# On the cone conjecture for Enriques manifolds

Gianluca Pacienza Institut Elie Cartan de Lorraine, Université de Lorraine et
CNRS, 54000 Nancy, France<EMAIL_ADDRESS>and  Alessandra Sarti UMR
7348 du CNRS, Université de Poitiers, Bât. H3 - Site du Futuroscope, TSA
61125, 11 bd Marie et Pierre Curie, 86073 POITIERS Cedex 9, France
<EMAIL_ADDRESS>
###### Abstract.

Enriques manifolds are non simply connected manifolds whose universal cover is
irreducible holomorphic symplectic, and as such they are natural
generalizations of Enriques surfaces. The goal of this note is to prove the
Morrison-Kawamata cone conjecture for such manifolds when the degree of the
cover is prime using the analogous result (established by Amerik-Verbitsky)
for their universal cover. We also verify the conjecture for the known
examples having non-prime degree.

###### Key words and phrases:

holomorphic symplectic varieties, ample cone, automorphisms, Enriques
manifolds.

###### 2022 Mathematics Subject Classification:

14C99, 14J28, 14J35, 14J40.

À Claire, avec gratitude et admiration

##  1\. Introduction

An Enriques manifold is a connected complex manifold $X$ which is not simply
connected and whose universal covering ${\widetilde{X}}$ is an irreducible
holomorphic symplectic (IHS) manifold. Enriques manifolds were simultaneously
and independently introduced in [BNWS11] and [OS11a] as a natural
generalization of Enriques surfaces. The fundamental group of an Enriques
manifold is a cyclic and finite group $G=\langle g\rangle$ and its order is
called the index of $X$, which is also the order of the torsion canonical
class $K_{X}\in\mathop{\rm Pic}\nolimits(X)$. From the definition it follows
that an Enriques manifold is even dimensional. Moreover, as
$X={\widetilde{X}}/G$, it is compact and, since $h^{2,0}(X)=h^{0,2}(X)=0$, it
turns out that an Enriques manifold is always projective (cf [BNWS11,
Proposition 2.1, (4)] and [OS11a, Corollary 2.7]) and so is its universal
cover. In those papers examples of Enriques manifolds of index 2, 3 and 4 are
constructed, while their periods are studied in [OS11b].

The Morrison-Kawamata cone conjecture (see [Mor93, Ka97]) concerns the action
of the automorphism group of manifolds (and more generally pairs) with
numerically trivial canonical class on the cone of nef classes which are
effective and predicts the existence of a rational, polyhedral fundamental
domain for such action. More precisely, and more generally, we have the
following.

######  Conjecture 1.1 (The Morrison-Kawamata Cone conjecture).

Let $X\to S$ be a $K$-trivial fiber space, that is a proper surjective
morphism $f:X\to S$ with connected fibers between normal varieties such that
$X$ has ${\mathbb{Q}}$-factorial and terminal singularities and $K_{X}$ is
zero in $N^{1}(X/S)$.

  1. (1)

There exists a rational polyhedral cone $\Pi$ which is a fundamental domain
for the action of $\mathop{\rm Aut}\nolimits(X/S)$ on $\mathop{\rm
Nef}\nolimits^{e}(X/S):=\mathop{\rm Nef}\nolimits(X/S)\cap\mathop{\rm
Eff}\nolimits(X/S)$ in the sense that

    1. (a)

$\mathop{\rm Nef}\nolimits^{e}(X/S)=\cup_{g\in\mathop{\rm
Aut}\nolimits(X/S)}g^{*}\Pi$.

    2. (b)

$\textrm{int}(\Pi)\cap\textrm{int}(g^{*}\Pi)=\emptyset$, unless
$g^{*}=\textrm{id}$ in $\mathop{\rm GL}\nolimits(N^{1}(X/S))$.

  2. (2)

There exists a rational polyhedral cone $\Pi^{\prime}$ which is a fundamental
domain in the sense above for the action of $\mathop{\rm Bir}\nolimits(X/S)$
on $\overline{\mathop{\rm Mov}\nolimits}^{e}(X/S):=\overline{\mathop{\rm
Mov}\nolimits}(X/S)\cap\mathop{\rm Eff}\nolimits(X/S)$.

Item (2) above is also known as the birational Cone conjecture.

The conjecture has been proved in dimension 2 by Sterk-Looijenga, Namikawa,
Kawamata, and Totaro (see [Ste85, Na85, Ka97, Tot10]), by Prendergast-Smith
[PS12] for abelian varieties, by Amerik-Verbitsky [AV17, AV20] for IHS
manifolds, building upon [Mar11], see also Markman-Yoshioka [MY15] and by
Lehn-Mongardi-Pacienza [LMP22] for singular IHS varieties. For a recent
extension to a not necessarily closed field of characteristic 0 see [Ta21,
Theorem 1.0.5]. Notice that for IHS varieties ${\widetilde{X}}$ in the
conjecture $\mathop{\rm Nef}\nolimits^{e}({\widetilde{X}})$ is replaced by the
convex hull $\mathop{\rm Nef}\nolimits^{+}({\widetilde{X}})$ of $\mathop{\rm
Nef}\nolimits({\widetilde{X}})\cap\mathop{\rm
Pic}\nolimits({\widetilde{X}})_{{\mathbb{Q}}}$ inside $\mathop{\rm
Pic}\nolimits({\widetilde{X}})_{{\mathbb{R}}}$ (same for ${\mathop{\rm
Mov}\nolimits}^{e}(X)$ which is replaced by the convex hull ${\mathop{\rm
Mov}\nolimits}^{+}(X)$ of ${\mathop{\rm Mov}\nolimits}(X)\cap\mathop{\rm
Pic}\nolimits({\widetilde{X}})_{{\mathbb{Q}}}$ inside $\mathop{\rm
Pic}\nolimits({\widetilde{X}})_{{\mathbb{R}}}$). One can see that $\mathop{\rm
Nef}\nolimits^{e}({\widetilde{X}})\subset\mathop{\rm
Nef}\nolimits^{+}({\widetilde{X}})$ (cf. [MY15, Remark 1.4]). This statement
is analogous to the classical one for the 4 known deformations type (and
equivalent in general modulo the SYZ-conjecture). For Calabi-Yau varietes the
conjecture is open in general. For a recent result in this direction, see
[GLW22] and the reference therein. For recent results in the relative case,
establishing in particular the conjecture for families of $K3$ surfaces, see
[LZ22]. We refer the reader to [Tot12, LOP18] for nice introductions to this
topic. The conjecture is deeply related with birational geometry. Item (1) of
the conjecture yields the finiteness, up to automorphisms, of birational
contractions and fiber space structures of the initial variety, while item (2)
implies, modulo standard conjectures of the MMP, the finiteness of minimal
models, up to isomorphisms, of any ${\mathbb{Q}}$-factorial and terminal
variety with non-negative Kodaira dimension (cf. [CL14, Theorem 2.4]).

From now on we will restrict ourselves to the absolute and smooth case. By the
Beauville-Bogomolov decomposition theorem (see [Be83a]) we know that any
$K$-trivial variety $V$ admits a finite étale cover $\widetilde{V}\to V$,
where $\widetilde{V}$ is a product of Calabi-Yau manifolds, IHS manifolds and
an abelian variety. A general question is: suppose we know Conjecture 1.1 for
$\widetilde{V}$, can we deduce for $V$? Our main result provides a positive
answer when we have only one factor of IHS type.

######  Theorem 1.2.

The Morrison-Kawamata cone conjecture (respectively the birational Morrison-
Kawamata cone conjecture) holds for the $\mathop{\rm Aut}\nolimits(X)$-action
on the $\mathop{\rm Nef}\nolimits^{+}$-cone (respectively for the $\mathop{\rm
Bir}\nolimits(X)$-action on $\overline{\mathop{\rm Mov}\nolimits}^{+}(X)$) on
any Enriques manifold $X$ of prime index $p$.

The index 4 Enriques manifolds are therefore left out by our result.
Nevertheless, the proof of Theorem 1.2 and the explicit computation of the
action of $G$ on the cohomology of the IHS cover allow us to verify the
conjecture in the index 4 known cases, see Section 4.1 for more details. Of
course it would be nice to have a general argument valid for any index.

As for the proof, if $\pi:{\widetilde{X}}\to X={\widetilde{X}}/G$ is the
covering map, by Amerik-Verbitsky [AV17, AV20] there exists a rational
polyhedral convex domain ${\widetilde{D}}$ which is a fundamental domain for
the action of $\mathop{\rm Aut}\nolimits({\widetilde{X}})$ on $\mathop{\rm
Nef}\nolimits^{+}({\widetilde{X}})$. We set

$D:={\widetilde{D}}\cap\pi^{*}N^{1}(X).$

Where recall that $N^{1}(X)$ denotes the Néron-Severi group of $X$ which
coincides with the Picard group of $X$. The proof of Theorem 1.2 then consists
in showing that $\pi_{*}(D)$ is a fundamental domain for the action of
$\mathop{\rm Aut}\nolimits(X)$ on $\mathop{\rm Nef}\nolimits^{+}(X)$: the
rationality and polyhedrality of $D$ are implied by those of
${\widetilde{D}}$. The main point is to show that if $\xi\in\pi^{*}N^{1}(X)$
and $\varphi$ is an automorphism of ${\widetilde{X}}$ such that
$\varphi^{*}(\xi)\in{\widetilde{D}}$, then $\varphi$ “descends” to an
automorphism of $X$, i.e. $\varphi$ commutes with $G$. The idea to show this
commutativity is to check it only on cohomology and then use Global Torelli to
conclude. Some care has to be taken, as it is well known that there are non-
trivial automorphisms acting trivially on cohomology for the 2 deformation
classes coming from abelian surfaces (see Remark 2.2 for details). Let us
finally point out that the commutativity is shown first on the general point
in the moduli space of Enriques manifolds, for which we check (under the
hypothesis that the index is prime) that $\mathop{\rm
Pic}\nolimits({\widetilde{X}})=H^{2}({\widetilde{X}},\mathbb{Z})^{G}$ and
then, via a limiting process, we extend the result to special points.

Acknowledgments. We thank A. Chiodo for his kind invitation to Jussieu, where
this project started and L. Kamenova, G. Mongardi and A. Oblomkov, the
organizers of the workshop “Hyperkähler quotients, singularities and quivers”
at the Simons Center for Geometry and Physics, where it was completed. We
thank S. Boissière, C. Bonnafé, E. Floris, A. Garbagnati and E. Markman for
very useful exchanges. G.P. is partially supported by the CNRS International
Emerging Actions project “Birational and arithmetic aspects of orbifolds”.
A.S. is partially supported by the ANR project No. ANR-20-CE40-0026-01
(SMAGP).

##  2\. Preliminaries

###  2.1. Basic facts on IHS manifolds

An irreducible holomorphic symplectic manifold is a compact Kähler manifold
${\widetilde{X}}$ which is simply connected and carries a holomorphic
symplectic 2–form $\sigma$ everywhere non–degenerate, such that
$H^{0}({\widetilde{X}},\Omega^{2}_{\widetilde{X}})={\mathbb{C}}\cdot\sigma$.
For a general introduction of the subject, we refer to [Huy99].

Let ${\widetilde{X}}$ be an irreducible holomorphic symplectic manifold of
dimension $2n\geq 2$. Let $\sigma\in
H^{0}({\widetilde{X}},\Omega^{2}_{\widetilde{X}})$ such that
$\int_{{\widetilde{X}}}\sigma^{n}\bar{\sigma}^{n}=1$. Then, following [Be83a],
the second cohomology group $H^{2}({\widetilde{X}},\mathbb{C})$ is endowed
with a quadratic form $q=q_{{\widetilde{X}}}$ defined as follows

$q(a):=\frac{n}{2}\int_{{\widetilde{X}}}(\sigma\bar{\sigma})^{n-1}a^{2}+(1-n)\bigg{(}\int_{{\widetilde{X}}}\sigma^{n}\bar{\sigma}^{n-1}a\bigg{)}\cdot\bigg{(}\int_{{\widetilde{X}}}\sigma^{n-1}\bar{\sigma}^{n}a\bigg{)},\
a\in H^{2}({\widetilde{X}},\mathbb{C}),$

which is non-degenerate and, up to a positive multiple, is induced by an
integral nondivisible quadratic form on $H^{2}({\widetilde{X}},\mathbb{Z})$ of
signature $(3,b_{2}({\widetilde{X}})-3)$. The form $q$ is called the
Beauville-Bogomolov-Fujiki quadratic form of ${\widetilde{X}}$. By Fujiki
[Fuj87], there exists a positive rational number $c=c_{\widetilde{X}}$ (the
Fujiki constant of ${\widetilde{X}}$) such that

$c\cdot q^{n}(\alpha)=\int_{{\widetilde{X}}}\alpha^{2n},\ \forall\alpha\in
H^{2}({\widetilde{X}},\mathbb{Z}).$

We recall also the following Hodge theoretic version of Torelli theorem.

######  Theorem 2.1.

[Mar11, Theorem 1.3] Let $Z$ and $Y$ be two irreducible holomorphic symplectic
manifolds deformation equivalent one to each other. Then:

  * (1)

$Z$ and $Y$ are bimeromorphic if and only if there exists a parallel transport
operator $f\\!:\\!H^{2}(Z,\mathbb{Z})\longrightarrow H^{2}(Y,\mathbb{Z})$ that
is an isomorphism of Hodge structures;

  * (2)

if this is the case, there exists an isomorphism $\bar{f}:Y\longrightarrow Z$
inducing $f$ if and only if $f$ preserves a Kähler class.

######  Remark 2.2.

As usual ${\widetilde{X}}$ denotes an IHS manifold. Recall that
$\nu:\mathop{\rm Aut}\nolimits({\widetilde{X}})\longrightarrow{\rm
Aut}(H^{2}({\widetilde{X}},\mathbb{Z}))$ is finite, by [Huy99, Proposition
9.1]. Notice moreover that by a result of Hassett and Tschinkel, [HT13,
Theorem 2.1] the kernel of the homomorphism $\nu$ is invariant under smooth
deformations of the manifold ${\widetilde{X}}$. This allows to compute it for
all 4 known deformation types of IHS manifolds, thanks to [Be83a, Proposition
10], [BNWS11, Corollary 3.3] and [MW17, Theorems 3.1 and 5.2 ]. We have that
$\nu$ is injective for deformations of punctual Hilbert schemes of $K3$
surfaces and OG10 manifolds, while the kernel is generated by the group of
translations by points of order $n$ on an abelian surface and by $-{\rm id}$
(respectively equal to $(\mathbb{Z}/2\mathbb{Z})^{\times 8}$) if
${\widetilde{X}}$ is a deformation of a generalized Kummer of dimension $2n$
(respectively of an OG6 manifold).

###  2.2. Basic facts on Enriques manifolds.

Consider now the quotient $\pi:{\widetilde{X}}\to X={\widetilde{X}}/G$, where
$|G|=d$. Recall that the automorphisms of $X$ identify to those of
${\widetilde{X}}$ commuting with $g$, i.e.

(1) $\mathop{\rm Aut}\nolimits(X)=\\{{\widetilde{\tau}}\in\mathop{\rm
Aut}\nolimits({\widetilde{X}}):{\widetilde{\tau}}\circ
g=g\circ{\widetilde{\tau}}\\}/G.$

Moreover the action of $g$ on ${\widetilde{X}}$ cannot be symplectic, i.e. we
cannot have $g^{*}\sigma_{\widetilde{X}}=\sigma_{\widetilde{X}}$, otherwise
$X$ would have $h^{2,0}(X)\not=0$. Therefore there exists a $d$-th root of
unity $\lambda\not=1$ such that
$g^{*}\sigma_{\widetilde{X}}=\lambda\sigma_{\widetilde{X}}$. The action must
be moreover purely non–symplectic, i.e. $\lambda$ is a primitive root of unity
of the same order of $g$, otherwise, if $G$ contains symplectic automorphisms,
there are points with a non trivial stabilizer and the quotient would be
singular. Observe that if $G$ is of prime order then a non–symplectic action
is the same as a purely non–symplectic one.

######  Lemma 2.3.

For any Enriques manifold $X$ we have $H^{2}(X,{\mathbb{Z}})=\mathop{\rm
Pic}\nolimits(X)$. Moreover we have
$\pi^{*}H^{2}(X,{\mathbb{Z}})=H^{2}({\widetilde{X}},{\mathbb{Z}})^{G}\subset\mathop{\rm
Pic}\nolimits({\widetilde{X}})$. In particular

$\mathop{\rm dim}\nolimits_{\mathbb{R}}\pi^{*}N^{1}(X)=\mathop{\rm
dim}\nolimits_{\mathbb{R}}\pi^{*}\mathop{\rm Pic}\nolimits(X)=\mathop{\rm
rk}\nolimits H^{2}({\widetilde{X}},{\mathbb{Z}})^{G}.$

###### Proof.

Consider $\xi\in H^{2}(X,{\mathbb{Z}})$, we have

$q(\pi^{*}\xi,\sigma)=q(g^{*}\pi^{*}\xi,g^{*}\sigma)=q(\pi^{*}\xi,\lambda\sigma)=\lambda
q(\pi^{*}\xi,\sigma)$

with $\lambda\not=1$ as $g$ is not a symplectic automorphism. Then
$q(\pi^{*}\xi,\sigma)=0$, which implies that $\pi^{*}\xi\in\mathop{\rm
Pic}\nolimits({\widetilde{X}})$, as $\mathop{\rm
Pic}\nolimits({\widetilde{X}})=\sigma^{\perp_{q}}\cap
H^{2}({\widetilde{X}},{\mathbb{Z}})$ so that $\xi\in\mathop{\rm
Pic}\nolimits(X)$. We have shown that $H^{2}(X,\mathbb{Z})\subset\mathop{\rm
Pic}\nolimits(X)\subset H^{2}(X,\mathbb{Z})$ so we get the first equality
(notice that this equality also follows immediately from the fact that
$H^{2,0}(X)=0$, but we will need the argument below). By construction
$\pi^{*}H^{2}(X,{\mathbb{Z}})\subset H^{2}({\widetilde{X}},{\mathbb{Z}})^{G}$
with finite index and moreover by [OS11a, proof of Proposition 2.8, and proof
of Proposition 5.1] we get in fact equality. Take now a $G$–invariant class
$\eta$ in $H^{2}({\widetilde{X}},\mathbb{Z})$. Since $\pi^{*}\mathop{\rm
Pic}\nolimits(X)\subset\mathop{\rm Pic}\nolimits({\widetilde{X}})$ the
previous equality implies that
$H^{2}({\widetilde{X}},\mathbb{Z})^{G}\subset\mathop{\rm
Pic}\nolimits({\widetilde{X}})$ (which is in fact a more general fact for
non–symplectic automorphisms acting on IHS manifold, but this gives an easy
proof). ∎

###  2.3. Examples

We recall the following examples from [BNWS11] and [OS11a] which to our
knowledge are the only known examples so far.

####  2.3.1. From $K3^{[n]}$ manifolds

Consider a K3 surface $S$ with an Enriques involution $\iota$. On the Hilbert
scheme $S^{[n]}$ consider the natural involution $\iota^{[n]}$. Notice that
$\iota^{[n]}$ has no fixed points if $n$ is odd and
$S^{[n]}/\langle\iota^{[n]}\rangle$ is an Enriques manifold of dimension $2n$
and index 2. Moreover in this case we know the invariant sublattice for the
action of $\iota^{[n]}$. In fact $H^{2}(S,\mathbb{Z})^{\iota}=U(2)\oplus
E_{8}(2)$ where $U$ is the hyperbolic plane with bilinear form multiplied by
$2$ and $E_{8}(2)$ is the negative definite lattice associated to the
corresponding Dynkin diagram and bilinear form multiplied by $2$. Since
$\iota^{[n]}$ is a natural automorphism, we have

$H^{2}(S^{[n]},\mathbb{Z})^{\iota^{[n]}}=H^{2}(S,\mathbb{Z})^{\iota}\oplus\langle-2(n-1)\rangle=U(2)\oplus
E_{8}(2)\oplus\langle-2(n-1)\rangle.$

Notice that the punctual Hilbert scheme $(S/\iota)^{[n]}$ of the Enriques
surface is not an Enriques manifold, in the sense of our definition, as its
universal cover is a Calabi-Yau manifold, see [OS11a, Theorem 3.1]

####  2.3.2. From ${\rm Kum}_{n}$ manifolds

These examples arise from a bielliptic surface, i.e. a surface $S$ with
torsion canonical class of order $d\in\\{2,3,4\\}$ admitting a finite étale
covering by an abelian surface $A\to S$, and specifically, under certain
conditions on $d$ and $n$, from a free action of a finite group on $A^{[n+1]}$
which preserves ${\rm Kum}_{n}(A)\subset A^{[n+1]}$. We refer the reader to
[OS11a, Section 6] and [BNWS11, Section 4.2] for further details. We recall in
detail the example for $d=4$ in Section 4.1.

###  2.4. Moduli spaces of marked Enriques manifolds.

We follow the presentation in [BCS16]. See also [OS11b] for another equivalent
approach and for the study of the period map.

Let ${\widetilde{X}}$ be an IHS manifold. Recall that a marking for
${\widetilde{X}}$ is an isometry to a given lattice
$H^{2}({\widetilde{X}},{\mathbb{Z}})\to L$.

Let $M$ be an even non-degenerate lattice of rank $\rho\geq 1$ and signature
$(1,\rho-1)$. An _$M$ -polarized_ IHS manifold is a pair $({\widetilde{X}},j)$
where ${\widetilde{X}}$ is a projective IHS manifold and $j$ is a primitive
embedding of lattices $j:M\hookrightarrow\mathop{\rm
NS}\nolimits({\widetilde{X}})$. Two $M$-polarized IHS manifolds
$({\widetilde{X}}_{1},j_{1})$ and $({\widetilde{X}}_{2},j_{2})$ are called
_equivalent_ if there exists an isomorphism
$f:{\widetilde{X}}_{1}\to{\widetilde{X}}_{2}$ such that $j_{1}=f^{*}\circ
j_{2}$. As in [DK07, Section 10] and [D96] one can construct a moduli space of
_marked_ $M$-polarized IHS manifolds as follows. We fix a primitive embedding
of $M$ in $L$ and we identify $M$ with its image in $L$. A _marking_ of
$({\widetilde{X}},j)$ is an isomorphism of lattices $\eta:L\to
H^{2}({\widetilde{X}},\mathbb{Z})$ such that $\eta_{|M}=j$. As observed in
[D96, p.11], if the embedding of $M$ in $L$ is unique up to an isometry of $L$
then every $M$-polarization admits a compatible marking. Two $M$-polarized
marked IHS manifolds $({\widetilde{X}}_{1},j_{1},\eta_{1})$ and
$({\widetilde{X}}_{2},j_{2},\eta_{2})$ are called _equivalent_ if there exists
an isomorphism $f:{\widetilde{X}}_{1}\to{\widetilde{X}}_{2}$ such that
$\eta_{1}=f^{*}\circ\eta_{2}$ (this clearly implies that $j_{1}=f^{*}\circ
j_{2}$). Let $T:=M^{\perp}\cap L$ be the orthogonal complement of $M$ in $L$
and set

$\Omega_{M}:=\\{x\in\mathbb{P}(T\otimes\mathbb{C})\,|\,q(x)=0,\,q(x+\bar{x})>0\\}.$

Since $T$ has signature $(2,\mathop{\rm rk}\nolimits(T)-\rho)$ the period
domain $\Omega_{M}$ is a disjoint union of two connected components of
dimension $\mathop{\rm rk}\nolimits(T)-\rho$. For each $M$-polarized marked
IHS manifolds $(X,j,\eta)$, since $\eta(M)\subset\mathop{\rm
NS}\nolimits({\widetilde{X}})$, we have
$\eta^{-1}(H^{2,0}({\widetilde{X}}))\in\Omega_{M}$. On the other hand, by the
surjectivity of the period map [Huy99, Theorem 8.1] restricted to any
connected component $\mathfrak{M}^{0}_{L}$ of $\mathfrak{M}_{L}$ (the moduli
space of IHS manifolds with second integral cohomology isometric to a given
lattice $L$) we can associate to each point $\omega\in\Omega_{M}$ an
$M$-polarized IHS manifold as follows: there exists a marked pair
$({\widetilde{X}},\eta)\in\mathfrak{M}_{L}^{0}$ such that
$\eta^{-1}(H^{2,0}({\widetilde{X}}))=\omega\in\mathbb{P}(T\otimes\mathbb{C})$
so $M=T^{\perp}\subset\omega^{\perp}\cap L$, hence $\eta(M)\subset
H^{2,0}({\widetilde{X}})^{\perp}\cap
H^{1,1}_{\mathbb{Z}}({\widetilde{X}})=\mathop{\rm
NS}\nolimits({\widetilde{X}})$ and we take $({\widetilde{X}},\eta_{|M},\eta)$.
Recall that for a generic point in the moduli space we have $\mathop{\rm
NS}\nolimits(\widetilde{X})=\eta(M)$.

By the Local Torelli Theorem for IHS manifolds, an $M$-polarized IHS manifold
$({\widetilde{X}},j)$ has a local deformation space $\mathop{\rm
Def}\nolimits_{M}({\widetilde{X}})$ that is contractible, smooth of dimension
$\mathop{\rm rk}\nolimits(T)-\rho$, such that the period map
$P\colon\mathop{\rm Def}\nolimits_{M}({\widetilde{X}})\to\Omega_{M}$ is a
local isomorphism (see [D96]). By gluing all these local deformation spaces
one obtains a moduli space $K_{M}$ of marked $M$-polarized IHS manifolds that
is a non-separated analytic space, with a period map $P\colon
K_{M}\to\Omega_{M}$.

To construct a period domain for Enriques manifolds we have to take the
non–symplectic action into account.

Let $({\widetilde{X}},j)$ be an $M$-polarized IHS manifold and $G=\langle
g\rangle$ a cyclic group of prime order $p\geq 2$ acting non-symplectically on
${\widetilde{X}}$. Assume that the action of $G$ on $j(M)$ is the identity and
that there exists a group homomorphism $\rho:G\longrightarrow O(L)$ such that

$M=L^{\rho}:=\\{x\in L\,|\,\rho(g)(x)=x,\forall g\in G\\}.$

We define a $(\rho,M)$-polarization of $({\widetilde{X}},j)$ as a marking
$\eta\colon L\to H^{2}({\widetilde{X}},\mathbb{Z})$ such that $\eta_{|M}=j$
and $g^{\ast}=\eta\circ\rho(g)\circ\eta^{-1}$.

Two $(\rho,M)$-polarized IHS manifolds $({\widetilde{X}}_{1},j_{1})$ and
$({\widetilde{X}}_{2},j_{2})$ are isomorphic if there are markings,
$\eta_{1}\colon L\to H^{2}({\widetilde{X}}_{1},\mathbb{Z})$ and
$\eta_{2}\colon L\longrightarrow H^{2}({\widetilde{X}}_{2},\mathbb{Z})$ such
that $\eta_{i}{{}_{|M}}=j_{i}$, and an isomorphism
$f\colon{\widetilde{X}}_{1}\to{\widetilde{X}}_{2}$ such that
$\eta_{1}=f^{*}\circ\eta_{2}$.

Recall that by construction $\mathbb{C}\sigma_{\widetilde{X}}$ is identified
to the line in $L\otimes\mathbb{C}$ defined by
$\eta^{-1}(H^{2,0}({\widetilde{X}}))$. Let $\lambda\in\mathbb{C}^{*}$ such
that $\rho(g)(\sigma_{\widetilde{X}})=\lambda\sigma_{\widetilde{X}}$. Observe
that $\lambda\not=1$ since the action is non-symplectic and it is a primitive
$p$-th root of unity since $p$ is prime. By construction
$\sigma_{\widetilde{X}}$ belongs to the eigenspace of $T\otimes\mathbb{C}$
relative to the eigenvalue $\lambda$, where $T=M^{\perp}\cap L$. We denote it
by $T(\lambda)$ (if $p=2$, we have $\lambda=-1$ and we denote
$T(\lambda)=T_{\mathbb{R}}(\lambda)\otimes\mathbb{C}$, where
$T_{\mathbb{R}}(\lambda)$ is the real eigenspace relative to $\lambda=-1$).

Assume that $\lambda\not=-1$, then the period belongs to the space

$\Omega_{M}^{\rho,\lambda}:=\\{x\in\mathbb{P}(T(\lambda))\,|\,q(x+\bar{x})>0\\}$

of dimension $\mathop{\rm dim}\nolimits T(\lambda)-1$ which is a complex ball
if $\mathop{\rm dim}\nolimits T(\lambda)\geq 2$. By using the fact that
$\lambda\not=-1$ it is easy to check that every point
$x\in\Omega_{M}^{\rho,\lambda}$ satisfies automatically the condition
$q(x)=0$.

If $\lambda=-1$ we set
$\Omega_{M}^{\rho,\lambda}:=\\{x\in\mathbb{P}(T(\lambda))\,|\,q(x)=0,\,q(x+\bar{x})>0\\}$.
It has dimension $\mathop{\rm dim}\nolimits T(\lambda)-2$. Clearly in both
cases $\Omega_{M}^{\rho,\lambda}\subset\Omega_{M}$.

Let $X$ be a marked Enriques manifold of index $d$, that is the data of a
$(\rho,M)$-polarization of $({\widetilde{X}},j)$ with a marking $\eta\colon
L\to H^{2}(X,\mathbb{Z})$ such that $\eta_{|M}=j$ and
$g^{\ast}=\eta\circ\rho(g)\circ\eta^{-1}$, so that $X={\widetilde{X}}/G$.
Recall that by the Bogomolov-Tian-Todorov theorem the Kuranishi space
$\mathop{\rm Def}\nolimits(X)$ of an Enriques manifold $X$ of index $d$ is
smooth. Moreover Oguiso and Schröer verified in [OS11b, Proposition 1.2] that,
after possibly shrinking $\mathop{\rm Def}\nolimits(X)$, every point
parametrizes an index $d$ Enriques manifold.

If $\mathscr{X}\to B$ is a flat family of marked Enriques manifolds, then, as
remarked in [OS11b, Section 2] the universal covering
$\mathscr{{\widetilde{X}}}\to\mathscr{X}$ of the family is also the fiberwise
universal covering. The period map of the family $\mathscr{X}\to B$ is then
defined as

(2) ${\mathfrak{p}}_{B}:B\to\Omega_{M}^{\rho,\lambda},\
b\mapsto\eta^{-1}(H^{2,0}({\widetilde{X}}_{b})).$

By [OS11b, Theorem 2.4] the local Torelli theorem holds, namely the period map
${\mathfrak{p}}_{B}$ is a local isomorphism. Hence, as for IHS manifolds, we
can patch together the Kuranishi spaces via the Oguiso-Schröer local Torelli
theorem, to construct a (non-separated) moduli space of (marked) Enriques
manifolds in terms of $(\rho,M)$-polarized IHS manifolds.

Let us now briefly discuss the case when the order of $G$ is not necessarily a
prime, but the action of $G$ is purely non–symplectic (recall that it means
that $g$ acts on the holomorphic two-form by multiplication by a primitive
root of unity of the same order as $g$). So let $|G|=d$, where $d$ is not
necessarily a prime number and fix the action as
$g^{*}\sigma_{{\widetilde{X}}}=\lambda\sigma_{{\widetilde{X}}}$ where
$\lambda$ is a primitive $d$–root of unity. Then take $T(\lambda)$ the
eigenspace in $T\otimes\mathbb{C}$ relative to the eigenvalue $\lambda$ ,
where we use here the same notations as before, i.e. $M$ is the invariant
sublattice in $L$ for the action of $\rho(G)$ and $T$ its orthogonal
complement. As before the period $\sigma_{{\widetilde{X}}}$ of
${\widetilde{X}}$ belongs to $\Omega_{M}^{\rho,\lambda}$, but here the
Néron–Severi group of a generic ${\widetilde{X}}$ in the moduli space just
contains $M$ and it is not equal (as it is the case for the prime order, see
Lemma 3.2). This means concretely that all eigenspaces corresponding to
non–primitive $d$–roots of unity for the actions of $G$ on
$H^{2}({\widetilde{X}},\mathbb{Z})\otimes\mathbb{C}$ are contained in
$\mathop{\rm Pic}\nolimits({\widetilde{X}})\otimes\mathbb{C}$ and the
eigenspaces for the primitive roots are contained in
$T_{\widetilde{X}}\otimes\mathbb{C}$.

##  3\. Proof of the cone conjecture

First we prove the following.

######  Lemma 3.1.

$H^{2}({\widetilde{X}},{\mathbb{Q}})=\pi^{*}[H^{2}(X,{\mathbb{Q}})]\oplus[\pi^{*}(H^{2}(X,{\mathbb{Q}}))]^{\perp_{q}}$

###### Proof.

Let us take a non-zero class
$\beta\in[\pi^{*}(H^{2}(X,{\mathbb{Q}}))]^{\perp_{q}}$. Suppose by
contradiction that

(3) $\beta\in\pi^{*}H^{2}(X,{\mathbb{Q}}).$

Then $q(\beta,\beta)=0$. Let $A\in\mathop{\rm Pic}\nolimits({\widetilde{X}})$
an ample line bundle. Without loss of generality we may assume that
$A=\pi^{*}B$ with $B$ ample (so that $A$ is $G$-invariant). By the choice of
$\beta$ and the $G$-invariance of $A$ we have that $q(\beta,A)=0$ which, by
(3), contradicts the Hodge-index theorem for $q$. ∎

We start making the proof in the case that $|G|$ is cyclic of prime order and
we take $X$ generic in the moduli space, so that $\mathop{\rm
Pic}\nolimits({\widetilde{X}})=H^{2}({\widetilde{X}},\mathbb{Z})^{G}$ as shown
in Lemma 3.2.

###  3.1. The proof when the order is prime at the general point.

######  Lemma 3.2.

Under the hypothesis $|G|=p$ is a prime number for a general Enriques manifold
${\widetilde{X}}\to X$ of index $p$ we have that $\mathop{\rm
Pic}\nolimits({\widetilde{X}})=H^{2}({\widetilde{X}},{\mathbb{Z}})^{G}$.

###### Proof.

We use the notation introduced in Section 2.4. Set
$M=H^{2}({\widetilde{X}},{\mathbb{Z}})^{G}$ and $T=M^{\perp}\cap L$. We first
claim that, if $t\in T\setminus\\{0\\}$, then $t^{\perp}$ can not contain
$\Omega^{\rho,\lambda}_{M}$. To see this, first notice that $t^{\perp}\cap L$
is endowed with a weight 2 Hodge structure. If
$t^{\perp}\supset\Omega^{\rho,\lambda}_{M}$ then it must contain the full
projectivized eigenspace $\mathbb{P}T(\lambda)$. Therefore $t^{\perp}$ must
contain the smallest weight 2 Hodge sub-structure of
$H^{2}({\widetilde{X}},{\mathbb{Z}})$ containing $T(\lambda)$, which is $T$
since the natural representation $G\to\textrm{GL}(T\otimes\mathbb{Q})$ is
irreducible (since the eigenvectors of the representation of $G$ in
${\mathbb{C}}^{*}$ are primitive roots of unity by the hypothesis $d=p$). Then
$t\in T^{\perp}$, which is absurd, and the claim is proved.

Then for each $t\in T\setminus\\{0\\}$ consider the restriction $H_{t}$ of the
hyperplane ${t}^{\perp}$ to $\Omega^{\rho,\lambda}_{M}$ and let
$\mathscr{H}:=\bigcup_{t\in T\setminus\\{0\\}}H_{t}$. Each subset
$\Omega^{\rho,\lambda}_{M}\setminus{H}_{t}$ is open and dense in
$\Omega^{\rho,\lambda}_{M}$ hence by Baire’s Theorem the subset
$\Omega^{0}:=\Omega^{\rho,\lambda}_{M}\setminus\mathscr{H}$ is dense in
$\Omega^{\rho,\lambda}_{M}$ since $\mathscr{H}$ is a countable union of
complex closed subspaces. If $x=\mathfrak{p}(X,\eta)\in\Omega^{0}$ then
$\mathop{\rm Pic}\nolimits({\widetilde{X}})=\eta(L^{1,1}(x))=\eta(M)$, where
$L^{1,1}(x)=\\{\ell\in L:x\cdot\ell=0\\}$. Hence we conclude by local Torelli
[OS11b, Theorem 2.4]. ∎

Consider $\xi\in\mathop{\rm Nef}\nolimits^{+}(X)$. Since the Cone conjecture
holds on ${\widetilde{X}}$ we have the existence of
$\widetilde{\tau}\in\mathop{\rm Aut}\nolimits({\widetilde{X}})$ and
$\widetilde{\delta}\in\widetilde{D}$ such that

(4) $(\widetilde{\tau})^{*}\pi^{*}\xi=\widetilde{\delta}.$

We will actually not need condition (4) and prove more generally that, when
$X$ is general $\widetilde{\tau}^{*}$ commutes with $g^{*}$, for any
$\widetilde{\tau}\in\mathop{\rm Aut}\nolimits({\widetilde{X}})$.

######  Lemma 3.3.

Any automorphism $\widetilde{\tau}\in\mathop{\rm
Aut}\nolimits({\widetilde{X}})$ commutes with $g$ on
$H^{2}({\widetilde{X}},{\mathbb{Z}})^{G}=\pi^{*}H^{2}(X,{\mathbb{Z}})$.

###### Proof.

Consider $\eta\in H^{2}({\widetilde{X}},{\mathbb{Z}})^{G}$ and set
$e:=(\widetilde{\tau})^{*}(\eta)$. First notice that $e\in
H^{2}({\widetilde{X}},{\mathbb{Z}})^{G}$. Indeed by Lemma 2.3 the class $\eta$
lies in $\mathop{\rm Pic}\nolimits({\widetilde{X}})$ and automorphisms
preserve the Picard group, hence $e=(\widetilde{\tau})^{*}(\eta)\in\mathop{\rm
Pic}\nolimits({\widetilde{X}})=H^{2}({\widetilde{X}},{\mathbb{Z}})^{G}$.
Therefore $g^{*}(e)=e$ from which we deduce

$g^{*}((\widetilde{\tau})^{*}(\eta)):=g^{*}(e)=e=(\widetilde{\tau})^{*}(\eta)=(\widetilde{\tau})^{*}(g^{*}(\eta))$

where the last equality holds since $\eta\in
H^{2}({\widetilde{X}},{\mathbb{Z}})^{G}$. ∎

We then check the commutativity also on
$T(X):=(\pi^{*}H^{2}(X,{\mathbb{Z}}))^{\perp_{q}}\cap
H^{2}({\widetilde{X}},{\mathbb{Z}})$.

######  Lemma 3.4.

Any automorphism $\widetilde{\tau}\in\mathop{\rm
Aut}\nolimits({\widetilde{X}})$ commutes with $g$ on $T(X)$.

###### Proof.

Notice that $T(X)$ is the transcendental part of a weight two Hodge structure
of $K3$ type. Hence by [Huy16, Corollary 3.4, Chapter 3] the group of all
Hodge isometries of $T(X)$ is a finite cyclic group. The automorphisms
preserve $\mathop{\rm Pic}\nolimits({\widetilde{X}})$, which, by Lemma 3.2,
equals $H^{2}({\widetilde{X}},{\mathbb{Z}})^{G}$. Therefore any element of
$\mathop{\rm Aut}\nolimits({\widetilde{X}})$ preserves $T(X)$ and induces a
Hodge isometry of $T(X)$. Then the restrictions to $T(X)$ of
$(\widetilde{\tau})^{*}$ and $g^{*}$ are elements of a finite cyclic group and
as such commute. ∎

By Lemma 3.1, Lemma 3.3 and Lemma 3.4 the automorphism $\widetilde{\tau}$
commutes with $g$ on $H^{2}({\widetilde{X}},{\mathbb{Z}})$. By Torelli Theorem
2.1 if the map $\mathop{\rm Aut}\nolimits({\widetilde{X}})\longrightarrow
O(H^{2}({\widetilde{X}},\mathbb{Z}))$ is injective, then they commute also on
${\widetilde{X}}$ which by (1) implies that $\widetilde{\tau}$ descends to an
automorphism $\tau$ on $X$ and therefore
$\widetilde{\delta}=(\pi)^{*}\delta\in\widetilde{D}\cap(\pi)^{*}N^{1}(X)$ for
some class $\delta\in\mathop{\rm Nef}\nolimits^{+}(X)$.

If the map $\mathop{\rm Aut}\nolimits({\widetilde{X}})\longrightarrow
O(H^{2}({\widetilde{X}},\mathbb{Z}))$ is not injective, then by [Huy99,
Proposition 9.1, item (v)] the kernel $K$ is finite. Consider

$\Gamma:=\\{\varphi\in\mathop{\rm Aut}\nolimits({\widetilde{X}})\,|\,\varphi
g^{-1}\varphi^{-1}g\in K\\}$

and observe that $\Gamma$ is in fact equal to $\mathop{\rm
Aut}\nolimits({\widetilde{X}})$ as we have shown that each automorphism
induces an isometry that commutes on $H^{2}({\widetilde{X}},\mathbb{Z})$ with
$g$. Consider

$\Gamma_{0}:=\\{\varphi\in\mathop{\rm
Aut}\nolimits({\widetilde{X}})\,|\,\varphi g^{-1}\varphi^{-1}g=id\\}$

and notice that $\Gamma_{0}$ is a subgroup of $\mathop{\rm
Aut}\nolimits({\widetilde{X}})$ which is not necessarily normal. Let us show
that it has finite index in $\mathop{\rm Aut}\nolimits({\widetilde{X}})$.
Consider the map

$\beta:\mathop{\rm Aut}\nolimits({\widetilde{X}})\longrightarrow
K,\,\,\varphi\mapsto\varphi g^{-1}\varphi^{-1}g.$

Then one easily checks that $\beta(\varphi_{1})=\beta(\varphi_{2})$ if and
only if $\varphi_{2}^{-1}\varphi_{1}\in\Gamma_{0}$. Therefore if we take the
set $\mathop{\rm Aut}\nolimits({\widetilde{X}})/\Gamma_{0}$ of left cosets we
have an injective map $\mathop{\rm
Aut}\nolimits({\widetilde{X}})/\Gamma_{0}\longrightarrow K$ showing that
$\Gamma_{0}$ has finite index in $\mathop{\rm Aut}\nolimits({\widetilde{X}})$.
Let now $\bar{\gamma}_{1},\ldots,\bar{\gamma}_{r}$ be all the classes in
$\mathop{\rm Aut}\nolimits({\widetilde{X}})/\Gamma_{0}$ and $\varphi$ be the
automorphism such that $\varphi^{*}(\pi^{*}\xi)\in\widetilde{D}$. By
considering the class $\bar{\varphi}\in\mathop{\rm
Aut}\nolimits({\widetilde{X}})/\Gamma_{0}$ then
$\bar{\varphi}=\bar{\gamma}_{j}$ for a certain $j=1,\ldots,r$ so that
$\gamma_{j}^{-1}\varphi\in\Gamma_{0}$. We then modify the fundamental domain
$\widetilde{D}$ by taking

$\widetilde{D}^{\prime}:=\widetilde{D}\cup\bigcup_{i=1}^{r}(\gamma_{i}^{-1})^{*}(\widetilde{D}).$

Observe that $\widetilde{D}^{\prime}$ is obviously still rational and
polyhedral and
$(\gamma_{j}^{-1})^{*}\varphi^{*}(\pi^{*}\xi)\in\widetilde{D}^{\prime}$.
Moreover $\gamma_{j}^{-1}\varphi$ descends to an automorphism on $X$ and we
conclude as before. The theorem is then proved in the prime order case under
the genericity assumption.

###  3.2. The cone conjecture for prime order for all points in moduli space

In the previous section we have seen that Theorem 1.2 holds at the general
point of the moduli space. Via a limiting procedure we show here how to deduce
the same conclusion everywhere.

######  Theorem 3.5.

Let $X_{0}$ be a marked Enriques manifold and let
$\mathfrak{M}_{0}\subset\mathfrak{M}_{L}$ be an irreducible component of the
moduli space containing $[X_{0}]$. Assume that the Kawamata-Morrison cone
conjecture holds on manifolds $X$ corresponding to the general point of
$\mathfrak{M}_{0}$. Then the Kawamata-Morrison cone conjecture holds for
$X_{0}$.

###### Proof.

Let ${\widetilde{X}}_{0}\to X_{0}$ be the IHS universal cover. Consider a
smooth integral curve $B\subset\mathfrak{M}_{0}$ with $0=[X_{0}]\in B$ and
such that for the general $b\in B\setminus\\{0\\}$ the Kawamata-Morrison cone
conjecture holds on the corresponding manifold $X_{b}$. Let $\mathscr{X}\to B$
the associated family.

Consider a class $\xi_{0}\in\mathop{\rm Nef}\nolimits^{+}(X_{0})$. Notice that
by Lemma 2.3 there exists $\\{\xi_{b}\in\mathop{\rm
Nef}\nolimits^{+}(X_{b})\\}_{b\in B}$ a family of nef effective classes with
$\lim_{b\to 0}\xi_{b}=\xi_{0}$. Recall that the relative group of
automorphisms $\mathop{\rm Aut}\nolimits(\mathscr{X}/B)$ is an open subscheme
of $\mathop{\rm Hilb}\nolimits(\mathscr{X}\times_{B}\mathscr{X})$ (cf. [Kol96,
Theorem I.1.10]). By hypothesis, for the general $b\in B\setminus\\{0\\}$
there exists a rational polyhedral fundamental domain $D_{b}$ and an
automorphism $\tau_{b}\in\mathop{\rm Aut}\nolimits(X_{b})$ such that
$\tau^{*}\xi_{b}=\delta_{b}\in D_{b}$. We look at the $\tau_{b}$’s as
automorphisms ${\widetilde{\tau}}_{b}$ on the universal covers
$\widetilde{X}_{b}$ commuting with the $G$-action. By the regularity of the
action

$\mathop{\rm Aut}\nolimits(\mathscr{X}/B)\times\mathop{\rm
Nef}\nolimits(\mathscr{X}/B)\to\mathop{\rm Nef}\nolimits(\mathscr{X}/B)$

the automorphisms $\tau_{b}\in\mathop{\rm Aut}\nolimits(X_{b})$ form a family
over $B\setminus\\{0\\}$ i.e. there exists $\mathscr{T}\in\mathop{\rm
Aut}\nolimits(\mathscr{X}/(B\setminus\\{0\\}))$ such that
$\mathscr{T}_{|X_{b}}=\tau_{b}$.

Consider the induced Hodge isometries
${\widetilde{\tau}}_{b}^{*}:H^{2}({\widetilde{X}}_{0},{\mathbb{Z}})\cong
H^{2}({\widetilde{X}}_{b},{\mathbb{Z}})\to
H^{2}({\widetilde{X}}_{b},{\mathbb{Z}})\cong
H^{2}({\widetilde{X}}_{0},{\mathbb{Z}})$.

We set $\widetilde{t}_{0}:=\lim_{b\to 0}{\widetilde{\tau}}_{b}^{*}$.

Notice that $\widetilde{t}_{0}$ is a Hodge isometry of
$H^{2}({\widetilde{X}}_{0},{\mathbb{Z}})$. Moreover $\widetilde{t}_{0}$ is a
monodromy operator as the ${\widetilde{\tau}}_{b}^{*}$’s are. Finally remark
that $\widetilde{t}_{0}$ preserves an ample class again as the
${\widetilde{\tau}}_{b}^{*}$’s do. Therefore, by the Hodge-theoretic global
Torelli theorem [Mar11, Theorem 1.3] there exists an automorphism
${\widetilde{\tau}}_{0}:{\widetilde{X}}\to{\widetilde{X}}$ such that
${\widetilde{\tau}}_{0}^{*}=\widetilde{t}_{0}$. The condition of commuting
with $G$ being a closed condition we have that ${\widetilde{\tau}}_{0}^{*}$
descends to an automorphism $\tau_{0}\in\mathop{\rm Aut}\nolimits(X_{0})$
which, by construction, is the limit in zero of the $\tau_{b}$’s . ∎

###### Proof of Theorem 1.2.

The Morrison-Kawamata cone conjecture for the $\mathop{\rm
Aut}\nolimits(X)$-action on the $\mathop{\rm Nef}\nolimits^{+}(X)$-cone for an
Enriques manifold $X$ of prime index $p$ follows for the combination of
Section 3.1 (where we use that the index is prime) and of Theorem 3.5 (which
holds in general).

To obtain the birational Morrison-Kawamata cone conjecture for the
$\mathop{\rm Bir}\nolimits(X)$-action on $\overline{\mathop{\rm
Mov}\nolimits}^{+}(X)$ on any Enriques manifold $X$ of prime index $p$ the
same argument works, as by [Huy99, Proposition 9.1, item (iv)] the kernel of
the group homomorphism $\mathop{\rm Bir}\nolimits({\widetilde{X}})\to
O(H^{2}({\widetilde{X}},{\mathbb{Z}}))$ equals the kernel of $\mathop{\rm
Aut}\nolimits({\widetilde{X}})\to O(H^{2}({\widetilde{X}},{\mathbb{Z}}))$. ∎

##  4\. Further results

Recall that the Lefschetz number of an IHS manifold ${\widetilde{X}}$ of
dimension $2n$ with a purely non–symplectic action by an automorphism $g$ of
finite order is given by:

$L(g)=\sum_{j=0}^{2n}(-1)^{j}tr(g^{*}_{|H^{j}({\widetilde{X}},\mathscr{O}_{{\widetilde{X}}})})=1+\bar{\lambda}+\bar{\lambda}^{2}+\ldots+\bar{\lambda}^{n}$

where we fix the action
$g^{*}\sigma_{{\widetilde{X}}}=\lambda\sigma_{{\widetilde{X}}}$ and we have
used that
$H^{j}(\mathscr{O}_{{\widetilde{X}}})=\overline{H^{0}(\Omega^{j}_{{\widetilde{X}}})}$.
Moreover we know that for $j=2t$ then
$\overline{H^{0}(\Omega^{j}_{{\widetilde{X}}})}$ is generated by
$\overline{\omega_{{\widetilde{X}}}^{t}}$, whence the cohomology groups for
odd $j$ are zero. Since the fixed locus is empty we must have $L(g)=0$. This
means that $\lambda$ is at most a primitive $n+1$ root of unity and so we have
that $|G|=d\leq n+1$, more precisely $d$ divides $n+1$. By looking at the
Lefschetz number, we can a priori have an Enriques manifold of any index,
provided that the dimension of the IHS manifold is high enough. However until
now we know existence of Enriques manifolds only of index $2,3,4$ (in the case
of the index $2$ we have Enriques surfaces, a construction involving
$K3^{[n]}$-type manifolds for $n$ odd and a construction involving Kumn
manifolds of odd dimension), see [BNWS11, OS11a]. In general the proof of
Theorem 1.2 yields the following two results.

######  Proposition 4.1.

Let ${\widetilde{X}}\to X={\widetilde{X}}/G$ be an Enriques manifold. Assume
the action of $G=\langle g\rangle$ on $\mathop{\rm
Pic}\nolimits({\widetilde{X}})$ is the identity, i.e. $\mathop{\rm
Pic}\nolimits({\widetilde{X}})=H^{2}({\widetilde{X}},\mathbb{Z})^{G}$. Then
the cone conjecture holds for the Enriques manifold $X$ and all its
deformations.

###### Proof.

We consider a small variant of the proof of Theorem 1.2. We consider the
decomposition

$H^{2}({\widetilde{X}},{\mathbb{Q}})=\mathop{\rm
Pic}\nolimits({\widetilde{X}})_{\mathbb{Q}}\oplus
T({\widetilde{X}})_{\mathbb{Q}},\ T({\widetilde{X}}):=\mathop{\rm
Pic}\nolimits({\widetilde{X}})^{\perp_{q_{\widetilde{X}}}}.$

Then we obviously have commutativity on $\mathop{\rm
Pic}\nolimits({\widetilde{X}})$, while on $T({\widetilde{X}})$, which is
preserved by $\mathop{\rm Aut}\nolimits({\widetilde{X}})$, we have
commutativity by applying [Huy16, Corollary 3.4, Chapter 3] as in the proof of
Lemma 3.4. The conclusion now follows by Global Torelli, with the same care to
take into account automorphisms acting trivially on cohomology. Now we can
argue exactly as in the proof of Theorem 1.2 and use the limiting procedure as
in the proof of Theorem 3.5. ∎

###  4.1. The cone conjecture for the index 4 examples

As a first application of Proposition 4.1 we show that the cone conjecture
holds for the known example of index four constructed by using generalized
Kummer manifolds. First of all we recall the construction (see [BNWS11,
Section 4.2] and [OS11a, Section 6]). Take integers $m,n$ such that
$4m=(n+1)$. Let $E_{i}$ be the elliptic curve with complex multiplication by
$i$ and $F$ is another elliptic curve. Set $A:=E_{i}\times F$. For $F$ a
generic elliptic curve we have $\mathop{\rm rk}\nolimits\mathop{\rm
Pic}\nolimits(E_{i}\times F)=2$ (indeed $\mathop{\rm Pic}\nolimits(E_{i}\times
F)=\mathop{\rm Pic}\nolimits(X)\times\mathop{\rm Pic}\nolimits(F)\times
Hom(Jac(E_{i}),Jac(F))$, and for the genericity of $F$ we have
$Hom(Jac(E_{i}),Jac(F))=0$. Consider now $a:=(a_{1},a_{2})$ where $a_{2}\in A$
is a $4$-torsion point and $a_{1}\in E_{i}[n+1]$ is such that
$2ma_{1}\not\in\mathbb{Z}(1+i)/2$. Let $t_{a}$ be the translation by the point
$a$ on $E_{i}\times F$ and $h=\textrm{diag}(i\,1)$ is the multiplication by
$i$ on the first component. Set $\psi:=t_{a}\circ h$ then $\psi$ does not have
fixed points on Kum${}_{n}(E_{i}\times F)$ as shown in [BNWS11, Section 4.2]
and [OS11a, Section 6]. For $(z,w)\in E_{i}\times F$ we have
$t_{a}(h(z,w))=(iz+a_{1},w+a_{2})$. Let us now compute the action in
cohomology. Recall that $H^{2}(A,\mathbb{Z})=U\oplus U\oplus U$ and consider
$H^{2}(A,\mathbb{R})$. If we write $z=z_{1}+iz_{2}$ for the coordinate on
$E_{i}$ and $w=w_{1}+iw_{2}$ for the coordinate on $F$ then
$H^{2}(A,\mathbb{R})$ is generated by the 2-forms

$dz_{1}\wedge dw_{1},\,dz_{1}\wedge dw_{2},\,dz_{2}\wedge
dw_{1},\,dz_{2}\wedge dw_{2},\,dz_{1}\wedge dz_{2},\,dw_{1}\wedge dw_{2}.$

A translation acts trivially in cohomology and the action of the
multiplication by $i$ acts sending $z_{1}+iz_{2}$ to $-z_{2}+iz_{1}$ so that
the image under $\psi^{*}$ of the previous basis is

$\psi^{*}(dz_{1}\wedge dw_{1})=-dz_{2}\wedge dw_{1},\,\psi^{*}(dz_{1}\wedge
dw_{2})=-dz_{2}\wedge dw_{2},\,\psi^{*}(dz_{2}\wedge dw_{1})=dz_{1}\wedge
dw_{1},$ $\psi^{*}(dz_{2}\wedge dw_{2})=dz_{1}\wedge
dw_{2},\,\psi^{*}(dz_{1}\wedge dz_{2})=dz_{1}\wedge
dz_{2},\,\psi^{*}(dw_{1}\wedge dw_{2})=dw_{1}\wedge dw_{2}.$

Hence the matrix of $\psi^{*}$ in this basis is

$\left(\begin{array}[]{rrrrrr}0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\ -1&0&0&0&0&0\\\
0&-1&0&0&0&0\\\ 0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\ \end{array}\right).$

The eigenvalues of this matrix are $i$ and $-i$ with multiplicity $2$ and then
$1$ with multiplicity 2. This means that the invariant part for the action of
$\psi$ on the cohomology of $A$ is precisely $\mathop{\rm
Pic}\nolimits(E_{i}\times F)$. Consider the induced natural automorphism
$\psi^{[n]}$ on Kum${}_{n}(E_{i}\times F)$. By the choice of $a_{1}$ and
$a_{2}$ the induced automorphism $\psi^{[n]}$ acts freely on
Kum${}_{n}(E_{i}\times F)$ (see [OS11a, Theorem 6.4]). From the above
description we have that the invariant part for the action on the cohomology
of Kum${}_{n}(E_{i}\times F)$ of the induced automorphism $\psi^{[n]}$ is
exactly the Picard group (since the automorphism $\psi^{[n]}$ is natural it
acts as the identity on the exceptional divisor). Therefore we can invoke
Proposition 4.1 and conclude that Conjecture 1.1 holds in this case.

###  4.2. The cone conjecture for other possible indices

We now discuss Enriques manifolds that can arise from all the known examples
of IHS manifolds. The two examples of O’Grady $OG_{6}$ and $OG_{10}$ are six
and $10$–dimensional so that to produce Enriques manifolds we have to make the
quotient by a fixed point free automorphism of order $2$ or $4$, respectively
order $2$, $3$ or $6$. It is an interesting open question to understand
whether such automorphisms exist.

Consider now an Enriques manifold ${\widetilde{X}}\to X={\widetilde{X}}/G$
such that ${\widetilde{X}}$ is a deformation of $K3^{[n]}$ or of
${\textrm{Kum}}_{n}$. In this case the group $G=\langle g\rangle$ is forced to
have order $d$ that divides $n+1$. Recall that the respective second Betti
numbers are $23$ and $7$, so that, since a primitive root is an eigenvalue of
the action of $g^{*}$ on $H^{2}({\widetilde{X}},\mathbb{Z})\otimes\mathbb{C}$,
we must have for the Euler totient function $\varphi(d)\leq 22$ respectively
$\leq 6$ (in fact in general we have $\varphi(d)\leq
b_{2}({\widetilde{X}})-1$), see [OS11a, Proposition 2.9] for the details. It
is easy then to find the list of possible index $d$ and we can formulate:

######  Proposition 4.2.

Let ${\widetilde{X}}\to X={\widetilde{X}}/G$ be an Enriques manifold quotient
of an IHS manifold ${\widetilde{X}}$.

  * a)

If ${\widetilde{X}}$ is of $K3^{[n]}$-type, then, see [OS11a, Proposition
2.9], $d:=|G|\leq 66$ and

$d\in\\{2,\ldots,28,30,32,33,34,36,38,40,42,44,46,48,50,54,60,66\\}.$

In these cases the cone conjecture holds for all
$d\in\\{2,5,7,11,13,17,19,23,46\\}.$

  * b)

If ${\widetilde{X}}$ of ${\rm{Kum}}_{n}$-type, then, see [OS11a, Proposition
2.9], $d:=|G|\leq 18$ and

$d\in\\{2,3,4,5,6,7,8,9,10,12,14,18\\}.$

In these cases the cone conjecture holds for all $d\in\\{2,3,5,7,9,14,18\\}$.

###### Proof.

For the list of possible $d$ see [OS11a, Proposition 2.9] even if the cases
$d=48,60$ are missing in that list in the first case and in the second case
the case $d=24$ is included, but this is not possible since $\varphi(24)=8$
which is bigger then $b_{2}({\widetilde{X}})=7$ for ${\widetilde{X}}$ of Kumn-
type. For prime $d$ then one applies Theorem 1.2. In the case of $d=3$ and
${\widetilde{X}}$ of $K3^{[n]}$–type one uses the results of [BCS16, Table 1]
to show that a non–symplectic automorphism of order $3$ on a $K3^{[2]}$–type
manifold has never empty fixed locus. For the orders $46$, $9$, $14$ and $18$,
we have $\varphi(46)=22$, $\varphi(9)=6$, $\varphi(14)=6$ and $\varphi(18)=6$
and we remark that the rank of the Picard group is forced to be equal to one.
In fact in these cases the transcendental lattice has exactly rank equal to
$\varphi(d)$ (see [Be83b, Proposition 6]) and each primitive root of unity
appears with multiplicity one. Now we know that ${\widetilde{X}}$ always
contains an ample invariant class so that the action of $G$ is trivial on
$\mathop{\rm Pic}\nolimits({\widetilde{X}})$ and one concludes with
Proposition 4.1. ∎

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