Consider subspaces $V_{1}\subset V_{5}\subset V_{6}$ such that

(4.20) $[V_{1}]\in Y^{2}_{A},\quad[V_{5}]\in Y^{2}_{A^{\perp}},\quad
A\cap(V_{1}\wedge\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{2}}\hskip
0.56905pt{V}_{5})=0.$

Let $X$ and $X^{\prime}$ be the smooth ordinary GM threefolds corresponding to
the Lagrangian data sets $(V_{6},V_{5},A)$ and
$(V_{6}^{\vee},V_{1}^{\perp},A^{\perp})$, and let $L_{0}\subset X$ and
$L^{\prime}_{0}\subset X^{\prime}$ be the lines corresponding to the points
$[V_{1}]$ of $Y^{2}_{A,V_{5}}$ and $[V_{5}^{\perp}]$ of
$Y^{2}_{A^{\bot},V_{1}^{\bot}}$ via the maps (2.23).

There is a diagram of birational maps

(4.21)
$\textstyle{\operatorname{Bl}_{L_{0}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\scriptstyle{\beta}$$\scriptstyle{\psi}$$\textstyle{\operatorname{Bl}_{L^{\prime}_{0}}(X^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X^{\prime}}}$$\scriptstyle{\beta^{\prime}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\varpi}}$$\textstyle{\bar{X}}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{{\varpi^{\prime}}}$

where $\beta$ and $\beta^{\prime}$ are the respective blow ups of $X$ and
$X^{\prime}$ along the lines $L_{0}$ and $L^{\prime}_{0}$, the birational maps
${\varpi}$ and ${\varpi^{\prime}}$ are the respective linear projections of
$X$ from $L_{0}$, and of $X^{\prime}$ from $L^{\prime}_{0}$, the morphisms
$\rho_{X}$ and $\rho_{X^{\prime}}$ are small crepant extremal birational
contractions, the variety $\bar{X}$ is a normal, Gorenstein, cubic
hypersurface in
$\operatorname{\mathsf{Gr}}(2,V_{5}/V_{1})={\operatorname{\mathsf{Gr}}(2,V_{1}^{\perp}/V_{5}^{\perp})}$
with terminal singularities, and the map
$\psi=\rho_{X^{\prime}}^{-1}\circ\rho_{X}$ is a flop.

The proposition says that the birational isomorphism
${\varpi}^{\prime-1}\circ{\varpi}\colon X\dashrightarrow X^{\prime}$ is the
“elementary rational map with center along the line $L_{0}$” in the sense of
[IP99, (4.1.1)] (or elementary transformation along the line $L_{0}$) if the
GM threefold $X$ and the line $L_{0}$ are sufficiently general. We use the
proposition and the fact that the elementary transformation is defined for any
$X$ and $L_{0}$ to specify what the moduli point of the result of the
elementary transformation is in general (we use the description of the coarse
moduli space $\mathbf{M}_{3}^{\rm GM}$ for GM threefolds given in (2.8)).

######  Corollary 4.18.

Let $X$ be a smooth GM threefold with moduli point
$([A],[V_{5}])\in\mathbf{M}_{3}^{\rm GM}$ and let $L\subset X$ be any line.
The moduli point of the result $X^{\prime}_{L}$ of the elementary
transformation of $X$ along the line $L$ is
$([A^{\perp}],\sigma(L))\in\mathbf{M}_{3}^{\rm GM}$.

###### Proof.

Let $\mathfrak M_{3}^{\rm GM}$ be the moduli stack of smooth GM threefolds
(see [DK20b]), let $\mathscr{X}\to\mathfrak M_{3}^{\rm GM}$ be the universal
family of threefolds over it, and let $F_{1}(\mathscr{X}/\mathfrak M_{3}^{\rm
GM})$ be the relative Hilbert scheme of lines. As we already mentioned, by
[IP99, Section 4.1], the elementary transformation is defined for any line
contained in any smooth GM threefold $X$. Moreover, this transformation can be
performed for a family of lines and will produce a family of GM threefolds.
This defines a morphism

$\displaystyle F_{1}(\mathscr{X}/\mathfrak M_{3}^{\rm GM})$
$\displaystyle\longrightarrow$ $\displaystyle\mathbf{M}_{3}^{\rm GM}$
$\displaystyle([X],[L])$ $\displaystyle\longmapsto$
$\displaystyle[X^{\prime}_{L}].$

By [DK20b, Theorem 5.11] and [DK19, Theorem 4.7], the left side is irreducible
and birational to

$\\{(A,V_{5},V_{1})\in\operatorname{\mathsf{LGr}}_{\rm
ndv}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{3}}\hskip
0.56905pt{V}_{6})\times{\operatorname{\mathsf{Fl}}(1,5;V_{6})}\mid[V_{5}]\in
Y^{\geq 2}_{A^{\perp}},\ [V_{1}]\in Y^{\geq
2}_{A}\\}/\\!\\!/\operatorname{PGL}(V_{6}).$

By Proposition 4.17, over the dense open subset of triples $(A,V_{5},V_{1})$
satisfying condition (4.20), this map coincides with the projection

$(A,V_{5},V_{1})\mapsto({A^{\perp}},{V_{1}^{\perp}})\in\mathbf{M}_{3}^{\rm
GM}.$

Since the target is separated ([DK20b, Theorem 5.15]), by continuity, the two
maps coincide everywhere. ∎

To prove Proposition 4.17, we start with some preliminaries. First, subspaces
$V_{1}$ and $V_{5}$ satisfying the conditions (4.20) exist: this follows from
[DK18, Lemma B.5] (where $\widehat{Y}_{A}$ is defined in [DK18, (B.5)]). More
exactly, for $[V_{5}]$ general in $Y^{2}_{A^{\perp}}$ (so that $X$ is a
general GM threefold associated with the fixed Lagrangian $A$), these
conditions will be satisfied for a general $[V_{1}]\in Y^{2}_{A,V_{5}}$,
corresponding to a general line $L_{0}\subset X$. As explained in the proof of
[DK18, Theorem 4.20], the conditions (4.20) are equivalent to

$[V_{1}]\in
Y^{2}_{A,V_{5}}\smallsetminus{}_{1}(X)\qquad\textnormal{and}\qquad[V_{5}^{\perp}]\in
Y^{2}_{A^{\perp},V_{1}^{\perp}}\smallsetminus{}_{1}(X^{\prime}),$

hence the lines $L_{0}$ and $L^{\prime}_{0}$ are nice and the assumptions of
[DK18, Proposition 4.19] are satisfied. It was explained in the proof of that
proposition that these lines can be written as

(4.22) $L_{0}={\bf P}(V_{1}\wedge V_{3}),\qquad L^{\prime}_{0}={\bf
P}(V_{5}^{\perp}\wedge V_{3}^{\perp})$

for the same subspace $V_{3}$ such that $V_{1}\subset V_{3}\subset V_{5}$.

Following [DK18, Section 4.4], we introduce the second quadratic fibration

$\rho_{2}\colon{\bf
P}_{{X}}(V_{5}/\mathscr{U}_{X})\longrightarrow\operatorname{\mathsf{Gr}}(3,V_{5}),$

and analogously for $X^{\prime}$, and study the diagram [DK18, (4.5)]

(4.23)
$\textstyle{\widetilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\tilde{\rho}_{2}}$$\textstyle{\widetilde{X}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\rho}^{\prime}_{2}}$$\scriptstyle{f^{\prime}}$$\textstyle{X}$$\textstyle{\operatorname{\mathsf{Gr}}(2,V_{5}/V_{1})}$$\textstyle{X^{\prime},}$

where ${\tilde{\rho}_{2}}$ is obtained from $\rho_{2}$ by restriction to
$\operatorname{\mathsf{Gr}}(2,V_{5}/V_{1})\subset\operatorname{\mathsf{Gr}}(3,V_{5})$
and analogously for $\tilde{\rho}^{\prime}_{2}$. The next lemma is a
refinement of [DK18, Lemma 4.18].

######  Lemma 4.19.

The scheme $\widetilde{X}$ has two irreducible components,
$\operatorname{Bl}_{L_{0}}(X)$ and $f^{-1}(L_{0})$. They are both smooth of
dimension $3$ and meet transversely along the exceptional divisor of
$\operatorname{Bl}_{L_{0}}(X)$.

Moreover, the map
$\tilde{\rho}_{2}\colon\widetilde{X}\to\operatorname{\mathsf{Gr}}(2,V_{5}/V_{1})$
is induced by the linear projection from the line $L_{0}$.

###### Proof.

We defined in [DK18, Section 4.4] the EPW quartic hypersurface
$Z_{A}\subset\operatorname{\mathsf{Gr}}(3,V_{6})$. For any subspace
$V_{1}\subset V_{5}$, we denote by

$Z_{A,V_{5}}\subset\operatorname{\mathsf{Gr}}(3,V_{5})\quad\textnormal{and}\quad
Z_{A,V_{1},V_{5}}\subset\operatorname{\mathsf{Gr}}(2,V_{5}/V_{1})$

the subschemes obtained by intersecting $Z_{A}$ with the subvarieties
$\operatorname{\mathsf{Gr}}(3,V_{5})$ and
$\operatorname{\mathsf{Gr}}(2,V_{5}/V_{1})$ of
$\operatorname{\mathsf{Gr}}(3,V_{6})$.

Consider the commutative diagram

---
$\textstyle{{\bf
P}_{X}(V_{5}/\mathscr{U}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{2}}$$\textstyle{{\bf
P}_{M_{X}}(V_{5}/\mathscr{U}_{M_{X}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
P}_{\operatorname{\mathsf{Gr}}(2,V_{5})}(V_{5}/\mathscr{U})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{\mathsf{Fl}}(2,3;V_{5})}$$\textstyle{Z_{A,V_{5}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{\mathsf{Gr}}(3,V_{5}),}$

where $M_{X}$ was defined in (2.7) and the vertical arrow is the canonical
projection, which induces the diagonal arrows (the left diagonal arrow factors
through $Z_{A,V_{5}}$ by [DK18, Proposition 4.10]). Pulling this diagram back
by the inclusion
$\operatorname{\mathsf{Gr}}(2,V_{5}/V_{1})\subset\operatorname{\mathsf{Gr}}(3,V_{5})$,
we obtain the diagram

$\textstyle{\widetilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\rho}_{2}}$$\textstyle{\operatorname{Bl}_{L_{0}}(M_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Bl}_{{\bf
P}(V_{1}\wedge
V_{5})}(\operatorname{\mathsf{Gr}}(2,V_{5}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z_{A,V_{1},V_{5}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{\mathsf{Gr}}(2,V_{5}/V_{1}).}$

Indeed,
$\operatorname{\mathsf{Gr}}(2,V_{5}/V_{1})\subset\operatorname{\mathsf{Gr}}(3,V_{5})$
is the zero-locus of the section of the vector bundle $V_{5}/\mathscr{U}_{3}$
corresponding to $V_{1}$ and, by [Kuz16, Lemma 2.1], the zero-locus of the
corresponding section on ${\bf
P}_{\operatorname{\mathsf{Gr}}(2,V_{5})}(V_{5}/\mathscr{U})$ is the blow up of
$\operatorname{\mathsf{Gr}}(2,V_{5})$ along the zero-locus of induced section
of $V_{5}/\mathscr{U}$, which is equal to the locus ${\bf P}(V_{1}\wedge
V_{5})$ of 2-dimensional subspaces in $V_{5}$ containing $V_{1}$. Note also
that the map

$\operatorname{Bl}_{{\bf P}(V_{1}\wedge
V_{5})}(\operatorname{\mathsf{Gr}}(2,V_{5}))\to\operatorname{\mathsf{Gr}}(2,V_{5}/V_{1})$

is induced by the linear projection $V_{5}\to V_{5}/V_{1}$ from $V_{1}$, or
equivalently by the linear projection ${\bf
P}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{2}}\hskip
0.56905pt{V}_{5})\dashrightarrow{\bf
P}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{2}}\hskip
0.56905pt{(}V_{5}/V_{1}))$ from ${\bf P}(V_{1}\wedge V_{5})$.

Furthermore, $M_{X}\subset\operatorname{\mathsf{Gr}}(2,V_{5})$ is the linear
section by the subspace ${\bf P}(W)\subset{\bf
P}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{2}}\hskip
0.56905pt{V}_{5})$ which is transverse to ${\bf P}(V_{1}\wedge V_{5})$ by
Lemma 4.6, because the line $L_{0}$ is nice; the pullback of ${\bf
P}_{M_{X}}(V_{5}/\mathscr{U}_{M_{X}})$ is therefore
$\operatorname{Bl}_{L_{0}}(M_{X})$. Moreover, the map
$\operatorname{Bl}_{L_{0}}(M_{X})\to\operatorname{\mathsf{Gr}}(2,V_{5}/V_{1})$
is induced by the linear projection from ${\bf P}(W\cap(V_{1}\wedge
V_{5}))=L_{0}$.

To prove the lemma, it remains to invoke the equality

$\operatorname{Bl}_{L_{0}}(M_{X})\times_{M_{X}}X=\operatorname{Bl}_{L_{0}}(X)\cup
f^{-1}(L_{0}),$

which holds because the first component is the strict transform of $X$ and the
second component is the exceptional divisor of
$\operatorname{Bl}_{L_{0}}(M_{X})\to M_{X}$. ∎

It was proved in [DK18, Lemma 4.18 and Proposition 4.19] that

  * •

$\tilde{\rho}_{2}$ maps $f^{-1}(L_{0})$ birationally onto the Schubert
hyperplane divisor
${\mathsf{D}}\subset\operatorname{\mathsf{Gr}}(2,V_{5}/V_{1})$ parameterizing
subspaces intersecting $V_{3}/V_{1}$, where $V_{3}$ was defined in (4.22);

  * •

$\tilde{\rho}_{2}$ maps $\operatorname{Bl}_{L_{0}}(X)$ birationally onto a
cubic hypersurface $\bar{X}\subset\operatorname{\mathsf{Gr}}(2,V_{5}/V_{1})$;

  * •

the image of $\tilde{\rho}_{2}$ is the quartic hypersurface
$Z_{A,V_{1},V_{5}}$; it is therefore equal to $\bar{X}\cup{\mathsf{D}}$.

We denote by $\rho_{X}\colon\operatorname{Bl}_{L_{0}}(X)\to\bar{X}$ the
(birational) restriction of $\tilde{\rho}_{2}$ and we define
$\rho_{X^{\prime}}$ similarly.

###### Proof of Proposition 4.17.

We have already constructed the left part of the diagram (4.21). The right
part is constructed analogously. It remains to prove that $\psi$ is a flop.

As explained in [IP99, Lemma 4.1.1 and Corollary 4.3.2], $\rho_{X}$ is a
flopping contraction. The same is true for $\rho_{X^{\prime}}$, so $\psi$ is
either a flop or an isomorphism. If $\psi$ is an isomorphism, the maps $\beta$
and $\beta^{\prime}$ are the contractions of the same extremal ray, hence
$X\simeq X^{\prime}$. Let us show that this is impossible.

Indeed, we can perform this construction on a fixed $X$ (that is, with $A$ and
$V_{5}$ fixed) but with $[V_{1}]$ varying in the curve $Y^{2}_{A,V_{5}}$.
Locally, the map $\psi$ will remain an isomorphism and the threefolds
$X^{\prime}$ obtained by the construction will be all isomorphic to $X$. But
this is impossible: by definition, $X^{\prime}$ is the ordinary GM threefold
corresponding to the Lagrangian data set
$(V_{6}^{\vee},V_{1}^{\perp},A^{\perp})$, hence its moduli point describes the
curve

$Y^{2}_{A,V_{5}}/\\!\\!/\operatorname{PGL}(V_{6})_{A}\subset{\mathfrak{p}}_{3}^{-1}([A^{\perp}])\subset\mathbf{M}_{3}^{\rm
GM}$

(recall that the map $\mathfrak{p}_{3}$ was defined in (2.9) and that the
group $\operatorname{PGL}(V_{6})_{A}$ is finite). It follows that $\psi$ is a
flop and the proof of the proposition is complete. ∎

######  Remark 4.20.

One can describe the maps $\rho_{X}$ and $\rho_{X^{\prime}}$ in (4.21)
further: in fact, the intersections with
$\operatorname{\mathsf{Gr}}(2,V_{5}/V_{1})$ of the subscheme
${}_{2}(X)\subset\operatorname{\mathsf{Gr}}(3,V_{5})$ defined in [DK18, (4.2)]
and of the analogous subscheme
${}_{2}(X^{\prime})\subset\operatorname{\mathsf{Gr}}(2,V_{6}/V_{1})$ are cubic
surfaces (cubic scrolls or cones) ${}_{2,V_{1}}(X)$ and
${}_{2,V_{5}}(X^{\prime})$ such that

$\bar{X}\cap{\mathsf{D}}={}_{2,V_{1}}(X)\cup{}_{2,V_{5}}(X^{\prime}).$

The morphisms $\rho_{X}$ and $\rho_{X^{\prime}}$ are the blow ups of $\bar{X}$
along the Weil divisors ${}_{2,V_{1}}(X)$ and ${}_{2,V_{5}}(X^{\prime})$,
respectively.

## 5\. Intermediate Jacobians of GM fivefolds

In this section, we perform, for GM fivefolds, a construction analogous to
what we did in Section 4 for threefolds. The curves in the construction are
replaced by surfaces: lines by planes, elliptic quintic curves by quintic del
Pezzo surfaces, and rational quartic curves by rational quartic surface
scrolls. In Section 5.5, we give an alternative proof of the main result.

### 5.1. Family of surfaces

Given an arbitrary GM fivefold $X$ with associated Lagrangian $A$, we begin by
choosing an arbitrary $\sigma$-plane ${}_{0}\subset X$ (that is, a point of
$F^{2}_{\sigma}(X)$; see (2.2)). We consider the open surface $S_{0}\subset
Y_{A}^{\geq 2}$ defined by (2.14) and the family of quadrics
$\mathscr{Q}_{0}\to S_{0}$ obtained by restricting to $S_{0}$ the universal
family (2.11) of 8-dimensional quadrics containing $X$. We denote by
$F^{5}(\mathscr{Q}_{0}/S_{0})\to S_{0}$ the relative Hilbert scheme of linear
5-spaces in the fibers of $\mathscr{Q}_{0}\to S_{0}$ and by
$F^{5}_{{}_{0}}(\mathscr{Q}_{0}/S_{0})\to S_{0}$ the subscheme parameterizing
those 5-spaces which contain the plane 0. Applying Corollary 2.9, we obtain an
isomorphism

(5.1)
$F^{5}_{{}_{0}}(\mathscr{Q}_{0}/S_{0})\simeq\widetilde{S}_{0}:={\widetilde{Y}_{A}^{2}}\times_{{Y_{A}^{2}}}S_{0}$

of schemes over $S_{0}$. In particular, the canonical map
$F^{5}_{{}_{0}}(\mathscr{Q}_{0}/S_{0})\to S_{0}$ is the double étale covering
$\pi\colon\widetilde{S}_{0}\to S_{0}$ induced by the double covering
$\pi_{A}$.

Note that $\widetilde{S}_{0}$ is a smooth surface. Let

$\widetilde{\mathscr{Q}}_{0}:=\mathscr{Q}_{0}\times_{S_{0}}\widetilde{S}_{0}$

be the base change of the family of quadrics $\mathscr{Q}_{0}\to S_{0}$ along
$\pi$. We have a canonical map

$\widetilde{S}_{0}\to
F^{5}_{{}_{0}}(\mathscr{Q}_{0}/S_{0})\times_{S_{0}}\widetilde{S}_{0}\hookrightarrow
F^{5}(\mathscr{Q}_{0}/S_{0})\times_{S_{0}}\widetilde{S}_{0}\simeq
F^{5}(\widetilde{\mathscr{Q}}_{0}/\widetilde{S}_{0}),$

where the first map is the product of the isomorphism (5.1) with the identity
map. By construction, it is a section of the projection
$F^{5}(\widetilde{\mathscr{Q}}_{0}/\widetilde{S}_{0})\to\widetilde{S}_{0}$.

Let $\mathscr{P}^{5}\subset\widetilde{\mathscr{Q}}_{0}\subset{\bf
P}(W)\times\widetilde{S}_{0}$ be the pullback of the universal family of
linear 5-spaces over $F^{5}(\widetilde{\mathscr{Q}}_{0}/\widetilde{S}_{0})$
along this section. Set

(5.2) $Z_{0}:=\mathscr{P}^{5}\cap(M_{X}\times\widetilde{S}_{0}),$

where the Grassmannian hull $M_{X}\subset{\bf P}(W)$ was defined in (2.7).

######  Proposition 5.1.

The map $Z_{0}\to\widetilde{S}_{0}$ is a flat family of surfaces in $X$
containing 0 with Hilbert polynomial

(5.3) $h(t)=\frac{5}{2}(t^{2}+t)+1.$

In particular, $Z_{0}\subset X\times\widetilde{S}_{0}$.

###### Proof.

Let $y\in\widetilde{S}_{0}$ and set $[v]:=\pi(y)\in{\bf
P}(V_{6})\smallsetminus{\bf P}(V_{5})$. The fiber of $Z_{0}$ over $y$ is

$Z_{0,y}:=M_{X}\cap\mathscr{P}^{5}_{y}={\operatorname{\mathsf{CGr}}}(2,V_{5})\cap\mathscr{P}^{5}_{y}.$

Since the cone ${\operatorname{\mathsf{CGr}}}(2,V_{5})\subset{\bf P}({\bf
C}\oplus\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{2}}\hskip
0.56905pt{V}_{5})$ has codimension 3 and degree 5, the intersection
${\operatorname{\mathsf{CGr}}}(2,V_{5})\cap\mathscr{P}^{5}_{y}$ has dimension
at least 2 and degree at most 5 (and if the dimension is 2, the degree is 5).
Furthermore, $\mathscr{P}^{5}_{y}$ is contained in $Q_{v}$, hence

$Z_{0,y}\subset M_{X}\cap Q_{v}=X.$

Since $X$ contains no divisors of degrees less than 10, we have
$\dim(Z_{0,y})\leq 3$ and, moreover, if $\dim(Z_{0,y})=3$, any irreducible
3-dimensional component $Z_{0,y}$ has even degree ([DK19, Corollary 3.5]). By
Lemma 2.3, its image in $\operatorname{\mathsf{Gr}}(2,V_{5})$ must be a
hyperplane section of $\operatorname{\mathsf{Gr}}(2,V_{4})$ and Lemma 2.7
gives a contradiction. Therefore $Z_{0,y}$ is a surface. This argument also
proves the inclusion

$Z_{0}\subset X\times\widetilde{S}_{0}.$

Since the surface $Z_{0,y}$ is a dimensionally transverse linear section of
${\operatorname{\mathsf{CGr}}}(2,V_{5})$, we obtain from Lemma 2.4 a
resolution

$0\to\mathscr{O}_{\mathscr{P}^{5}_{y}}(-5)\to\mathscr{O}_{\mathscr{P}^{5}_{y}}(-3)^{\oplus
5}\to\mathscr{O}_{\mathscr{P}^{5}_{y}}(-2)^{\oplus
5}\to\mathscr{O}_{\mathscr{P}^{5}_{y}}\to\mathscr{O}_{Z_{0,y}}\to 0.$

It follows that the Hilbert polynomial of $Z_{0,y}$ is given by (5.3). Since
it is independent of $y$, the family of surfaces $Z_{0}$ is flat over
$\widetilde{S}_{0}$. Finally, since ${}_{0}\subset M_{X}$ and
${}_{0}\subset\mathscr{P}^{5}_{y}$ by construction, we obtain ${}_{0}\subset
Z_{0,y}$. ∎

Applying to the family $Z_{0}\to\widetilde{S}_{0}$ the Hilbert closure
construction from Definition 4.2, we obtain the following result.

######  Lemma 5.2.

There is a subscheme

(5.4) $Z\subset X\times\widetilde{Y}^{\geq 2}_{A}$

such that, away from a finite subset of the surface $\widetilde{Y}^{\geq
2}_{A}$, the map $Z\to\widetilde{Y}^{\geq 2}_{A}$ is a flat family of surfaces
with Hilbert polynomial (5.3) containing the plane 0. Moreover, we have

$Z\times_{\widetilde{Y}^{\geq 2}_{A}}\widetilde{S}_{0}=Z_{0}$

as subschemes of $X\times\widetilde{S}_{0}$.

By Proposition 5.1, the schemes $Z_{0}$ and $Z$ have pure dimension 4.

The main result of this section is the following theorem (recall from
Propositions 2.5 and 2.6 that the abelian groups $H_{1}(\widetilde{Y}^{\geq
2}_{A(X)},{\bf Z})$ and $H_{5}(X,{\bf Z})$ are both free of rank 20 and from
Theorem 4.4 that the abelian variety $\operatorname{Alb}(\widetilde{Y}^{\geq
2}_{A(X)})$ is endowed with a canonical principal polarization when $Y^{\geq
3}_{A(X)}=\varnothing$).

######  Theorem 5.3.

Let $X$ be a smooth GM fivefold and let 0 be a $\sigma$-plane contained in
$X$. Let $Z\subset X\times\widetilde{Y}^{\geq 2}_{A(X)}$ be the subscheme
defined by (5.4). If $Y_{A(X)}^{\geq 3}=\varnothing$, the Abel–Jacobi map

$\operatorname{\mathsf{A\\!J}}_{Z}\colon H_{1}(\widetilde{Y}^{\geq
2}_{A(X)},{\bf Z})\longrightarrow H_{5}(X,{\bf Z})$

is an isomorphism of integral Hodge structures. It induces an isomorphism

(5.5) $\operatorname{Alb}(\widetilde{Y}^{\geq
2}_{A(X)})\operatorname{\stackrel{{\scriptstyle{}_{\scriptstyle\sim}}}{{\longrightarrow}}}\operatorname{Jac}(X)$

of principally polarized abelian varieties from the Albanese variety of the
surface $\widetilde{Y}^{\geq 2}_{A(X)}$ to the intermediate Jacobian of $X$.

The proof of this theorem takes up Sections 5.2–5.4.

### 5.2. The boundary of the family

Let $X$ be a smooth GM fivefold. To prove Theorem 5.3, we study the family of
surfaces $Z$ described in Lemma 5.2 over the boundary $\widetilde{Y}^{\geq
2}_{A}\smallsetminus\widetilde{S}_{0}$. We assume in this section that $X$ is
general (in particular it is ordinary), so that the curve $Y_{A,V_{5}}^{\geq
2}$ is smooth.

Consider the Hilbert scheme $F^{2}_{\sigma}(X)$ of $\sigma$-planes on $X$; we
identify it with the smooth connected curve $\widetilde{Y}^{\geq 2}_{A,V_{5}}$
via the isomorphism (2.19). For $y\in\widetilde{Y}^{\geq 2}_{A,V_{5}}$, we
denote by ${}_{y}\subset X$ the corresponding $\sigma$-plane.

We denote by $y_{0}\in\widetilde{Y}^{\geq 2}_{A,V_{5}}$ the point such that
${}_{y_{0}}={}_{0}$ is the plane chosen in Section 5.1. Set
$[v_{0}]:=\pi_{A}(y_{0})$. By Lemma 2.1(b), we have, for an appropriate
hyperplane $V_{4}\subset V_{5}$,

(5.6) ${}_{0}={\bf P}(v_{0}\wedge V_{4}).$

We set

$Y^{\geq 2}_{A,V_{4}}:=Y^{\geq 2}_{A}\cap{\bf P}(V_{4}).$

This is a hyperplane section of the smooth curve ${Y_{A,V_{5}}^{2}}$ hence is
finite; the induced double coverings of this set parameterizes $\sigma$-planes
on $X$ that intersect 0.

Denote by $\widetilde{\mathscr{Q}}$ the pullback of the family of quadrics
$\mathscr{Q}$ along the map $\widetilde{Y}^{\geq 2}_{A}\to Y_{A}^{\geq 2}$.
Recall that a section $\widetilde{S}_{0}\to
F^{5}_{{}_{0}}(\widetilde{\mathscr{Q}}_{0}/\widetilde{S}_{0})$ was constructed
in Section 5.1. Since the surface $\widetilde{Y}^{\geq 2}_{A}$ is smooth and
the Hilbert scheme $F^{5}_{{}_{0}}(\widetilde{\mathscr{Q}}/\widetilde{Y}^{\geq
2}_{A})$ is proper over $\widetilde{Y}^{\geq 2}_{A}$, this section extends to
an open subset

(5.7) $\widetilde{S}_{0+}\subset\pi_{A}^{-1}(Y^{\geq
2}_{A}\smallsetminus(Y^{\geq 2}_{A,V_{4}}\cup Y^{\geq
2}_{A,V^{\prime}_{4}}))\subset\widetilde{Y}^{\geq 2}_{A}$

which contains a general point of the curve
$\widetilde{S}_{0,V_{5}}:=\widetilde{S}_{0+}\smallsetminus\widetilde{S}_{0}\subset\widetilde{Y}^{\geq
2}_{A,V_{5}}$. We denote by

$\mathscr{P}^{5}_{+}\subset{\bf
P}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{2}}\hskip
0.56905pt{V}_{5})\times\widetilde{S}_{0+}$

the corresponding family of 5-spaces and, for $y\in\widetilde{S}_{0+}$, by
$\mathscr{P}^{5}_{+y}\subset{\bf
P}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{2}}\hskip
0.56905pt{V}_{5})$ the corresponding linear 5-space. By definition, we have
${}_{0}\subset\mathscr{P}^{5}_{+y}$ for each $y\in\widetilde{S}_{0,V_{5}}$.

######  Lemma 5.4.

For each point $y\in\widetilde{S}_{0,V_{5}}$, we have

(5.8) $\mathscr{P}^{5}_{+y}=\langle{}_{y},{}_{0}\rangle.$

###### Proof.

Set $[v]:=\pi_{A}(y)\in Y^{\geq 2}_{A,V_{5}}$ and let
$W_{6}\subset\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{2}}\hskip
0.56905pt{V}_{5}$ be the 6-dimensional subspace corresponding to the 5-space
$\mathscr{P}^{5}_{y}\subset{\bf
P}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{2}}\hskip
0.56905pt{V}_{5})$. By definition, we have

${\bf P}(W_{6})\subset Q_{v}={\operatorname{Cone}}_{{\bf P}(v\wedge
V_{5})}(\operatorname{\mathsf{Gr}}(2,V_{5}/v)).$

Since $\operatorname{\mathsf{Gr}}(2,V_{5}/v)$ is a smooth 4-dimensional
quadric, the maximal dimension of a linear subspace that it contains is 2,
hence the subspace

$W_{y}:=W_{6}\cap(v\wedge V_{5})$

is at least 3-dimensional. We claim that ${\bf P}(W_{y})$ is contained in $X$.

Let $\\{w_{i}\\}$ be a basis of $W_{y}$, let $q_{v}$ be an equation of
$Q_{v}$, and consider a line $\langle[v],[v^{\prime}]\rangle\subset{\bf
P}(V_{6})$ tangent to $Y^{\geq 2}_{A}$ at $[v]$, with $[v^{\prime}]\in{\bf
P}(V_{6})\smallsetminus{\bf P}(V_{5})$. Let $\operatorname{Spec}({\bf
C}[\epsilon]/\epsilon^{2})\to Y^{\geq 2}_{A}$ be the corresponding morphism
that takes the closed point to $[v]$. Since $\widetilde{S}_{0+}$ is étale over
$Y^{\geq 2}_{A,V_{5}}$, the morphism can be lifted to a morphism

$\operatorname{Spec}({\bf
C}[\epsilon]/\epsilon^{2})\longrightarrow\widetilde{S}_{0+}$

that takes the closed point to $y$. This implies that there are vectors
$w^{\prime}_{i}$ in
$\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{2}}\hskip
0.56905pt{V}_{5}$ such that the subspace in
$\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{2}}\hskip
0.56905pt{V}_{5}$ generated by $w_{i}+\epsilon w^{\prime}_{i}$ is isotropic
for the quadratic form $q_{v}+\epsilon q_{v^{\prime}}$, where $q_{v^{\prime}}$
is an equation of $Q_{v^{\prime}}$. We have therefore

$0=(q_{v}+\epsilon q_{v^{\prime}})(w_{i}+\epsilon
w^{\prime}_{i},w_{j}+\epsilon
w^{\prime}_{j})=\epsilon(q_{v}(w_{i},w^{\prime}_{j})+q_{v}(w_{j},w^{\prime}_{i})+q_{v^{\prime}}(w_{i},w_{j})).$

Note that $q_{v}(w_{i},w^{\prime}_{j})=q_{v}(w_{j},w^{\prime}_{i})=0$, since
$w_{i},w_{j}\in v\wedge V_{5}=\operatorname{Ker}(q_{v})$ for all $i,j$. It
follows that $q_{v^{\prime}}(w_{i},w_{j})=0$ for all $i,j$, hence $W_{y}$ is
isotropic for $q_{v^{\prime}}$, that is, ${\bf P}(W_{y})\subset
Q_{v^{\prime}}$. Since ${\bf P}(W_{y})\subset{\bf P}(v\wedge
V_{5})\subset\operatorname{\mathsf{Gr}}(2,V_{5})$, we conclude that

${\bf P}(W_{y})\subset\operatorname{\mathsf{Gr}}(2,V_{5})\cap
Q_{v^{\prime}}=X,$

thus proving the claim.

Since ${\bf P}(W_{y})\subset{\bf P}(v\wedge V_{5})\cap X$ and $X$ contains no
linear 3-spaces ([DK19, Theorem 4.2]), it follows that $\dim(W_{y})=3$ and
${\bf P}(W_{y})$ is a $\sigma$-plane on $X$. Moreover, the induced map

$\displaystyle\widetilde{S}_{0,V_{5}}$ $\displaystyle\longrightarrow$
$\displaystyle F^{2}_{\sigma}(X)$ $\displaystyle y$ $\displaystyle\longmapsto$
$\displaystyle{\bf P}(W_{y})$

is a $Y^{\geq 2}_{A,V_{5}}$-morphism. But
$F^{2}_{\sigma}(X)\simeq\widetilde{Y}^{\geq 2}_{A,V_{5}}$, while
$\widetilde{S}_{0,V_{5}}$ is an open subscheme in $\widetilde{Y}^{\geq
2}_{A,V_{5}}$, and $\widetilde{Y}^{\geq 2}_{A,V_{5}}$ is a connected étale
covering of $Y^{\geq 2}_{A,V_{5}}$. Therefore, replacing if necessary the
isomorphism (2.19) by its composition with the involution of the double
covering, we may assume that the above map $\widetilde{S}_{0,V_{5}}\to
F^{2}_{\sigma}(X)$ coincides with the embedding
$\widetilde{S}_{0,V_{5}}\hookrightarrow\widetilde{Y}^{\geq 2}_{A,V_{5}}$,
hence ${\bf P}(W_{y})={}_{y}$ for all $y\in\widetilde{S}_{0,V_{5}}$.

This means that there is an inclusion ${}_{y}\subset\mathscr{P}^{5}_{+y}$ for
all $y\in\widetilde{S}_{0,V_{5}}$. Since ${}_{0}\subset\mathscr{P}^{5}_{+y}$
by definition, the right side of (5.8) is contained in the left side. Finally,
since ${}_{0}\cap{}_{y}=\varnothing$ by definition of $\widetilde{S}_{0+}$
(since planes intersecting 0 are parameterized by the double cover of the
subscheme $Y^{\geq 2}_{A,V_{4}}$), the inclusion is an equality. ∎

The family $\mathscr{P}^{5}_{+}$ of projective 5-spaces over
$\widetilde{S}_{0+}$ agrees by construction with the family $\mathscr{P}^{5}$
over $\widetilde{S}_{0}\subset\widetilde{S}_{0+}$. We set

(5.9) $Z_{0+}:=\mathscr{P}^{5}_{+}\cap(M_{X}\times\widetilde{S}_{0+}).$

Comparing this with (5.2), we obtain

$Z_{0+}\times_{\widetilde{S}_{0+}}\widetilde{S}_{0}=Z_{0}.$

We denote by

$Z_{0,V_{5}}:=Z_{0,+}\times_{\widetilde{S}_{0+}}\widetilde{S}_{0,V_{5}}\subset
M_{X}\times\widetilde{S}_{0,V_{5}}$

the restriction of the family (5.9) to the curve
$\widetilde{S}_{0,V_{5}}={\widetilde{S}_{0+}\smallsetminus\widetilde{S}_{0}}$.

######  Proposition 5.5.

Let $X$ be a general GM fivefold. The map
$Z_{0,V_{5}}\to\widetilde{S}_{0,V_{5}}$ is a flat family of $3$-dimensional
cubic scrolls ${\bf P}^{1}\times{\bf P}^{2}$.

###### Proof.

Let $y\in\widetilde{S}_{0,V_{5}}$ and set again $[v]:=\pi(y)\in{\bf
P}(V_{5})$. By (5.8), the linear 5-space $\mathscr{P}^{5}_{+y}$ is the linear
span of the planes 0 and y, that is, a hyperplane in ${\bf P}(V_{2}\wedge
V_{5})$, where $V_{2}\subset V_{5}$ is the subspace spanned by $v_{0}$ and
$v$. Therefore, $\operatorname{\mathsf{Gr}}(2,V_{5})\cap\mathscr{P}^{5}_{+y}$
is a hyperplane section of the cone $\operatorname{Cone}_{{\bf
P}(\wedge^{2}V_{5})}({\bf P}(V_{2})\times{\bf P}(V_{5}/V_{2}))$ (see Lemma
2.2).

The vertex $[\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{2}}\hskip
0.56905pt{V}_{2}]={[v_{0}\wedge v]}$ of the cone does not belong to
$\mathscr{P}^{5}_{+y}$: if it did, 0 and y would intersect at the point
$[v_{0}\wedge v]$ and this would contradict the definition of $S_{0+}$.
Therefore, the fiber of $Z_{0,V_{5}}$ over $y$ is isomorphic to the
3-dimensional cubic scroll ${\bf P}(V_{2})\times{\bf P}(V_{5}/V_{2})$. ∎

Propositions 5.1 and 5.5 show that all the components of $Z_{0}$ and
$Z_{0,V_{5}}$ have dimension $4$. They are components of the scheme $Z_{0+}$.
We also consider the Hilbert closure

(5.10) $Z_{F}\subset M_{X}\times F^{2}_{\sigma}(X)$

of $Z_{0,V_{5}}$ in $M_{X}\times F^{2}_{\sigma}(X)$, constructed as in
Definition 4.2.

### 5.3. A relation between the subschemes

Let $X$ be a general GM fivefold. In (5.4) and (5.10), we have constructed
subschemes $Z\subset X\times\widetilde{Y}^{\geq 2}_{A}$ and $Z_{F}\subset
M_{X}\times F^{2}_{\sigma}(X)$. The proof of Theorem 5.3 is based on a
relation between the schemes $Z\cap(X\times\widetilde{S}_{0,V_{5}})$ and
$Z_{F}\cap(X\times\widetilde{S}_{0,V_{5}})$, where the curve
$\widetilde{S}_{0,V_{5}}$ is considered as a subscheme of both the surface
$\widetilde{Y}^{\geq 2}_{A}$ and the curve $F^{2}_{\sigma}(X)$.

Consider the commutative diagram

(5.11) |  |
---|---|---
$\textstyle{X\times\widetilde{S}_{0,V_{5}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$openloc.
closed$\scriptstyle{i}$$\textstyle{\mathscr{L}^{2}_{\sigma}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\times
F^{2}_{\sigma}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{{\tilde{\sigma}}}$$\textstyle{X\times{\widetilde{Y}^{\geq
2}_{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{X}\times\widetilde{S}_{0,V_{5}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$openloc.
closed$\textstyle{Z_{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{X}\times
F^{2}_{\sigma}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\tilde{\sigma}}}$$\textstyle{M_{X}\times{\widetilde{Y}^{\geq
2}_{A}},}$

where $\mathscr{L}^{2}_{\sigma}(X)\subset X\times F^{2}_{\sigma}(X)$ is the
universal family of $\sigma$-planes, $i\colon X\hookrightarrow M_{X}$ is the
embedding, and ${\tilde{\sigma}}$ is the isomorphism (2.19).

######  Proposition 5.6.

We have an equality

(5.12)
$Z_{F}\cap(X\times\widetilde{S}_{0,V_{5}})={({\tilde{\sigma}}^{-1}(Z)\cap(X\times\widetilde{S}_{0,V_{5}}))+(\mathscr{L}^{2}_{\sigma}(X)\cap(X\times\widetilde{S}_{0,V_{5}}))}$

of cycles.

###### Proof.

The left side of (5.12) can be rewritten as

$Z_{F}\cap(X\times\widetilde{S}_{0,V_{5}})=Z_{0,V_{5}}\cap(X\times\widetilde{S}_{0,V_{5}})=Z_{0,V_{5}}\times_{M_{X}}X.$

By Proposition 5.5, the morphism $Z_{0,V_{5}}\to\widetilde{S}_{0,V_{5}}$ is a
flat family of smooth 3-dimensional cubic scrolls. Since $X$ contains no such
threefolds ([DK19, Corollary 3.5]), it contains no fibers of $Z_{0,V_{5}}$.
Therefore, the morphism

$Z_{0,V_{5}}\times_{M_{X}}X\longrightarrow\widetilde{S}_{0,V_{5}}$

is a flat family of surfaces whose fiber over $y\in\widetilde{S}_{0,V_{5}}$ is
the dimensionally transverse intersection

$S_{y}:=Z_{0,V_{5},y}\cap{Q_{0}}$

of the smooth 3-dimensional cubic scroll $Z_{0,V_{5},y}\simeq{\bf
P}^{2}\times{\bf P}^{1}$ with any non-Plücker quadric ${Q_{0}}$ containing
$X$. Such an intersection is a surface of class $2f_{2}+2f_{1}$ in
$Z_{0,V_{5},y}$, where $f_{i}$ is the preimage of the hyperplane class on
${\bf P}^{i}$ under the projection $Z_{0,V_{5},y}\simeq{\bf P}^{2}\times{\bf
P}^{1}\to{\bf P}^{i}$.

By (5.8) and (5.9), the planes 0 and y are contained in the scroll
$Z_{0,V_{5},y}$. Since they are also contained in $X$, they are contained in
the quadric $Q_{0}$. It follows that they are components of $S_{y}$, each of
class $f_{1}$. Therefore,

$S_{y}={}_{0}+{}_{y}+S^{\prime}_{y},$

where $S^{\prime}_{y}\subset Z_{0,V_{5},y}$ is a surface of class $2f_{2}$,
that is, the product of a conic in ${\bf P}^{2}$ with ${\bf P}^{1}$. In
particular, it has degree 4 and contains no planes. Since $Z_{y}\subset
Z_{0,V_{5},y}$ is a surface of degree 5 that contains 0, we have
$Z_{y}=S^{\prime}_{y}+{}_{0}$ for all $y\in\widetilde{S}_{0,V_{5}}$; this
proves (5.12). ∎

### 5.4. Abel–Jacobi maps

Let $X$ be a smooth ordinary GM fivefold with associated Lagrangian $A$.
Assume $Y^{3}_{A}=\varnothing$, so that $\widetilde{Y}^{\geq 2}_{A}$ is a
smooth surface and the curve $\widetilde{Y}^{\geq 2}_{A,V_{5}}$, hence also
the Hilbert scheme $F^{2}_{\sigma}(X)$ of $\sigma$-planes in $X$, is a smooth
curve. Let $\mathscr{L}^{2}_{\sigma}(X)\subset X\times F^{2}_{\sigma}(X)$
denote the universal family of $\sigma$-planes on $X$. Consider the
Abel–Jacobi maps

$\operatorname{\mathsf{A\\!J}}_{Z}\colon H_{1}(\widetilde{Y}^{\geq 2}_{A},{\bf
Z})\to H_{5}(X,{\bf
Z})\quad\textnormal{and}\quad\operatorname{\mathsf{A\\!J}}_{\mathscr{L}^{2}_{\sigma}(X)}\colon
H_{1}(F^{2}_{\sigma}(X),{\bf Z})\to H_{5}(X,{\bf Z})$

and recall the isomorphism ${\tilde{\sigma}}\colon
F^{2}_{\sigma}(X)\operatorname{\stackrel{{\scriptstyle{}_{\scriptstyle\sim}}}{{\to}}}\widetilde{Y}^{\geq
2}_{A,V_{5}}$ from (2.19).

######  Proposition 5.7.

Let $X$ be a smooth ordinary GM fivefold with associated Lagrangian $A$.
Assume that $Y^{3}_{A}=\varnothing$ and $F_{2}^{\sigma}(X)$ is smooth. The
composition of maps

$H_{1}(F^{2}_{\sigma}(X),{\bf Z})\xrightarrow{\ {\tilde{\sigma}}_{*}\
}H_{1}(\widetilde{Y}^{\geq 2}_{A},{\bf Z})\xrightarrow{\
\operatorname{\mathsf{A\\!J}}_{Z}\ }H_{5}(X,{\bf Z})$

is equal to the map
$-\operatorname{\mathsf{A\\!J}}_{\mathscr{L}^{2}_{\sigma}(X)}$.

###### Proof.

Analogous to the proof of Proposition 4.15. ∎

The above proposition connects the Abel–Jacobi maps
$\operatorname{\mathsf{A\\!J}}_{Z}$ and
$\operatorname{\mathsf{A\\!J}}_{{\mathscr{L}^{2}_{\sigma}(X)}}$. The next
lemma uses the Clemens–Tyurin argument (Section 3.3) to show that the latter
is surjective.

######  Lemma 5.8.

Let $X$ be a general GM fivefold. The Abel–Jacobi map

$\operatorname{\mathsf{A\\!J}}_{\mathscr{L}^{2}_{\sigma}(X)}\colon
H_{1}(F^{2}_{\sigma}(X),{\bf Z})\longrightarrow H_{5}(X,{\bf Z})$

is surjective.

###### Proof.

Let $Y$ be a general GM sixfold and let $X\subset Y$ be a general hyperplane
section, so that $X$ is a general GM fivefold. Set $F_{Y}:=F^{2}_{\sigma}(Y)$,
the Hilbert scheme of $\sigma$-planes contained in $Y$. We check that the
assumptions of Proposition 3.3 (with $m=2$) are satisfied.

Assumption (a) holds because $F^{2}_{\sigma}(Y)$ is a smooth irreducible
fourfold by [DK19, Corollary 5.13]. Furthermore, the map
$q\colon\mathscr{L}^{2}_{\sigma}(Y)\to Y$ is generically finite of degree $12$
([DK19, Lemma 5.15]), hence (b) holds as well. Next, $F^{2}_{\sigma}(X)$ is a
smooth curve by Lemma 2.10, hence (c) holds. Finally, $H_{5}(Y,{\bf
Z})=H_{7}(Y,{\bf Z})=0$ by [DK19, Proposition 3.1], hence (d) holds.

Applying Proposition 3.3, we deduce the surjectivity of
$\operatorname{\mathsf{A\\!J}}_{\mathscr{L}^{2}_{\sigma}(X)}$. ∎

Combining the above results, we can now prove Theorem 5.3.

###### Proof of Theorem 5.3.

Assume first that the GM fivefold $X$ is general. A combination of Proposition
5.7 and Lemma 5.8 proves that the map $\operatorname{\mathsf{A\\!J}}_{Z}$ is
surjective. By Propositions 2.5 and 2.6, its source and target are free
abelian groups of rank 20. Therefore, the Abel–Jacobi map is an isomorphism.

Since the Abel–Jacobi map is defined by the cohomology class of an algebraic
cycle, it preserves the Hodge structures, hence induces an isomorphism of the
corresponding abelian varieties: the Albanese variety of $\widetilde{Y}^{\geq
2}_{A(X)}$ and the intermediate Jacobian of $X$.

Since the scheme $Z\subset X\times\widetilde{Y}^{\geq 2}_{{A(X)}}$ was defined
in Section 5.1 for all smooth $X$ and all $\sigma$-planes ${}_{0}\subset X$
and since this definition works in families, these two statements follow by
continuity for any $X$ such that $Y^{3}_{A(X)}=\varnothing$.

It remains to prove that the isomorphism (5.5) respects the principal
polarizations. For $X$ very general, the Picard number of
$\operatorname{Jac}(X)$ is $1$ by Corollary 3.6, hence any two principal
polarizations on $\operatorname{Jac}(X)$ coincide. This proves the claim for
very general $X$ and, by continuity, for any smooth $X$ such that
$\widetilde{Y}^{\geq 2}_{A(X)}$ is also smooth. ∎

### 5.5. Simplicity argument

We give an alternative argument for the isomorphism (5.5) for a smooth GM
fivefold $X$, based on a simplicity result of independent interest, analogous
to the one proved in Proposition 3.4.

Let $S$ be a smooth connected projective surface and let $\jmath\colon
C\hookrightarrow S$ be a smooth (irreducible) ample curve. By the Lefschetz
theorem, the morphism $\jmath_{*}\colon H_{1}(C,{\bf Z})\to H_{1}(S,{\bf Z})$
is surjective, hence the induced morphsim

$\operatorname{Jac}(C)\simeq\operatorname{Alb}(C)\longrightarrow\operatorname{Alb}(S)$

is surjective with connected kernel. We denote this kernel by $K(C,S)$.

Consider now a connected double étale cover $\pi\colon\widetilde{S}\to S$ and
set ${\widetilde{C}}:=\pi^{-1}(C)$, a smooth ample curve on the surface
$\widetilde{S}$.

######  Lemma 5.9.

There is a commutative diagram

(5.13)
---
$\textstyle{K({\widetilde{C}},C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P({\widetilde{C}},C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P(\widetilde{S},S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K({\widetilde{C}},\widetilde{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\pi_{*}}}$$\textstyle{\operatorname{Jac}({\widetilde{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\pi_{*}}}$$\textstyle{\operatorname{Alb}(\widetilde{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\pi_{*}}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K(C,S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Jac}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Alb}(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0,}$

where $K({\widetilde{C}},C)$, $P({\widetilde{C}},C)$ (the Prym variety of the
double cover ${\widetilde{C}}\to C$), and $P(\widetilde{S},S)$ are the neutral
components of the respective kernels of the vertical maps ${\pi_{*}}$ induced
by $\pi$.

###### Proof.

The surjectivity of the maps
$\operatorname{Jac}({\widetilde{C}})\to\operatorname{Jac}(C)$ and
$\operatorname{Alb}(\widetilde{S})\to\operatorname{Alb}(S)$ is obvious. The
only thing we have to prove is the surjectivity of the map
$K({\widetilde{C}},\widetilde{S})\to K(C,S)$ or, equivalently, the
surjectivity of the map $P({\widetilde{C}},C)\to P(\widetilde{S},S)$. On the
level of cotangent spaces, the surjectivity of this second map corresponds to
the injectivity of the restriction $H^{1}(S,\eta)\to H^{1}(C,\eta|_{C})$,
where $\eta$ is the line bundle of order 2 on $S$ corresponding to the double
étale covering $\pi$. Its kernel is controlled by $H^{1}(S,\eta(-C))$, which
vanishes by Kodaira vanishing and Serre duality because $\eta(C)$ is ample on
$S$. This proves the injectivity of the morphism $H^{1}(S,\eta)\to
H^{1}(C,\eta|_{C})$, hence the lemma. ∎

The next statement is the main result of this section. The definition of
“trivial endomorphism ring” can be found in Section 3.4.

######  Theorem 5.10.

Let $S\subset{\bf P}^{N}$ be a smooth connected projective surface and let
$\pi\colon\widetilde{S}\to S$ be a connected double étale cover. Let
$H\subset{\bf P}^{N}$ be a very general hyperplane and set $C:=S\cap H$. With
the notation above, the endomorphism ring of the abelian variety
$K({\widetilde{C}},C)$ is trivial.

###### Proof.

We use the notation of Section 3.4. Choose a Lefschetz pencil $f\colon
S\dashrightarrow{\bf P}^{1}$ of hyperplane sections of $S$. The connected
double étale cover $\pi\colon\widetilde{S}\to S$ induces for each $t\in{\bf
P}^{1}$ a connected double étale cover $\pi_{t}\colon{\widetilde{C}}_{t}\to
C_{t}$ between fibers. Denote by $\jmath_{t}\colon C_{t}\hookrightarrow S$ and
${\tilde{\jmath}}_{t}\colon{\widetilde{C}}_{t}\hookrightarrow\widetilde{S}$
the embeddings.

For $t\in{\bf P}^{1}\smallsetminus\\{t_{1},\dots,t_{r}\\}$, the involution
$\tau$ of $\widetilde{S}$ attached to $\pi$ acts on each summand of the
orthogonal direct sum decomposition

$H^{1}({\widetilde{C}}_{t},{\bf Q})=H^{1}({\widetilde{C}}_{t},{\bf
Q})_{\textnormal{van}}\oplus{\tilde{\jmath}}_{t}^{\,*}H^{1}(\widetilde{S},{\bf
Q})$

from (3.3) and it preserves the symplectic form given by cup-product. The
$\tau$-invariant subspaces are

$H^{1}(C_{t},{\bf Q})=H^{1}(C_{t},{\bf
Q})_{\textnormal{van}}\oplus\jmath_{t}^{*}H^{1}(S,{\bf Q}).$

The isogeny class of the abelian variety $K(C_{t},S)$ defined in (5.13) is
obtained from the Hodge structure of $H^{1}(C_{t},{\bf
Q})_{\textnormal{van}}$, hence its endomorphism ring is trivial by Proposition
3.4. Therefore, to study the isogeny class of the neutral component
$K({\widetilde{C}}_{t},C_{t})$ of the kernel of the surjection

$K({\widetilde{C}}_{t},\widetilde{S})\longrightarrow K(C_{t},S),$

we need to study the rational Hodge structure on the $\tau$-antiinvariant
subspace $H^{1}({\widetilde{C}}_{t},{\bf Q})_{\textnormal{van}}^{-}$.

For each $i\in\\{1,\dots,r\\}$, the curve ${\widetilde{C}}_{t_{i}}$ has two
nodes over the node of $C_{t_{i}}$, hence there are two disjoint vanishing
cycles $\delta^{\prime}_{i}$ and
$\delta^{\prime\prime}_{i}=\tau^{*}(\delta^{\prime}_{i})$. Since the vanishing
cycles span the vector space $H^{1}({\widetilde{C}}_{t},{\bf
Q})_{\textnormal{van}}$, the cycles
$\delta^{\prime}_{1}-\delta^{\prime\prime}_{1},\dots,\delta^{\prime}_{r}-\delta^{\prime\prime}_{r}$
span the antiinvariant subspace $H^{1}({\widetilde{C}}_{t},{\bf
Q})_{\textnormal{van}}^{-}$. The image of the monodromy representation

$\tilde{\rho}\colon\pi_{1}({\bf
P}^{1}\smallsetminus\\{t_{1},\dots,t_{r}\\},t)\longrightarrow\operatorname{Sp}(H^{1}({\widetilde{C}}_{t},{\bf
Q}))$

consists of automorphisms that are $\tau$-equivariant and, reasoning as in the
proof of [Voi03, Proposition 3.23], we see that, up to changing signs, the
classes
$\delta^{\prime}_{1}-\delta^{\prime\prime}_{1},\dots,\delta^{\prime}_{r}-\delta^{\prime\prime}_{r}$
are all in the same monodromy orbit. Moreover, as in the proof of Proposition
3.4, there is for each $i\in\\{1,\dots,r\\}$ an element of $\pi_{1}({\bf
P}^{1}\smallsetminus\\{t_{1},\dots,t_{r}\\},0)$ that acts on
$H^{1}({\widetilde{C}}_{t},{\bf Q})$ by

$T_{i}:=T_{\delta^{\prime\prime}_{i}}\circ T_{\delta^{\prime}_{i}}\colon
x\longmapsto
x-(x\cdot\delta^{\prime}_{i})\delta^{\prime}_{i}-(x\cdot\delta^{\prime\prime}_{i})\delta^{\prime\prime}_{i}.$

If $x$ is $\tau$-antiinvariant, we have

$(x\cdot\delta^{\prime\prime}_{i})=(x\cdot\tau^{*}(\delta^{\prime}_{i}))=(\tau^{*}(x)\cdot\delta^{\prime}_{i})=-(x\cdot\delta^{\prime}_{i}),$

hence

$T_{i}(x)=x-(x\cdot\delta^{\prime}_{i})(\delta^{\prime}_{i}-\delta^{\prime\prime}_{i})=x-\frac{1}{2}(x\cdot(\delta^{\prime}_{i}-\delta^{\prime\prime}_{i}))(\delta^{\prime}_{i}-\delta^{\prime\prime}_{i}).$

One then deduces from that and [PS03, Lemma 4] that the monodromy action on
$H^{1}({\widetilde{C}}_{t},{\bf Q})_{\textnormal{van}}^{-}$ is big; it follows
that the Zariski closure of the monodromy group for
$K({\widetilde{C}}_{t},C_{t})$ is the full symplectic group
$\operatorname{Sp}(H^{1}({\widetilde{C}}_{t},{\bf
Q})_{\textnormal{van}}^{-})$. As in the proof of [PS03, Theorem 17], for
$t\in{\bf P}^{1}$ very general, any endomorphism of
$K({\widetilde{C}}_{t},C_{t})$ intertwines every element of the monodromy
group, hence every element of the symplectic group. It must therefore be a
multiple of the identity.

The endomorphism ring of the abelian variety $K({\widetilde{C}}_{t},C_{t})$ is
therefore trivial. ∎

We now apply the theorem to GM fivefolds. Let $X$ be a general GM fivefold
with Lagrangian data set $(V_{6},V_{5},A)$. Our starting point is again the
surjectivity, proved in Lemma 5.8, of the Abel–Jacobi map

$\operatorname{\mathsf{A\\!J}}_{{\mathscr{L}^{2}_{\sigma}}(X)}\colon
H_{1}(F^{2}_{\sigma}(X),{\bf Z})\longrightarrow H_{5}(X,{\bf Z})$

associated with the Hilbert scheme $F^{2}_{\sigma}(X)$ that parametrizes
$\sigma$-planes contained in $X$. This Hilbert scheme is isomorphic to the
smooth curve $\widetilde{Y}^{\geq 2}_{A,V_{5}}$ (Lemma 2.10) defined as the
inverse image by the double cover

$\pi_{A}\colon\widetilde{Y}^{\geq 2}_{A}\longrightarrow Y^{\geq 2}_{A}$

of the hyperplane section $Y^{\geq 2}_{A,V_{5}}=Y^{\geq 2}_{A}\cap{\bf
P}(V_{5})$. The surjectivity of the map
$\operatorname{\mathsf{A\\!J}}_{{\mathscr{L}^{2}_{\sigma}}(X)}$ is therefore
equivalent to the connectedness of the kernel of the induced surjective
morphism

(5.14) $\Phi\colon\operatorname{Jac}(\widetilde{Y}^{\geq
2}_{A,V_{5}})\longrightarrow\operatorname{Jac}(X)$

between Jacobians. By Lemma 2.10 and Proposition 2.6, the dimension of this
kernel is

$g(\widetilde{Y}^{\geq
2}_{A,V_{5}})-\dim(\operatorname{Jac}(X))={161}-10=151.$

######  Corollary 5.11.

Let $X$ be a smooth GM fivefold with Lagrangian data set $(V_{6},V_{5},A)$.
Assume that the surface $\widetilde{Y}^{\geq 2}_{A}$ and the curve
$\widetilde{Y}^{\geq 2}_{A,V_{5}}$ are smooth. The morphism from (5.14) then
factors as

$\Phi\colon\operatorname{Jac}(\widetilde{Y}^{\geq
2}_{A,V_{5}})\twoheadrightarrow\operatorname{Alb}(\widetilde{Y}^{\geq
2}_{A})\operatorname{\stackrel{{\scriptstyle{}_{\scriptstyle\sim}}}{{\longrightarrow}}}\operatorname{Jac}(X),$

where the left arrow is the Albanese map
$\operatorname{Jac}(\widetilde{Y}^{\geq
2}_{A,V_{5}})=\operatorname{Alb}(\widetilde{Y}^{\geq
2}_{A,V_{5}})\to\operatorname{Alb}(\widetilde{Y}^{\geq 2}_{A})$.

###### Proof.

Since $\operatorname{Alb}(Y^{\geq 2}_{A})=0$ (Proposition 2.5), the diagram
(5.13) reads

(5.15) |
---|---
$\textstyle{K_{V_{5}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{A,V_{5}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Alb}(\widetilde{Y}^{\geq
2}_{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{K}_{V_{5}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Jac}(\widetilde{Y}^{\geq
2}_{A,V_{5}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Alb}(\widetilde{Y}^{\geq
2}_{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\operatorname{Jac}(Y^{\geq
2}_{A,V_{5}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Jac}(Y^{\geq
2}_{A,V_{5}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0,}$

where $K_{V_{5}}:=K(\widetilde{Y}^{\geq 2}_{A,V_{5}},Y^{\geq 2}_{A,V_{5}})$,
$\widetilde{K}_{V_{5}}:=K(\widetilde{Y}^{\geq 2}_{A,V_{5}},\widetilde{Y}^{\geq
2}_{A})$, and $P_{A,V_{5}}$ is the Prym variety of the double covering
$\widetilde{Y}^{\geq 2}_{A,V_{5}}\to Y^{\geq 2}_{A,V_{5}}$. The genus of the
curve $Y^{\geq 2}_{A,V_{5}}$ is 81 by (2.20), hence the dimension of the
variety $P_{A,V_{5}}$ is 80. Also, the dimension of
$\operatorname{Alb}(\widetilde{Y}^{\geq 2}_{A})$ is 10 by Proposition 2.5.
Therefore, $\dim(K_{V_{5}})=70$.

When $V_{5}$ is a very general hyperplane in $V_{6}$, the abelian varieties
$\operatorname{Jac}(Y^{\geq 2}_{A,V_{5}})$ and $K_{V_{5}}$ are simple by
Proposition 3.4 and Theorem 5.10, hence they are the only two simple factors
of the abelian variety $\widetilde{K}_{V_{5}}$. Since $\operatorname{Jac}(X)$
has dimension 10, the abelian variety $\widetilde{K}_{V_{5}}$ must therefore
be contained in the kernel of .

In other words, the composition
$\widetilde{K}_{V_{5}}\hookrightarrow\operatorname{Jac}(\widetilde{Y}^{\geq
2}_{A,V_{5}})\stackrel{{\scriptstyle\Phi}}{{\twoheadrightarrow}}\operatorname{Jac}(X)$
vanishes for $V_{5}$ very general. By continuity, it vanishes for all
hyperplanes $V_{5}$ such that $\widetilde{Y}^{\geq 2}_{A,V_{5}}$ is smooth.
The kernel of , being connected of dimension $151$, must then be equal to
$\widetilde{K}_{V_{5}}$, which implies the corollary. ∎

Note that this argument cannot be applied to GM threefolds, because the
corresponding hyperplane sections $Y^{\geq 2}_{A,V_{5}}$ are very far from
being general.

## 6\. Period maps

In this section, we prove Theorems 1.1 and 1.3. We will use the description
(2.8) of the coarse moduli space $\mathbf{M}_{n}^{\rm GM}$ of smooth GM
varieties of dimension $n$. Let

(6.1) $\mathbf{M}^{\rm EPW}_{\mathrm{ndv}}=\operatorname{\mathsf{LGr}}_{\rm
ndv}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{3}}\hskip
0.56905pt{V}_{6})/\\!\\!/\operatorname{PGL}(V_{6})$

be the coarse quasiprojective moduli space of EPW sextics defined by
Lagrangian subspaces $A$ with no decomposable vectors (see (2.3) for the
definition). This is an open subset of the coarse moduli space
$\mathbf{M}^{\rm EPW}$ of EPW sextics defined in (1.6). We denote by
$\mathbf{r}$ the involution of $\mathbf{M}^{\rm EPW}_{\mathrm{ndv}}$ defined
by $\mathbf{r}([A])=([A^{\perp}])$. The morphism $\mathfrak{p}_{n}$ was
defined in (2.9).

######  Lemma 6.1.

There exists a regular morphism $\bar{\wp}\colon\mathbf{M}^{\rm
EPW}_{\mathrm{ndv}}/\mathbf{r}\to\mathbf{A}_{10}$ such that for
$n\in\\{3,5\\}$, the composition

(6.2) $\mathbf{M}_{n}^{\rm GM}\xrightarrow{\ \mathfrak{p}_{n}\
}\mathbf{M}^{\rm EPW}_{\mathrm{ndv}}\xrightarrow{\ \ \ }\mathbf{M}^{\rm
EPW}_{\mathrm{ndv}}/\mathbf{r}\xrightarrow{\ \bar{\wp}\ }\mathbf{A}_{10}$

is equal to the period map

$\displaystyle\wp_{n}\colon\mathbf{M}_{n}^{\rm GM}$
$\displaystyle\longrightarrow\mathbf{A}_{10}$ $\displaystyle[X]$
$\displaystyle\longmapsto[\operatorname{Jac}(X)].$

###### Proof.

Consider the universal EPW variety

$\mathscr{Y}^{5-n}_{\mathscr{A}^{\perp}}:=\\{(A,V_{5})\in\operatorname{\mathsf{LGr}}_{\rm
ndv}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{3}}\hskip
0.56905pt{V}_{6})\times{\bf
P}(V_{6}^{\vee})\mid\dim(A\cap\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{3}}\hskip
0.56905pt{V}_{5})=5-n\\},$

so that
$\mathscr{Y}^{5-n}_{\mathscr{A}^{\perp}}/\\!\\!/\operatorname{PGL}(V_{6})$ is
the coarse moduli space of ordinary GM varieties of dimension $n$, an open
subscheme of $\mathbf{M}_{n}^{\rm GM}$. For $n\in\\{3,5\\}$, the projection

$\mathrm{pr}\colon\mathscr{Y}^{5-n}_{\mathscr{A}^{\perp}}\longrightarrow\operatorname{\mathsf{LGr}}_{\rm
ndv}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{3}}\hskip
0.56905pt{V}_{6})$

is a smooth surjective morphism. We show that the map

$\displaystyle\tilde{\wp}_{n}\colon\mathscr{Y}^{5-n}_{\mathscr{A}^{\perp}}$
$\displaystyle\longrightarrow\mathbf{A}_{10}$ $\displaystyle(A,V_{5})$
$\displaystyle\longmapsto[\operatorname{Jac}(X_{A,V_{5}})]$

factors through $\mathrm{pr}$. By smooth descent, it is enough for this to
show that the two maps

$\textstyle{\mathscr{Y}^{5-n}_{\mathscr{A}^{\perp}}\times_{\operatorname{\mathsf{LGr}}_{\rm
ndv}({\bigwedgeop\displaylimits\hskip-2.56073pt^{3}}\hskip
0.28453pt{V}_{6})}\mathscr{Y}^{5-n}_{\mathscr{A}^{\perp}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{A}_{10}}$

defined as compositions of the projections to the factors with the map
$\tilde{\wp}_{n}$ are equal. Since the fiber product is a smooth variety, it
is enough to verify the equality pointwise on an open subset. In other words,
we need to check that for general GM varieties $X,X^{\prime}$ of the same
dimension $n\in\\{3,5\\}$ with $A(X)=A(X^{\prime})$, there is an isomorphism
$\operatorname{Jac}(X)\simeq\operatorname{Jac}(X^{\prime})$ of principally
polarized abelian varieties. For $n=3$, this holds by Theorem 4.4, and for
$n=5$, by Theorem 5.3; in both cases, the intermediate Jacobians are
isomorphic to $\operatorname{Alb}(\widetilde{Y}^{\geq 2}_{A})$ as soon as
$A=A(X)=A(X^{\prime})$ is such that $Y^{\geq 3}_{A}=\varnothing$. This proves
that in both cases, the morphism $\tilde{\wp}_{n}$ factors as

$\mathscr{Y}^{5-n}_{\mathscr{A}^{\perp}}\xrightarrow{\ \mathrm{pr}\
}\operatorname{\mathsf{LGr}}_{\rm
ndv}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{3}}\hskip
0.56905pt{V}_{6})\longrightarrow\mathbf{A}_{10}.$

The maps $\tilde{\wp}_{n}$ are $\operatorname{PGL}(V_{6})$-invariant, hence so
is the map

$\operatorname{\mathsf{LGr}}_{\rm
ndv}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{3}}\hskip
0.56905pt{V}_{6})\longrightarrow\mathbf{A}_{10}$

constructed via factorization. Therefore, it factors through a regular map

$\mathbf{M}^{\rm EPW}_{\mathrm{ndv}}=\operatorname{\mathsf{LGr}}_{\rm
ndv}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{3}}\hskip
0.56905pt{V}_{6})/\\!\\!/\operatorname{PGL}(V_{6})\longrightarrow\mathbf{A}_{10}.$

Similarly, this map is $\mathbf{r}$-invariant by Theorem 4.4, hence we obtain
a regular morphism

$\bar{\wp}\colon\mathbf{M}^{\rm
EPW}_{\mathrm{ndv}}/\mathbf{r}\longrightarrow\mathbf{A}_{10}.$

The composition (6.2) agrees with the period map by construction. ∎

Recall that we denoted by $\operatorname{Alb}(\widetilde{Y}_{A}^{\geq 2})$ the
Albanese variety of (any desingularization of) the double EPW surface
$\widetilde{Y}_{A}^{\geq 2}$.

######  Proposition 6.2.

For any Lagrangian
$[A]\in\operatorname{\mathsf{LGr}}_{\mathrm{ndv}}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{3}}\hskip
0.56905pt{V}_{6})$, we have
$\bar{\wp}([A])=[\operatorname{Alb}(\widetilde{Y}_{A}^{\geq 2})]$.

###### Proof.

If $Y^{3}_{A}=\varnothing$, the equality holds by Theorems 4.4 and 5.3, and
Lemma 6.1.

Assume $Y^{3}_{A}\neq\varnothing$. Consider a neighborhood
$U\subset\operatorname{\mathsf{LGr}}_{\mathrm{ndv}}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{3}}\hskip
0.56905pt{V}_{6})$ of $[A]$ such that the determinant of the tautological
bundle $\mathscr{A}$ is trivial over $U$. Consider a universal family
$\mathscr{Y}^{\geq 2}_{\mathscr{A}}\to U$ of EPW surfaces over it and the
composition

$p\colon\widetilde{\mathscr{Y}}^{\geq 2}_{\mathscr{A}}\xrightarrow{\ \pi\
}\mathscr{Y}^{\geq 2}_{\mathscr{A}}\longrightarrow{U},$

where $\pi$ is the double covering constructed as in the proof of [DK20a,
Theorem 5.2(2)]. This proof implies that $\widetilde{\mathscr{Y}}^{\geq
2}_{\mathscr{A}}$ is a smooth variety. The fiber of $p$ over $[A]$ is the
surface $\widetilde{Y}^{\geq 2}_{A}$, which is smooth away from a finite
number of ordinary double points ([DK20a, Theorem 5.2(2)]).

Let $C\subset U$ be a general smooth affine curve passing through the point
$[A]$, so that its tangent space at $[A]$ lies outside the finitely many
hyperplanes which are the images of the differential of $p$ at the various
singular points of the fiber $p^{-1}([A])$. Upon shrinking $U$, the base
change

$\widetilde{\mathscr{Y}}^{\geq 2}_{C}:=\widetilde{\mathscr{Y}}^{\geq
2}_{\mathscr{A}}\times_{U}C$

is then a smooth threefold and the morphism
$p_{C}\colon\widetilde{\mathscr{Y}}^{\geq 2}_{C}\to C$ is smooth over the
complement of the point $[A]\in C$.

Consider a double covering ${\widetilde{C}}\to C$ branched at $[A]$. The base
change

$\widetilde{\mathscr{Y}}^{\geq
2}_{\widetilde{C}}:=\widetilde{\mathscr{Y}}^{\geq
2}_{C}\times_{C}{\widetilde{C}}$

is a threefold with finitely many ordinary double points in the central fiber.
By [Ati58], there is an analytic simultaneous resolution
${(\widetilde{\mathscr{Y}}^{\geq
2}_{\widetilde{C}})^{\prime}}\to\widetilde{\mathscr{Y}}^{\geq
2}_{\widetilde{C}}$ such that the composition

$p^{\prime}\colon(\widetilde{\mathscr{Y}}^{\geq
2}_{\widetilde{C}})^{\prime}\longrightarrow\widetilde{\mathscr{Y}}^{\geq
2}_{\widetilde{C}}\longrightarrow{\widetilde{C}}$

is a smooth morphism with central fiber isomorphic to the (smooth) blow up
${(\widetilde{Y}^{\geq 2}_{\widetilde{C}})^{\prime}}\to\widetilde{Y}^{\geq
2}_{A}$ of the singular points of $\widetilde{Y}^{\geq 2}_{A}$. The sheaf
$R^{1}p^{\prime}_{*}{(\underline{{\bf Z}})}$ is then locally constant and its
stalk at $[A]$ is isomorphic to $H^{1}({(\widetilde{Y}^{\geq
2}_{\widetilde{C}})^{\prime}},{\bf Z})$.

By Theorems 4.4 and 5.3, this sheaf carries, away from the point $[A]$, a
variation of Hodge structure that comes from the middle cohomology of a family
of smooth projective varieties of odd dimension, hence it has a canonical
principal polarization. Since the sheaf $R^{1}p^{\prime}_{*}(\underline{{\bf
Z}})$ is locally constant on the whole ${\widetilde{C}}$, this polarization
extends across this point. In particular, the natural Hodge structure on the
stalk $H^{1}({(\widetilde{Y}^{\geq 2}_{\widetilde{C}})^{\prime}},{\bf Z})$ at
$[A]$ has a principal polarization, hence provides a principal polarization on
the Albanese variety $\operatorname{Alb}({(\widetilde{Y}^{\geq
2}_{\widetilde{C}})^{\prime}})$ and the map
${\widetilde{C}}\to\mathbf{A}_{10}$ defined by the above variation takes the
point $[A]$ to $[\operatorname{Alb}({(\widetilde{Y}^{\geq
2}_{\widetilde{C}})^{\prime}})]$. Since this map agrees on
${\widetilde{C}}\smallsetminus\\{[A]\\}$ with the composition

${\widetilde{C}}\longrightarrow C\longrightarrow
U\longrightarrow\mathbf{M}^{\rm
EPW}_{\mathrm{ndv}}\longrightarrow\mathbf{M}^{\rm
EPW}_{\mathrm{ndv}}/\mathbf{r}\xrightarrow{\ \bar{\wp}\ }\mathbf{A}_{10},$

it agrees everywhere, hence
$\bar{\wp}([A])=[\operatorname{Alb}({(\widetilde{Y}^{\geq
2}_{\widetilde{C}})^{\prime}})]$. ∎

We now use these results to prove Theorem 1.3.

###### Proof of Theorem 1.3.

The factorization of the period map $\wp_{n}$ is proved in Lemma 6.1 and the
equality $\wp_{n}([X])=[\operatorname{Alb}(\widetilde{Y}^{\geq 2}_{A(X)})]$
follows from this factorization and Proposition 6.2. ∎

######  Remark 6.3.

Consider the natural action of $\operatorname{PGL}(V_{6})$ on
$\operatorname{\mathsf{LGr}}_{\mathrm{ndv}}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{3}}\hskip
0.56905pt{V}_{6})\times{\bf P}(V_{6}^{\vee})$, linearized as in [DK20b,
Section 5.4]. For each $n\in\\{3,4,5,6\\}$, there is by [DK20b, Theorem 5.15]
a canonical embedding

(6.3) $\mathbf{M}_{n}^{\rm
GM}\subset(\operatorname{\mathsf{LGr}}_{\mathrm{ndv}}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{3}}\hskip
0.56905pt{V}_{6})\times{\bf
P}(V_{6}^{\vee}))/\\!\\!/\operatorname{PGL}(V_{6})$

and
$(\operatorname{\mathsf{LGr}}_{\mathrm{ndv}}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{3}}\hskip
0.56905pt{V}_{6})\times{\bf
P}(V_{6}^{\vee}))/\\!\\!/\operatorname{PGL}(V_{6})\to\mathbf{M}^{\rm
EPW}_{\mathrm{ndv}}$ is generically a ${\bf P}^{5}$-fibration (the fiber over
any point $[A]$ is isomorphic to ${\bf
P}(V_{6}^{\vee})/\\!\\!/\operatorname{PGL}(V_{6})_{A}$). The inclusion (6.3)
is an open embedding for $n=5$, a closed embedding for $n=3$, and

$(\operatorname{\mathsf{LGr}}_{\mathrm{ndv}}(\textstyle{\bigwedgeop\displaylimits\hskip-2.56073pt^{3}}\hskip
0.56905pt{V}_{6})\times{\bf
P}(V_{6}^{\vee}))/\\!\\!/\operatorname{PGL}(V_{6})=\mathbf{M}_{5}^{\rm
GM}\sqcup\mathbf{M}_{3}^{\rm GM}$

by (2.10). This property is reminiscent of the Satake compactification.

We can also complete the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

By Proposition 6.2, the morphism $\bar{\wp}$ defines for each Lagrangian $A$
with no decomposable vectors a principal polarization on
$\operatorname{Alb}(\widetilde{Y}^{\geq 2}_{A})$ such that (1.2) holds; this
proves the first part of the theorem.

By Lemma 6.1, the isomorphism (1.4) of principally polarized abelian varieties
holds for all smooth GM varieties $X$ of dimension $n\in\\{3,5\\}$; this
proves the last part of the theorem.

Finally, the isomorphism (1.3) for $A$ with $Y^{3}_{A}=\varnothing$ was
established in Theorems 4.4 and 5.3. For $A$ with $Y^{3}_{A}\neq\varnothing$,
the proof of Proposition 6.2 gives an isomorphism

$H_{n}(X,{\bf Z})\simeq H_{1}(({\widetilde{Y}^{\geq 2}_{A(X)}})^{\prime},{\bf
Z}),$

where $({\widetilde{Y}^{\geq 2}_{A(X)}})^{\prime}$ is a desingularization of
$\widetilde{Y}^{\geq 2}_{A(X)}$. Since the only singularities of
$\widetilde{Y}^{\geq 2}_{A}$ are ordinary double points, there is a canonical
isomorphism $H_{1}((\widetilde{Y}^{\geq 2}_{A}{})^{\prime},{\bf
Z})\operatorname{\stackrel{{\scriptstyle{}_{\scriptstyle\sim}}}{{\to}}}H_{1}(\widetilde{Y}^{\geq
2}_{A},{\bf Z})$. This proves the second part of the theorem. ∎

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