of
parity $k$) over $\mathrm{X}$, called a _Hermitian VHS_. In every case the
Hodge decomposition is induced by $d\rho(\text{Ad}(g){\tt E})$; that is, a
weight subspace $V_{\xi}\subset V^{p,q}$ $\iff$ ${\tt E}(\xi)=p-q$.

For groups other than $E_{6}$ and $E_{7}$, most555see $\S 3.4$ for discussion
of one possible exception Hermitian $\mathbb{Q}$-PVHS arise from the relative
cohomology of various canonical families of abelian varieties; in some cases
(cf. [FL, R]) one has also families of Calabi-Yau varieties. While these
structures may matter for questions of geometric origin (i.e. algebraic
cycles), the behavior of the $\\{\mathcal{H}^{k}(j)\\}$ depends only on
$\mathcal{V}_{\mathbb{R}}$. Moreover, any Hermitian PVHS of weight $-1$ on
$\mathrm{X}$ decomposes _over $\mathbb{R}$_ into a direct sum of the
irreducible homogeneous real variations
$\tilde{\mathcal{V}}_{\mathbb{R}}^{\lambda}$ arising (as above) from the PHS
on $(\tilde{V}^{\lambda},\tilde{\rho}^{\lambda})$ described at the end of $\S
2.2$. Indeed, since $\Gamma\leq G(\mathbb{R})$ is Zariski-dense, we have an
equivalence between real Hermitian VHS $\mathcal{V}_{\mathbb{R}}$ over
$\mathrm{X}$ and representations of $G_{\mathbb{R}}$.

Before turning to our main calculation, we remind the reader what forms $D$
and ${\tt E}$ can take when $G_{\mathbb{C}}$ is simple. First, (2.11) forces
$\Delta_{n}\cap\Sigma$ to be a singleton $\\{\sigma_{{\tt I}}\\}$, which must
additionally be a _special_ simple root. That is, if we write
$\mathfrak{g}=V^{\lambda_{\text{ad}}}$,
$\lambda_{\text{ad}}=\sum_{i=1}^{n}M_{i}^{\text{ad}}\sigma_{i}$, then
$\sigma_{{\tt I}}$ must be one of the simple roots $\sigma_{i}$ for which
$M_{i}^{\text{ad}}=1$. Further, the choice of $\sigma_{{\tt I}}$ determines:
the decomposition $\Delta=\Delta_{c}\amalg\Delta_{n}$, and thus the real form
$\mathfrak{g}_{\mathbb{R}}$; and the Hodge structure at the base point
$\varphi_{0}$, by

(2.12) ${\tt{E}}(\sigma_{\tt{I}})=-2\;,\;\;{\tt{E}}(\sigma_{j})=0\;(\forall
j\neq\tt{I}).$

This leads to the classification (up to isomorphism) of the irreducible
Hermitian symmetric domains of noncompact type:

$D$ | $(R,\sigma_{{\tt I}})$ | $G_{\mathbb{R}}$ | $d$ | range
---|---|---|---|---
$\mathrm{I}_{p,n-p+1}$ | $(A_{n},\sigma_{p})$ | $\mathit{SU}(p,n-p+1)$ | $p(n-p+1)$ | $\tiny\begin{array}[]{c}n\geq 2\\\ 1\leq p\leq\left\lfloor\tfrac{n+1}{2}\right\rfloor\end{array}$
$\mathrm{II}_{n}$ | $(D_{n},\sigma_{n})$ | $\mathit{Spin}^{*}(2n)$ | $\tfrac{1}{2}n(n-1)$ | $n\geq 4$
$\mathrm{III}_{n}$ | $(C_{n},\sigma_{n})$ | $\mathit{Sp}(2n,\mathbb{R})$ | $\tfrac{1}{2}n(n+1)$ | $n\geq 1$
$\mathrm{IV}_{2n-1}$ | $(B_{n},\sigma_{1})$ | $\mathit{Spin}(2,2n-1)$ | $2n-1$ | $n\geq 3$
$\mathrm{IV}_{2n-2}$ | $(D_{n},\sigma_{1})$ | $\mathit{Spin}(2,2n-2)$ | $2n-2$ | $n\geq 4$
$\mathrm{EIII}$ | $(E_{6},\sigma_{1})$ | $E_{6}(-14)$ | $16$ | —–
$\mathrm{EVII}$ | $(E_{7},\sigma_{7})$ | $E_{7}(-25)$ | $27$ | —–

Here $R$ is the root system, $G_{\mathbb{R}}$ is the simply connected group
(which has all $\tilde{V}^{\lambda}$ as representations), and
$d=\dim_{\mathbb{C}}D=\dim_{\mathbb{C}}\mathrm{X}$. (See [LZ] for more
details.)

######  Remark 2.10.

We have omitted some cases to avoid redundancy due to exceptional isomorphisms
and conjugate-isomorphisms. The latter are induced by the action of $\tau$ on
$\sigma_{{\tt I}}$. For $A_{n}$, $\tau$ exchanges $\sigma_{i}$ and
$\sigma_{n+1-i}$ $(\forall i)$; for $D_{n}$, $n$ odd, $\tau$ exchanges
$\sigma_{n-1}$ and $\sigma_{n}$; for $E_{6}$, $\tau$ exchanges
$\sigma_{6}\leftrightarrow\sigma_{1}$ and
$\sigma_{5}\leftrightarrow\sigma_{3}$; and in all other cases the action is
trivial.

The exceptional isomorphisms are
$\mathrm{III}_{1}\cong\mathrm{I}_{1,1}\cong\mathrm{IV}_{1}\cong\mathrm{II}_{2}$,
$\mathrm{III}_{2}\cong\mathrm{IV}_{3}$,
$\mathrm{I}_{2,2}\cong\mathrm{IV}_{4}$,
$\mathrm{II}_{3}\cong\mathrm{I}_{1,3}$, $\mathrm{II}_{4}\cong\mathrm{IV}_{6}$
(triality for $D_{4}$), and
$\mathrm{IV}_{2}\cong\mathrm{III}_{1}\times\mathrm{III}_{1}$. For this reason
we consider only $A_{n\geq 2}$, $B_{n\geq 3}$, $C_{n\geq 1}$, and $D_{n\geq
4}$.

In each case, the real variations arising from $H^{1}$ of the canonical
families of abelian varieties over $\mathrm{X}$ are (half-twists of666this is
only relevant for $A_{n}$, cf. $\S 4$) the $\tilde{\mathcal{V}}^{\lambda}$
with $-\tfrac{1}{2}\left({\tt E}(\lambda)+{\tt E}(\tau(\lambda))\right)=1$
(cf. (2.10)). All such $\lambda$ take the form $\omega_{i}$, with the
possibilities corresponding to so-called “symplectic nodes” (cf. [LZ]). This
will be recalled case by case where relevant in $\S\S$3-4. The variations of
Calabi-Yau type are even simpler to describe: they are (again, up to half-
twist for $A_{n}$, $D_{n\text{ odd}}$, $E_{6}$) precisely the
$\tilde{\mathcal{V}}^{k\omega_{{\tt I}}}$ for $k\geq 1$ [FL].

###  2.4. The main calculation

We now apply Proposition 2.7 to compute the $\\{\mathcal{H}^{k}(j)\\}$ for the
Hermitian variations $\tilde{\mathcal{V}}^{\lambda}$ (over $\mathrm{X}$ or
$D$) with ${\tt E}(\lambda)$ odd. Of course, we can work with (its summands)
the _complex_ variations $\mathcal{V}_{\mathbb{C}}^{\lambda}$, and do the
computation at one point $\varphi_{0}\in D$. Write

(2.13) $W^{0}(k,j):=\left\\{w\in
W^{0}(k)\left|\tfrac{1}{2}\left({\tt{E}}(w\cdot\lambda)-1\right)=j\right.\right\\}.$

######  Proposition 2.11.

For the $\mathcal{V}_{\mathbb{C}}^{\lambda}$,
$\mathcal{H}^{k}(j)|_{\varphi_{0}}\cong\oplus_{w\in
W^{0}(k,j)}V_{0}^{w\cdot\lambda}$.

###### Proof.

First, we note that

(2.14) $\oplus_{j}\mathcal{H}^{k}(j)|_{\varphi_{0}}\cong
H^{k}(\mathfrak{n},V^{\lambda})$

follows from the identification of the Lie algebra cohomology complex
$\left(\wedge^{\bullet}\mathfrak{n}^{\vee}\otimes V^{\lambda},d\right)$ with
the associated graded
$\left(\oplus_{j}Gr_{F}^{j}C^{\bullet},\bar{\nabla}\right)$ under the
isomorphism $\mathfrak{n}^{\vee}\cong\Omega_{D,\varphi_{0}}^{1}$. To see this
identification, extend $v\in(V^{\lambda})^{j,-j-1}$ to a section of
$(\mathcal{V}^{\lambda})^{j,-j-1}$ by $\tilde{v}=\rho^{\lambda}(g).v$, and
write (for $X\in\mathfrak{n}$)
$(\nabla_{X}\tilde{v})|_{\varphi_{0}}=\lim_{\epsilon\to
0}\tfrac{1}{\epsilon}(e^{\epsilon
d\rho^{\lambda}(X)}v-v)=d\rho^{\lambda}(X).v$ ($=$“$X(v)$”).

Next, we compare Hodge gradings on the two sides of (2.14). For
$X^{*}\in\mathfrak{n}^{\vee}$ and $v\in
Gr_{F}^{j-1}V^{\lambda}=(V^{\lambda})^{j-1,-j}$, we have ${\tt
E}(X^{*})=2X^{*}$ and ${\tt E}(v)=(2j-1)v$ $\implies$ ${\tt E}(X^{*}\otimes
v)=(2j+1)X^{*}\otimes v$. So the image of $\mathcal{H}^{k}(j)|_{\varphi_{0}}$
in $H^{k}(\mathfrak{n},V^{\lambda})=\oplus_{w\in
W^{0}(k)}V_{0}^{w\cdot\lambda}$ consists of all the weight spaces with weights
$\xi$ for which ${\tt E}(\xi)=2j+1$. Since $\mathfrak{g}^{0}$ commutes with
${\tt E}$, these are just the $V_{0}^{w\cdot\lambda}$ for which ${\tt
E}(w\cdot\lambda)=2j+1$. The result follows. ∎

Obviously $W^{0}(0)=\\{\text{id}\\}$, and always ${\tt E}(\lambda)\leq-1$; so
immediately we have $\mathcal{H}^{0}(j)=\\{0\\}$ for $j\geq 0$. We claim that
$W^{0}(1)=\\{{\tt s}\\}$, where ${\tt s}$ is the reflection in the
distinguished simple root $\sigma_{{\tt I}}$. Indeed, elements of $W^{0}(1)$
have length one, so must be the reflection in one simple root. But the
reflection $w_{i}$ in $\sigma_{i\neq{\tt I}}$ doesn’t satisfy
$w_{i}(\Delta^{+})\supseteq\Delta_{0}^{+}$ (as $\sigma_{i}\notin
w_{i}(\Delta^{+})$), whereas ${\tt s}(\Delta_{0}^{+})\subseteq\Delta^{+}$
since ${\tt s}(\sigma_{{\tt I}})=-\sigma_{{\tt I}}$ and $|{\tt
s}(\Delta^{+})\cap\Delta^{-}|=1$. So we have the basic invariant

(2.15)
$\mu(\lambda):=\tfrac{1}{2}\left({\tt{E}}({\tt{s}}\cdot\lambda)-1\right)$

which depends upon the choice of $\sigma_{{\tt I}}$ (hence $D$ and
$\mathrm{X}$). Propositions 2.3 and 2.11 yield at once the

######  Theorem 2.12.

If $\mu(\lambda),\mu(\tau(\lambda))<0$, then
$\mathrm{NF}_{\mathscr{T}}(\jmath^{*}\mathcal{V})=\\{0\\}$ for any étale
neighborhood $\jmath:\mathscr{T}\to\mathrm{X}$ and any variation $\mathcal{V}$
on $\mathrm{X}$ with the canonical Hermitian variation
$\tilde{\mathcal{V}}_{\mathbb{R}}^{\lambda}$ as underlying $\mathbb{R}$-VHS.
(In particular, we have $\overline{\mathit{AJ}}(Z)=0$ if $\mathcal{V},Z$ are
as in (2.4), with $\mathscr{S}=\mathrm{X}$.)

One can of course replace $\mathrm{X}$ by a Zariski open subset in these
statements.

######  Remark 2.13.

If $\mu(\lambda)=\mu(\tau(\lambda))=0$, then (arguing as in [Fa]) one can show
that M. Saito’s canonical MHS on $H^{1}(\mathrm{X},\mathbb{V})$ is pure of
type $(0,0)$. (Naturally, $H^{1}(\mathrm{X},\mathbb{V})$ could still be
$\\{0\\}$.)

Suppose that for $\mathbb{V}$ as above we have $\mu(\lambda)\geq 0$, so that
$\mathcal{E}_{1}^{1-\mu(\lambda),\mu(\lambda)}=\mathcal{H}_{\bar{\nabla}}^{1}(\mu(\lambda))\neq\\{0\\}$.
To conclude the stronger result that
$\mathcal{E}_{\infty}^{1-\mu(\lambda),\mu(\lambda)}$ hence
$\mathcal{H}_{\nabla}^{1}(F^{0}C^{\bullet})$ is nonzero, we would need to
compute the nonlinear
$d_{\ell}:\mathcal{E}_{\ell}^{1-\mu(\lambda),\mu(\lambda)}\to\mathcal{E}_{\ell}^{2-\mu(\lambda)-\ell,\mu(l)+\ell}$.
The present methods are only of use here if they show that all
$\mathcal{H}_{\bar{\nabla}}^{2}(j)=\\{0\\}$ for $j>\mu(\lambda)$. But this is
asking too much: even in the key case
$\\{D=\mathrm{III}_{3},\lambda=\omega_{3},\mu(\lambda)=0\\}$ where we have a
nonzero _geometric_ normal function coming from the Ceresa cycle, we compute
that $\mathcal{H}^{2}(1)\neq\\{0\\}$. So with regard to predicting normal
functions it is probably more useful to stick to $\mathcal{H}^{1}(j)$ and make
the provisional

######  Definition 2.14.

The complex (resp. real) variation $\mathcal{V}^{\lambda}$ (resp.
$\tilde{\mathcal{V}}_{\mathbb{R}}^{\lambda}$) has an _infinitesimal normal
function_ if $\mu(\lambda)\geq 0$.

##  3\. Analysis of (mostly) tube domain cases

In this section we study the real variations
$\tilde{\mathcal{V}}_{\mathbb{R}}^{\lambda}$ arising from dominant integral
$\lambda$ with ${\tt E}(\lambda)$ an odd integer. Though we (more or less)
carry this out for all the domains in the table 2.3, the results are of
interest mainly when $D$ is of tube type ($\mathrm{I}_{p.p}$,
$\mathrm{II}_{2m}$, $\mathrm{III}_{n}$, $\mathrm{IV}_{m}$, or
$\mathrm{EVIII}$). As above, we have ${\tt E}(\sigma_{{\tt I}})-2$, ${\tt
E}(\sigma_{j\neq{\tt I}})=0$, and write
$\lambda=\sum_{i=1}^{n}m_{i}\omega_{i}$; note that ${\tt s}$ sends
$\sigma_{{\tt I}}\mapsto-\sigma_{{\tt I}}$ and fixes all $\omega_{j\neq{\tt
I}}$. To streamline the discussion of examples, we make the

######  Definition 3.1.

A family $\mathscr{A}\overset{\pi}{\to}\mathscr{S}$ of abelian varieties
_admits no reduced normal functions_ if
$\mathrm{NF}_{\mathscr{S}}(\mathcal{V})=\\{0\\}$ for every $r\in\mathbb{N}$
and irreducible VHS $\mathcal{V}\subset
R^{2r-1}\pi_{*}\mathbb{Q}_{\mathscr{A}}(r)\otimes\mathcal{O}_{\mathscr{S}}$ of
level $>1$.

In all the cases below where we obtain such an assertion, one knows that all
level-one real sub-VHS are in fact defined over $\mathbb{Q}$.

###  3.1. Case $\mathrm{III}_{n}$

This is the case analyzed by Nori [No1] ($\mathcal{H}^{1}(j)$) and Fakhruddin
[Fa] ($\mathcal{H}^{k}(j)$ for $k>1$). We have

$\Sigma=\\{e_{1}-e_{2},e_{2}-e_{3},\ldots,e_{n-1}-e_{n},2e_{n}\\},$
$\Omega=\\{e_{1},e_{1}+e_{2},e_{1}+e_{2}+e_{3},\ldots,e_{1}+\cdots+e_{n}\\},$

and so ${\tt I}=n$ $\implies$ ${\tt E}(\lambda)=-\sum_{i=1}^{n}im_{i}$; the
only level-one variation is thus $\mathcal{V}_{\mathbb{R}}^{\omega_{1}}$
(which arises from $H^{1}$ of the universal abelian variety). There are no
complex or quaternionic irreps, and (taking ${\tt E}(\lambda)$ odd) we have
${\tt s}(\omega_{n})=2\omega_{n-1}-\omega_{n}$ $\implies$

$\mu(\lambda)=-\tfrac{1}{2}\sum_{i<n}im_{i}+(1-\tfrac{n}{2})m_{n}+\tfrac{1}{2}.$

So for $\ell(V^{\lambda})=-{\tt E}(\lambda)>1$ odd, $\mu(\lambda)\geq 0$ for
only

$n=1\text{ and }\lambda=a\omega_{1}\;(a>1\text{ odd}),$ $n=2\text{ and
}\lambda=\omega_{1}+a\omega_{2}\;(a\geq 1),$ $n=3\text{ and
}\lambda=\omega_{3};$

in particular, no étale pullback of the universal abelian variety
$\mathscr{A}^{\mathrm{III}_{g}}\to\mathcal{A}_{g}(\Gamma)=\Gamma\backslash\mathrm{III}_{g}$
admits reduced normal functions for $g\geq 4$. These are the results of Nori
[No1].

###  3.2. Case $\mathrm{EVIII}$

We have ${\tt I}=7$ and

$\Sigma=\left\\{\tfrac{1}{2}(e_{8}-e_{7}-\cdots-
e_{2}+e_{1}),e_{2}+e_{1},e_{2}-e_{1},e_{3}-e_{2},\ldots,e_{6}-e_{5}\right\\};$

$\Omega$ is written in terms of the $\\{\sigma_{i}\\}$ in [Kn, p. 688]. This
yields

${\tt E}(\lambda)=-2m_{1}-3m_{2}-4m_{3}-6m_{4}-5m_{5}-4m_{6}-3m_{7},$

${\tt E}({\tt s}(\omega_{7}))=-1$, and

$\mu(\lambda)=-m_{1}-\tfrac{3}{2}m_{2}-2m_{3}-3m_{4}-\tfrac{5}{2}m_{5}-2m_{6}-\tfrac{1}{2}m_{7}+\tfrac{1}{2}.$

There are no complex or quaternionic irreps, and $\mu(\lambda)\neq 0$ (with
${\tt E}(\lambda)$ odd) for

$\lambda=\omega_{7}.$

###  3.3. Case $\mathrm{EIII}$

Here ${\tt I}$=1, $\Omega$ is expressed in terms of

$\Sigma=\left\\{\tfrac{1}{2}(e_{8}-e_{7}-\cdots-
e_{2}+e_{1}),e_{1}+e_{2},e_{2}-e_{1},\ldots e_{5}-e_{4}\right\\}$

in [Kn, p. 687], and $\tau=(16)(35)$ on both the $\\{\sigma_{i}\\}$ and
$\\{\omega_{i}\\}$. Since

${\tt E}(\lambda)=-\tfrac{2}{3}(4m_{1}+5m_{3}+4m_{5}+2m_{6})-2m_{2}-4m_{4},$

we find that ${\tt E}(\lambda)\in\mathbb{Z}$ $\iff$
$3|(4m_{1}+5m_{3}+4m_{5}+2m_{6})$ $\iff$ ${\tt E}(\lambda)\in 2\mathbb{Z}$. So
there are no odd-level VHS without half-twists, and further analysis is
deferred to $\S 4$.

We can now turn to the richer remaining classical cases.

###  3.4. Case $\mathrm{II}_{n}$

With ${\tt I}=n$ (note that $n\geq 4$),

$\Sigma=\\{e_{1}-e_{2},\ldots,e_{n-1}-e_{n},e_{n-1}+e_{n}\\},$
$\Omega=\\{e_{1},e_{1}+e_{2},\ldots,e_{1}+\cdots+e_{n-2},\tfrac{1}{2}(e_{1}+\cdots+e_{n-1}-e_{n}),\tfrac{1}{2}(e_{1}+\cdots
e_{n})\\}$

we obtain

${\tt
E}(\lambda)=-\sum_{i=1}^{n-2}im_{i}-\tfrac{n-2}{2}m_{n-1}-\tfrac{n}{2}m_{n}.$

Note that $\tau$ swaps $\sigma_{n-1}\leftrightarrow\sigma_{n}$ and
$\omega_{n-1}\leftrightarrow\omega_{n}$ if $n$ is odd, and is trivial for $n$
even. The only complex irreps are therefore the $V^{\lambda},$with $n$ odd and
$m_{n-1}\neq m_{n}$. The quaternionic ones are the $V^{\lambda}$ with

$\sum_{i=1}^{n-2}im_{i}+m_{n-1}\text{ odd (}n\text{
even),}\;\;\text{or}\;\;\left\\{\begin{array}[]{c}\sum_{i=1}^{n-2}im_{i}\text{
odd}\\\ m_{n-1}=m_{n}\end{array}\right.\text{(}n\text{ odd).}$

We have $\ell(\tilde{V}^{\lambda})=1$ for $\lambda=\omega_{1},\omega_{3}$
($n=4$) and $\lambda=\omega_{1}$ ($n>4$); these $V^{\lambda}$ are all
quaternionic (i.e. $\tilde{V}^{\lambda}=(V^{\lambda})^{\oplus 2}$ over
$\mathbb{C}$).

######  Definition 3.2.

By a _universal quaternionic abelian variety_ , we shall mean any family
$\mathscr{A}^{\mathrm{II}_{n}}\to\Gamma\backslash\mathrm{II}_{n}$ of abelian
$2n$-folds whose $H^{1}$ recovers
$\tilde{\mathcal{V}}_{\mathbb{R}}^{\omega_{1}}$.

Such families admit an embedding of a definite rational quaternion algebra
$\mathcal{Q}$ into $End(\mathscr{A})_{\mathbb{Q}}$. (The Mumford-Tate group
$G$ is a $\mathbb{Q}$-form of $G_{\mathbb{R}}$ and so, by considering its
fixed 2-tensors, $\mathcal{Q}$ is a $\mathbb{Q}$-form of $\mathbb{H}$. See
[vGV] for more details on quaternionic abelian varieties.) There are natural
embeddings $\mathrm{II}_{n}\hookrightarrow\mathrm{III}_{2n}$ which yield
countably many “quaternionic subfamilies” of
$\mathscr{A}^{\mathrm{III}_{2n}}$.

Now from ${\tt s}(\omega_{n})=\omega_{n-2}-\omega_{n}$ we find

$\mu(\lambda)=-\tfrac{1}{2}\sum_{i=1}^{n-2}im_{i}-\tfrac{n-2}{4}m_{n-1}-\tfrac{n-4}{4}m_{n}+\tfrac{1}{2}.$

Imposing $-{\tt E}(\lambda)$ odd $>1$, $n\geq 4$, and $\mu(\lambda)\geq 0$
yields

$\lambda=\omega_{1}+a\omega_{4},\;\omega_{3}+a\omega_{4}\;\;(n=4,a>0)$

which is quaternionic, and

$\lambda=\omega_{6}\;\;(n=6)$

which is real.

######  Theorem 3.3.

No étale pullback of a universal quaternionic abelian variety of (relative)
dimension $2n$ admits reduced normal functions outside the case $n=4$.

###### Proof.

It remains to deal with $n=6$. The point is that only
$\mathcal{V}^{\omega_{5}+\omega_{6}}$ and
$\mathcal{V}^{2\omega_{5}},$$\mathcal{V}^{2\omega_{6}}$ occur in
$H_{rel}^{*}(\mathscr{A}^{\mathrm{II}_{6}})\cong\bigwedge^{*}\left(\mathcal{V}^{\omega_{1}}\oplus\mathcal{V}^{\omega_{1}}\right)$;
the half-spin variations777In all the $D_{n}$ cases, $V^{\omega_{n-1}}$ and
$V^{\omega_{n}}$ identify with ${\rm spin}^{-}$ and ${\rm spin}^{+}$ (with
order depending on parity of $n$). These are representations of $Spin^{*}(12)$
but not of $SO^{*}(12)$, the Mumford-Tate group of
$\mathscr{A}^{\mathrm{II}_{6}}$.
$\mathcal{V}^{\omega_{5}},\mathcal{V}^{\omega_{6}}$ do not. (See [FH] for the
equivalent fact on representations of $SO(12)$.) ∎

What is special about the quaternionic $8$-folds that might yield
$\overline{\mathit{AJ}}$-nontrivial elements of the Griffiths group? According
to [vGV], there exist families $\mathscr{A}^{\mathrm{II}_{4}}$ whose general
member arises as a quaternionic Prym variety, associated to a certain $8:1$
unramified cover of a general genus $3$ curve (plausible as
$\dim\mathcal{M}_{3}=\dim\mathrm{II}_{4}=6$). Since
$\bigwedge^{3}V^{\omega_{1}}=V^{\omega_{3}+\omega_{4}}$, the (non-Calabi-Yau)
variation $\tilde{\mathcal{V}}_{\mathbb{R}}^{\omega_{3}+\omega_{4}}$ occurs in
$H^{3}$ of $\mathscr{A}^{\mathrm{II}_{4}}$, and it seems reasonable to expect
that one can construct $\overline{\mathit{AJ}}$-nontrivial 1-cycles from the
Abel-Prym image of the genus $17$ cover curves. For $n=6$, it seems to be an
open problem to give a simple motivic construction of
$\mathcal{V}_{\mathbb{R}}^{\omega_{6}}$, so we cannot speculate about cycles
in this case.

###  3.5. Case $\mathrm{I}_{p,n-p+1}$

We have $n\geq 2$, $1\leq p\leq\left\lfloor\tfrac{n+1}{2}\right\rfloor$, ${\tt
I}=p$,

$\Sigma=\\{e_{1}-e_{2},e_{2}-e_{3},\ldots,e_{n}-e_{n+1}\\},$
$\Omega=\\{\omega_{i}=e_{1}+\cdots+e_{i}-\tfrac{i}{n+1}(e_{1}+\cdots+e_{n})\\}_{i=1,\ldots,n},$

with $\tau=\prod_{j=1}^{\left\lfloor\frac{n}{2}\right\rfloor}(j,n+1-j)$ on the
$\\{\sigma_{i}\\}$ and $\\{\omega_{i}\\}$, and ${\tt
s}(\omega_{p})=\omega_{p-1}-\omega_{p}+\omega_{p+1}$ (or
$-\omega_{1}+\omega_{2}$ if $p=1$). This yields

(3.1) $\left\\{\begin{array}[]{c}{\tt
E}(\lambda)=\tfrac{2}{n+1}\left(\sum_{i=1}^{p}(n+1-p)im_{i}+\sum_{i=p+1}^{n}p(n+1-i)m_{i}\right)\\\
{\tt E}({\tt s}\cdot\lambda)=2+2m_{p}+{\tt
E}(\lambda),\;\;\mu(\lambda)=\tfrac{1}{2}+m_{p}+\tfrac{1}{2}{\tt
E}(\lambda)\end{array}\right.$

and for $n$ odd there are quite a few (mostly complex) representations with
${\tt E}(\lambda)$ an odd integer and $\mu(\lambda)\geq 0$. For example,
$p=2,n=7$ produces $\tilde{V}^{\omega_{6}}=V^{\omega_{2}}\oplus
V^{\omega_{6}}$ and $\tilde{V}^{2\omega_{7}}=V^{2\omega_{7}}\oplus
V^{2\omega_{1}}$, which both have level $3$ (and
$\mu(2\omega_{7})=\mu(\omega_{6})=\mu(\omega_{2})=0$). We have not carried out
an exhaustive search, because the resulting variations seem obscure and the
situation drastically improves with the introduction of half-twists ($\S 4$).

So restricting to $p=\tfrac{n+1}{2}$ ($n$ odd), i.e. the case
$\mathrm{I}_{p,p}$ ($p\geq 2$), we have

(3.2)
$\left\\{\begin{array}[]{c}{\tt{E}}(\lambda)=-\sum_{i=1}^{p}im_{i}-\sum_{i=p+1}^{2p-1}(2p-i)m_{i}\\\
\mu(\lambda)=-\tfrac{1}{2}\sum_{i<p}im_{i}-(\tfrac{p}{2}-1)m_{p}-\sum_{i>p}(p-\tfrac{i}{2})m_{i}+\tfrac{1}{2}.\end{array}\right.$

We remark that all representations are either real ($m_{j}=m_{n+1-j}$ $\forall
j$) or complex, and that the sole level $1$ variation is
$\tilde{\mathcal{V}}^{\omega_{1}}=\mathcal{V}^{\omega_{1}}\oplus\mathcal{V}^{\omega_{n}}$.

######  Definition 3.4.

A universal Weil abelian variety is a family
$\mathscr{A}^{\mathrm{I}_{p,p}}\to\Gamma\backslash\mathrm{I}_{p,p}$ of abelian
$2p$-folds (of dimension $p^{2}$) whose $H^{1}$ recovers
$\tilde{\mathcal{V}}_{\mathbb{R}}^{\omega_{1}}$.

Such families admit an embedding of an imaginary quadratic field into
$End(\mathscr{A})_{\mathbb{Q}}$, and produce countably many subfamilies of
$\mathscr{A}^{\mathrm{III}_{2p}}$.

######  Theorem 3.5.

No étale pullback of a universal Weil abelian variety of (relative) dimension
$2p$ admits reduced normal functions outside the cases $p=2,3$.

###### Proof.

For $p>3$, there are no solutions to $\mu(\lambda)\geq 0$ (with the usual
constraints). ∎

Indeed, the only solutions are

$\lambda=\omega_{1}+a\omega_{2},\;\omega_{3}+a\omega_{2}\;\;(p=2)$

which are conjugate complex, and

$\lambda=\omega_{3}\;\;(p=3)$

which is real (with $\mathcal{V}^{\omega_{3}}$ of Calabi-Yau type). One easily
shows that $\tilde{\mathcal{V}}^{\omega_{1}+\omega_{2}}\subset
H_{rel}^{3}(\mathscr{A}^{\mathrm{I}_{2,2}})$ and
$\tilde{\mathcal{V}}^{\omega_{3}}\subset
H_{rel}^{3}(\mathscr{A}^{\mathrm{I}_{3,3}})$, and these are the two cases
where there exist families $\mathscr{A}^{\mathrm{I}_{p,p}}$ with generic
member arising from a generalized (3 to 1) Prym construction. More precisely,
let $\mathcal{B}_{p,p}$ denote the moduli space of abelian $2p$-folds $A$ with
: (a) a degree-$3$ automorphism $\alpha$, having $\zeta_{3}$\- and
$\bar{\zeta}_{3}$-eigenspaces in $T_{0}A$ both of dimension $p$; and (b) a
polarization $\Sigma$ with $\ker(\Sigma)\subset A[3]$ of order $3^{2p}$. Then
denoting by $\mathcal{R}_{g}^{3}$ the moduli space of unramified $3:1$ curve
covers $\tilde{C}\to C$ with $g_{C}=g$, the Prym map
$\mathcal{P}:\,\mathcal{R}_{g}^{3}\to\mathcal{B}_{g-1,g-1}$ defined by888Note
that via
$pr:=2\text{id}-\alpha_{*}-\alpha_{*}^{2}:\,J(\tilde{C})\twoheadrightarrow A$,
we may view $A$ as the quotient $J(\tilde{C})/J(C)$.
$A:=\ker\left(J(\tilde{C})\twoheadrightarrow J(C)\right)^{\circ}$ is dominant
for $g=3,4$ [F]. This was used by Schoen [Sc] to prove the Hodge Conjecture
for the specific Weil 4- and 6-folds parametrized by $\mathcal{B}_{2,2}$ and
$\mathcal{B}_{3,3}$.

Let $\tilde{C}^{+}$ [resp. $\tilde{C}^{-}$] denote the Abel image of
$\tilde{C}$ in $J(\tilde{C})$ [resp. its image under $-\text{id}$], defined up
to algebraic equivalence, and write
$\mathcal{O}\subseteq\mathcal{O}_{\mathbb{Q}(\sqrt{-3})}$ for an order. We now
demonstrate that in the $g_{C}=4$ ($g_{\tilde{C}}=10$) case, the _Prym-Ceresa
cycle_

$Z:={pr}_{*}\left(\tilde{C}^{+}-\tilde{C}^{-}\right)\in{Griff}^{5}(A)_{\mathbb{Q}}$

generates a nontrivial reduced normal function on
$\mathcal{R}_{4}^{3}(\mathcal{O})$ (over
$\mathcal{B}_{3,3}(\mathcal{O})$).999moduli spaces with $\mathcal{O}$-level
structure (This is in contrast to the more standard 2:1 Prym setting, where
such cycles are algebraically equivalent to zero.) This is accomplished by
degenerating $C$ to a union $D\cup E$ ($D$ of genus $3$, $E$ elliptic, meeting
at a single node), so that the cover $\tilde{C}$ becomes $(D_{1}\sqcup
D_{2}\sqcup D_{3})\cup\tilde{E}$ (each $D_{i}\cong D$ meeting $\tilde{E}\simeq
E$ once). The cyclic automorphism $\alpha$ permutes the $\\{D_{i}\\}$, and the
specialized Prym variety $A$ is the quotient of
$J(E)\oplus\bigoplus_{i=1}^{3}J(D_{i})$ by the invariant part $J(E)\oplus
J(D)_{\Delta}$ ($\Delta=$ diagonal). We have $H^{1}(A)\cong
V^{\omega_{1}}\oplus V^{\omega_{5}}$, and $H^{3}(A)^{\alpha}\cong
V^{\omega_{3}}\oplus V^{\omega_{3}}$. Writing
$\\{\omega_{i}^{j}\\}_{j=1,2,3}\subset\Omega^{1}(J(D_{i}))$ for bases, the
invariant form

$\omega:=(\omega_{1}^{1}+\zeta_{3}\omega_{2}^{1}+\bar{\zeta}_{3}\omega_{3}^{1})\wedge(\omega_{1}^{2}+\zeta_{3}\omega_{2}^{2}+\bar{\zeta}_{2}\omega_{3}^{2})\wedge(\omega_{1}^{3}+\zeta_{3}\omega_{2}^{3}+\bar{\zeta}_{3}\omega_{3}^{3})$

is the pullback of a class from $H^{3,0}(A)^{\alpha}$. The projection of the
Ceresa cycle $D_{1}^{+}-D_{1}^{-}=\partial\Gamma$ (for $D_{1}$) to $A$ has
$\int_{\Gamma}\omega=\int_{\Gamma}\omega_{1}^{1}\wedge\omega_{1}^{2}\wedge\omega_{1}^{3}$
generically nonzero (otherwise $\overline{\mathit{AJ}}$ would not detect the
usual generic Ceresa cycle). But the degeneration of the Prym-Ceresa cycle is
the sum of the projections of the $D_{i}^{+}-D_{i}^{-}$ to $A$, and (by
invariance under $\alpha$) each has the same nonzero image under the
composition of $\overline{\mathit{AJ}}^{5}$ with the projection
$J\left(H^{3}(A)^{\vee}\right)\twoheadrightarrow
J\left((H^{3}(A)^{\alpha})^{\vee}\right)$. So we have the desired geometric
realization of $(\mathrm{I}_{3,3},\omega_{3})$.

Applying the (algebraic) Lefschetz operator to $Z$, we obtain cycles (over
$\mathcal{R}_{4}^{3}(\mathcal{O})$) $Z^{k}\in{Griff}^{k}(A)_{\mathbb{Q}}$ for
$k=2,3,4,5$ with nontorsion $\overline{AJ}^{k}$ image. By [Ra] (see $\S 1$
above), $\mathcal{R}_{4}^{3}(\mathcal{O})$101010More precisely, the minimal
quotient of $\mathcal{R}_{4}^{3}(\mathcal{O})$ over
$\mathcal{B}_{3,3}(\mathcal{O})$ which supports $Z$ (modulo algebraic
equivalence). cannot be an $\mathrm{I}_{3,3}(\Gamma)$ ($\Gamma$ arithmetic),
and so has a nonempty branch locus over $\mathcal{B}_{3,3}(\mathcal{O})$.
Moreover, the push-forward of $\overline{AJ}^{k}(Z^{k})$ to
$\mathcal{B}_{3,3}(\mathcal{O})$ must be zero, so that the Galois group of the
cover acts nontrivially. Now the last paragraph of [No2] applies, with the
obvious substitutions: we push forward $Z^{k}$ along the isogenies of the
universal Weil $6$-fold over the automorphisms of $\mathrm{I}_{3,3}$ induced
by $G(\mathbb{Q})$ (regarding this as taking place in the inverse limit over
level structures $\mathcal{O}$). Since the action of $G(\mathbb{Q})$ produces
infinitely many translates of the original base locus, the absolute Galois
group of
$\underset{\mathcal{O}}{\underrightarrow{\lim}}\mathbb{C}(\mathcal{B}_{3,3}(\mathcal{O}))$
acts through infinitely many distinct representations upon the push-forwards
of $Z^{k}$. So we have the

######  Theorem 3.6.

Let $A$ be a very general Weil abelian 6-fold in the family parametrized by
$\mathcal{B}_{3,3}$. Then the groups ${Griff}^{k}(A)_{\mathbb{Q}}$, and their
$\overline{AJ}^{k}$-images, are countably infinite-dimensional for
$k=2,3,4,5$.

###  3.6. Cases $\mathrm{IV}_{2n-2}$ and $\mathrm{IV}_{2n-1}$

Beginning with $\mathrm{IV}_{2n-2}$ (the $D_{n}$ case), we note that $n\geq
4$, ${\tt I}=1$, $\Sigma$ and $\Omega$ are as in $\S 3.4$, and

${\tt E}(\lambda)\left(={\tt
E}(\tau(\lambda))\right)=-2(m_{1}+\cdots+m_{n-2})-(m_{n-1}+m_{n}).$

$V^{\lambda}$ is complex iff $n$ is odd and $m_{n-1}\neq m_{n}$, and
quaternionic iff $4|n$ and $m_{n-1}+m_{n}$ is odd. So the level-one real
Hermitian VHS are ($\otimes\mathbb{C}$)

$\begin{array}[]{ccc}(n\text{
odd})&\tilde{\mathcal{V}}^{\omega_{n}}=\mathcal{V}^{\omega_{n}}\oplus\mathcal{V}^{\omega_{n-1}}\\\
(4|n)&\tilde{\mathcal{V}}^{\omega_{n}}=\left(\mathcal{V}^{\omega_{{}_{n}}}\right)^{\oplus
2},&\tilde{\mathcal{V}}^{\omega_{n-1}}=\left(\mathcal{V}^{\omega_{n-1}}\right)^{\oplus
2}\\\
(4|n+2)&\tilde{\mathcal{V}}^{\omega_{n}}=\mathcal{V}^{\omega_{n}},&\tilde{\mathcal{V}}^{\omega_{n-1}}=\mathcal{V}^{\omega_{n-1}}.\end{array}$

From ${\tt s}(\omega_{1})=\omega_{2}-\omega_{1}$, we define

$\mu(\lambda)=\tfrac{1}{2}-m_{2}-\cdots-
m_{n-2}-\tfrac{1}{2}m_{n-1}-\tfrac{1}{2}m_{n},$

so that $-{\tt E}(\lambda)>1$ odd and $\mu(\lambda)\geq 0$ $\implies$

$\lambda=a\omega_{1}+\omega_{n-1}\;\text{or}\;a\omega_{1}+\omega_{n}\;\;(a>0),$

which works for arbitrary $n$!

A similar (but simpler) story unfolds in the $B_{n}$ case
$\mathrm{IV}_{2n-1}$, where (with $n\geq 3$, ${\tt I}=1$)

$\Sigma=\\{e_{1}-e_{2},\ldots,e_{n-1}-e_{n},e_{n}\\},$
$\Omega=\\{e_{1},e_{1}+e_{2},\ldots,e_{1}+\cdots+e_{n-1},\tfrac{1}{2}(e_{1}+\cdots+e_{n})\\},$
${\tt E}(\lambda)=-2(m_{1}+\cdots+m_{n-1})-m_{n},$

and (with no complex irreps, as $\tau=\text{id}$) the only quaternionic irreps
are the $V^{(2k+1)\omega_{n}}$ for $\left\lfloor\tfrac{n-1}{2}\right\rfloor$
odd. The level one Hermitian $\mathbb{R}$-VHS are thus ($\otimes\mathbb{C}$)
the

$\tilde{\mathcal{V}}^{\omega_{n}}=\left(\tilde{\mathcal{V}}^{\omega_{n}}\right)^{\oplus
2}\;(n=3,4;7,8;\ldots)\;\;\text{or}\;\;\mathcal{V}^{\omega_{n}}\;(\text{otherwise}).$

We have ${\tt s}(\omega_{1})=\omega_{2}-\omega_{1}$, hence

$\mu(\lambda)=\tfrac{1}{2}-m_{2}-\cdots-m_{n-1}-\tfrac{1}{2}m_{n},$

so that the variations associated to

$\lambda=a\omega_{1}+\omega_{n}\;\;(a>0)$

are (for any $n$) the ones with infinitesimal reduced normal functions.

######  Definition 3.7.

A universal spin abelian variety is an abelian family
$\mathscr{A}\to\Gamma\backslash\mathrm{IV}_{m}$ with $H^{1}$ (over
$\mathbb{R}$) a number of copies of
$\tilde{\mathcal{V}}_{\mathbb{R}}^{\omega_{n-1}}$ and
$\tilde{\mathcal{V}}_{\mathbb{R}}^{\omega_{n}}$ ($m=2n-2$ even) resp.
$\tilde{\mathcal{V}}_{\mathbb{R}}^{\omega_{n}}$ ($m=2n-1$ odd).

The minimum possible (relative) dimension of $\mathscr{A}$ is clearly $2^{n}$
($m$ and $\left\lfloor\tfrac{n-1}{2}\right\rfloor$ odd), $2^{n-1}$ ($m$ and
$\left\lfloor\tfrac{n+1}{2}\right\rfloor$ odd; $m$ even and $4\nmid n+2$), or
$2^{n-2}$ ($m$ even and $4|n+2$). However, the main natural source of spin
abelian varieties is the Kuga-Satake construction (cf. [vG2]), which produces
varieties of much higher dimension: for $m\leq 19$, one has families of $K3$
surfaces $\mathscr{X}$ with
$H_{tr}^{2}(\mathscr{X})\cong\mathcal{V}^{\omega_{1}}$, and Clifford algebras
produce an embedding $H_{tr}^{2}(\mathscr{X})\hookrightarrow
H^{1}(\mathscr{A})^{\oplus 2}$ with

$H^{1}(\mathscr{A})\cong\left\\{\begin{array}[]{cc}\left(\mathcal{V}^{\omega_{n-1}}\right)^{\oplus
2^{n-1}}\oplus\left(\mathcal{V}^{\omega_{n}}\right)^{\oplus 2^{n-1}}&(m\text{
even})\\\ \left(\mathcal{V}^{\omega_{n}}\right)^{\oplus 2^{n}}&(m\text{
odd})\end{array}\right..$

So the situation is in marked contrast to those encountered above:111111We
haven’t ruled out that for some “minimal” spin abelian varieties one might
have a vanishing result analogous to the above ones, except for
$\mathrm{IV}_{m}$ with $m=7,8,9;13,15;\text{etc.}$

######  Proposition 3.8.

The relative cohomology of any Kuga-Satake family of spin abelian varieties
over $\mathrm{IV}_{m}$ (any $m\geq 7$) admits infinitesimal (reduced) normal
functions.

###### Proof.

We only need to show that (say) $\mathcal{V}^{\omega_{1}+\omega_{n}}$ occurs
in $H^{3}(\mathscr{A})=\bigwedge^{3}H^{1}(\mathscr{A})$. This is done by
considering the decomposition of $(V^{\omega_{n}})^{\otimes 2}$ or
$V^{\omega_{n}}\otimes V^{\omega_{n-1}}$; e.g. for $m$ odd,

$\left(V^{\omega_{n}}\right)^{\otimes 2}\cong V^{2\omega_{n}}\oplus
V^{\omega_{n-1}}\oplus\cdots\oplus V^{\omega_{1}}\oplus\mathbf{1}$
$\implies\;\left(V^{\omega_{n}}\right)^{\otimes 3}\supseteq
V^{\omega_{1}}\otimes V^{\omega_{n}}\supseteq V^{\omega_{1}+\omega_{n}}.$

(There are enough copies of $\mathcal{V}^{\omega_{n}}$ that we can consider
tensor rather than wedge powers.) ∎

Finally, we remark that the triality isomorphism for $D_{4}$ exhibits the
universal quaternionic abelian $8$-folds as “minimal” spin $8$-folds, by
equating $\tilde{\mathcal{V}}_{\mathbb{R}}^{\omega_{4}}\to\mathrm{IV}_{6}$ and
$\tilde{\mathcal{V}}_{\mathbb{R}}^{\omega_{1}}\to\mathrm{II}_{4}$.
Specialization under $\mathrm{IV}_{5}\hookrightarrow\mathrm{IV}_{6}$ shrinks
the Mumford-Tate group to $\mathit{Spin}(2,5)$ and gives geometric
realizations of
$\tilde{\mathcal{V}}_{\mathbb{R}}^{\omega_{3}}\to\mathrm{IV}_{5}$ (as noticed
by [vGV]). All these cases admit infinitesimal normal functions.

##  4\. Half-twists and non-tube cases

In this section we consider a slight generalization of the homogeneous
variations described in $\S 2.3$, by enlarging our simple $G_{\mathbb{C}}$ to
$\tilde{G}_{\mathbb{C}}=\mathbb{G}_{m}\cdot G_{\mathbb{C}}$ (or
$G_{\mathbb{R}}$ to $U(1)\cdot G_{\mathbb{R}}$) and taking $\tilde{{\tt E}}$
to have a component in the abelian factor of $\tilde{\mathfrak{g}}$. This
allows us to shift the Hodge grading of a complex summand (e.g. to make it
integral), which is necessary in order to study the cohomology of abelian
varieties of generalized Weil type and to obtain all Calabi-Yau variations. We
restrict our investigation to these examples and proceed with a minimum of
formality.

Given an irrep $V^{\lambda}$ of $\mathfrak{g}$ and ${\tt E}\in\mathfrak{t}$ as
before, we take $\tilde{{\tt E}}=({\tt
E},1)\in\mathfrak{g}\oplus\mathbb{C}=\tilde{\mathfrak{g}}$, and define
representations of $\tilde{\mathfrak{g}}$ by

$V^{\lambda}\left\\{\tfrac{a}{2}\right\\}^{p,-p-1}:=\left(V^{\lambda}\right)^{p+\frac{a}{2},-p-\frac{a}{2}-1}.$

We write

(4.1)
$\tilde{V^{\lambda}}\left\\{\tfrac{a}{2}\right\\}:=V^{\lambda}\left\\{\tfrac{a}{2}\right\\}\oplus
V^{\tau(\lambda)}\left\\{\tfrac{-a}{2}\right\\},$

whether or not $V^{\lambda}$ or $V^{\lambda}\left\\{\tfrac{a}{2}\right\\}$ is
complex; this is a variant of van Geemen’s half-twist [vG1] that preserves the
weight. As a real Hodge structure, (4.1) has level

(4.2)
$\ell\left(\tilde{V}^{\lambda}\left\\{\tfrac{a}{2}\right\\}\right)=\text{max}\left\\{-{\tt{E}}(\lambda)+a,-{\tt{E}}(\tau(\lambda))-a\right\\}.$

When (4.2) is odd, define

(4.3)
$\mu(\lambda,a):=\tfrac{1}{2}\left\\{{\tt{E}}({\tt{s}}\cdot\lambda)-a-1\right\\}.$

######  Proposition 4.1.

Assume (4.2) is odd. If either $\mu(\lambda,a)$ or $\mu(\tau(\lambda),-a)$ is
$\geq 0$, then the corresponding weight-$(-1)$ $\mathbb{R}$-VHS
$\tilde{\mathcal{V}}_{\mathbb{R}}^{\lambda}\left\\{\tfrac{a}{2}\right\\}$ over
$\mathrm{X}=\Gamma\backslash D$ has a nonzero $\mathcal{H}^{1}(j)$ ($j\geq
0$). (As above, we shall say it has an infinitesimal normal function.) If both
invariants are negative, any underlying $\mathbb{Q}$-VHS admits no nontrivial
reduced normal function on an étale neighborhood of $\mathrm{X}$.

###  4.1. Case $\mathrm{I}_{p,n-p+1}$ (bis)

As before, we assume $p\leq\left\lfloor\tfrac{n+1}{2}\right\rfloor$. Referring
to (3.1), the only choices that make (2.10) equal $1$ are $\lambda=\omega_{1}$
or $\omega_{n}$, unless $p=1$ (in which case all $\omega_{i}$ work). We shall
focus attention on the variations arising in the odd exterior powers of

(4.4)
$\mathcal{H}_{p,n-p+1}:=\tilde{\mathcal{V}}^{\omega_{1}}_{\mathbb{R}}\left\\{\tfrac{2p-n-1}{2(n+1)}\right\\}\;\;(\text{i.e.
}a=\tfrac{2p-n-1}{n+1})$

over $\mathrm{X}=\Gamma\backslash\mathrm{I}_{p,n-p+1}$. Notice that (4.4) has
level one, with
$\underline{h}(\mathcal{V}^{\omega_{1}}\left\\{\tfrac{a}{2}\right\\})=(p,n-p+1)$
and
$\underline{h}(\mathcal{V}^{\omega_{n}}\left\\{\tfrac{-a}{2}\right\\})=(n-p+1,p)$
for its two complex summands. (These are both the Hodge numbers and the
signatures of the relevant Hermitian forms.) Our focus on these cases is
motivated by the

######  Definition 4.2.

(cf. [Iz]) A _universal $k$-Weil abelian $(n+1)$-fold_ is an abelian variety
$\mathscr{A}\to\mathrm{X}$ whose $H^{1}$ recovers $\mathcal{H}_{p,n-p+1}$ with
$n-p+1=p+k$.

As in the ($0$-)Weil case, $\mathit{End}(\mathscr{A})_{\mathbb{Q}}$ contains
an imaginary quadratic field; but the generic Mumford-Tate group is $U(p,p+k)$
instead of $\mathit{SU}(p,p)$. For the $H_{\mathit{prim}}^{2d+1}$ of
$\mathscr{A}$, an easy argument with Young diagrams shows that121212where
$\omega_{0}+\omega_{n-2d}$ [resp. $\omega_{2d+1}+\omega_{0}$] means
$\omega_{n-2d}$ [resp. $\omega_{2d+1}$]

(4.5)
$\frac{\bigwedge^{2d+1}\mathcal{H}_{p,n-p+1}}{\bigwedge^{2d-1}\mathcal{H}_{p,d-p+1}}\cong\bigoplus_{j=0}^{2d+1}\mathcal{V}^{\omega_{j}+\omega_{n-2d+j}}\left\\{\tfrac{(n-2p+1)(2d-2j+1)}{2(n+1)}\right\\}$

for $1\leq d\leq\tfrac{n}{2}$. There are no terms of level one in (4.5),
except at $j=0,2d+1$ when $2d+1=n+1$ in case $\mathrm{I}_{p,p+1}$ and when
$2d+1=n$ in case $\mathrm{I}_{p,p}$. For all the other terms on the right-hand
side of (4.5), we compute using (3.1)

(4.6)
$\mu\left(\omega_{j}+\omega_{n-2d+j},\tfrac{(n-2p+1)(2n-2j+1)}{n+1}\right)=\\\
\left\\{\begin{array}[]{cc}j-d-p,&p<j\\\ 1-d,&p=j\text{ or }n-2d+j\\\
-d,&j<p<n-2d+j\\\ p+d-j-n,&n-2d+j<p\end{array}\right..$

Since $p+d-j-n\geq 0$ (with $d\leq\tfrac{n}{2}$, $p\leq\tfrac{n+1}{2}$) boils
down to the level one cases we identified, the last two entries can be
ignored. We have $1-d\geq 0$ and $p=j$ or $n-2d+j$ in the cases $(d,p)=(1,1)$,
$(1,2)$, $(1,3)$; and $j-d-p\geq 0$ for some $j$ $\implies$ $2d+1\geq 2p-1$.
Hence we conclude that the variations of level $>1$ in $H^{\mathit{odd}}$ of
$\mathscr{A}$ admitting infinitesimal normal functions therefore lie in

$\left\\{\begin{array}[]{cc}\text{all possible degrees }(3\text{ thru
}2n-1),&\text{if }p\leq 3\\\ \text{degrees }2p-1\text{ thru }2n-2p+3,&\text{if
}3<p<\tfrac{n+1}{2}\\\ \text{no degrees,}&\text{if
}3<p=\tfrac{n+1}{2}\end{array}\right..$

Note that $p=\tfrac{n+1}{2}$ ($k=0$) is the Weil abelian setting dispensed
with in $\S 3.5$. Omitting this case, we can restate our conclusion as
follows:

######  Theorem 4.3.

For $k\leq n-7$, no étale pullback of a universal $k$-Weil abelian
$(n+1)$-fold admits reduced normal functions arising from cycles of dimension
less than $\tfrac{n-k-1}{2}$ or codimension less than $\tfrac{n-k+1}{2}$.

As to why (or whether) greater disparity in the signature of the Hermitian
form should lead to more $\overline{\mathit{AJ}}$-nontrivial cycles in the
Griffiths group, we can say nothing yet.

###  4.2. Comments on the Hermitian Calabi-Yau VHS

For the tube domain cases, the CY variations are nothing but the
$\mathcal{V}_{\mathbb{R}}^{k\omega_{{\tt I}}}$ (described for $k=1$ by Gross
[Gro]). In the remaining cases (after [SZ, FL]), where $V^{k\omega_{{\tt I}}}$
is complex, we choose the shift in $\tilde{\mathcal{V}}^{k\omega_{{\tt
I}}}\left\\{\tfrac{a}{2}\right\\}$ to minimize the (odd) level while having
the CY property.

Writing $\epsilon(m)=1$ ($m$ even) resp. $2$ ($m$ odd), to accomplish this we
need (cf. (2.10),(4.2))

$\ell\left(\tilde{V}^{k\omega_{{\tt
I}}}\left\\{\tfrac{a}{2}\right\\}\right)=\ell\left(V^{k\omega_{{\tt
I}}}\right)+\epsilon\left(\ell\left(V^{k\omega_{{\tt
I}}}\right)\right)\;\;\implies$ $a=\tfrac{1}{2}\left\\{{\tt E}(k\omega_{{\tt
I}})-{\tt E}(k\tau(\omega_{{\tt
I}}))\right\\}+\epsilon\left(\tfrac{1}{2}\left\\{{\tt E}(k\omega_{{\tt
I}})+{\tt E}(k\tau(\omega_{{\tt I}}))\right\\}\right).$

This yields $a=\tfrac{2kp^{2}}{n+1}-kp+\epsilon(kp)$ (for
$\mathrm{I}_{p,n-p+1(\neq p)}$), $a=-\tfrac{k}{2}+\epsilon(mk)$ (for
$\mathrm{II}_{2m+1}$), and $a=-\tfrac{2k}{3}+1$ (for
$\mathrm{EIII}$).131313For $k=1$, this gives the complex examples in [FL],
where we note that the sign on our half-twist is the opposite of theirs. Note
that [op. cit.] omits the $(p,k)=(1,2)$ cases
$\mathrm{I}_{1,n}\left(A_{n},\sigma_{1};2\omega_{1}\right)\left\\{\tfrac{3-n}{2(n+1)}\right\\}$,
which also have level 3. Plugging into (4.3) gives

$\mu(k\omega_{{\tt
I}},a)=k-\left\lfloor\tfrac{kp+1}{2}\right\rfloor,\;1-\left\lfloor\tfrac{mk+1}{2}\right\rfloor,\;\text{resp.
}1-k,$

which is non-negative when $p=1$ or $2$ ($\mathrm{I}_{p,n-p+1}$),
$(m,k)=(2,1)$ ($\mathrm{II}_{2m+1}$), resp. $k=1$ ($\mathrm{EIII}$). This
includes all the level 3 cases, but also two infinite series of examples
(since $k$ is arbitrary) for $\mathrm{I}_{1,n}$ and $\mathrm{I}_{2,n-1}$.
Putting this together with the results in $\S 3$, we have the

######  Proposition 4.4.

Amongst the minimal-level Calabi-Yau variations over irreducible Hermitian
symmetric domains other than $\mathrm{I}_{1,n}$ or $\mathrm{I}_{2,n-1}$, only
those of level three admit infinitesimal normal functions.

## References

  * [Ce] G. Ceresa,_ $C$ is not equivalent to $C^{-}$ in its Jacobian_, Ann. of Math. (2) 117 (1983), no. 2, 285-291.
  * [CNP] A. Collino, J. Naranjo, and G. Pirola, _The Fano normal function_ , J. Math. Pures Appl. (9) 98 (2012), no. 3, 346-366.
  * [F] C. Faber, _Prym varieties of triple cyclic covers_ , Math. Z. 199 (1988), 61-78.
  * [Fa] N. Fakhruddin, _Algebraic cycles on generic abelian varieties_ , Compositio Math. 100 (1996), no. 1, 101-119.
  * [FL] R. Friedman and R. Laza, _Semialgebraic horizontal subvarieties of Calabi-Yau type_ , Duke Math. J. 162 (2013), no. 12, 2077-2148.
  * [FH] W. Fulton and J. Harris, “Representation Theory: A First Course”, Springer-Verlag, New York, 1991.
  * [vG1] B. van Geemen, _Half-twists of Hodge structures of CM-type_ , J. Math. Soc. Japan 53 (2001), no. 4, 813-833.
  * [vG2] ———, _Kuga-Satake varieties and the Hodge conjecture_ , in “The arithmetic and geometry of algebraic cycles (Banff, AB, 1998),”, 51-82, NATO Sci. Ser. C Math. Phys. Sci. 548, Kluwer Acad. Publ., Dordrecht, 2000.
  * [vGV] B. van Geemen and A. Verra, _Quaternionic Pryms and Hodge classes_ , Topology 42 (2003), no. 1, 35-53.
  * [GGK] M. Green, P. Griffiths and M. Kerr, “Mumford-Tate groups and domains: their geometry and arithmetic”, Annals of Math. Studies 183, Princeton Univ. Press, Princeton, NJ, 2012.
  * [Gre] M. Green, _Griffiths’s infinitesimal invariant and the Abel-Jacobi map_ , J. Differ. Geom. 29 (1989), 545-555.
  * [GG] M. Green and P. Griffiths, _Algebraic cycles and singularities of normal functions_ , in “Algebraic cycles and motives (Vol. I)”, 206-263, London Math. Soc. Lect. Not. Ser. 343, Cambridge Univ. Press, Cambridge, 2007.
  * [Gro] B. Gross, _A remark on tube domains_ , Math. Res. Lett. 1 (1994), no. 1, 1-9.
  * [Ha] R. Hain, _Torelli groups and geometry of moduli spaces of curves_ , in “Current topics in complex algebraic geometry (Clemens and Kollar, eds.)”, 97-143, MSRI Publications, No. 28, Cambridge Univ. Press, 1995.
  * [Iz] E. Izadi, _Some remarks on the Hodge conjecture for abelian varieties_ , Ann. Mat. Pura Appl. (4) 189 (2010), no. 3, 487-495.
  * [KP] M. Kerr and G. Pearlstein,_An exponential history of functions with logarithmic growth_ , in “Topology of stratified spaces”, 281-374, Math. Sci. Res. Inst. Publ. 58, Cambridge Univ. Press, 2011.
  * [Kn] A. Knapp, “Lie groups: beyond an introduction (2nd ed.)”, Progress in Math. 140, Birkhäuser, Boston, 2002.
  * [Ko] B. Kostant, _Lie algebra cohomology and the generalized Borel-Weil theorem_ , Ann. of Math. (2) 74 (1961), 329-387.
  * [LZ] R. Laza and Z. Zhang, _Classical period domains_ , to appear in “Recent Advances in Hodge Theory (Kerr and Pearlstein, eds.)”, Cambridge Univ. Press, 2015.
  * [MO] B. Moonen and F. Oort, _The Torelli locus and special subvarieties_ , in “Handbook of moduli (vol. II)”, 549-594, Adv. Lect. Math. (ALM) 25, Intl. Press, Somerville, MA, 2013.
  * [No1] M. Nori, _Algebraic cycles and Hodge-theoretic connectivity_ , Invent. Math. 111(1993), no. 2, 349-373.
  * [No2] ———, _Cycles on the generic abelian threefold_ , Proc. Indian Acad. Sci. (Math. Sci.), Vol. 99, no. 3, Dec. 1989, pp. 191-196.
  * [Ra] M. Raghunathan, _Cohomology of arithmetic subgroups of algebraic groups: I_ , Ann. of Math. 86 (1967), 409-424.
  * [Ro] C. Robles, _Schubert varieties as variations of Hodge structure_ , Selecta Math. (N.S.) 20 (2014), no. 3, 719-768.
  * [R] C. Rohde, “Cyclic coverings, Calabi-Yau manifolds, and complex multiplication”, Lect. Notes Math. 1975, Springer-Verlag, New York, 2009.
  * [S] W. Schmid, _Variation of Hodge structure: the singularties of the period mapping_ , Invent. Math. 22 (1973), 211-319.
  * [Sc] C. Schoen, _Hodge classes on self-products of a variety with an automorphism_ , Compositio Math. 65 (1988), no. 1, 3-32; Addendum, Compositio Math. 114 (1998), no. 3, 329-336.
  * [SZ] M. Sheng and K. Zuo, _Polarized variations of Hodge structures of Calabi-Yau type and characteristic subvarieties over bounded symmetric domains_ , Math. Ann. 348 (2010), no. 1, 211-236.