# Fourfolds of Weil type and the spinor map

Bert van Geemen  Dipartimento di Matematica, Università di Milano, Via Saldini
50, I-20133 Milano, Italia<EMAIL_ADDRESS>
###### Abstract.

Recent papers by Markman and O’Grady give, besides their main results on the
Hodge conjecture and on hyperkähler varieties, surprising and explicit
descriptions of families of abelian fourfolds of Weil type with trivial
discriminant. They also provide a new perspective on the well-known fact that
these abelian varieties are Kuga Satake varieties for certain weight two Hodge
structures of rank six.

In this paper we give a pedestrian introduction to these results. The spinor
map, which is defined using a half-spin representation of $SO(8)$, is used
intensively. For simplicity, we use basic representation theory and we avoid
the use of triality.

## Introduction

The recent papers [Mar], [O’G] by Markman and O’Grady provide new descriptions
of families of abelian fourfolds of Weil type. Markman uses these to prove
that certain Hodge classes on these fourfolds are algebraic. Both show that
these abelian varieties are isogeneous to the intermediate Jacobians of
algebraic hyperkähler varieties of Kummer type. O’Grady further relates this
to the Kuga Satake construction for the (primitive) second cohomology group of
algebraic Kummer type varieties. See also [V] for further developments.

An abelian fourfold of Weil type has an imaginary quadratic field
$K={\mathbb{Q}}(\sqrt{-d})$ in its endomorphism algebra. These fourfolds
define two subspaces of the complexification of their first homology group
$H_{1}$, a free ${\mathbb{Z}}$-module of rank $8$. They are the
$+i$-eigenspace of the complex structure on $H_{1}\otimes{\mathbb{R}}$ defined
by $A$ and one of the two eigenspaces of the $K$-action. Markman obtains the
polarization on the abelian fourfold, an alternating form on $H_{1}$, from a
symmetric(!) form on $H_{1}$ and the $K$-action. The two subspaces turn out to
be maximally isotropic subspaces for this symmetric form.

In this paper we will mainly follow Markman’s approach. He considers a free,
rank $8$, ${\mathbb{Z}}$-module $V$ equipped with a bilinear form. This $V$
will become the first cohomology group of the fourfolds of Weil type. The
maximally isotropic subspaces of the complexification $V_{\mathbb{C}}$ of $V$
are well-known to be parametrized by two copies of a Legendrian Grassmannian,
a complex manifold of dimension six. The spinor map is a natural embedding of
this Grassmannian in ${\bf P}^{7}$, the image is a quadric hypersurface
$Q^{+}$. This map already made several appearances in algebraic geometry, for
example in the study of vector bundles over hyperelliptic curves in [vG1], of
K3 surfaces in [Muk], of secant varieties in [Man] and of integrable systems
[BHH].

The spinor map is best constructed using the representation theory of
$Spin(V)$, a double cover of the orthogonal group $SO(V)$ defined by the
bilinear form on $V$. The spin group has a half-spin representation whose
projectivization is ${\bf P}^{7}$. The spinor map is equivariant for the
action of $Spin(V)$. A natural integral structure on the half-spin
representation allows one to identify it with the complexification of a free
${\mathbb{Z}}$ module $S^{+}$ of rank $8$. There is a non-degenerate bilinear
form on $S^{+}$ which defines the quadric $Q^{+}$.

An analytic open subset $\Omega\subset Q^{+}$ parametrizes complex structures
on $V_{\mathbb{R}}$ that preserve the bilinear form on $V$. Fixing a general
element $s\in S^{+}$ and considering only the complex structures on
$V_{\mathbb{R}}$ corresponding to $\ell\in\Omega\cap s^{\perp}$ produces a
five dimensional family of complex tori ${\mathcal{T}}_{\ell}$, not algebraic
in general, that have a Hodge class, called the Cayley class,

$c_{s}\,\in\,H^{2,2}({\mathcal{T}}_{\ell},{\mathbb{Z}})\,=\,H^{4}({\mathcal{T}}_{\ell},{\mathbb{Z}})\cap
H^{2,2}({\mathcal{T}}_{\ell}).$

The idea of using these tori and the associated action of
$Spin(7)=Spin(s^{\perp})$ to study the Hodge conjecture for fourfolds of Weil
type is due to V. Muñoz [Mun]. In §3.3 we observe that the existence of the
Cayley classes can be deduced from a relation between the spinor and the
Plücker map. Using representation theory we then compute the class $c_{s}$ for
certain $s$ that are relevant for Markman’s results in Proposition 6.14.

For any $h\in S^{+}$ such that the sublattice $\langle h,s\rangle$ of $S^{+}$
spanned by $h$ and $s$ has rank two and is positive definite, the tori
parametrized by the four dimensional domain $\Omega\cap\langle
h,s\rangle^{\perp}$ turn out to be abelian fourfolds of Weil type. The
imaginary quadratic field $K$ depends on $h,s$, but fixing $s$ and choosing
$h$ suitably, any such field occurs. The polarization is determined by $K$ and
the bilinear form on $V$. A further discrete invariant, the discriminant of a
polarized abelian variety of Weil type, is always trivial for the fourfolds
constructed in this way. See Theorem 4.6 for these results of Markman and
O’Grady.

The Hodge conjecture for an abelian fourfold $A$ of Weil type is non-trivial.
There is a natural 2-dimensional subspace $W_{K}\subset
H^{2,2}(A,{\mathbb{Q}})$ of Hodge classes. It is not known in general if this
subspace is spanned by classes of algebraic cycles. If $c_{s}$ is algebraic,
then all classes in $W_{K}$ are also algebraic. Markman makes important
progress in the study of the Hodge conjecture by showing that $c_{s}$ is
algebraic for all abelian fourfolds appearing in his construction, which are
all fourfolds of Weil type with trivial discriminant. For this he uses
deformation theory of sheaves on hyperkähler manifolds, see §5 for a brief
outline.

Triality, an automorphism of order three of $Spin(V)$, allows one to relate
the standard representation of $Spin(V)$ (via $(SO(V)$ on $V$) and the two
half-spin representations, one of which is $S^{+}$. While it is prominent in
[Mar], we use instead an ‘ad hoc’ Lemma 6.9. It is of importance for instance
in the results on the Cayley class and for the Kuga Satake varieties.

We limit ourselves to a basic exposition of the constructions of Markman and
O’Grady of the abelian fourfolds of Weil type with trivial discriminant and of
the Cayley classes of Muñoz and Markman. Some details of the representation
theory involved in the construction of the spinor map and the Cayley classes
can be found in the Appendix, §6. The relation with the Kuga Satake
construction is indicated in §6.15 \- 6.18.

### Acknowledgements

Discussions with E. Markman and K.G. O’Grady were very helpful.

##  1\. Tori with an orthogonal structure

###  1.1. The lattice $V$

The complex tori we consider are all quotients of a fixed real vector space,
with a varying complex structure, by a fixed lattice. Whereas one might expect
an alternating form, a polarization, on the first cohomology group to be
important, Markman instead fixes a symmetric, non-degenerate, bilinear form
$(\bullet,\bullet)_{V}$ on a rank eight free ${\mathbb{Z}}$-module $V$ of
signature $(4+,4-)$. He fixes a rank four free ${\mathbb{Z}}$-module $W$,
defines $W^{*}:={\operatorname{Hom}}_{\mathbb{Z}}(W,{\mathbb{Z}})$ and

$V\,:=\,W\,\oplus\,W^{*},\qquad\big{(}(w_{1},w_{1}^{*}),(w_{2},w_{2}^{*})\big{)}_{V}\,:=\,w_{1}^{*}(w_{2})\,+\,w_{2}^{*}(w_{1})~{}.$

If $e_{1},\ldots,e_{4}$ is a ${\mathbb{Z}}$-basis of $W$ and
$e_{i+4}:=e_{i}^{*}$, where $e_{1}^{*},\ldots,e_{4}^{*}$ is the dual basis of
$W^{*}$ so that $e_{i}^{*}(e_{j})=\delta_{ij}$ (Kronecker’s delta), then

$(v_{1},v_{2})_{V}\,:=\,\sum_{i=1}^{4}x_{i}y_{i+4}+x_{i+4}y_{i},\qquad\big{(}v_{1}:=\sum_{i=1}^{8}x_{i}e_{i},\quad
v_{2}:=\sum_{i=1}^{8}y_{i}e_{i}\;\in V\big{)}~{},$

hence $(V,(\bullet,\bullet)_{V})\cong U^{\oplus 4}$, the direct sum of four
copies of the hyperbolic plane
$U=({\mathbb{Z}}^{2},\small{\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}})$.

In [Mar] one finds $W:=H^{1}(X,{\mathbb{Z}})$ for an abelian surface $X$, but
for the basic properties of the complex tori this is not needed.

###  1.2. Complex structures on $V_{\mathbb{R}}$

Let $V_{\mathbb{R}}:=V\otimes_{\mathbb{Z}}{\mathbb{R}}$, it is an eight
dimensional vector space over the real numbers. A complex structure on
$V_{\mathbb{R}}$ is a linear map $J:V_{\mathbb{R}}\rightarrow V_{\mathbb{R}}$
with $J^{2}=-I$. Such a map has two (complex) eigenspaces $Z_{+},Z_{-}\subset
V_{\mathbb{C}}:=V\otimes_{\mathbb{Z}}{\mathbb{C}}$ corresponding to the
eigenvalues $i,-i\in{\mathbb{C}}$ of $J$. These eigenspaces are complex
conjugate, $\overline{Z_{+}}=Z_{-}$, where the complex conjugation on
$V_{\mathbb{C}}$ is defined as $\overline{v\otimes z}=v\otimes\bar{z}$ for
$v\in V$ and $z\in{\mathbb{C}}$.

$V_{\mathbb{C}}\,=\,Z_{+}\oplus Z_{-}=Z_{+}\oplus\overline{Z_{+}},\qquad
J\,=\,(i,-i)\,\in\,End(Z_{+})\oplus End(Z_{-})~{}.$

Conversely, given two complex conjugate subspaces $Z_{\pm}\subset
V_{\mathbb{C}}$ such that $V_{\mathbb{C}}=Z_{+}\oplus Z_{-}$ one can define a
linear map $\tilde{J}:V_{\mathbb{C}}\rightarrow V_{\mathbb{C}}$ by
$\tilde{J}(v_{+}+v_{-})=iv_{+}-iv_{-}$. Then there is a linear map
$J:V_{\mathbb{R}}\rightarrow V_{\mathbb{R}}$ whose ${\mathbb{C}}$-linear
extension to $V_{\mathbb{C}}$ is $\tilde{J}$. In fact, the inclusion
$V_{\mathbb{R}}\hookrightarrow V_{\mathbb{C}}$ identifies $V_{\mathbb{R}}$
with the $(v_{+},v_{-})\in Z_{+}\oplus Z_{-}$ with $\overline{v_{+}}=v_{-}$.
Writing $v\in V_{\mathbb{R}}$ as $v=v_{+}+v_{-}$, with
$v_{-}=\overline{v_{+}}$, one has $\tilde{J}v=iv_{+}+\overline{iv_{+}}\in
V_{\mathbb{R}}$, so $J$ is just the restriction of $\tilde{J}$ to
$V_{\mathbb{R}}$.

###  1.3. Orthogonal complex structures and isotropic subspaces

The ${\mathbb{R}}$-bilinear extension of $(\bullet,\bullet)_{V}$ defines a
bilinear form on $V_{\mathbb{R}}$, denoted by the same symbol. We consider now
the complex structures $J$ that preserve this bilinear form, so
$(Jv_{1},Jv_{2})_{V}=(v_{1},v_{2})_{V}$ for all $v_{1},v_{2}\in
V_{\mathbb{R}}$. Equivalently, $J\in SO(V_{\mathbb{R}},(\bullet,\bullet)_{V})$
and we will call $J$ an orthogonal complex structure. Notice that for such a
complex structure $J$ and for eigenvectors $v_{1+},v_{2,+}\in Z_{+}$ we have,
for the ${\mathbb{C}}$-bilinear extension of the bilinear form,

$(v_{1+},v_{2+})_{V}\,=\,(Jv_{1+},Jv_{2+})_{V}\,=\,(iv_{1+},iv_{2+})_{V}\,=\,i^{2}(v_{1+},v_{2+})_{V}\,=\,-(v_{1+},v_{2+})_{V}~{}.$

Hence the restriction of $(\bullet,\bullet)_{V}$ to $Z_{+}$ is identically
zero. Thus $Z_{+}$ is an isotropic subspace of $V_{\mathbb{C}}$ (and since
$\dim Z_{+}=4=(1/2)\dim V_{\mathbb{C}}$ it is a maximally isotropic, or
Legendrian, subspace of $V_{{\mathbb{C}}}$). Similarly $Z_{-}$ is a maximally
isotropic subspace of $V_{\mathbb{C}}$ (and since the bilinear form is non-
degenerate it induces a duality $Z_{+}\cong Z_{-}^{*}$).

One easily verifies that, conversely, an isotropic subspace $Z_{+}\subset
V_{\mathbb{C}}$ such that $V_{\mathbb{C}}=V_{+}\oplus\overline{V_{+}}$ defines
a complex structure $J$ on $V_{\mathbb{R}}$ that preserves
$(\bullet,\bullet)_{V}$. We summarize this in the following lemma.

###  1.4. Lemma

There is a natural bijection between the following two sets:

  * •

the orthogonal complex structures $J\in
SO(V_{\mathbb{R}},(\bullet,\bullet)_{V})$ on $V_{\mathbb{R}}$,

  * •

the maximally isotropic subspaces $Z$ of $V_{\mathbb{C}}$ such that
$V_{\mathbb{C}}=Z\oplus\overline{Z}$.

##  2\. The Legendrian Grassmannian and the spinor map

### 2.1.

In this section we recall that a connected component
$IG(4,V_{\mathbb{C}})^{+}$ of the Grassmannian of maximally isotropic
subspaces of $V_{\mathbb{C}}$ is isomorphic to a smooth six dimensional
quadric $Q^{+}\subset{\bf P}S^{+}_{\mathbb{C}}\cong{\bf P}^{7}$, where
$(S^{+},(\bullet,\bullet)_{S^{+}})$ is a certain lattice of rank eight. This
isomorphism is induced by the spinor map, which is equivariant for the action
of the double cover $Spin(V)$ of $SO(V)$ on $V$ and $S^{+}$ respectively. We
refer to the Appendix §6 for more details.

###  2.2. The Grassmannian $IG(4,V_{\mathbb{C}})^{+}$

The (complex) four dimensional subspaces of $V_{\mathbb{C}}$ are parametrized
by the Grassmannian $G(4,V_{\mathbb{C}})$, which has dimension
$4\cdot(8-4)=16$. The maximally isotropic subspaces for
$(\bullet,\bullet)_{V}$ (which are also those for the associated quadratic
form) are parametrized by two (isomorphic, disjoint, connected) complex
submanifolds of dimension six of $G(4,V_{\mathbb{C}})$, denoted by
$IG(4,V_{\mathbb{C}})^{+}$ and $IG(4,V_{\mathbb{C}})^{-}$. (See [GH, Chapter
6] for linear subspaces of quadrics.) This generalizes the two rulings
(families of lines) on a smooth quadric $Q\cong{\bf P}^{1}\times{\bf P}^{1}$
in ${\bf P}^{3}$. We denote by $IG(4,V_{\mathbb{C}})^{+}$ the connected
component which contains the maximally isotropic subspace
$W_{\mathbb{C}}^{*}$. A complex maximally isotropic subspace $Z$ defines a
point $[Z]\in IG(4,V_{\mathbb{C}})^{+}$ if and only if the dimension of $Z\cap
W_{\mathbb{C}}^{*}$ is even. In particular, also $[W_{\mathbb{C}}]\in
IG(4,V_{\mathbb{C}})^{+}$.

We recall a local parametrization of $IG(4,V_{\mathbb{C}})^{+}$ by alternating
$4\times 4$ complex matrices. A basis of $W^{*}$ is given by the last four
basis vectors of $V$ in §1.1. Thus $W_{\mathbb{C}}^{*}$ is spanned by the
columns of the $8\times 4$ matrix $({}^{0}_{I})$. Slightly deforming
$W_{\mathbb{C}}$, we obtain another subspace spanned by the columns of an
$8\times 4$ matrix. Since $detI=1\neq 0$ we may assume that the lower $4\times
4$ submatrix is still invertible. Then we can find a basis of the same
subspace given by the columns of a matrix $({}^{B}_{I})$, the corresponding
subspace will be denoted by $Z_{B}$. Thus we found a Zariski open subset
$G(4,V_{\mathbb{C}})_{0}$ of $G(4,V_{\mathbb{C}})$ of dimension $4^{2}=16$
parametrized by $4\times 4$ complex matrices.

In general $Z_{B}$ will not be isotropic, but one easily verifies that

${(\bullet,\bullet)_{V}}_{|Z_{B}\times
Z_{B}}\,=\,0\quad\Longleftrightarrow\quad\begin{pmatrix}{}^{t}B&I\end{pmatrix}\begin{pmatrix}0&I\\\
I&0\end{pmatrix}\begin{pmatrix}B\\\
I\end{pmatrix}\,=\,0\quad\Longleftrightarrow\quad{}^{t}B\,+\,B\,=\,0~{}.$

Hence the vector space of alternating $4\times 4$ matrices $Alt_{4}$ provides
us with a parametrization of a Zariski open subset of
$IG(4,V_{\mathbb{C}})^{+}$ of dimension $4(4-1)/2=6$ which we denote by
$IG(4,V_{\mathbb{C}})^{+}_{0}$:

$Alt_{4}\,\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\,IG(4,V_{\mathbb{C}})^{+}_{0}\;\hookrightarrow\;IG(4,V_{\mathbb{C}})^{+},\qquad
B\,\longmapsto\,[Z_{B}]\,=\,\left[\begin{pmatrix}B\\\
I\end{pmatrix}\right]~{}.$

The isotropic subspace $Z_{B}$ is also the graph of the (alternating) map
$W^{*}\rightarrow W$, $w^{*}\mapsto Bw^{*}$.

###  2.3. The Plücker map

The Grassmannian $G(4,V_{\mathbb{C}})$ has a natural embedding, the Plücker
map $\pi$, into a projective space ${\bf P}^{N}={\bf
P}\wedge^{4}V_{\mathbb{C}}$ of dimension $N+1=\binom{8}{4}=70$:

$\pi:\,G(4,V_{\mathbb{C}})\,\longrightarrow\,{\bf
P}\wedge^{4}V_{\mathbb{C}},\qquad Z\,\longmapsto\,[\wedge^{4}Z]~{}.$

The Plücker map is equivariant for the action of $GL(V_{\mathbb{C}})$.

On the open subset $G(4,V_{\mathbb{C}})_{0}$ of $G(4,V_{\mathbb{C}})$ the
Plücker map is thus given by the determinants of the $4\times 4$ submatrices
of the $8\times 4$ matrix $P:=(^{B}_{I})$. Using the basis of $V$ from §1.1,
the coefficient of $e_{i_{1}}\wedge\ldots\wedge e_{i_{4}}$ in

$[\wedge^{4}Z_{B}]\,=\,[r_{1}\wedge\ldots\wedge
r_{4}],\qquad\big{(}r_{j}\,=\,\sum_{k=1}^{8}P_{kj}e_{k}\,\in\,Z_{B},\quad
P:=(^{B}_{I})\big{)}$

is the determinant of the $4\times 4$ submatrix of $P$ with rows
$i_{1},\ldots,i_{4}$.

###  2.4. The spinor map

The Picard group of $G(4,V_{\mathbb{C}})$ is generated by the Plücker line
bundle $\pi^{*}{\mathcal{O}}_{{\bf P}^{N}}(1)$. The restriction of this line
bundle to $IG(4,V_{\mathbb{C}})^{+}$ does not generate the Picard group of
$IG(4,V_{\mathbb{C}})^{+}$, but there is a line bundle ${\mathcal{L}}$ on
$IG(4,V_{\mathbb{C}})^{+}$ such that

$(\pi^{*}{\mathcal{O}}_{{\bf
P}^{N}}(1))_{|IG(4,V_{\mathbb{C}})^{+}}\,\cong\,{\mathcal{L}}^{\otimes 2}~{},$

and $Pic(IG(4,V_{\mathbb{C}})^{+})\cong{\mathbb{Z}}$ is generated by
${\mathcal{L}}$. One has $h^{0}(IG(4,V_{\mathbb{C}})^{+},{\mathcal{L}})=8$ and
the natural spinor (or Cartan) map

$\gamma:\,IG(4,V_{\mathbb{C}})^{+}\,\longrightarrow\,{\bf
P}S^{+}_{\mathbb{C}}\,\cong\,{\bf
P}H^{0}(IG(4,V_{\mathbb{C}})^{+},{\mathcal{L}})^{*}$

is an embedding whose image is a smooth quadric $Q^{+}\subset{\bf
P}S^{+}_{\mathbb{C}}$. Here $S^{+}_{\mathbb{C}}$ is the complexification of a
lattice $S^{+}$ that will be defined below (in an ad hoc manner), see also
§5.2 and §6.2.

For $Z_{B}$ in the open subset $IG(4,V_{\mathbb{C}})^{+}_{0}$, where
$B=(b_{ij})$ is an alternating $4\times 4$ matrix, this map is given, in a
suitable basis of $S^{+}$, by (see Theorem 6.8.4):

$\gamma:\,Z_{B}\,\longmapsto\,(z_{1}:\ldots:z_{7})\,=\,\big{(}1:b_{12}:b_{13}:b_{14}:b_{12}b_{34}-b_{13}b_{24}+b_{14}b_{23}:-b_{34}:b_{24}:-b_{23}\big{)}~{}.$

The coordinate functions are, up to signs, the Pfaffians of the alternating
submatrices of $B$ with an even number of rows and columns. The closure of the
image of $\gamma$ is the spinor variety, a smooth quadric:

$Q^{+}\,=\,\gamma(IG(4,V_{\mathbb{C}})^{+})\,=\,\\{\,(z_{1}:\ldots:z_{8})\,\in\,{\bf
P}S^{+}_{\mathbb{C}}:\quad
z_{1}z_{5}+z_{2}z_{6}+z_{3}z_{7}+z_{4}z_{8}\,=\,0\,\\}~{}.$

In fact the signs and the order of the coordinate functions on
$S^{+}_{\mathbb{C}}$ were chosen in such a way as to obtain this simple
equation.

The homogeneous coordinates above define a ${\mathbb{Z}}$-module
$S^{+}\cong{\mathbb{Z}}^{8}\subset S^{+}_{\mathbb{C}}$ with bilinear form
$(\bullet,\bullet)_{S^{+}}$ such that for $z=(z_{1},\ldots,z_{8})\in S^{+}$
one has $(z,z)_{S^{+}}=2(z_{1}z_{5}+z_{2}z_{6}+z_{3}z_{7}+z_{4}z_{8})$. In
particular, $S^{+}\cong U^{4}$ and for $z\in S^{+}_{\mathbb{C}}$ one has $z\in
Q^{+}$ iff $(z,z)_{S^{+}}=0$ where we use the ${\mathbb{C}}$-bilinear
extension of the bilinear form.

###  2.5. Orthogonal complex structures and their period space $\Omega$

An orthogonal complex structure $J$ on $V_{\mathbb{R}}$ is determined by (and
determines) a maximally isotropic subspace $Z_{+}$ such that
$V_{\mathbb{C}}=Z_{+}\oplus\overline{Z_{+}}$. Using the spinor map we see that
$\ell:=\gamma([Z_{+}])$ is a point of the quadric $Q^{+}\subset{\bf
P}S^{+}_{\mathbb{C}}$, that is $(\ell,\ell)_{S^{+}}=0$. Since the spinor map
is defined over ${\mathbb{Q}}$, we get $[\overline{Z_{+}}]=\bar{\ell}$, the
complex conjugate of the point $\ell$ in ${\bf P}S^{+}_{\mathbb{C}}$. The
condition that $Z_{+}\cap\overline{Z_{+}}=0$ is equivalent to the fact that
the complex line spanned by $\ell,\bar{\ell}$ is not contained in $Q^{+}$ (see
Lemma 6.9). This again is equivalent to $(\ell,\bar{\ell})_{S^{+}}\neq 0$ and
since $(\ell,\bar{\ell})_{S^{+}}\in{\mathbb{R}}$ we see that
$(\ell,\bar{\ell})_{S^{+}}$ is either positive or negative.

We define an open (six dimensional, connected) analytic subset of $Q^{+}$ by

$\Omega\,=\,\Omega_{S^{+}}\,:=\,\\{\,\ell\,\in\,{\bf
P}S^{+}_{\mathbb{C}}:\;(\ell,\ell)_{S^{+}}\,=\,0,\quad(\ell,\overline{\ell})_{S^{+}}\,>\,0\,\\}~{}.$

Then any $\ell\in\Omega$ defines a maximal isotropic subspace $Z_{\ell}$ of
$V_{\mathbb{C}}$ such that $V_{\mathbb{C}}=Z_{\ell}\oplus\overline{Z_{\ell}}$
and thus it defines an orthogonal complex structure $J_{\ell}$ on
$V_{\mathbb{R}}$.

The complex structure $J_{\ell}$ on $V_{\mathbb{R}}$ defines a complex torus
${\mathcal{T}}_{\ell}$ of dimension four by requiring an isomorphism of weight
1 Hodge structures

$H^{1}({\mathcal{T}}_{\ell},{\mathbb{Z}})\,=\,(V,J_{\ell}),\qquad\mbox{i.e.}\quad
H^{1,0}({\mathcal{T}}_{\ell})\,=\,Z_{\ell}~{}.$

This complex torus can also be defined as
${\mathcal{T}}_{\ell}=V_{\mathbb{C}}/(Z_{\ell}+V)$.

##  3\. Tori with an orthogonal structure and a Cayley class

### 3.1.

Using representation theory (explained in more detail in the Appendix), we
recall the relation between the spinor and the Plücker map. We also find a
natural map from $S^{+}$ to $\wedge^{4}V$, the image of $s\in S^{+}$ is
denoted by $c_{s}\in\wedge^{4}V$. For $\ell\in\Omega$ the complex torus
${\mathcal{T}}_{\ell}$ has
$H^{1}({\mathcal{T}}_{\ell},{\mathbb{Z}})=(V,J_{\ell})$. Thus we can also
identify the Hodge structures
$\wedge^{4}V=H^{4}({\mathcal{T}}_{\ell},{\mathbb{Z}})$ and for $s\in S^{+}$ we
obtain a cohomology class $c_{s}\in H^{4}({\mathcal{T}}_{\ell},{\mathbb{Z}})$
which is Markman’s Cayley class of $s$.

In §3.4 we recall Markman’s result that the Cayley class is a Hodge class, so
$c_{s}\in H^{2,2}({\mathcal{T}}_{\ell},{\mathbb{Z}})$, if and only if
$\ell\in\Omega_{s^{\perp}}:=s^{\perp}\cap\Omega$ where $s^{\perp}$ is the
hyperplane in $S^{+}_{\mathbb{C}}$ defined by $s$ using the bilinear form on
$S^{+}$. Hence the five dimensional complex manifold $\Omega_{s^{\perp}}$
parametrizes the four dimensional complex tori with an orthogonal structure
and Hodge class $c_{s}$.

###  3.2. The spinor and the Plücker map

From the isomorphism $\pi^{*}{\mathcal{O}}_{{\bf
P}^{N}}\cong{\mathcal{L}}^{\otimes 2}$ over $IG(4,V_{\mathbb{C}})^{+}$, one
can deduce that the Plücker map on $IG(4,V_{\mathbb{C}})^{+}$ is the
composition of the spinor map $\gamma$ with the second Veronese map $\nu$ on
${\bf P}S^{+}_{\mathbb{C}}$. The Veronese map is induced by

$\nu:\,S^{+}\,\longrightarrow\,Sym^{2}(S^{+}),\qquad s\,\longmapsto\,s\odot
s~{}.$

More precisely, the group $Spin(V_{\mathbb{C}})$, a double cover of
$SO(V_{\mathbb{C}})$, has a natural (half-spin) representation on
$S^{+}_{\mathbb{C}}$ and on the 36 dimensional vector space
$Sym^{2}(S^{+}_{\mathbb{C}})$. This latter representation is reducible, due to
the $Spin(V)$-invariant quadric on $S^{+}$ which dually defines an invariant
one dimensional subspace $\Gamma_{0}$ of $Sym^{2}(S_{\mathbb{C}}^{+})$. A
complement of this subspace turns out to be an irreducible
$Spin(V_{\mathbb{C}})$-representation and is denoted by $\Gamma_{2\alpha}$:

$Sym^{2}(S^{+}_{\mathbb{C}})\,\cong\,\Gamma_{2\alpha}\,\oplus\,\Gamma_{0}~{}.$

The subspace $\Gamma_{2\alpha}$ is spanned by the symmetric tensors $z\odot
z\in Sym^{2}(S^{+}_{\mathbb{C}})$ with $[z]\in Q^{+}\subset{\bf
P}S^{+}_{\mathbb{C}}$.

There is a decomposition of the $70$-dimensional $\wedge^{4}V_{\mathbb{C}}$ in
two irreducible $Spin(V_{\mathbb{C}})$-representations of dimension $35$ (it
corresponds to the decomposition of $\wedge^{4}V_{\mathbb{C}}$ into dual and
anti-selfdual $4$-forms for the Hodge star operator defined by
$(\bullet,\bullet)_{V}$):

$\wedge^{4}V_{\mathbb{C}}\,=\,\Gamma_{2\alpha}\,\oplus\,\Gamma_{2\beta}~{}.$

The image of $Q^{+}$ by the second Veronese map spans the linear subspace
${\bf P}\Gamma_{2\alpha}\subset{\bf P}^{N}={\bf P}\wedge^{4}V_{\mathbb{C}}$.

###  3.3. The Cayley classes

Another consequence of the relation between the $Spin(V)$-representations
$Sym^{2}(S^{+})$ and $\wedge^{4}V$ is that any element $s\in S^{+}$ defines a
4-form $c_{s}\in\wedge^{4}V$, which is called the Cayley class of $s$ ([Mar,
Remark 12.4], [Mun, §2.1]). It is obtained as the composition

$S^{+}\,\stackrel{{\scriptstyle\nu}}{{\longrightarrow}}\,Sym^{2}(S^{+})\,\cong\,\Gamma_{2\alpha}\,\oplus\,\Gamma_{0}\longrightarrow\,\Gamma_{2\alpha}\,\longrightarrow\,\wedge^{4}V~{},\qquad
s\longmapsto c_{s}~{}.$

This map is equivariant for the action of $Spin(V)$. The stabilizers in
$Spin(V)$ of $s$ and $c_{s}$ thus have the same Lie subalgebra. If
$(s,s)_{S_{+}}\neq 0$ the complexification of this Lie algebra is isomorphic
to $so(7)_{\mathbb{C}}$.

###  3.4. The Cayley class and Hodge classes

Let $\ell\in\Omega\subset Q^{+}$ and let ${\mathcal{T}}_{\ell}$ be the
associated complex torus. The Hodge decomposition of the first cohomology
group $H^{1}({\mathcal{T}}_{\ell},{\mathbb{Z}})=(V,J_{\ell})$ is given by the
eigenspaces $Z_{\ell},\overline{Z}_{\ell}=Z_{\overline{\ell}}$ of the
orthogonal complex structure $J_{\ell}$ in $V_{\mathbb{C}}$:

$H^{1}({\mathcal{T}}_{\ell},{\mathbb{C}})\,=\,V_{\mathbb{C}}\,=\,Z_{\ell}\,\oplus\,Z_{\overline{\ell}},\qquad
J_{\ell}\,=\,(i,-i)\,\in\,End(Z_{l})\oplus End(Z_{\overline{\ell}})~{}.$

To describe the Hodge structure on $H^{k}({\mathcal{T}}_{\ell},{\mathbb{Z}})$
we use the homomorphism

$h_{V,\ell}:\;U(1)\,:=\,\\{z\in{\mathbb{C}}:z\overline{z}=1\\}\,\longrightarrow\,GL(V_{\mathbb{R}}),\qquad
h_{V,\ell}(a+bi)\,:=\,aI+bJ_{\ell},$

where $a,b\in{\mathbb{R}}$, $a^{2}+b^{2}=1$. Notice that
$aI+bJ_{\ell}=(a+bi,a-bi)\in End(Z_{l})\oplus End(Z_{\overline{\ell}})$.

Since
$H^{k}({\mathcal{T}}_{\ell},{\mathbb{Z}})=\wedge^{k}H^{1}({\mathcal{T}}_{\ell},{\mathbb{Z}})=\wedge^{k}V$,
the Hodge decomposition $H^{k}({\mathcal{T}}_{\ell},{\mathbb{C}})=\oplus
H^{p,q}({\mathcal{T}}_{\ell})$ is defined by

$H^{p,q}({\mathcal{T}}_{\ell})\,=\,\big{(}\wedge^{p}Z_{\ell}\big{)}\otimes\big{(}\wedge^{q}Z_{\overline{\ell}}\big{)}\,=\,\\{x\,\in\,\wedge^{k}V_{\mathbb{C}}:\;h_{V,\ell}(a+bi)\cdot
x\,=\,(a+bi)^{p}(a-bi)^{q}x\quad\forall a+bi\in U(1)\,\\}~{},$

In particular, the Hodge classes in
$H^{2p}({\mathcal{T}}_{\ell},{\mathbb{Z}})$ are the invariants of the one-
parameter subgroup $h_{V,\ell}$ of $SO(V_{\mathbb{R}})$. The following
proposition is essentially [Mar, Lemma 12.2].

###  3.5. Proposition

Let $c_{s}\in\wedge^{4}V$ be the Cayley class defined by $s\in S^{+}$, the
integral lattice, and let $\ell\in\Omega_{S^{+}}$. Then $c_{s}$ is an integral
Hodge class in $H^{4}({\mathcal{T}}_{\ell},{\mathbb{Z}})$ exactly when
$(\ell,s)_{S^{+}}=0$:

$c_{s}\in\,H^{2,2}({\mathcal{T}}_{\ell},{\mathbb{Z}})\,:=\,H^{4}({\mathcal{T}}_{\ell},{\mathbb{Z}})\cap
H^{2,2}({\mathcal{T}}_{\ell})\qquad\mbox{if and only
if}\qquad\ell\,\in\,\Omega_{s^{\perp}}\,:=\,\\{\ell\in\Omega:\,(\ell,s)_{S^{+}}\,=\,0\,\\}~{}.$

Proof. First we observe that $h_{V,\ell}(z)\in SO(V_{\mathbb{R}})$ for all
$z\in U(1)$. In fact, for $v,w\in V_{\mathbb{R}}$ we have

$\big{(}(aI+bJ_{\ell})v,(aI+bJ_{\ell})w\big{)}_{V}\,=\,a^{2}\big{(}v,w\big{)}_{V}\,+\,ab\Big{(}\big{(}v,J_{\ell}w\big{)}_{V}\,+\,\big{(}J_{\ell}v,w\big{)}_{V}\Big{)}\,+\,b^{2}\big{(}J_{\ell}v,J_{\ell}w\big{)}_{V}\,=\,\big{(}v,w\big{)}_{V}~{},$

because $(J_{\ell}v,J_{\ell}w)_{V}=(v,w)_{V}$ implies
$(v,J_{\ell}w)=(J_{\ell}v,J_{\ell}^{2}w)_{V}$ and $J_{\ell}^{2}=-I$.

The homomorphism lifting the one-parameter subgroup
$h_{V,\ell}:U(1)\rightarrow SO(V_{\mathbb{C}})$ to $Spin(V_{\mathbb{C}})$ is
denoted by

$h_{\ell}:\,U(1):=\\{z\in{\mathbb{C}}:z\overline{z}=1\\}\,\longrightarrow\,Spin(V_{\mathbb{C}})~{}.$

The action of $h_{\ell}(z)\in Spin(V_{\mathbb{C}})$ in the half-spin
representation $\rho^{+}$ on $S^{+}_{\mathbb{C}}$ is (see Lemma 6.9):

$\rho^{+}(h_{\ell}(z))\ell\,=\,z^{2}\ell,\quad\rho^{+}(h_{\ell}(z))\overline{\ell}\,=\,\bar{z}^{2}\overline{\ell},\qquad\rho^{+}(h_{\ell}(z))s\,=\,s,\quad\forall
s\in\langle\ell,\overline{\ell}\rangle^{\perp}~{}.$

Using the induced action of $Spin(V_{\mathbb{C}})$ on $s\odot s\in
Sym^{2}(S^{+}_{\mathbb{C}})$ and its image
$c_{s}\in\wedge^{4}V_{\mathbb{C}}=H^{4}({\mathcal{T}}_{\ell},{\mathbb{C}})$ we
see that $c_{s}$ is invariant under $h_{\ell}(z)$ for all $z\in U(1)$ if and
only if $s$ is invariant, so $s\in\langle\ell,\overline{\ell}\rangle^{\perp}$.
For $s\in S^{+}$ the condition $(s,\ell)_{S^{+}}=0$ implies, by complex
conjugation, that also $(s,\overline{\ell})_{S^{+}}=0$, which proves the
proposition. ∎

##  4\. Abelian varieties of Weil type

###  4.1. The complex tori ${\mathcal{T}}_{\ell}$ and abelian varieties

For a point $\ell\in\Omega$, an open subset of the spinor quadric $Q^{+}$, we
defined a complex torus ${\mathcal{T}}_{\ell}$ of dimension four whose first
cohomology group is identified with $V$ and whose Hodge structure is
determined by $H^{1,0}({\mathcal{T}}_{\ell})=Z_{\ell}$, the maximal isotropic
subspace of $V_{\mathbb{C}}$ corresponding to $\ell$.

Fixing an $s\in S^{+}$ we also found that for $\ell\in\Omega_{s^{\perp}}$ this
complex torus has an integral Hodge class (the Cayley class) $c_{s}\in
H^{2,2}({\mathcal{T}}_{\ell},{\mathbb{Z}})$. Now we assume that
$(s,s)_{S^{+}}>0$ and we fix another, non-isotropic, class $h\in s^{\perp}$
with $(h,h)_{S^{+}}>0$. Hence the rank two sublattice $\langle
h,s\rangle\subset S^{+}$ generated by $h,s$ is positive definite for the
bilinear form on $S^{+}$. For $\ell\in\langle h,s\rangle^{\perp}\cap\Omega$
the torus $T_{\ell}$ turns out to be an abelian variety of Weil type and the
Cayley class $c_{s}$ is a non-trivial Hodge class. This result, Theorem 4.6
below, is due to O’Grady [O’G, Theorem 5.1] and Markman [Mar, Corollary 12.9,
Theorem 13.4]. First we recall the basic facts on abelian varieties of Weil
type.

###  4.2. Abelian varieties of Weil type

Let $A$ be an abelian variety and let $K={\mathbb{Q}}(\sqrt{-d})$, with
$d\in{\mathbb{Z}}_{>0}$, be an imaginary quadratic field. An abelian variety
of Weil type (with field $K$) is a pair $(A,K)$, where $A$ is an abelian
variety and $K\hookrightarrow End(A)_{\mathbb{Q}}$ is a subalgebra of the
endomorphism algebra of $A$, such that for all $x\in K$,
$x\notin{\mathbb{Q}}$, the endomorphism of $T_{0}A$ defined by the
differential of $x=a+b\sqrt{-d}\in K$, with $a,b\in{\mathbb{Q}}$, has
eigenvalues $x=a+b\sqrt{-d}$ and $\bar{x}=a-b\sqrt{-d}$ with the same
multiplicity. Equivalently, the eigenvalues of $x^{*}$ on $H^{1,0}$ have the
same multiplicity. In particular, if $(A,K)$ is of Weil type, then $\dim A$ is
even.

Given an abelian variety of Weil type $(A,K)$, there exists a polarization
$\omega_{K}\in H^{1,1}(A,{\mathbb{Z}})$ on $A$ such that for all $x\in K$ its
pull-back is

$x^{*}\omega_{K}\,=\,\mbox{Nm}(x)\omega_{K},\qquad\mbox{Nm}(x)\,=\,x\bar{x}~{},$

where $\mbox{Nm}(x)$ is the norm of $x\in K$ (see [vG2, Lemma 5.2.1]). We call
such a 2-form a polarization of Weil type and $(A,K,\omega_{K})$ is called a
polarized abelian variety of Weil type.

###  4.3. The Weil classes.

For a general abelian variety of Weil type $(A,K)$ of dimension $2n$, the
spaces of Hodge classes

$B^{p}(A)\,:=\,H^{p,p}(A,{\mathbb{Q}})\,:=\,H^{2p}(A,{\mathbb{Q}})\cap
H^{p,p}(A)$

have dimensions ([W], see also [vG2, Theorem 6.12]):

$\dim B^{p}(A)\,=\,1,\quad(p\neq n),\qquad\dim B^{n}(A)\,=\,3~{}.$

Since $\dim B^{1}(A)=1$, any $\omega\in B^{1}(A)$, $\omega\neq 0$, defines (up
to sign) a polarization on $A$ which will be of Weil type.

The action of the multiplicative group $K^{\times}:=K-\\{0\\}$ on
$H^{1}(A,K):=H^{1}(A,{\mathbb{Q}})\otimes_{\mathbb{Q}}K$ has an eigenspace
decomposition into two $2n$-dimensional $K$ subspaces

$H^{1}(A,K)\,=\,Z_{\kappa}\,\oplus\,Z_{\overline{\kappa}},\qquad
x^{*}(v,w)\,=\,(xv,\overline{x}w)$

that are conjugate over $K$. Since $A$ is of Weil type, the complexifications
of these eigenspaces both have Hodge numbers $h^{1,0}=h^{0,1}=n$. Thus in
$H^{2n}(A,K)=\wedge^{2n}H^{1}(A,K)$ there are two 1-dimensional subspaces
$\wedge^{2n}Z_{\kappa}$, $\wedge^{2n}Z_{\overline{\kappa}}$ of Hodge type
$(n,n)$. Since they are conjugate, their direct sum is defined over
${\mathbb{Q}}$. This defines a 2-dimensional subspace of Hodge classes

$W_{K}\,\subset\,H^{n,n}(A,{\mathbb{Q}}),\qquad
W_{K}\otimes_{\mathbb{Q}}K\,=\,\wedge^{2n}Z_{\kappa}\,\oplus\,\wedge^{2n}Z_{\overline{\kappa}}~{}.$

(There is also a natural identification of $W_{K}$ with
$\wedge^{2n}_{K}H^{1}(A,{\mathbb{Q}})$ where $H^{1}(A,{\mathbb{Q}})$ is viewed
as a $2n$-dimensional $K$ vector space.) The subspace $W_{K}$ is called the
space of Weil classes. For any $A$ of Weil type one has

${\mathbb{Q}}\omega_{K}^{n}\;\oplus\;W_{K}\;\subseteq\;B^{n}(A)$

where $\omega_{K}^{n}$, is the $n$-fold exterior product of $\omega_{K}$ with
itself. For a general $A$ of Weil type one has
$B^{n}(A)={\mathbb{Q}}\omega_{K}^{n}\,\oplus\,W_{K}$.

An element $x\in K$ acts with eigenvalues $(x\bar{x})^{n}$, $x^{2n}$,
$\bar{x}^{2n}$ on ${\mathbb{Q}}\omega_{K}^{n}\,\oplus\,W_{K}$. Thus if a non-
zero element $c$ in the three dimensional ${\mathbb{Q}}$ vector space
${\mathbb{Q}}\omega_{K}^{n}\,\oplus\,W_{K}$ is algebraic and it is not an
eigenvector for the $K$-action (so it is not a multiple of $\omega^{n}_{K}$)
then all classes in ${\mathbb{Q}}\omega_{K}^{n}\,\oplus\,W_{K}$ are algebraic
since $\omega_{K}^{n}$ is and so is $x^{*}c$ for all $x\in K$.

###  4.4. The Hermitian form

The ${\mathbb{Q}}$ vector space $H_{1}(A,{\mathbb{Q}})$ is also a $K$ vector
space for the action of $K$ given by $x_{*}$ for $x\in K\subset
End(A)_{\mathbb{Q}}$. A polarization of Weil type $\omega_{K}\in
H^{2}(A,{\mathbb{Q}})$ defines an alternating form on $H_{1}(A,{\mathbb{Q}})$
and it also defines a $K$-valued Hermitian form $H$ on this $K$-vector space
by:

$H\,:\,H_{1}(A,{\mathbb{Q}})\times
H_{1}(A,{\mathbb{Q}})\,\longrightarrow\,K,\qquad
H(x,y)\,:=\,\omega_{K}(x,(\sqrt{-d})_{*}y)\,+\,\sqrt{-d}\omega_{K}(x,y).$

If $\Psi\in M_{n}(K)$ is the Hermitian matrix defining $H$ w.r.t. some
$K$-basis of $H_{1}(A,{\mathbb{Q}})$ then
$\det(\Psi)\in{\mathbb{Q}}^{\times}={\mathbb{Q}}-\\{0\\}$ and the class of
$\det(\Psi)\in{\mathbb{Q}}^{\times}/\mbox{Nm}(K^{\times})$, called the
discriminant of $H$, is independent of the choice of the basis. Given two non-
degenerate Hermitian forms $H,H^{\prime}$ on $K^{n}$, there is a $K$-linear
map $M:K^{n}\rightarrow K^{n}$ such that $H(x,y)=H(Mx,My)$ for all $x,y\in
K^{n}$ if and only if $H$, $H^{\prime}$ have the same signature and the same
discriminant.

The discriminant of a polarized abelian variety of Weil type
$(A,K,\omega_{K})$ is the discriminant of $H$.

In Markman’s approach, the real part of $H$, which is a bilinear form, is (up
to the duality between $H_{1}(A,{\mathbb{Z}})$ and $H^{1}(A,{\mathbb{Z}})$ and
up to a scalar multiple) the bilinear form $(\cdot,\cdot)_{V}$, cf. §4.8. In
particular, it is the same for all families of Weil type, for all fields,
considered in [Mar] and in Theorem 4.6 below.

###  4.5. Complete families

Given a $K$ vector space $U$ of dimension $2n$ and a Hermitian form $H:U\times
U\rightarrow K$, any $2n$-dimensional abelian variety of Weil type $A$ with
field $K$ and discriminant equal to the discriminant of $H$ is obtained by
choosing a free ${\mathbb{Z}}$-module $\Lambda\subset U$ of rank $4n$ and a
complex structure on $J$ on
$\Lambda_{\mathbb{R}}:=\Lambda\otimes_{\mathbb{Z}}{\mathbb{R}}$ such that $J$
commutes with $K$, the two eigenspaces of $x\in K$, $x\notin{\mathbb{Q}}$, on
$(\Lambda_{\mathbb{R}},J)$ have the same dimension and finally the imaginary
part $\omega_{K}$ of $H$ defines a polarization on the complex torus
$(\Lambda_{\mathbb{R}},J)/\Lambda$.

The unitary group $U(H)_{\mathbb{R}}\cong U(n,n)$ of the Hermitian form $H$ on
the ${\mathbb{C}}=K\otimes_{\mathbb{Q}}{\mathbb{R}}$ vector space
$\Lambda_{\mathbb{R}}$ acts by conjugation $g\cdot J:=gJg^{-1}$ on these
complex structures. From this one obtains a complete family of abelian
$2n$-folds of Weil type parametrized by a Hermitian symmetric domain
isomorphic to $U(n,n)/(U(n)\times U(n))$, so of complex dimension $n^{2}$. The
unitary group $SU(H)\subset GL(\Lambda_{\mathbb{Q}})$, viewed as algebraic
group over ${\mathbb{Q}}$, is the special Mumford Tate group of the general
abelian variety in the family, see [vG2].

We discuss the proof of the following theorem in the remainder of this
section.

###  4.6. Theorem

Let $h,s\in S^{+}$ be perpendicular and such that $\langle h,s\rangle\subset
S^{+}$ is a positive definite rank two sublattice. Let
$d:=(h,h)_{S^{+}}(s,s)_{S^{+}}\in{\mathbb{Q}}_{>0}$ and let
$\ell\in\Omega_{\\{h,s\\}^{\perp}}$, where

$\Omega_{\\{h,s\\}^{\perp}}\,:=\,\\{\ell\in\Omega_{s^{\perp}}\,:\quad(\ell,h)_{S^{+}}\,=\,0\,\\}\;=\;\\{\ell\in\Omega\,:\quad(\ell,s)_{S^{+}}\,=\,(\ell,h)_{S^{+}}\,=\,0\,\\}$

is a complex manifold of dimension four. Then we have:

  1. a)

The complex four dimensional torus ${\mathcal{T}}_{\ell}$ has endomorphisms by
$K={\mathbb{Q}}(\sqrt{-d})$, that is
$K\subset\mbox{End}({\mathcal{T}}_{\ell})_{\mathbb{Q}}$.

  2. b)

The complex torus ${\mathcal{T}}_{\ell}$ has a polarization $\omega_{K}\in
H^{2}({\mathcal{T}}_{\ell},{\mathbb{Z}})$ and
$({\mathcal{T}}_{\ell},K,\omega_{K})$ is polarized abelian fourfold of Weil
type.

  3. c)

The discriminant of the polarization $\omega_{K}\in
H^{2}({\mathcal{T}}_{\ell},{\mathbb{Z}})$ is trivial.

  4. d)

The Cayley class $c_{s}\in H^{2,2}({\mathcal{T}}_{\ell},{\mathbb{Z}})$ is not
contained in the subspace ${\mathbb{Q}}\omega_{K}^{2}$ where
$\omega_{K}^{2}=\omega_{K}\wedge\omega_{K}$.

  5. e)

The four dimensional family of these fourfolds of Weil type parametrized by
$\Omega_{\\{h,s\\}^{\perp}}$ is complete.

###  4.7. Endomorphisms of ${\mathcal{T}}_{\ell}$

Since the sublattice $\langle h,s\rangle$ is positive definite, we may assume
that the restriction $q$ of the quadratic form on $S^{+}$ is given by
$q(xh+ys)=ax^{2}+by^{2}$, with both
$a=(h,h)_{S^{+}},b=(s,s)_{S^{+}}\in{\mathbb{Q}}$ positive. Hence $d=ab>0$. The
zero locus of $q$ is defined by $a^{-1}((ax)^{2}+aby^{2})=0$, showing that
there are two isotropic lines in $\langle h,s\rangle_{\mathbb{C}}$ defined by
$ax\pm\sqrt{-d}y=0$. These two lines are conjugate over $K$ where the
conjugation on $K$ is $\overline{x+y\sqrt{-d}}=x-y\sqrt{-d}$ with
$x,y\in{\mathbb{Q}}$. In ${\bf P}S^{+}_{\mathbb{C}}$ they correspond to the
two points of intersection of the line ${\bf P}\langle
h,s\rangle_{\mathbb{C}}$ with the spinor quadric $Q^{+}\cong
IG(4,V_{\mathbb{C}})^{+}$, which we denote by $\kappa,\overline{\kappa}$:

$\\{\kappa,\,\overline{\kappa}\\}\,=\,Q^{+}\,\cap\,{\bf P}\langle
h,s\rangle_{\mathbb{C}}\qquad(\subset{\bf P}S^{+}_{\mathbb{C}})~{}.$

As $Q^{+}=\gamma(IG(4,V_{\mathbb{C}})^{+})$, these two points define two
maximal isotropic subspaces in $V_{K}:=V\otimes_{\mathbb{Q}}K$ denoted by
$Z_{\kappa},Z_{\overline{\kappa}}$. Since the points
$\kappa,\overline{\kappa}$ are conjugate over $K$, so are these subspaces: if
$w=v+\sqrt{-d}v^{\prime}\in Z_{\kappa}$ with $v,v^{\prime}\in V_{\mathbb{Q}}$
then $\overline{w}=v-\sqrt{-d}v^{\prime}\in Z_{\overline{\kappa}}$.

The plane $\langle h,s\rangle_{\mathbb{C}}$ is not contained in $Q^{+}$, hence
these two subspaces have trivial intersection (Lemma 6.9, [Ch, III.1.12]):

$V_{K}\,=\,Z_{\kappa}\,\oplus\,Z_{\overline{\kappa}},\qquad\overline{(v_{1},v_{2})}=(\overline{v_{2}},\overline{v_{1}})\qquad(v_{1}\in
Z_{\kappa},\,v_{2}\in Z_{\overline{\kappa}})~{}.$

We identify the ${\mathbb{Q}}$ vector space $V_{\mathbb{Q}}$ with the image of
$V_{\mathbb{Q}}\hookrightarrow V_{K}$, it consists of the points
$(v_{1},\overline{v_{1}})$ with $v_{1}\in Z_{\kappa}$. Now we define an action
of $K$ on $V_{\mathbb{Q}}\,(\subset V_{K})$ by

$K\times V_{\mathbb{Q}}\,\longrightarrow\,V_{\mathbb{Q}},\qquad
x\cdot(v_{1},\overline{v_{1}})\,:=\,(xv_{1},\bar{x}\overline{v_{1}})\,=\,(xv_{1},\overline{xv_{1}})\qquad(\in\,V_{\mathbb{Q}}\,\subset\,Z_{\kappa}\,\oplus\,Z_{\overline{\kappa}})~{},$

where $\bar{x}$ is the conjugate of $x\in K$.

To show that this induces an inclusion
$K\subset\mbox{End}({\mathcal{T}}_{\ell})_{\mathbb{Q}}$, it suffices to verify
that any $x\in K$ commutes with the complex structure $J_{\ell}$ on
$V_{\mathbb{R}}$. Since $\ell\in\Omega_{{h,s}^{\perp}}$ we have
$(\ell,\kappa)_{S^{+}}=0$ and similarly the scalar products of any one of
$\ell,\overline{\ell}$ and any one of $\kappa,\overline{\kappa}$ are zero.
Therefore the intersection of $Z_{\ell},Z_{\overline{\ell}}$ with the
complexifications of $Z_{\kappa},Z_{\overline{\kappa}}$ is not zero by Lemma
6.9. Since these spaces are parametrized by the same connected component
$IG(4,V_{\mathbb{C}})^{+}$, their intersection is even dimensional and thus it
is two dimensional. From the eigenspace decomposition for $J_{\ell}$,
$V_{\mathbb{C}}=Z_{\ell}\oplus Z_{\overline{\ell}}$, we obtain the
decomposition

$V_{\mathbb{C}}\,=\,(Z_{\ell}\cap
Z_{\kappa,{\mathbb{C}}})\;\oplus\;(Z_{\ell}\cap
Z_{\overline{\kappa},{\mathbb{C}}})\;\oplus\;(Z_{\overline{\ell}}\cap
Z_{\kappa,{\mathbb{C}}})\;\oplus\;(Z_{\overline{\ell}}\cap
Z_{\overline{\kappa},{\mathbb{C}}})~{}.$

The action of $J_{\ell}$ and $x\in K$ on these four summands are scalar
multiplications (by $\pm i$ and $x,\bar{x}$ respectively), hence the action of
$K$ indeed commutes with $J_{\ell}$. Since each summand has dimension $2$, the
eigenvalues of $x\in K$, $x\not\in{\mathbb{Q}}$, on
$Z_{\ell}=H^{1,0}({\mathcal{T}}_{\ell})$ have the same dimension.

###  4.8. The polarization

The combination of the $K$-action on
$V_{\mathbb{Q}}=H^{1}({\mathcal{T}}_{\ell},{\mathbb{Q}})$ with the bilinear
form $(\bullet,\bullet)_{V}$ leads a polarization $\omega_{K}\in
H^{2}({\mathcal{T}}_{\ell},{\mathbb{Q}})$ on ${\mathcal{T}}_{\ell}$. We define
a bilinear form $E$ on $V_{\mathbb{Q}}$ by:

$E:\,V\times V\,\longrightarrow\,{\mathbb{Q}},\qquad
E(v,w)\,=\,(\sqrt{-d}\cdot v,w)_{V}~{}.$

The duality $V=H_{1}({\mathcal{T}}_{\ell},{\mathbb{Q}})^{dual}$ implies that
$E$ defines an element
$\omega_{K}\in\wedge^{2}V=H^{2}({\mathcal{T}}_{\ell},{\mathbb{Q}})$. Similar
to the computations for Kähler forms and metrics we establish the basic
properties of $E$ which imply that $({\mathcal{T}}_{\ell},K,\omega_{K})$ is a
polarized abelian fourfold of Weil type.

First of all, we have for all $v,w\in V_{\mathbb{Q}}$ and all $x\in K$ that

$E(x\cdot v,x\cdot w)\,=\,x\bar{x}E(v,w)~{}.$

To verify this, we extend $E$ $K$-bilinearly to $V_{K}$ and we use that
$Z_{\kappa},Z_{\overline{\kappa}}$ are isotropic subspaces. Thus, with
$v=v_{1}+\overline{v_{1}}$, $w=w_{1}+\overline{w_{1}}\in Z_{\kappa}\oplus
Z_{\overline{\kappa}}$ we get

$\begin{array}[]{rcl}E(x\cdot v,x\cdot
w)&=&\big{(}x\sqrt{-d}v_{1}+\overline{x\sqrt{-d}v_{1}},xw_{1}+\overline{xw_{1}})\big{)}_{V}\\\
&=&\big{(}x\sqrt{-d}v_{1},\bar{x}\overline{w_{1}}\big{)}_{V}\,+\,\big{(}\bar{x}\overline{\sqrt{-d}v_{1}},xw_{1}\big{)}_{V}\\\
&=&x\bar{x}\big{(}(\sqrt{-d}v_{1},\overline{w_{1}})_{V}\,+\,(\overline{\sqrt{-d}v_{1}},w_{1})_{V}\big{)}\\\
&=&x\bar{x}E(v,w)~{}.\end{array}$

Next we show that $E$ is alternating:

$E(v,w)\,=\,(\sqrt{-d}\cdot v,w)_{V}\,=\,(w,\sqrt{-d}\cdot
v)_{V}\,=\,\mbox{$\frac{1}{d}$}(\sqrt{-d}\cdot v,\sqrt{-d}^{2}\cdot
w)_{V}\,=\,-(\sqrt{-d}\cdot v,w)_{V}\,=\,-E(w,v)~{}.$

To show that the 2-form $\omega_{K}$ is of type $(1,1)$ it suffices to show
that $E(J_{\ell}v,J_{\ell}w)=E(v,w)$ for all $v,w\in V_{\mathbb{R}}$:

$E(J_{\ell}v,J_{\ell}w)\,=\,\big{(}\sqrt{-d}\cdot
J_{\ell}v,J_{\ell}w\big{)}_{V}\,=\,\big{(}J_{\ell}(\sqrt{-d}\cdot
v),J_{\ell}w)_{V}\,=\,\big{(}\sqrt{-d}\cdot v,w)_{V}\,=\,E(v,w)~{}.$

Finally we verify that $E(J_{\ell}v,v)>0$ for non-zero $v\in V_{\mathbb{R}}$.
That is, we must show that $(\sqrt{-d}\cdot J_{\ell}v,w)>0$. The endomorphisms
$\sqrt{-d},J_{\ell}$ of $V_{\mathbb{R}}$ are both constructed from
decompositions of $V_{\mathbb{C}}$ with two conjugate isotropic subspaces
$Z_{\kappa},Z_{\overline{\kappa}}$ and $Z_{\ell},Z_{\overline{\ell}}$
respectively. The corresponding points
$\kappa,\overline{\kappa},\ell,\overline{\ell}\in Q^{+}=IG(4,V)^{+}$ span a
${\bf P}^{3}\in{\bf P}S^{+}_{\mathbb{C}}$ which is the projectivization of the
complexification of the four dimensional subspace
$<h,s,\ell+\overline{\ell},(\ell-\overline{\ell})/i>\subset
S^{+}_{\mathbb{R}}$ (here ${\mathbb{C}}={\mathbb{R}}+i{\mathbb{R}}$). Notice
that this basis consists of perpendicular vectors for
$(\bullet,\bullet)_{S^{+}}$ and that the subspace is positive definite.

The group $Spin(V_{\mathbb{R}})$ acts via $SO(S^{+}_{\mathbb{R}})$ on
$S^{+}_{\mathbb{R}}$ and this action is transitive on such subspaces. As
$Spin(V_{\mathbb{R}})$ also acts via $SO(V_{\mathbb{R}})$ on $V_{\mathbb{R}}$,
we see that it suffices to show that $(J_{1}J_{2}v,v)>0$ for all non-zero
$v\in V_{\mathbb{R}}$ where the linear maps $J_{1},J_{2}$ are defined by any
two orthogonal positive definite 2-dimensional subspaces of
$S^{+}_{\mathbb{R}}$. (Markman shows that the map $J_{1}J_{2}$ is already
determined, up to a scalar multiple, by the direct sum of these subspaces.)

We use the conventions from §2.4. A point $z=(z_{1},\ldots,z_{8})\in
S^{+}\cong U^{4}$ will be written as

$z\,=\,\left(\begin{pmatrix}z_{1}\\\
z_{5}\end{pmatrix},\begin{pmatrix}z_{2}\\\
z_{6}\end{pmatrix},\begin{pmatrix}z_{3}\\\
z_{7}\end{pmatrix},\begin{pmatrix}z_{4}\\\
z_{8}\end{pmatrix}\right),\qquad(z,z)_{S^{+}}=2(z_{1}z_{5}+\ldots+z_{4}z_{8})~{}.$

The following four points $\nu_{1},\ldots,\nu_{4}$, where $\nu:=(^{1}_{1})\in
U$, in $S^{+}$ are perpendicular and span a positive 4-plane in
$S^{+}_{\mathbb{R}}$ since $(\nu_{i},\nu_{i})_{S^{+}}=8$ and we also define
$\ell_{1},\ell_{2}\in S^{+}_{\mathbb{C}}$:

$\begin{array}[]{rcl}\nu_{1}&=&(\nu,\phantom{-}\nu,\phantom{-}\nu,\phantom{-}\nu),\\\
\nu_{2}&=&(\nu,\phantom{-}\nu,-\nu,-\nu),\\\
\nu_{3}&=&(\nu,-\nu,\phantom{-}\nu,-\nu),\\\
\nu_{4}&=&(\nu,-\nu,-\nu,\phantom{-}\nu),\\\
\end{array}\qquad{\begin{array}[]{rccccl}\ell_{1}&:=&(\nu_{1}+i\nu_{2})/(1+i)&=&(\nu,\phantom{-}\nu,-i\nu,-i\nu)\\\
\ell_{2}&:=&(\nu_{3}+i\nu_{4})/(1+i)&=&(\nu,-\nu,-i\nu,\phantom{-}i\nu)\\\
\end{array}}$

Then $\ell_{1},\overline{\ell_{1}}$ and $\ell_{2},\overline{\ell_{2}}$ are all
isotropic vectors and they span $\langle\nu_{1},\nu_{2}\rangle_{\mathbb{C}}$
and $\langle\nu_{3},\nu_{4}\rangle_{\mathbb{C}}$ respectively. Isotropic
vectors are in $Q^{+}=\gamma(IG(4,V_{\mathbb{C}})^{+})$ and since these four
all have first coordinate $z_{1}=1$ they are in the image of the open set
$IG(4,V_{\mathbb{C}})^{+}_{0}$ parametrized by the alternating $4\times 4$
matrices. Using the explicit description of $\gamma$ one finds

$\ell_{k}\,=\,\gamma(Z_{B_{k}})\quad(k=1,2),\qquad
B_{1}\,=\,\begin{pmatrix}0&1&-i&-i\\\ -1&0&i&-i\\\ i&-i&0&-1\\\
i&i&1&0\end{pmatrix},\qquad B_{2}\,=\,\begin{pmatrix}0&-1&-i&i\\\ 1&0&-i&-i\\\
i&i&0&1\\\ -i&i&-1&0\end{pmatrix}~{}.$

The eigenspace with eigenvalue $-1=i^{2}=(-i)^{2}$ of the endomorphism
$J_{1}J_{2}$ of $V_{\mathbb{R}}$ is the direct sum of $Z_{\ell_{1}}\cap
Z_{l_{2}}$ and its complex conjugate. Let $c_{k},d_{k}$ denote the $k$-th
column of the matrix $(^{B_{1}}_{\,I})$, $(^{B_{2}}_{\,I})$ respectively, then
$Z_{\ell_{1}},Z_{\ell_{2}}$ are spanned by the $c_{k}$ and the $d_{k}$
($k=1,\ldots,4$) respectively. Their intersection is spanned by

$c_{1}-ic_{3}\,=\,{d_{1}-id_{3}},\quad
c_{2}-ic_{4}\,=\,{d_{2}-id_{4}}\qquad(\in Z_{\ell_{1}}\cap Z_{\ell_{2}})~{}.$

Considering $(c_{1}-ic_{3})\pm\overline{(c_{1}-ic_{3}})$ etc., one finds a
basis of the $-1$-eigenspace of $J_{1},J_{2}$. Its perpendicular is the
$+1$-eigenspace. Recall that $e_{1},\ldots,e_{8}$ are the basis vectors of $V$
as in 1.1, then the eigenspace decomposition is:

$V_{\mathbb{R}}\,=\,V_{+}\,\oplus\,V_{-}\,=\,\langle
e_{1}+e_{5},e_{2}+e_{6},e_{3}+e_{7},e_{4}+e_{8}\rangle_{\mathbb{R}}\,\oplus\,\langle
e_{1}-e_{5},e_{2}-e_{6},e_{3}-e_{7},e_{4}-e_{8}\rangle_{\mathbb{R}}~{}.$

Notice that $(\bullet,\bullet)_{V}$ is positive definite on $V_{+}$ and
negative definite on $V_{-}$. Writing $v=v_{+}+v_{-}$ as sum of $J_{1}J_{2}$
eigenvectors, one has
$(J_{1}J_{2}v,v)_{V}=(v_{+},v_{+})_{V}-(v_{-},v_{-})_{V}$ and thus indeed
$(J_{1}J_{2}v,v)_{V}>0$ for all non-zero $v\in V_{\mathbb{R}}$.

###  4.9. The discriminant

We refer to [Mar, Lemma 12.11] (cf. [O’G, Theorem 5.1]) for the computation of
the discriminant. See also Proposition 6.19 for a proof of the triviality of
the discriminant using results from Lombardo [Lo].

###  4.10. The Cayley class and the Weil classes

We define two subgroups of $Spin(V)$. Let $Spin(V)_{s}$ be the subgroup which
fixes $s\in S^{+}$ and let $Spin(V)_{h,s}$ be the subgroup which fixes all
elements in $\langle h,s\rangle$. Then one can show that the Cayley class
$c_{s}$ is the unique $Spin(V)_{s}$-invariant in $\wedge^{4}V$ and that
$\omega_{K}$ is the unique $Spin(V)_{s,h}$-invariant in $\wedge^{2}V$. This
implies that $c_{s}\not\in{\mathbb{Q}}\omega_{K}^{2}$ (cf. [Mun, Prop 2],
[Mar, Thm 13.4] and §6.12).

One can also use that the
$K\otimes_{\mathbb{Q}}{\mathbb{C}}\cong{\mathbb{C}}\times{\mathbb{C}}$-action
on $V_{\mathbb{C}}$ has the eigenspaces
$(Z_{\kappa})_{\mathbb{C}},(Z_{\bar{\kappa}})_{\mathbb{C}}$. The one parameter
subgroup $h_{R}$ of $Spin(V_{\mathbb{C}})$ which acts as multiplication by
$t,t^{-1}$ respectively on these eigenspaces fixes $E$, and thus it fixes
$\omega_{K}\in\wedge^{2}V$ and also $\omega^{2}_{K}\in\wedge^{4}V$. On the
other hand, $h_{R}$ has eigenvalues $t^{2},t^{-2}$ on
$\langle\kappa,\bar{\kappa}\rangle_{\mathbb{C}}=\langle
h,s\rangle_{\mathbb{C}}\subset S^{+}_{\mathbb{C}}$ by Lemma 6.9. Therefore
$c_{s}$, the image of $s\odot s$ in $\wedge^{4}V$, is not invariant under
$h_{R}$ and thus it cannot be a multiple of $\omega^{2}_{K}$.

###  4.11. Complete families

The Lie group $Spin(V_{\mathbb{R}})_{h,s}$ acts on
$\Omega_{\\{h,s\\}^{\perp}}$. This action induces an action of
$Spin(V_{\mathbb{R}})_{h,s}$ on the orthogonal complex structures on
$V_{\mathbb{R}}$ by $J_{g\cdot\ell}=gJ_{\ell}g^{-1}$. The fixed points
$\kappa,\bar{\kappa}\in Q^{+}\cap\langle h,s\rangle_{\mathbb{C}}$ of the
action of $Spin(V_{\mathbb{R}})_{h,s}$ on $Q^{+}$ correspond to the
eigenspaces $Z_{\kappa,{\mathbb{C}}},Z_{\bar{\kappa},{\mathbb{C}}}$ of the
$K$-action, which are thus mapped into themselves. This implies that the image
of $Spin(V_{\mathbb{R}})_{h,s}$ in $SO(V_{\mathbb{R}})$ commutes with the $K$
action on $V_{\mathbb{R}}$. This image thus preserves the Hermitian form $H$
and therefore $Spin(V)_{h,s}$ maps to the algebraic group $SU(H)$ which is the
Mumford Tate group of the general ${\mathcal{T}}_{\ell}$ with
$\ell\in\Omega_{\\{h,s\\}^{\perp}}$. For dimension reasons this map is
surjective on the real points of these groups and thus the family of abelian
fourfolds of Weil type is complete.

##  5\. Moduli spaces of sheaves on an abelian surface

### 5.1.

The constructions considered thus far have a natural geometrical
interpretation in terms of moduli spaces of sheaves on abelian surfaces. We
now briefly recall the basic definitions and results, due to Mukai and
Yoshioka. The notation used thus far is now adapted to this context, for
example, the free ${\mathbb{Z}}$-module $W$ of rank four will become
$W=H^{1}(X,{\mathbb{Z}})$ for an abelian surface $X$ etc.

We conclude with a brief outline of Markman’s proof of the Hodge conjecture
for the general abelian fourfolds of Weil type with trivial discriminant.

###  5.2. The Mukai lattice of an abelian surface

Let $X$ be an abelian surface and let $\hat{X}=Pic^{0}(X)$ be the dual abelian
surface. Let

$W\,=\,H^{1}(X,{\mathbb{Z}}),\quad
W^{*}\,=\,H^{1}(\hat{X},{\mathbb{Z}})\,=\,H^{1}(X,{\mathbb{Z}})^{*},\qquad
V\,:=\,W\oplus W^{*}~{}.$

The Chern character of a coherent sheaf on $X$ takes values in

$S^{+}\,:=\,\wedge^{even}H^{*}(X,{\mathbb{Z}})\,=\,H^{0}(X,{\mathbb{Z}})\,\oplus\,H^{2}(X,{\mathbb{Z}})\,\oplus\,H^{4}(X,{\mathbb{Z}})~{},$

and we will identify $H^{0}(X,{\mathbb{Z}})$, $H^{4}(X,{\mathbb{Z}})$ with
${\mathbb{Z}}$, using the generators $1$ and a volume form compatible with the
orientation on the complex manifold $X$.

The Mukai lattice of $X$ is the (free, rank $8$) ${\mathbb{Z}}$-module $S^{+}$
with the bilinear form given by (this bilinear form coincides up to sign with
$(\bullet,\bullet)_{S^{+}}$):

$(r,c,s)\cdot(r^{\prime},c^{\prime},s^{\prime})\,:=\,-(rs^{\prime}+r^{\prime}s)+\,c\wedge
c^{\prime}~{}.$

For $v=(r,c,s)\in S^{+}$, with $r>0$, $c\in NS(X)\subset
H^{2}(X,{\mathbb{Z}})$ and $v^{2}\geq 6$ the moduli space of sheaves $E$ on
$X$ with $ch(E)=v$, denoted by ${\mathcal{M}}(v)$, is a smooth holomorphic
symplectic manifold of dimension $v^{2}+2$.

###  5.3. The case $v=s_{n}$

We now take $v=s_{n}=(1,0,-n)$, so that $v^{2}=2n\geq 6$ and $\dim M(v)=2n+2$.
Let $Z\subset X$ be a subscheme of length $n$, then its ideal sheaf
${\mathcal{I}}_{Z}$ has $ch({\mathcal{I}}_{Z})=v$ (for an abelian surface, the
Chern character $ch(E)$ is the Mukai vector $v(E)$ of the sheaf $E$). This
induces an inclusion of complex manifolds

$Hilb^{n}(X)\,=\,X^{[n]}\;\hookrightarrow\;{\mathcal{M}}(v)\qquad(v=s_{n}=(1,0,-n))~{}.$

For ${\mathcal{L}}\in\hat{X}$ and ${\mathcal{I}}_{Z}\in X^{[n]}$ one also has
${\mathcal{L}}\otimes{\mathcal{I}}_{Z}\in{\mathcal{M}}(v)$.

The Albanese map $\alpha:X^{[n]}\rightarrow X$ of $X^{[n]}$ fits in a diagram:

$\begin{array}[]{rcl}X^{[n]}&&\\\ \downarrow&\searrow\alpha&\\\
X^{(n)}&\stackrel{{\scriptstyle\Sigma}}{{\longrightarrow}}&X\end{array}\qquad\Sigma([p_{1},\ldots,p_{n}])\,:=\,p_{1}+\ldots+p_{n}~{},$

here $X^{(n)}$ is the $n$-th symmetric power of $X$ and
$[p_{1},\ldots,p_{n}]\in X^{(n)}$ is the image of $(p_{1},\ldots,p_{n})\in
X^{n}$ in $X^{(n)}$.

The generalized Kummer variety $K_{n-1}(X)$, of dimension $2n-2$, is the
irreducible holomorphic symplectic manifold obtained as

$K_{n-1}(X)\,=\,\alpha^{-1}(0)\subset X^{[n]}~{}.$

Using locally free resolutions of sheaves one defines a determinant map
$\det:M(v)\rightarrow\hat{X}$ and one has
$\det({\mathcal{L}}\otimes{\mathcal{I}}_{Z})={\mathcal{L}}$ for
${\mathcal{L}}\in\hat{X}$. Yoshioka [Y] showed that

$M(v)\,\cong\,\hat{X}\times({\det}^{-1})({\mathcal{O}}_{X})\,\cong\,\hat{X}\,\times\,X^{[n]}\,\cong\,\hat{X}\times\Big{(}(X\times
K_{n-1}(X))/X[n]\Big{)}$

where $X[n]\subset X$ is the subgroup of $n$-torsion points. In particular,
the Bogomolov decomposition of $M(v)$ is the product of the abelian fourfold
$X\times\hat{X}$ and the irreducible holomorphic symplectic manifold
$K_{n-1}(X)$.

###  5.4. The cohomology of the generalized Kummer variety

The composition of the Mukai homomorphism and the restriction map

$v^{\perp}\,\longrightarrow\,H^{2}(M(v),{\mathbb{Z}})\,\longrightarrow\,H^{2}(K_{n-1}(X),{\mathbb{Z}})$

induces a Hodge isometry (for the weight two Hodge structure on $v^{\perp}$
defined by $(v^{\perp})^{2,0}=H^{2,0}(X)$ and with the BBF quadratic form on
$H^{2}(K_{n-1}(X),{\mathbb{Z}})$) [Y, Thm. 0.2].

This implies, by the surjectivity of the period map and with $v=s_{n}=s$, that
$\Omega_{s^{\perp}}$ is the period space of deformations of $K_{n-1}(X)$,
these deformations are called Kummer type varieties.

Moreover, $h^{3,0}(K_{n-1}(X))=0$ so that
$H^{3}(K_{n-1}(X),{\mathbb{C}})=H^{2,1}\oplus H^{1,2}$ is essentially the
first cohomology group of its intermediate Jacobian
$H^{3}({\mathbb{C}})/(H^{2,1}\oplus H^{3}({\mathbb{Z}}))$ and one has ([Y,
Prop. 4.20]):

$H^{3}(K_{n-1}(X),{\mathbb{Z}})\,=\,H^{1}(X,{\mathbb{Z}})\,\oplus\,H^{3}(X,{\mathbb{Z}})\,\cong\,H^{1}(X,{\mathbb{Z}})\,\oplus\,H^{1}(\hat{X},{\mathbb{Z}})\,=\,V~{}.$

O’Grady and Markman showed that for $\ell\in\Omega_{s^{\perp}}$ and any
deformation $Y_{\ell}$ of $K_{n-1}(X)$ with period
$H^{2,0}(Y_{\ell})={\mathbb{C}}\ell\subset(s^{\perp})_{\mathbb{C}}$, there is
an isomorphism of Hodge structures (up to Tate twist and isogeny)
$H^{3}(Y_{\ell},{\mathbb{Z}})=H^{1}({\mathcal{T}}_{\ell},{\mathbb{Z}})$. In
case the complex manifold $Y_{\ell}$ is algebraic and $h\in
H^{2}(Y,{\mathbb{Z}})=s^{\perp}$ is the class of an ample divisor, hence
$\ell\in\Omega_{\\{h,s\\}^{\perp}}$, O’Grady [O’G] showed that the torus
${\mathcal{T}}_{\ell}$ is an abelian variety of Weil type. Moreover, he showed
that for algebraic $Y_{\ell}$ the Kuga Satake variety of the weight two
polarized Hodge structure of rank six $h^{\perp}\subset
H^{2}(Y_{\ell},{\mathbb{Z}})$ is (isogeneous to) ${\mathcal{T}}_{\ell}^{4}$
(see also §6.18 where $h^{\perp}\cong H^{2}_{\ell}$).

O’Grady also makes a detailed study of the cohomology of generalized Kummer
varieties and in particular he observes that there is a natural map (recall
$\dim Y_{\ell}=\dim K_{n-1}(X)=2n-2$):

$H^{3}(Y_{\ell},{\mathbb{Z}})\,\longrightarrow\,H^{4n-6}(Y_{\ell},{\mathbb{Z}})\,\longrightarrow\,H^{2}(Y_{\ell},{\mathbb{Z}})^{\vee},$

the last map is Poincaré duality, which relates the Hodge structures on
$H^{3}(Y_{\ell})$ and $H^{2}(Y_{\ell})$.

###  5.5. Markman’s theorem

Given a sheaf $F^{\prime}\in{\mathcal{M}}(v)$ ($v=s_{n}$ as in §5.3), there is
a natural map

$\iota_{F^{\prime}}\,:\,X\times\hat{X}\,\longrightarrow\,{\mathcal{M}}(v),\qquad(x,{\mathcal{L}})\,\longmapsto\,(t_{x}^{*}F^{\prime})\otimes{\mathcal{L}}$

where $t_{x}:X\rightarrow X$, $y\mapsto x+y$ is the translation by $x$.
Deforming $K_{n-1}(X)$ to $Y_{\ell}$, with $\ell\in\Omega_{s^{\perp}}$, this
map deforms to a map

$\iota\,:\,{\mathcal{T}}_{\ell}\,\longrightarrow\,Y_{\ell}.$

A universal sheaf ${\mathcal{E}}$ on $X\times{\mathcal{M}}(v)$ defines a sheaf
$E$ on $M(v)\times M(v)$ by
$E:=\mathcal{E}xt^{1}_{\pi_{13}}(\pi^{*}_{12}{\mathcal{E}},\pi_{23}^{*}{\mathcal{E}})$
where $\pi_{ij}$ are the projections from $M(v)\times X\times M(v)$. For $F\in
M(v)$ let $E_{F}$ the restriction of $E$ to $\\{F\\}\times M(v)=M(v)$. This
defines a sheaf on $X\times\hat{X}$ whose second Chern class is exactly the
Cayley class defined by $v=s_{n}\in S^{+}$ ([Mar, Prop. 11.2], see also Prop.
6.14):

$c_{2}(\iota_{F^{\prime}}^{*}{\mathcal{E}nd}(E_{F}))\,=\,c_{v}\,\in\,\wedge^{4}V\,=\,H^{4}(X\times\hat{X},{\mathbb{Z}})~{}.$

Markman, using results of Verbitsky, shows that the sheaf $E_{F}$ on $M(v)$
deforms to a sheaf over any deformation $Y_{\ell}$ of $M(v)$. Thus $c_{v}\in
H^{4}({\mathcal{T}}_{\ell},{\mathbb{Z}})$ is an algebraic class whenever
${\mathcal{T}}_{\ell}$ is an abelian variety. From Theorem 4.6.d we have that
$c_{v}$ is not an eigenvector for the action of the multiplicative group
$K^{\times}$ on the Hodge classes in ${\mathbb{Q}}\omega^{2}_{K}\oplus
W_{K}\subset H^{2,2}({\mathcal{T}}_{\ell},{\mathbb{Z}})$. Thus
$\omega_{K}^{2},c_{v}$ and the images of $c_{v}$ under the $K^{\times}$ action
span ${\mathbb{Q}}\omega^{2}_{K}\oplus W_{K}$. Since any fourfold of Weil type
with trivial discriminant is isogeneous to a $T_{\ell}$, for any such fourfold
the space $W_{K}$ is spanned by algebraic classes.

##  6\. Appendix: The spinor map

###  6.1. Background

The spinor map was defined by Cartan [Ca] (see also [BHH]). The description
given by Chevalley in [Ch] was used by Markman [Man, §2]. We define the spinor
map using the representation theory of orthogonal groups as in [FH, Chapter
20] (but our $(v,w)_{V}$ is $2Q(v,w)$ in [FH]).

We change the notation: in this Appendix $V$ stands for $V_{\mathbb{C}}$,
$S^{+}$ for $S^{+}_{\mathbb{C}}$ etc. so all ${\mathbb{Z}}$-modules are
replaced by their complexifications. Whenever convenient we will also write
${\mathbb{C}}^{2n}$ for $V$ and $SO(2n)$ for $SO(V)$ etc.

###  6.2. The Clifford algebra of $V$

The Clifford algebra $C(V)$ of the complex vector space $V$, of dimension
$2n$, with the bilinear form $(\bullet,\bullet)_{V}$ is the quotient of the
tensor algebra

$C(V)\,:=\,\oplus_{k\geq 0}V^{\otimes k}/\langle v\otimes w\,+\,w\otimes
v\,-\,(v,w)_{V}\cdot 1\,\rangle~{},$

by the two sided ideal generated by the $v\otimes w+w\otimes v-(v,w)_{V}$ with
$v,w\in V$, or equivalently, by the two sided ideal generated by the $v\otimes
v-(1/2)(v,v)_{V}$ for $v\in V$.

The Clifford algebra has dimension $2^{2n}$. We identify $V$ with its image in
$C(V)$. The even Clifford algebra $C(V)^{+}$ is the image of $\oplus_{k\geq
0}V^{\otimes 2k}$.

Let $V=W\oplus W^{*}$ be the complexification of the lattices in §1.1. Since
$W,W^{*}$ are isotropic one has $vw=-wv\in C(V)$ for all $v,w\in W$ and also
for all $v,w\in W^{*}$. The subalgebras of $C(V)$ generated by $W,W^{*}$ are
isomorphic to the exterior algebras $\wedge^{\bullet}W$ and
$\wedge^{\bullet}W^{*}$ respectively.

Let $e^{*}:=e_{n+1}\cdots e_{2n}\in C(V)$ be the product of the elements in a
basis of $W^{*}$. Then the left ideal $S:=C(V)e^{*}$ of $C(V)$ is isomorphic,
as a ${\mathbb{C}}$ vector space, to $\wedge^{\bullet}W$,

$\sigma:\,\wedge^{\bullet}W\,\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\,S\,:=\,C(V)e^{*},\qquad
w_{1}\wedge w_{2}\wedge\ldots\wedge w_{r}\,\longmapsto\,w_{1}w_{2}\ldots
w_{r}e^{*}~{},$

([Ch, II.2.2], [FH, Exercise 20.12]). Under this isomorphism, left
multiplication by $w\in W$ and $w^{*}\in W^{*}$ on $S$ correspond to the
following endomorphisms of $\wedge^{\bullet}W$:

$w\sigma(\eta)\,=\,\sigma(w\wedge\eta),\qquad
w^{*}\sigma(\eta)\,=\,\sigma(D_{w^{*}}\eta)~{},\qquad(\eta\in\,\wedge^{\bullet}W)~{},$

where $D_{w^{*}}$ is the derivation on $\wedge^{\bullet}W$ defined by

$D_{w^{*}}(1)=0,\qquad D_{w^{*}}(w_{1}\wedge\ldots\wedge
w_{r})\,=\,\sum_{i=1}^{r}(-1)^{i-1}w^{*}(w_{i})(w_{1}\wedge\ldots\wedge\widehat{w_{i}}\wedge\ldots\wedge
w_{r})~{},$

(here $w^{*}(w)=(w,w^{*})_{V}$ for $w\in W,w^{*}\in W^{*}$).

These operations of $W,W^{*}$ on $\wedge^{\bullet}W$ define a $C(V)$-module
structure and $\sigma$ is a homomorphism of $C(V)$-modules. It induces an
isomorphism of ${\mathbb{C}}$-algebras between the even Clifford algebra and
the direct sum of two matrix algebras (cf. [FH, (20.13)])

$C(V)^{+}\,\cong\,\mbox{End}(S^{+})\,\oplus\,\mbox{End}(S^{-}),\qquad
S^{+}\,:=\,\wedge^{even}W,\qquad S^{-}\,:=\,\wedge^{odd}W~{}.$

Since $\dim W=n$ one has $\dim S^{\pm}=2^{n-1}$.

###  6.3. The spin group of $V$

The conjugation on $C(V)$ is the anti-involution given by

$x\,:=\,x_{1}\cdots x_{r}\,\longmapsto\,x^{*}\,:=\,(-1)^{r}x_{r}\cdots x_{1},$

notice that it maps $C(V)^{+}$ into itself. The spin group of $V$ is

$Spin(V)\,:=\,\\{x\in C(V)^{+}:\;xx^{*}\,=\,1,\quad
xVx^{*}\,\subset\,V\,\\}~{}.$

Elements in $Spin(V)$ thus induce linear maps on $V$ and one has the following
result.

###  6.4. Theorem

There is a surjective homomorphism of complex Lie groups

$\rho_{V}:\,Spin(V)\,\longrightarrow\,SO(V),\qquad
x\,\longmapsto[v\,\longmapsto\,xvx^{*}]$

with kernel $\\{\pm 1\\}$.

Proof. For a proof see [FH, Thm 20.28]. ∎

###  6.5. The half-spin representations

Besides this ‘standard representation’ of $Spin(V)$ on $V$, one also has the
two half-spin representations $\rho^{+},\rho^{-}$ of $Spin(V)$ on $S^{+}$ and
$S^{-}$ respectively (vector spaces of dimension $2^{n-1}$), given by left
multiplication in $C(V)$:

$\rho^{\pm}:\,Spin(V)\,\longrightarrow\,GL(S^{\pm}),\qquad
x\,\longmapsto[\eta\,\longmapsto\,x\eta]~{}.$

See [FH, Exercise 20.38] for the fact that for $n\equiv 0\mod 4$ the image of
$Spin(V)$ lies in $SO(2^{n-1})$ (for a certain bilinear form $\beta$ on
$S^{+}\subset\wedge^{\bullet}W$ also considered in [Ch, 3.2]). The center of
$Spin(V)$, $\dim V>2$, is isomorphic to $({\mathbb{Z}}/2{\mathbb{Z}})^{2}$ if
$n$ is even and is cyclic of order four otherwise (cf. [FH, Exercise 20.36]).
For $n$ even, $n>2$, the three quotients of $Spin(V)$ by the order two
subgroups of the center are $SO(V)$ and the images of $Spin(V)$ in the two
half-spin representations.

###  6.6. The Lie algebra $spin(V)=so(2n)$

The Lie algebra $spin(V)$ of the subgroup $Spin(V)$ of the multiplicative
group of $C(V)^{+}$ consists of the $x\in C(V)^{+}$ such that $x+x^{*}=0$ and
$xv-vx\in V$ for all $v\in V$ (cf. [Ch, p.67-68]). It has a basis consisting
of the following $n(n-1)/2+n(n-1)/2+n^{2}=n(2n-1)$ elements:

$e_{i}e_{j},\quad e_{i+n}e_{j+n}\quad\mbox{with}\quad 1\leq i\leq j\leq
n;\qquad e_{i}e_{j+n}\,-\,\mbox{$\frac{1}{2}$}1,\quad 1\leq i,j\leq n~{}.$

To see that these elements are in $spin(V)$ (and to find their action on $V$)
one can use that for $x,y,v\in V$ one has

$[xy,v]\,:=\,xyv\,-\,vxy\,=\,x(-vy+(y,v)_{V})\,-\,(-xv+(x,v)_{V})y\,=\,(y,v)_{V}x\,-\,(x,v)_{V}y~{}.$

The Lie algebra $spin(V)$ is isomorphic to the Lie algebra $so(2n)$ of the
orthogonal group $SO(V)=SO(2n)$. This Lie algebra consists of the
$X\in\mbox{End}(V)$ such that $(Xv,w)_{V}+(v,Xw)_{V}=0$ for all $v,w\in V$.
One finds that

$so(2n)\,=\,\left\\{\,X\,=\,\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\in\mbox{End}(V)\;:\quad
A\,=\,-{}^{t}D,\quad{}^{t}B\,=\,-B,\quad{}^{t}C\,=\,-C\;\right\\}~{}.$

An isomorphism $spin(V)\rightarrow so(2n)$ is given by the differential of
$\rho_{V}$, so by the representation of $spin(V)$ on $V$ given by $x\cdot
v:=xv-vx$. Using the computation of $[xy,v]$ above, one verifies that this
isomorphism is given by

$spin(V)\,\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\,so(2n),\qquad\left\\{\begin{array}[]{rclcl}e_{i}e_{n+j}&\longmapsto&X_{i,j},&&X_{i,j}\,:=\,E_{i,j}\,-\,E_{n+j,n+i},\\\
e_{i}e_{j}&\longmapsto&Y_{i,j},&&Y_{i,j}\,:=\,E_{i,n+j}\,-\,E_{j,n+i},\\\
e_{i+n}e_{j+n}&\longmapsto&Z_{i,j},&&Z_{i,j}\,:=\,E_{n+i,j}\,-\,E_{n+j,i}~{}.\\\
\end{array}\right.$

We choose the Cartan subalgebra of $so(2n)$ to be the diagonal matrices in
$so(2n)$ (as in [FH, §18.1]):

$\mathfrak{h}\,:=\,\oplus_{i=1}^{n}\,{\mathbb{C}}H_{i},\qquad
H_{i}\,:=\,E_{i,i}\,-\,E_{n+i,n+i}~{}.$

The dual $\mathfrak{h}^{*}$ of $\mathfrak{h}$ then consists of the linear maps
(weights)

$\mathfrak{h}^{*}\,:=\,\oplus_{i=1}^{n}{\mathbb{C}}L_{i},\qquad
L_{i}(\sum_{j=1}^{n}t_{j}H_{j})\,:=\,t_{i}~{}.$

###  6.7. The spinor map

As before, we identify

$V\,=\,W\,\oplus\,W^{*}\,\qquad W\,:=\,\langle e_{1},\ldots,e_{n}\rangle,\quad
W^{*}\,:=\,\langle e_{n+1},\ldots,e_{2n}\rangle,$

with $W^{*}=\mbox{Hom}(W,{\mathbb{C}})$ the dual of $W$, where
$w^{*}(w):=Q(w,w^{*})$ for $w\in W,w^{*}\in W^{*}$. We denote by
$IG(n,2n)^{+}$ the connected component of the Grassmannian of maximal
isotropic subspaces of $V={\mathbb{C}}^{2n}$ that contains $W^{*}$. A complex
maximally isotropic subspace $Z$ defines a point $[Z]\in IG(n,2n)^{+}$ if and
only if $\dim(Z\cap W_{\mathbb{C}}^{*})\equiv n\mod 2$ is even.

We denote by $Z_{B}$, for an alternating $n\times n$ matrix $B$, the maximal
isotropic subspace spanned by the columns of $(^{B}_{I})$ analogous to §2.4,
notice that $W^{*}=Z_{0}$.

The Grassmannian $IG(n,2n)^{+}=SO(V)/P$ is a homogeneous space where
$P=P_{W^{*}}$ is the stabilizer of $W^{*}$ in the group $SO(2n)$. The Lie
algebra of $P$, which are the $X\in so(2n)$ with $XW^{*}\subset W^{*}$,
consists of the $X\in so(2n)$ with $B=0$.

We recall that the Pfaffian of an alternating $2m\times 2m$ matrix $A$ is the
complex number $\mbox{Pfaff}(A)$ defined by the following identity in
$\wedge^{2m}{\mathbb{C}}^{2m}$:

$\mbox{Pfaff}(A)e_{1}\wedge\ldots\wedge
e_{2m}\,=\,m!\omega_{A}^{m},\qquad(\omega_{A}\,:=\,\sum_{i<j}a_{ij}e_{i}\wedge
e_{j})~{}.$

###  6.8. Theorem

Let $\rho^{+}:Spin(V)\rightarrow GL(S^{+})$ be the half-spin representation of
$Spin(V)$ on $S^{+}=\wedge^{even}W:=\oplus_{k}\wedge^{2k}W$.

  1. (1)

In case $n$ is even, the highest weight of $S^{+}$ is $(L_{1}+\ldots+L_{n})/2$
and it is $(L_{1}+\ldots+L_{n-1}-L_{n})/2$ if $n$ is odd.

  2. (2)

The one dimensional subspace

$\langle 1\rangle\,=\langle\wedge^{0}W\rangle\,\subset\,\wedge^{even}W$

is invariant under the Lie algebra of $P$. Thus there is a $Spin(V)$
equivariant map

$\gamma:\,IG(n,2n)^{+}\,\longrightarrow\,{\bf
P}S^{+},\qquad\gamma([\rho_{V}(\tilde{g})W^{*}])\,=\,\rho^{+}(\tilde{g})1$

for $\tilde{g}\in Spin(V)$.

  3. (3)

For an alternating matrix $B\in M_{n}({\mathbb{C}})$, let

$X_{B}\,:=\,\begin{pmatrix}0&B\\\ 0&0\end{pmatrix}\in
so(2n),\qquad\tilde{g}_{B}\,:=\,exp(X_{B})\in Spin(V)~{}.$

In the standard representation $\rho_{V}:Spin(V)\rightarrow SO(2n)$ one has

$\rho_{V}(\tilde{g}_{B})\,=\,\begin{pmatrix}I&B\\\
0&I\end{pmatrix}\;(\in\,SO(2n))~{}\qquad\mbox{and}\quad\rho_{V}(\tilde{g}_{B})Z_{0}\,=\,Z_{B}~{}.$

In the half-spin representation on $S^{+}$ the action of $\tilde{g}_{B}$ is
given by a left multiplication:

$\rho^{+}(\tilde{g}_{B}):S^{+}\,\longrightarrow\,S^{+},\qquad\omega\,\longmapsto\,exp(\omega_{B})\wedge\omega~{},$

and one has

$exp(\omega_{B})\,=\,\sum_{I,\sharp I\equiv 0\mod
2}\mbox{Pfaff}(B_{I})e_{I}~{},$

where $I$ runs of over the subsets of $\\{1,\ldots,n\\}$ with an even number
of elements and $e_{I}=e_{i_{1}}\wedge\ldots\wedge
e_{i_{2k}}\in\wedge^{even}W=S^{+}$ with $i_{1}<\ldots<i_{2k}$.

  4. (4)

In the basis of $S^{+}$ consisting of the $e_{I}$, the spinor map $\gamma$ on
the open subset $IG(n,2n)^{+}_{0}$ is given by

$\gamma:\,IG(n,2n)^{+}_{0}\,\longrightarrow\,{\bf
P}S^{+},\qquad[Z_{B}]\,\longmapsto\,(\ldots:\mbox{Pfaff}(B_{I}):\ldots)~{}.$

The image of $\gamma$ is defined by quadrics.

Proof. The highest weight of the half-spin representation $S^{+}$ is
determined in [FH, Proposition 20.15].

The Lie algebra of $P$ is generated by the $X_{i,j}$ (matrices with $B=C=0$)
and the $Z_{i,j}$ (matrices with $A=B=D=0$). The images of these elements in
$End(S^{+})$ (as well as those of the $Y_{i,j}$) are:

$so(2n)\,\longrightarrow\,End(S^{+}),\qquad\left\\{\begin{array}[]{rclcl}X_{i,j}&\longmapsto&e_{i}e_{n+j}-\mbox{$\frac{1}{2}\delta_{ij}$}&\longmapsto&L_{e_{i}}\circ
D_{e_{n+j}}-\mbox{$\frac{1}{2}$}\delta_{ij},\\\
Y_{i,j}&\longmapsto&e_{i}e_{j}&\longmapsto&L_{e_{i}}\circ L_{e_{j}},\\\
Z_{i,j}&\longmapsto&e_{i+n}e_{j+n}&\longmapsto&D_{e_{i+n}}\circ
D_{e_{j+n}}~{}.\end{array}\right.$

Since $D_{w^{*}}(1)=0$ for all $w^{*}\in W^{*}$, we see that $X_{i,j}$ and
$Z_{i,j}$ map $1$ to an element in $\langle 1\rangle$. Hence $Lie(P)$ maps
$\langle 1\rangle$ into itself and thus also the inverse image of $P$ in
$Spin(V)$ maps this line into itself.

The element $X_{B}\in so(2n)$ determined by $B$ is
$X_{B}=\sum_{i<j}b_{ij}Y_{i,j}$. It acts as left multiplication by
$\omega_{B}:=\sum b_{ij}e_{i}\wedge e_{j}$ on $\wedge^{even}W$ and thus
$exp(X_{B})$ is left multiplication by $exp(\omega_{B})\in\wedge^{even}W$. The
exponential map of an endomorphism $\alpha$ is $\sum\alpha^{n}/n!$ and since
the left multiplication 2-forms generate a commutative subalgebra of nilpotent
elements, $exp(X_{B})$ is actually a finite sum and one also has

$exp(\omega_{B})\,=\,\prod_{i<j}exp(b_{ij}e_{i}\wedge
e_{j})\,=\,\prod_{i<j}(1+b_{ij}e_{i}\wedge e_{j})~{}.$

We now show that, with $B_{I}$ the submatrix of $B$ with coefficients
$b_{i,j}$ with $i,j\in I$,

$exp(\omega_{B})\,=\,\sum_{I,\sharp I\,even}\,\mbox{Pfaff}(B_{I})e_{I}~{}.$

In fact, $exp(\omega_{B})\in\wedge^{even}W$ is a linear combination of the
$e_{I}=e_{i_{1}}\wedge\ldots\wedge e_{i_{2k}}$, where $i_{1}\leq\ldots\leq
e_{i_{2k}}$ $I=\\{i_{1},\ldots,i_{2k}\\}\subset\\{1,\ldots,n\\}$ is a subset
with an even number of elements. Since for an integer $p$ one has that
$\omega_{B}^{p}\in\wedge^{2p}W$, the coefficient of $e_{I}$ is homogeneous of
degree $2k$, with $2k=\sharp I$, in the coefficients $b_{ij}$ of $B$ and only
those with $i,j\in I$ contribute. So the coefficient of $e_{I}$ is determined
by the $2k\times 2k$ alternating submatrix $B_{I}$ of $B$ with rows and
columns indexed by $I$. Moreover this coefficient is
$(\sum_{i_{k}<i_{l},i_{k},i_{l}\in I}b_{i{{}_{k}}i{{}_{l}}}e_{i_{k}}\wedge
e_{i_{l}})^{k}/k!$, which is indeed $\mbox{Pfaff}(B_{I})$.

Since $\rho_{V}(\tilde{g}_{B})Z_{0}\,=\,Z_{B}$ and $\gamma([Z_{0}])=1\in
S^{+}$ we get $\gamma([Z_{B}])=\rho^{+}(\tilde{g})1=exp(\omega_{B})\in
S^{+}=\wedge^{even}W$. The description of the spinor map follows immediately.
For the equations defining the image see [Ch, III.3.2] or [Li]. ∎

The following lemma is used several times in this paper, for example to relate
complex structures on $V_{\mathbb{R}}$ to elements of $S^{+}_{\mathbb{C}}$ or
to weight two Hodge structures on $S^{+}$ as in the Kuga Satake construction.
For $\dim V\neq 8$ however, $Spin(V)$ only allows one to relate polarized
weight two Hodge structures on $V$ to complex structures on the spin
representation. The special feature in the case $\dim V=8$ is triality, an
automorphism of order three of $Spin(V)$, which allows one to permute the
three irreducible 8-dimensional representations $V,S^{+},S^{-}$, see [FH,
§20.3], [Ch, Chapter 4], and which is implicit in the proof of the lemma.

###  6.9. Lemma

  1. a)

Let $V=U\oplus U^{*}$ be a decomposition of $V={\mathbb{C}}^{8}$ with two
maximally isotropic subspaces with $[U],[U^{*}]\in IG(4,V_{\mathbb{C}})^{+}$.
For $t\in{\mathbb{C}}$, $t\neq 0$, the orthogonal transformation
$(t\mbox{id}_{U},t^{-1}\mbox{id}_{U^{*}})\in(\mbox{End}(U)\oplus\mbox{End}(U^{*}))\cap
SO(V)$ has a lift $h(t)\in Spin(V)$ which acts as follows on $S^{+}$:

$\rho^{+}(h(t))\,\ell_{U}\,=\,t^{2}\ell_{U},\quad\rho^{+}(h(t))\,\ell_{U^{*}}\,=\,t^{2}\ell_{U^{*}},\qquad\rho^{+}(h(t))\,s\,=\,s,\quad\forall
s\in\langle\ell_{U},\,\ell_{U^{*}}\rangle^{\perp}~{},$

where $\ell_{U},\ell_{U^{*}}\in S^{+}$ are (any) representatives of
$\gamma([{U}]),\gamma([U^{*}])\in{\bf P}S^{+}$.

  2. b)

Let $Z_{1},Z_{2}$ be two distinct maximally isotropic subspaces of
$V_{\mathbb{C}}$ in the family parametrized by $IG(4,V_{\mathbb{C}})^{+}$.
Then $Z_{1}\cap Z_{2}=\\{0\\}$ if and only if the complex plane
$\langle[Z_{1}],[Z_{2}]\rangle$ is not contained in the spinor quadric
$Q^{+}$.

Proof. We use that the spinor map is equivariant for the action of $Spin(V)$.
There is an element of $Spin(V)$ mapping $U$ to $W$ since
$IG(4,8)^{+}=SO(V)/P$. Then $U^{*}$ is mapped to $Z_{B}$ for some $B\in
Alt_{4}$ and it is easy to see that there is another element in $Spin(V)$
fixing $W$ (so with $C=0$) and mapping $Z_{B}$ to $Z_{0}=W^{*}$. We thus may
replace $W,W^{*}$ with $U,U^{*}$. The one parameter subgroup $h$ acts as
multiplication by $t$ on $U\subset V$, hence $h$ is generated by an
$X\in{\mathfrak{h}}\subset spin(V)$ with $L_{i}(X)=1$ for $i=1,\ldots,4$ (and
thus $X=\sum H_{i}$). The weights of $S^{+}$ are $(\pm L_{1}\pm L_{2}\pm
L_{3}\pm L_{4})/2$ with an even number of $-$ signs, hence their values on $X$
are $2,-2$, with multiplicity one, and $0$ with multiplicity six. Thus
$\rho^{+}(h(t))$ is semisimple with eigenvalues $t^{2},t^{-2}$ and $1$, the
last with multiplicity six. The eigenvalue $t^{-2}$, the lowest weight of
$S^{+}$, is on $Z_{U^{*}}$, see Theorem 6.8. The element $g\in SO(V)$ that
maps $e_{i}\mapsto e_{i+4}$, $e_{i+4}\mapsto e_{i}$ for $i=1,\ldots,4$
interchanges $U$ and $U^{*}$ and acts (in the Adjoint representation) as $-id$
on $\mathfrak{h}$, hence the eigenvalue $t^{2}$ must be on $Z_{U}$. As
$Spin(V)$ preserves $(\bullet,\bullet)_{S^{+}}$, the decomposition into these
eigenspaces is orthogonal. (For any $n$, the one parameter subgroup of $SO(V)$
that acts as multiplication by $t^{2},t^{-2}$ on $e_{1},e_{n+1}$ respectively
and is the identity on $\langle e_{1},e_{n+1}\rangle^{\perp}$ is generated by
an $X\in spin(V)$ with $L_{1}(X)=2$, $L_{i}(X)=0$ for $i\geq 2$ and thus
$(1/2)(\pm L_{1}\pm L_{2}\ldots\pm L_{n})(X)=\pm(1/2)X$, showing that the lift
of this subgroup to $Spin(V)$ has only eigenvalues $t,t^{-1}$ on $S^{+}$, with
the same multiplicities, and the same holds for $S^{-}$. A similar result
holds for $SO(V)$ and its spin representation if $\dim V=2n+1$.)

Using the action of the orthogonal group, if $Z_{1}\cap Z_{2}=\\{0\\}$, then
we can map $Z_{1},Z_{2}$ to $W,W^{*}$. As $[W]=e_{*},[W^{*}]=1\in S^{+}$ and
$(e_{*},1)_{S^{+}}\neq 0$ it follows that the plane
$\langle[Z_{1}],[Z_{2}]\rangle$ is not contained in $Q^{+}$. On the other
hand, if $Z_{1}\cap Z_{2}\neq 0$, then we may assume $Z_{1}=W^{*}$ and
$Z_{1}=Z_{B}$ with $B$ the rank two alternating $4\times 4$ matrix with
$\omega_{B}=e_{1}\wedge e_{2}$. Then $[Z_{1}]=1$ and $[Z_{2}]=1+e_{1}\wedge
e_{2}$ so that $\langle[Z_{1}],[Z_{2}]\rangle\subset Q^{+}$. ∎

###  6.10. The spinor map and the Plücker map

We relate the spinor and Plücker maps on $IG(n,2n)^{+}$. Even if the theory of
line bundles on homogeneous spaces provides a natural setting for the results
below (cf. [FH, §23.3, p.393], [BHH, §II]), we only use representation theory.

Let $\Gamma_{\lambda}$ be the irreducible $so(2n)$-representation with highest
weight $\lambda$. The irreducible $so(2n)$-representation $S^{\epsilon}$,
$\epsilon\in\\{+,\,-\\}$ has highest weight
$\omega_{n}:=(L_{1}+\ldots+L_{n})/2$, where $\epsilon=+$ if $n$ is even and
$\epsilon=-$ else, with highest weight vector $e_{*}:=e_{1}\wedge\ldots\wedge
e_{n}\in S^{\epsilon}$, [FH, Prop. 20.15] (for $n=4$ we wrote $\alpha$ for
$\omega_{n}$ in §3.2).

The highest weight of $Sym^{2}(S^{\epsilon})$ is thus
$2\omega_{n}=L_{1}+\ldots+L_{n}$, with highest weight vector $e_{*}\odot
e_{*}$. In particular, $\Gamma_{2\omega_{n}}$ is an irreducible component of
$Sym^{2}(S^{+})$. In case $n=4$ we have $\dim\Gamma_{2\omega_{n}}=35=\dim
Sym^{2}(\Gamma_{\omega_{n}})-1$ (see below for dimension formula) and thus
(see [FH, Exercise 19.6] for general $n$):

$Sym^{2}(S^{+})\,=\,Sym^{2}(\Gamma_{\omega_{n}})\,=\,\Gamma_{2\omega_{n}}\,\oplus\,\Gamma_{0}~{},\qquad(n\,=\,4),$

where $\Gamma_{0}$ is the trivial 1-dimensional representation (for $n=4$ the
representation $\Gamma_{\omega_{n}}$ has an invariant quadratic form and thus
is self-dual [FH, Exercise 20.38]), this quadratic form produces
$\Gamma_{0}$).

Now we consider the representation of $so(2n)$ on $\wedge^{n}V$. In the
standard representation $V$ of $so(2n)$ the basis vector $e_{i}$ has weight
$L_{i}$ (and the basis vector $e_{i+n}$ has weight $-L_{i}$). Thus
$e_{1}\wedge\ldots\wedge e_{n}\in\wedge^{n}V$ has weight
$L_{1}+\ldots+L_{n}=2\omega_{n}$ and it is the highest weight vector in
$\wedge^{n}V$ (cf. [FH, §19.2]). Therefore $\Gamma_{2\omega_{n}}$ is a
subrepresentation of $\wedge^{n}V$. The $so(2n)$-representation $\wedge^{n}V$
is in fact reducible and it has two irreducible components of the same
dimension ([FH, Remarks p.289-290; Exercise 24.43]),

$\wedge^{n}V\,=\,\Gamma_{2\omega_{n}}\,\oplus\,\Gamma_{2\omega_{n-1}},\qquad\dim\Gamma_{2\omega_{n}}\,=\,\dim\Gamma_{2\omega_{n-1}}\,=\,\mbox{$\frac{1}{2}$}\binom{2n}{n}~{},$

where $\omega_{n-1}:=(L_{1}+\ldots+L_{n-1}-L_{n})/2$ is the highest weight of
$S^{\epsilon^{\prime}}$ (where $\\{\epsilon,\epsilon^{\prime}\\}=\\{+,-\\}$).
This splitting is also obtained as the eigenspace decomposition for the Hodge
star operator defined by the bilinear form $(\bullet,\bullet)_{V}$, cf. [J,
Example 3.5.2]. The two summands are known as the selfdual and anti-selfdual
forms.

The Plücker map, restricted to $IG(n,2n)^{+}$, is the natural map
$\pi:IG(n,2n)^{+}\rightarrow{\bf P}\wedge^{n}V$ and for $n$ even it is thus
the composition

$IG(n,2n)^{+}\,\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}\,{\bf
P}S^{+}\,\stackrel{{\scriptstyle\nu}}{{\longrightarrow}}\,{\bf
P}\Gamma_{2\omega_{n}}\,\subset\,{\bf P}\wedge^{n}V,$

where $\nu$ is the second Veronese map (for odd $n$ replace $\omega_{n}$ by
$\omega_{n-1}$, the highest weight of $S^{+}$).

Since the spinor map is given by Pfaffians and the Plücker map is given by
minors on the open subset of $IG(n,2n)^{+}$ parametrized by alternating
matrices, this result implies that any quadratic expression in Pfaffians is a
linear combination of minors, see [BHH].

###  6.11. Cayley classes and $Spin(7)$

We now restrict ourselves to the case $n=4$. The half-spin representation
$\rho^{+}$ on $S^{+}$ maps the group $Spin(V)$ onto the orthogonal group
$SO(S^{+})$ of the bilinear form $(\bullet,\bullet)_{S^{+}}$. For any $s\in
S^{+}$ with $(s,s)_{S^{+}}\neq 0$, the stabilizer of $s$ in $SO(S^{+})$ is the
orthogonal group $SO(s^{\perp})\cong SO(7)$.

The inverse image of $SO(s^{\perp})$ in $Spin(V)$ is denoted by $Spin(V)_{s}$
and it is isomorphic to $Spin(7)$. In the standard representation $\rho_{V}$
of $Spin(V)$ on $V$, the subgroup $Spin(V)_{s}$ still acts irreducibly, in
fact $V$ is isomorphic with the (unique, irreducible) spin representation of
$Spin(7)$.

###  6.12. Representations of $spin(V)_{s}=so(7)$

Recall from §3.3 that the image of $s\odot s$ under the composition
$Sym^{2}(S^{+})\rightarrow\Gamma_{2\alpha}\hookrightarrow\wedge^{4}V$ is
called the Cayley class $c_{s}$ of $s$. Since $s$ is fixed by $Spin(V)_{s}$,
the 4-form $c_{s}$ is also fixed by the Lie algebra $spin(V)_{s}\cong so(7)$.
We now show that $c_{s}$ is the unique $spin(V)_{s}$-invariant in
$\wedge^{4}V$ by considering the restriction to $so(7)$ of the
$so(V)=so(8)$-representations considered in §6.10.

Multiplication by $s$ gives an inclusion of $spin(V)_{s}$-representations

$\begin{array}[]{rcl}S^{+}\,=\,\langle
s\rangle\,\oplus\,s^{\perp}\,\hookrightarrow\,Sym^{2}(S^{+})&=&\Gamma_{0}\,\oplus\,\Gamma_{2\omega_{n}}\\\
&=&\Gamma_{0}\,\oplus\,\langle c_{s}\rangle\,\oplus\,s\odot
s^{\perp}\,\oplus\,\Gamma_{(2,0,0)}\\\
&=&\Gamma_{0}\,\oplus\,\Gamma_{(0,0,0)}\,\oplus\,\Gamma_{(1,0,0)}\,\oplus\,\Gamma_{(2,0,0)}~{},\end{array}$

where $\Gamma_{0}$ and $\Gamma_{(0,0,0)}$ are trivial
$spin(V)_{s}$-representations, $\Gamma_{(1,0,0)}\cong s\odot s^{\perp}\cong
s^{\perp}$ is the standard seven dimensional representation of
$spin(V)_{s}\cong so(7)$ and $\Gamma_{(2,0,0)}$ is irreducible of dimension
$35-1-7=27$ (the notation $\Gamma_{(a,b,c)}$ for $so(7)$-representations is as
in [FH]).

The representation of $spin(V)_{s}$ on the $spin(V)$-representation
$\Gamma_{2\omega_{n}}$ is thus a direct sum of three irreducible
representations. Its representation on the other irreducible component
$\Gamma_{2\omega_{n-1}}$ of $\wedge^{4}V$ is irreducible and it is isomorphic
to $\Gamma_{(0,0,2)}$. Thus one has the $spin(7)=so(7)$-decomposition into
irreducible representations (cf. [Mun, Prop 2], [J, Prop. 10.5.4]):

$\wedge^{4}V\,=\,\Gamma_{(0,0,0)}\,\oplus\,\Gamma_{(1,0,0)}\,\oplus\,\Gamma_{(2,0,0)}\,\oplus\,\Gamma_{(0,0,2)}~{},$

since there is a unique copy of the trivial representation of $so(7)$ in
$\wedge^{4}V$, the Cayley class is the unique $spin(V)_{s}$ invariant in
$\wedge^{4}V$.

### 6.13.

The following proposition computes the 4-form $c_{s}$, which spans the trivial
$spin(V)_{s}$-subrepresentation $\Gamma_{(0,0,0)}$ in $\wedge^{4}V$,
explicitly in a case of interest in Markman’s paper, cf. [Mar, 1.4.1,
Proposition 11.2]. There $s$ is called $w=s_{n}$. We consider in fact
$\frac{1}{n+1}c_{w}$ and we write $n$ for his $n+1$. Notice that the
computation below uses only representation theory.

###  6.14. Proposition

Let $n\in{\mathbb{Z}}$, $n\neq 0$, and let $s=s_{n}=1-ne_{*}\in S^{+}$. where
$e_{*}:=e_{1}\wedge e_{2}\wedge e_{3}\wedge e_{4}\in\wedge^{even}W=S^{+}$.
Then we have, up to a scalar multiple,

$c_{s}\,=\,-n\alpha^{2}\,+\,4n^{2}\beta\,+4\gamma\quad(\in\wedge^{4}V)~{},$

where the forms, now in $\wedge^{*}V$, involved are:

$\alpha\,:=\,e_{1}\wedge e_{5}+\ldots+e_{4}\wedge
e_{8},\qquad\beta\,:=\,e_{1}\wedge\ldots\wedge
e_{4},\qquad\gamma\,:=\,e_{5}\wedge\ldots\wedge e_{8}~{}.$

Proof. The space of $spin(V)_{s}$-invariants in $\wedge^{4}V$ is one
dimensional and it is spanned by $c_{s}$, see §6.12. So it suffices to show
that the right hand side is a non-zero $spin(V)_{s}$-invariant form.

The Lie algebra $spin(V)_{1,e_{*}}$ that acts trivially on the two dimensional
subspace of $S^{+}$ spanned by $1,e_{*}$ is isomorphic to $so(6)\cong sl(4)$.
The representation of $sl(4)$ on $V=W\oplus W^{*}$ is reducible and $W$ is the
standard representation of $sl(4)$ whereas $W^{*}$ is the dual of the standard
representation. This implies that $\beta\in\wedge^{4}W\subset\wedge^{4}V$ and
$\gamma\in\wedge^{4}W^{*}\subset\wedge^{4}V$ as well as the 2-form $\alpha$,
which is the $sl(4)$-invariant in $W\otimes W^{*}\subset\wedge^{2}V$
corresponding to the symplectic form
$((w_{1},w_{1}^{*}),(w_{2},w_{2}^{*}))=w_{1}^{*}(w_{2})-w_{2}^{*}(w_{1})$ on
$V$, are $spin(V)_{1,e_{*}}$-invariants. On the other hand,

$\wedge^{4}(W\oplus W^{*})\,=\,\wedge^{4}W\,\oplus
W\otimes\wedge^{3}W^{*}\,\oplus\,\wedge^{2}W\otimes\wedge^{2}W^{*}\,\oplus\,\wedge^{3}W\otimes
W^{*}\,\oplus\,\wedge^{4}W^{*}~{}.$

Since $W,W^{*}$ have dimension four, $\wedge^{3}W^{*}\cong W$ and it is well-
known that there are no $sl(4)$-invariants in $W\otimes W$ nor in
$W^{*}\otimes W^{*}$. Also $\wedge^{2}W$ is irreducible and thus the
$sl(4)$-invariants in $\wedge^{2}W\otimes\wedge^{2}W^{*}\cong
End(\wedge^{2}W)$ are a one dimensional subspace spanned by the trace. Hence
the subspace of $sl(4)$-invariants in $\wedge^{4}V$ has dimension three. Since
$\alpha^{2},\beta,\gamma$ are linearly independent invariants, the invariant
subspace is

$(\wedge^{4}V)^{spin(V)_{1,e_{*}}}\,=\,(\wedge^{4}V)^{sl(4)}\,=\,\langle\alpha^{2},\,\beta,\gamma\rangle~{}.$

Since $spin(V)_{1,e_{*}}\subset spin(V)_{s}$ it remains to show that $c_{s}$
is the linear combination of $\alpha^{2},\beta,\gamma$ that is $spin(V)_{s}$
invariant. This 21-dimensional Lie algebra is defined by

$spin(V)_{s}\,=\,\\{X\,\in\,spin(V):\;Xs\,=\,0\,\\}$

and the action of $spin(V)$ on $S^{+}$ is given in the proof of Theorem 6.8.
It is then easy to check that the following elements (of $so(2n)\cong
spin(V)$) span $spin(V)_{s}$:

${\mathfrak{h}}_{s}\,:=\,\\{\sum a_{i}X_{i,i}:\;\sum a_{i}\,=\,0\,\\},\quad
X_{i,j}\quad(i\neq j),\quad
nY_{i,j}\,\pm\,Z_{k,l}\quad(\\{i,j,k,l\\}=\\{1,\ldots,4\\}),$

where the sign depends on $i,\ldots,l$. In particular, $X:=nY_{1,2}+Z_{3,4}\in
spin(V)_{s}$ (in fact $X$ acts as $e_{1}e_{2}+D_{e_{3}}D_{e_{4}}$ on $S^{+}$
and $X(1)=ne_{1}e_{2}$, $X(e_{*})=-e_{1}e_{2}$, so $Xs=0$). The action of $X$
on $V$ is given by

$\begin{array}[]{rclrclrclrcl}X(e_{1})&=&0,&X(e_{2})&=&0,&X(e_{3})&=&-e_{8},&X(e_{4})&=&e_{7},\\\
X(e_{5})&=&-ne_{7},&X(e_{6})&=&ne_{8},&X(e_{7})&=&0,&X(e_{8})&=&0~{}.\end{array}$

Since the Lie algebra element $X$ acts a derivation on $\wedge^{4}V$ we have

$X(\alpha)\,=\,X(e_{1})\wedge e_{5}+e_{1}\wedge
X(e_{5})+\ldots\,=\,-2ne_{1}\wedge e_{2}\,+\,2e_{7}\wedge e_{8}~{}.$

Thus

$X(\alpha^{2})\,=\,2\alpha\wedge X(\alpha)\,=\,-4n(e_{1}\wedge
e_{2})\wedge(e_{3}\wedge e_{7}+e_{4}\wedge e_{8})+4(e_{1}\wedge
e_{5}+e_{2}\wedge e_{6})\wedge(e_{7}\wedge e_{8})~{}.$

Similarly one finds

$X(\beta)\,=\,(e_{1}\wedge e_{2})\wedge(e_{4}\wedge e_{8}\,+\,e_{3}\wedge
e_{7}),\qquad X(\gamma)\,=\,-n(e_{2}\wedge e_{6}\,+\,e_{1}\wedge
e_{5})\wedge(e_{7}\wedge e_{8})~{}.$

Therefore the only non-trivial linear combination of $\alpha^{2},\beta,\gamma$
that is mapped to zero by $X$ is $-n\alpha^{2}+4n^{2}\beta+4\gamma$. Hence
this must be the unique $spin(V)_{s}$-invariant in $\wedge^{4}V$. ∎

###  6.15. Kuga Satake varieties

Let $S^{+}$ be the lattice introduced in §2.4 (and not its complexification).
As in Theorem 4.6, let $h,s\in S^{+}\cong U^{\oplus 4}$ be two perpendicular
elements such that their span is a positive definite sublattice. Let
$H=H_{h,s}$ be the rank $6$ sublattice of signature $(2+,4-)$ orthogonal to
$\langle h,s\rangle$:

$H\,:=\,\langle h,s\rangle^{\perp}\,=\,\\{t\in
S^{+}:\,(t,h)\,=\,(t,s)\,=\,0\,\\}~{}.$

With this notation we have

$\Omega_{\\{h,s\\}^{\perp}}\,=\,\\{\ell\in{\bf
P}H_{\mathbb{C}}:\quad(\ell,\ell)_{S^{+}}\,=\,0,\quad(\ell,\bar{\ell})_{S^{+}}\,>\,0\;\\}~{}.$

Recall that any $\ell\in\Omega_{\\{h,s\\}^{\perp}}$ defines an abelian
fourfold of Weil type with underlying torus ${\mathcal{T}}_{\ell}$ by Theorem
4.6. Such an $\ell$ also defines a weight two Hodge structure on $H$ denoted
by $H_{\ell}$ as follows:

$H_{\ell,{\mathbb{C}}}\,=\,H_{\mathbb{C}}\,=\,\oplus_{p+q=2}H^{p,q}_{\ell},\qquad
H^{2,0}_{\ell}\,:=\,{\mathbb{C}}\ell,\qquad
H^{0,2}_{\ell}\,:=\,{\mathbb{C}}\bar{\ell},\qquad
H^{1,1}_{\ell}\,=\,\big{(}H^{2,0}_{\ell}\oplus
H^{0,2}_{\ell}\big{)}^{\perp}~{}.$

This Hodge structure is polarized since the restriction of
$(\bullet,\bullet)_{S^{+}}$ to the two dimensional real subspace
$(H^{2,0}_{\ell}\oplus H^{0,2}_{\ell})\cap H_{\mathbb{R}}$ is positive
definite.

As $\dim H^{2,0}_{\ell}=1$, there is a Kuga Satake (abelian) variety
$A_{\ell}$, of dimension $16$, associated to $H_{\ell}$ (see [KS], [D],
[vG3]). In general, it has the property that $H_{\ell}$ is a Hodge
substructure of $H^{2}(A_{\ell}^{2},{\mathbb{Q}})$, but in this case there are
actually several copies of $H_{\ell}$ in $H^{2}(A_{\ell},{\mathbb{Q}})$, see
§6.18. The even Clifford algebra $C(H)^{+}$ of $H$ is a lattice in the real
vector space $C(H)^{+}\otimes_{\mathbb{Z}}{\mathbb{R}}$ of dimension
$2^{5}=32$. A complex structure on $C(H)^{+}_{\mathbb{R}}$ is defined by left
multiplication by $f_{1}f_{2}\in C(H)^{+}_{\mathbb{R}}$, with $f_{1},f_{2}\in
H_{\mathbb{R}}$ such that $(f_{1},f_{1})_{S^{+}}=1$ and
$H^{2,0}_{\ell}=\langle f_{1}+if_{2}\rangle$ (cf. [vG3, §5.6]). The abelian
variety $A_{\ell}$ is the quotient
$(C(H)^{+}_{\mathbb{R}},f_{1}f_{2})/C(H)^{+}$.

In [Lo, Cor. 6.3, Thm 6.4] it is shown that $A_{\ell}$ is isogeneous to
$B_{\ell}^{4}$, where $B_{\ell}$ is an abelian fourfold of Weil type with
trivial discriminant. The following proposition, due to O’Grady ([O’G, §5.3]),
shows that $B_{\ell}$ and ${\mathcal{T}}_{\ell}$ are isogeneous. In [O’G] one
finds a more explicit description of this result, as well as applications to
generalized Kummer varieties.

###  6.16. Proposition

For $\ell\in\Omega_{\\{h,s\\}^{\perp}}$ the Kuga Satake variety $A_{\ell}$ of
the polarized weight two Hodge structure $H_{\ell}$ is isogeneous to
${\mathcal{T}}_{\ell}^{4}$, where ${\mathcal{T}}_{\ell}$ is the abelian
fourfold of Weil type defined by $\ell$.

Proof. The right multiplication on $C(H)^{+}_{\mathbb{R}}$ by an element of
$C(H)^{+}$ preserves the lattice, commutes with the complex structure and thus
defines an element in $\mbox{End}(A_{\ell})$. The ${\mathbb{Q}}$ vector space
$H_{\mathbb{Q}}$ is not a direct sum of two maximally isotropic subspaces and,
whereas $C(H)^{+}_{\mathbb{C}}\cong M_{4}({\mathbb{C}})\oplus
M_{4}({\mathbb{C}})$ (as in §6.2), one now has an isomorphism of algebras
([Lo, Thm. 6.2]), where $M_{4}(K)$ are the $4\times 4$ matrices with
coefficients in $K$,

$C(H)^{+}_{\mathbb{Q}}\,:=\,C(H)^{+}\otimes_{\mathbb{Z}}{\mathbb{Q}}\,\cong\,M_{4}(K)\,\subseteq\,\mbox{End}(A_{\ell})_{\mathbb{Q}},\qquad
K\,:=\,{\mathbb{Q}}(\sqrt{-ab}).$

This implies that any $A_{\ell}$ is isogeneous to $B_{\ell}^{4}$, where
$B_{\ell}$ is an abelian fourfold with
$K\subset\mbox{End}(B_{\ell})_{\mathbb{Q}}$ ($B_{\ell}$ is only determined up
to isogeny).

It remains to show that $B_{\ell}$ and ${\mathcal{T}}_{\ell}$ are isogeneous.
The inclusion $Spin(H)\subset Spin(S^{+})=Spin(V)$ defines a representation of
$Spin(H)$ on $V$ which is its spin representation. The isomorphism
$C(H)^{+}_{\mathbb{Q}}\cong M_{4}(K)$ implies that

$C(H)^{+}_{\mathbb{Q}}\,\cong\,V_{\mathbb{Q}}^{\oplus 4}$

as $Spin(H)$-representations. The same holds with ${\mathbb{Q}}$ replaced by
${\mathbb{R}}$. The weight two Hodge structure on the $Spin(H)$-representation
$H_{\ell}$ is defined by the one parameter subgroup $h_{\ell}$ of
$Spin(H)_{\mathbb{R}}\subset Spin(S^{+})_{\mathbb{R}}$ introduced in the proof
of Proposition 3.5. In fact, $h_{\ell}(t)$ acts on $S^{+}$ as multiplication
by $t^{2}$ on ${\mathbb{C}}\ell$, by $t^{-2}$ on ${\mathbb{C}}\bar{\ell}$ and
it is trivial on $\langle\ell,\bar{\ell}\rangle^{\perp}$. The complex
structure on $C(H)^{+}_{\mathbb{R}}\cong V_{\mathbb{R}}^{\oplus 4}$, which
defines the Kuga Satake variety $A_{\ell}\sim B_{\ell}^{\oplus 4}$, is also
defined by $h_{\ell}$ ([vG3, Prop. 6.3]), now acting on $V_{\mathbb{R}}^{4}$.
As $\rho_{V}(h_{\ell})=h_{V,\ell}$, the complex structure is $J_{\ell}$ on
$V_{\mathbb{R}}$. It follows that $B_{\ell}$ and ${\mathcal{T}}_{\ell}$ are
isogeneous. ∎

###  6.17. Remark

The proof of Proposition 6.16 uses the (algebraic) subgroup
$Spin(H)=Spin_{h,s}$ of $Spin(S^{+})=Spin(V)$. The decomposition
$S^{+}_{\mathbb{Q}}=H_{\mathbb{Q}}\oplus R_{\mathbb{Q}}$, with $R:=\langle
h,s\rangle$, implies that we actually have two commuting subgroups
$Spin(H),Spin(R)\subset Spin(S^{+})$.

Recall from §4.7 that
$R_{\mathbb{C}}={\mathbb{C}}\kappa\oplus{\mathbb{C}}\bar{\kappa}$ with
$\kappa,\bar{\kappa}\in Q^{+}$. The decomposition of
$V_{\mathbb{C}}=Z_{\kappa,{\mathbb{C}}}\oplus Z_{\bar{\kappa},{\mathbb{C}}}$
in the two isotropic eigenspaces for the $K$-action defines, as in Lemma 6.9,
a one parameter subgroup $h_{R}$ of $Spin(S^{+}_{\mathbb{R}})$. As
$h_{R}(t)\kappa=t^{2}\kappa$, $h_{R}(t)\overline{\kappa}=t^{-2}\bar{\kappa}$,
this identifies the subgroup $Spin(R_{\mathbb{R}})$ with this one parameter
subgroup, $h_{R}(U(1))=Spin(R_{\mathbb{R}})$. In particular, the $K$-action on
$V_{\mathbb{Q}}$ is generated by $Spin(R)$ and the scalar multiples of the
identity.

The fact that $Spin(H),Spin(R)\subset Spin(S^{+})$ commute implies that the
subspaces $Z_{\kappa,{\mathbb{C}}},Z_{\bar{\kappa},{\mathbb{C}}}$ are
$Spin(H_{\mathbb{C}})$-invariant subspaces. Thus the spin representation of
$Spin(H_{\mathbb{C}})$ on $V_{\mathbb{C}}$ is reducible. These two subspaces
are the two half-spin representations of $Spin(H_{\mathbb{C}})$.

There is an isomorphism $Spin(H_{\mathbb{C}})\cong SL(4,{\mathbb{C}})$ and the
half-spin representations are identified with the standard representation
${\mathbb{C}}^{4}$ of $SL(4,{\mathbb{C}})$ and its dual
$({\mathbb{C}}^{4})^{*}$. The representation $H_{\mathbb{C}}$ is identified
with $\wedge^{2}{\mathbb{C}}^{4}\cong\wedge^{2}({\mathbb{C}}^{4})^{*}$, the
isomorphism follows from the pairing, defined by the wedge product,
$(\wedge^{2}{\mathbb{C}}^{4})\times(\wedge^{2}{\mathbb{C}}^{4})\rightarrow\wedge^{4}{\mathbb{C}}^{4}\cong{\mathbb{C}}$.

###  6.18. The second cohomology group of ${\mathcal{T}}_{\ell}$

In [Lo] the Hodge structure on the second cohomology group
$H^{2}(B,{\mathbb{Q}})$ of an abelian fourfold of Weil type with field $K$ is
studied. This group has dimension $\binom{8}{2}=28$ and decomposes under the
$K$-action into a $16=1+15$-dimensional subspace $S_{B}^{\prime}$ on which
$x\in K$ acts as $x\bar{x}$, this subspace includes the polarization of Weil
type. There is a complementary subspace $S_{B}$ on which the eigenvalues of
$x$ are $x^{2},\bar{x}^{2}$ of dimension $12$. This subspace can be identified
with the six dimensional $K$ vector space $\wedge_{K}^{2}H^{1}(B,K)$.

$H^{2}(B,{\mathbb{Q}})\,=\,S_{B}\,\oplus\,S^{\prime}_{B},\qquad
S_{B}^{\prime}\,:=\,\\{\xi\in
H^{2}(B,{\mathbb{Q}}):\,x^{*}\xi\,=\,x\bar{x}\xi,\quad\forall x\in K\,\\}~{}.$

For a general fourfold of Weil type (so $SMT(B)_{\mathbb{R}}\cong SU(2,2)$)
the Hodge structure $S_{B}$ is a simple Hodge structure (so does not admit
non-trivial Hodge substructures) if and only if the discriminant of $B$ is
non-trivial [Lo, Cor. 3.6].

In case the discriminant is trivial, one finds that $S_{B}\cong H_{B}^{\oplus
2}$, for a weight two, rank six, polarized, Hodge structure $H_{B}$ which has
Hodge numbers $(1,4,1)$. Moreover, the Kuga Satake variety of $H_{B}$ is
isogeneous to $B^{4}$, so one recovers the weight two Hodge structure $H_{B}$
from its Kuga Satake variety.

The following proposition uses this result to show that the abelian fourfolds
of Weil type ${\mathcal{T}}_{\ell}$ have trivial discriminant.

###  6.19. Proposition

For $\ell\in\Omega_{\\{h,s\\}^{\perp}}$, with $h,s$ as in Theorem 4.6, the
polarized abelian fourfold of Weil type $({\mathcal{T}}_{\ell},K,\omega_{K})$
has trivial discriminant.

Proof. By [Lo, Cor. 3.6] it suffices to show that
$(H^{2}_{\ell,{\mathbb{Q}}})^{\oplus 2}$ is isomorphic to the Hodge
substructure $S_{{\mathcal{T}}_{\ell}}\subset
H^{2}({\mathcal{T}}_{\ell},{\mathbb{Q}})$.

As in the proof of Proposition 3.5, the (weight one) Hodge structure on
$V=H^{1}({\mathcal{T}}_{\ell},{\mathbb{Z}})$ defines a one parameter subgroup
$h_{\ell}$ in $Spin(V)$ (actually in $Spin(V)_{h,s}\subset
Spin(S^{+})=Spin(V)$). A representation $U$ of $Spin(V_{\mathbb{R}})$ on a
real vector space $U$ defines a Hodge decomposition $U_{\mathbb{C}}=\oplus
U^{p,q}$, with $\overline{U^{p,q}}=U^{q,p}$, given by the eigenspaces
$U^{p,q}=\\{u\in U:h_{\ell}(z)u=z^{a}\bar{z}^{b}u$ (but the weight is not
uniquely defined since $z\bar{z}=1$).

The representation $\rho^{+}$ on $S^{+}_{\mathbb{R}}$ has the Hodge
decomposition

$(S^{+})^{2,0}\,=\,H^{2,0}_{\ell}\,=\,{\mathbb{C}}\ell,\qquad(S^{+})^{0,2}\,=\,\overline{(S^{+})^{2,0}},\qquad\qquad(S^{+})^{1,1}\,=\,\big{(}(S^{+})^{2,0}\oplus(S^{+})^{0,2}\big{)}^{\perp}$

since these spaces are the eigenspaces for $h_{\ell}$ acting on
$S^{+}_{\mathbb{C}}$ (see Lemma 6.9). The Hodge structure $S^{+}_{\mathbb{Q}}$
is a direct sum of Hodge structures

$S^{+}_{\mathbb{Q}}\,=\,H^{2}_{\ell,{\mathbb{Q}}}\,\oplus\,R_{\mathbb{Q}},\qquad
R\,:=\,\langle h,s\rangle~{},$

where $R_{\mathbb{Q}}\cong{\mathbb{Q}}(-1)^{2}$ is a trivial Hodge
substructure with $R_{\mathbb{Q}}^{1,1}=R_{\mathbb{C}}$.

There is an isomorphism of $Spin(V)=Spin(S^{+})$-representations
$\wedge^{2}S^{+}=\wedge^{2}V$ (both are the irreducible $so(8)$-representation
with highest weight
$(L_{1}+L_{2}+L_{3}+L_{4})/2+(L_{1}+L_{2}-L_{3}-L_{4})/2=L_{1}+L_{2}$). Hence
we get a splitting of the Hodge structure on $\wedge^{2}S^{+}_{\mathbb{Q}}$
(which is again defined by $h_{\ell}$ eigenspaces) in three Hodge
substructures which have dimensions $\binom{6}{2}=15$, $6\cdot 2=12$ and $1$
respectively:

$\wedge^{2}S^{+}_{\mathbb{Q}}\,=\,(\wedge^{2}H^{2}_{\ell,{\mathbb{Q}}})\;\oplus\;(H^{2}_{\ell,{\mathbb{Q}}}\otimes
R_{\mathbb{Q}})\;\oplus\;(\wedge^{2}R_{\mathbb{Q}})~{}.$

(The Hodge structure $S^{+}$ has weight two, so the Hodge structure on
$\wedge^{2}S^{+}$ should have weight four. However, $(\dim S^{+})^{2,0}=1$, so
$\wedge^{2}S^{+}_{\mathbb{Q}}$ has trivial $(4,0)$ and $(0,4)$ summands and
thus it is the Tate twist of a weight two Hodge structure.)

Using the isomorphisms
$\wedge^{2}S^{+}_{\mathbb{Q}}=\wedge^{2}V=H^{2}({\mathcal{T}}_{\ell},{\mathbb{Z}})$
we see that

$H^{2}_{\ell,{\mathbb{Q}}}\otimes
R_{\mathbb{Q}}\,\cong\,(H^{2}_{\ell,{\mathbb{Q}}})^{\oplus
2}\;\hookrightarrow\;H^{2}({\mathcal{T}}_{\ell},{\mathbb{Q}})$

is a non-simple Hodge substructure of
$H^{2}({\mathcal{T}}_{\ell},{\mathbb{Q}})$.

It remains to check that $x\in K$ has eigenvalues $x^{2},\bar{x}^{2}$ on this
substructure. One can deduce this from the fact that representation
$\wedge^{2}V_{\mathbb{C}}$ of the complex Mumford Tate group
$SL(4,{\mathbb{C}})$ of ${\mathcal{T}}_{\ell}$ is isomorphic to

$\wedge^{2}({\mathbb{C}}^{4}\oplus({\mathbb{C}}^{4})^{*})\cong(\wedge^{2}{\mathbb{C}}^{4})^{\oplus
2}\,\oplus\,{\mathbb{C}}^{4}\otimes({\mathbb{C}}^{4})^{*}$

and the last summand is the direct sum of a trivial one dimensional
representation and an irreducible 15 dimensional representation. As the
complexification of a Hodge substructure is a subrepresentation, there is a
unique subrepresentation of dimension $12$. Hence
$S_{{\mathcal{T}}_{\ell}}=H^{2}_{\ell,{\mathbb{Q}}}\otimes R_{\mathbb{Q}}$ as
desired.

Alternatively, by Remark 6.17, the $K^{\times}$-action is essentially given by
the subgroup $Spin(R)$ of $Spin(S^{+})$. This subgroup acts trivially on
$\wedge^{2}H^{2}_{\ell,{\mathbb{Q}}}$ and $\wedge^{2}R_{\mathbb{Q}}$, so $K$
acts through the norm on these summands. Therefore
$S_{{\mathcal{T}}_{\ell}}=H^{2}_{\ell,{\mathbb{Q}}}\otimes R_{\mathbb{Q}}$ is
not simple. ∎

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