2.1 Hodge duality for Kähler spaces

For the definition and basic properties of Kähler complex spaces we refer to [Reference Varouchas63, Reference Bakker and Lehn3, Reference Bakker and Lehn4] and the references therein. The next result generalizes [Reference Greb, Kebekus and Peternell31, Proposition 6.9]. It should be well known to the experts, but we include it below nonetheless due to the lack of an explicit reference.

Proposition 2.3 (Hodge duality).

If $V$ is a normal, reduced, compact, Kähler, complex space of dimension $n$ with weakly rational singularities (e.g., klt singularities, see [Reference Kebekus and Schnell38, §1.4] and [Reference Kollár and Mori41, Theorem 5.22]), then for any integer $p \in \{ 0, \dots , n \}$ there are $\mathbb {C}$ -linear isomorphisms

$$\begin{align*}H^0 \big( V, \Omega_V^{[p]} \big) \simeq \overline{H^p \big( V,\mathcal{O}_V \big)}. \end{align*}$$

2.2 Primitive symplectic varieties

Recall that a normal variety $X$ of dimension at least two is called symplectic if it is endowed with a symplectic form $\sigma $ (i.e., a reflexive $2$ -form $\sigma \in H^0 \big ( X, \Omega _X^{[2]} \big )$ such that $\sigma |_{X_{\mathrm{reg}}}$ is closed and everywhere nondegenerate) such that $\sigma |_{X_{\mathrm{reg}}}$ extends to a holomorphic $2$ -form on some (or, equivalently, any) resolution of singularities of $X$ ; see [Reference Beauville8]. In particular, we have $\dim X = 2n$ for some $n \geq 1$ and $K_X \sim 0$ . For a thorough discussion of symplectic varieties we refer to [Reference Bakker and Lehn3, Reference Bakker and Lehn4, Reference Denisi, Ríos Ortiz, Tsakanikas and Xie17] and the numerous relevant references therein. In this paper we are mainly concerned with the following subclass of symplectic varieties.

Fix a primitive symplectic variety $X$ of dimension $2n$ . Recall that $X$ has rational singularities by [Reference Bakker and Lehn4, Theorem 3.4(1)] and that the torsion-free part $H^2(X,\mathbb {Z})_{\mathrm{tf}} := H^2(X,\mathbb {Z}) / \mathrm{(torsion)}$ of its second cohomology group is a free $\mathbb {Z}$ -module of finite rank by [Reference Bakker and Lehn3, Lemma 2.1] and carries also a pure Hodge structure of weight two by [Reference Bakker and Lehn4, Corollary 3.5]. Therefore, there exists a decomposition

(2.1) $$ \begin{align} H^2(X,\mathbb{C}) \simeq H^{2,0}(X) \oplus H^{1,1}(X) \oplus H^{0,2}(X) , \end{align} $$

where

$$ \begin{align*} H^{2,0}(X) &\simeq H^0 \big( X, \Omega_X^{[2]} \big) , \\ H^{1,1}(X) &\simeq H^1 \big( X, \Omega^{[1]}_X \big) , \text{ and } \\ \overline{H^{2,0}(X)} &\simeq H^{0,2}(X) \simeq H^2(X, \mathcal{O}_X) , \end{align*} $$

see [Reference Bakker and Lehn3, Corollary 2.3(1)].

2.2.1 The Beauville–Bogomolov–Fujiki form

If $\sigma \in H^{2,0}(X)$ is the cohomology class of a symplectic form on $X$ satisfying $\int _X (\sigma \overline {\sigma })^n=1$ , then there exists a quadratic form on $H^2(X,\mathbb {C})$ , called the Beauville–Bogomolov–Fujiki (BBF) form and defined as

(2.2) $$ \begin{align} q_X &\colon H^2(X,\mathbb{C}) \to \mathbb{C} , \nonumber \\[1mm] \alpha \mapsto q_X(\alpha) := \frac{n}{2} \int_X (\sigma \overline{\sigma})^{n-1} \, \alpha^2 &+ (1-n) \left( \int_X \sigma^n \, \overline{\sigma}^{n-1} \, \alpha \right) \left( \int_X \sigma^{n-1} \, \overline{\sigma}^n \, \alpha \right) , \end{align} $$

which does not depend on the choice of $\sigma $ and whose restriction to $H^2(X,\mathbb {R})$ is a real nondegenerate quadratic form with signature $\big (3,b_2(X)-3\big )$ ; see [Reference Schwald59, Section 3] and [Reference Bakker and Lehn4, Section 5], which are based on [Reference Namikawa48, Reference Matsushita46, Reference Kirschner39]. In particular, by (2.2) we infer that

(2.3) $$ \begin{align} q_X(\sigma) = 0 = q_X(\overline{\sigma}) , \quad q_X(\sigma+\overline{\sigma}) = 1 \end{align} $$

and

(2.4) $$ \begin{align} q_X(\sigma + v) = q_X(\overline{\sigma} +v) &= q_X(v) \\[1mm] &= \frac{n}{2} \int_X (\sigma \overline{\sigma})^{n-1} \, v^2 \quad \text{for any } \ v \in H^{1,1}(X). \nonumber \end{align} $$

Moreover, according to [Reference Bakker and Lehn4, Lemma 5.7], we can normalize $q_X$ so that it becomes an integral quadratic form $H^2(X,\mathbb {Z})\to \mathbb {Z}$ , and hence $q_X$ is invariant under locally trivial deformations of $X$ (see Definition 2.14). The BBF lattice $\big ( H^2(X,\mathbb {Z})_{\mathrm{tf}} \, , q_X \big )$ of $X$ is thus nondegenerate with signature of type $(3,*)$ .

Comment. We distinguish between the induced nondegenerate symmetric bilinear form $q_X ( \cdot , \cdot )$ on $H^2(X,\mathbb {C})$ defined by

(2.5) $$ \begin{align} q_X(\alpha,\beta) := \frac{1}{2} \big( q_X(\alpha+\beta)-q_X(\alpha) - q_X(\beta) \big) , \end{align} $$

see [Reference Schwald59, Remark 22], and the induced Hermitian form $q_X^h$ on $H^2(X,\mathbb {C})$ defined by

(2.6) $$ \begin{align} q_X^h (\alpha,\beta) := q_X(\alpha,\overline{\beta}) , \end{align} $$

see [Reference Oguiso and Schröer51, p. 1636]. For some useful properties of $q_X^h$ we refer to Lemma 2.7 below.

Remark 2.5. It follows from (2.3), (2.4) and (2.5) that, exactly as in the smooth case, the decomposition

$$\begin{align*}H^2(X,\mathbb{C}) \simeq \left( H^{2,0}(X) \oplus H^{0,2}(X) \right) \oplus H^{1,1}(X) \end{align*}$$

is orthogonal with respect to $q_X(\cdot , \cdot )$ .

2.2.2 Nonsymplectic finite group actions on PSVs

We now recall the following standard definitions.

Definition 2.6. Let $X$ be a primitive symplectic variety and denote by $\sigma \in H^0 \big ( X, \Omega _X^{[2]} \big )$ its (unique up to scalar) symplectic form.

1. (1) An automorphism $g \in \operatorname {\mathrm {Aut}}(X)$ is called nonsymplectic if $g^{[*]} \sigma \neq \sigma $ .

2. (2) A nonsymplectic automorphism $g \in \operatorname {\mathrm {Aut}}(X)$ is called purely nonsymplectic if there exists a primitive root of unity $\xi \neq 1$ of the same order as $g$ such that $g^{[*]} \sigma = \xi \sigma $ .

We also say that a finite subgroup $G \neq \{ \operatorname {\mathrm {Id}}_X \}$ of $\operatorname {\mathrm {Aut}}(X)$ acts nonsymplectically on $X$ if every element $g \in G \setminus \{ \operatorname {\mathrm {Id}}_X\}$ is a nonsymplectic automorphism of $X$ .

Assume now that the given primitive symplectic variety $(X, \sigma )$ admits a purely nonsymplectic automorphism $g \in \operatorname {\mathrm {Aut}}(X)$ of order $d \geq 2$ . Set $G := \langle g \rangle \subseteq \operatorname {\mathrm {Aut}}(X)$ and let $\xi $ be a primitive $d$ -th root of unity such that $g^{[*]} \sigma = \xi \sigma $ . As $h^{2,0}(X) = 1$ , observe that the natural map

$$\begin{align*}\rho \colon G \to \operatorname{\mathrm{GL}} \left( H^0 \big( X, \Omega^{[2]}_X \big) \right) , \ g \mapsto g^{[*]} \end{align*}$$

induces a bijection

(2.7) $$ \begin{align} G \to \mu_d(\mathbb{C}), \ g \mapsto \xi \end{align} $$

onto the multiplicative group of $d$ -th roots of unity; see [Reference Denisi, Ríos Ortiz, Tsakanikas and Xie17, Subsection 4.2]. Note also that the group of characters $\operatorname {\mathrm {Hom}} \big ( \mu _d(\mathbb {C}) , \mathbb {C}^* \big )$ is cyclic of order $d$ and contains a canonical generator, namely the identity character $\xi \in \mu _d(\mathbb {C}) \mapsto \xi \in \mathbb {C}^*$ . We use below the identification between $\operatorname {\mathrm {Hom}} \big ( \mu _d(\mathbb {C}) , \mathbb {C}^* \big )$ and $\mathbb {Z} / d\mathbb {Z}$ .

Consider next the linear representation

(2.8) $$ \begin{align} G \to \operatorname{\mathrm{GL}}\left( H^2(X,\mathbb{C}) \right) , \ g \mapsto g^*. \end{align} $$

The $\mathbb {C}$ -vector space $H^2(X,\mathbb {C})$ is endowed with a weight decomposition indexed by the characters $i \in \mathbb {Z} / d\mathbb {Z}$ , that is,

(2.9) $$ \begin{align} H^2(X,\mathbb{C}) = \bigoplus_{i\in \mathbb{Z}/ d\mathbb{Z}} H^2(X,\mathbb{C})_i, \end{align} $$

where the weight space $H^2(X,\mathbb {C})_i$ is the $\mathbb {C}$ -vector subspace of $H^2(X,\mathbb {C})$ on which the generator of $G$ acts via multiplication by $\xi ^i \in \mathbb {C}^*$ . In particular, $H^2(X,\mathbb {C})_0$ is the $G$ -invariant subspace of $H^2(X,\mathbb {C})$ and $H^2(X,\mathbb {C})_1$ is the weight space corresponding to the identity character, which clearly satisfies $H^2(X,\mathbb {C})_1 \supseteq H^{2,0}(X) \simeq \mathbb {C} \cdot \sigma $ . Recall also that by [Reference Schwald59, Lemmas 20 and 23] we have

(2.10) $$ \begin{align} q_X (\alpha) = q_X \big( g^* \alpha \big) \ \text{ for any } \, \alpha \in H^2(X,\mathbb{C}) , \end{align} $$

which shows that the representation (2.8) is orthogonal. In view of the bijection (2.7), the BBF lattice $\big ( H^2(X,\mathbb {Z})_{\mathrm{tf}} \, , q_X \big )$ of $X$ is also endowed with an orthogonal $\mu _d(\mathbb {C})$ -action.

We can now extend [Reference Oguiso and Schröer51, Lemma 2.1] to the singular setting.

Lemma 2.7. Let $X$ be a primitive symplectic variety and let $q_X$ be its BBF form. Assume that $X$ admits a purely nonsymplectic automorphism of order $d \geq 2$ and consider a weight decomposition of $H^2(X,\mathbb {C})$ as in (2.9) above. Then

$$\begin{align*}q_X^h \colon H^2(X,\mathbb{C}) \times H^2(X,\mathbb{C})\to \mathbb{C}, \ (\alpha,\beta) \mapsto q_X(\alpha,\overline{\beta}) \end{align*}$$

is a Hermitian form on $H^2(X,\mathbb {C})$ whose restriction to the weight space $H^2(X,\mathbb {C})_1$ is nondegenerate and has signature of type $(2,*)$ for $d=2$ and $(1,*)$ for $d\geq 3$ .

Proof. To prove the statement, we repeat verbatim the proof of [Reference Oguiso and Schröer51, Lemma 2.1], taking (2.3), (2.5), (2.6), (2.10) and [Reference Schwald59, Proposition 25] into account.

2.3 Irreducible symplectic varieties

We recall here the definition of a more special kind of primitive symplectic variety and include afterwards some relevant remarks for the sake of completeness.

Note that the above definition in the Kähler setting, which is [Reference Bakker, Guenancia and Lehn2, Definition 1.1], coincides with [Reference Greb, Kebekus and Peternell31, Definition 8.16], which was originally formulated in the projective setting; see the paragraph after [Reference Bakker, Guenancia and Lehn2, Definition 1.1] and [Reference Kollár and Mori41, Corollary 5.24].

By definition and by Proposition 2.3, irreducible symplectic varieties are automatically primitive symplectic, but the converse does not hold: counterexamples are mentioned, for instance, in [Reference Denisi, Ríos Ortiz, Tsakanikas and Xie17, Remark 4.7(ii)]. In fact, it follows readily from the Beauville–Bogomolov decomposition theorem in the singular Kähler setting [Reference Bakker, Guenancia and Lehn2, Theorem A] and the Künneth formula that an irreducible symplectic variety is precisely a primitive symplectic variety $X$ such that any quasi-étale cover $\widetilde {X}$ of $X$ is again a primitive symplectic variety; see [Reference Engel, Filipazzi, Greer, Mauri and Svaldi22, Remark 2.3].

2.3.1 Examples of ISVs: moduli spaces of sheaves on K3 surfaces

We first recall here some terminology that is relevant for the construction of moduli spaces of sheaves on K3 surfaces. These spaces will appear exclusively in Subsection 5.3. For further information about everything that follows we refer, for instance, to [Reference Huybrechts and Lehn37, Chapter 6] and [Reference Perego and Rapagnetta54, Subsections 1.1 and 2.1].

Fix henceforth a K3 surface $S$ and denote by $\sigma \in H^{2,0}(S)$ the (unique up to scalar) symplectic form on $S$ . Consider the even integral cohomology of $S$ , that is,

$$\begin{align*}\widetilde{H}(S,\mathbb{Z}) := H^{0}(S,\mathbb{Z}) \oplus H^{2}(S,\mathbb{Z}) \oplus H^{4}(S,\mathbb{Z}) , \end{align*}$$

and endow it with a lattice structure by defining a nondegenerate symmetric bilinear form, the so-called Mukai pairing, as follows:

$$ \begin{align*} \langle \cdot , \cdot \rangle \colon \widetilde{H}(S,\mathbb{Z}) \times \widetilde{H}(S,\mathbb{Z}) & \rightarrow \mathbb{Z} \\ (r_1, \ell_1, s_1), (r_2, \ell_2, s_2) &\mapsto -r_1s_2 + \ell_1 \ell_2 -s_1r_2 \, , \end{align*} $$

where $r_i \in H^{0}(S,\mathbb {Z}) \simeq \mathbb {Z}$ , $\ell _i \in H^{2}(S,\mathbb {Z})$ and $s_i \in H^{4}(S,\mathbb {Z}) \simeq \mathbb {Z}$ . The resulting nondegenerate lattice $\big ( \widetilde {H}(S,\mathbb {Z}) , \langle \cdot , \cdot \rangle \big )$ is called the Mukai lattice. An element $v = (r, \ell , s) \in \widetilde {H}(S,\mathbb {Z})$ is called a Mukai vector and satisfies

$$\begin{align*}v^2 := \langle v,v \rangle = \ell^2 - 2rs \in 2\mathbb{Z} , \end{align*}$$

since the intersection form on $S$ is even. It is said to be primitive if it is not divisible by any integer $m \geq 2$ .

Consider next the complexification $\widetilde {H}(S,\mathbb {C}) := \widetilde {H}(S,\mathbb {Z}) \otimes _Z \mathbb {C} $ of $\widetilde {H}(S,\mathbb {Z})$ , endowed with the $\mathbb {C}$ -linear extension $\langle \cdot , \cdot \rangle _{\mathbb {C}}$ of the Mukai pairing $\langle \cdot , \cdot \rangle $ . The natural Hodge structure of weight two on $H^2(S,\mathbb {Z})$ , which is given by the Hodge decomposition

$$\begin{align*}H^2(S,\mathbb{C}) = H^{2,0}(S) \oplus H^{1,1}(S) \oplus H^{0,2}(S) , \end{align*}$$

can be extended to a Hodge structure of weight two on $\widetilde {H}(S,\mathbb {Z})$ by setting

$$ \begin{align*} \widetilde{H}^{2,0}(S) & := H^{2,0}(S) \simeq \mathbb{C} \sigma , \\ \widetilde{H}^{0,2}(S) & := H^{0,2}(S) \simeq \mathbb{C} \overline{\sigma} , \text{ and} \\ \widetilde{H}^{1,1}(S) & := H^0(S,\mathbb{C})\oplus H^{1,1}(S)\oplus H^4(S,\mathbb{C}). \end{align*} $$

For any Mukai vector $v = (r, \ell , s) \in \widetilde {H}(S,\mathbb {Z})$ , we define its orthogonal sublattice in $\widetilde {H}(S,\mathbb {Z})$ with respect to the Mukai pairing as

$$\begin{align*}v^{\perp} := \big\{ w \in \widetilde{H}(S,\mathbb{Z}) \mid \langle v,w \rangle = 0 \big\} \end{align*}$$

and we denote by $(v^\perp )_{\mathbb {C}}$ its complexification, which is endowed with the restriction of $\langle \cdot , \cdot \rangle _{\mathbb {C}}$ to it. We also observe that $v^\perp $ inherits from $\widetilde {H}(S,\mathbb {Z})$ a Hodge structure of weight two in the obvious fashion. In addition, we claim that

(2.11) $$ \begin{align} \text{ if } \, \ell \in \operatorname{\mathrm{NS}}(S) \simeq \operatorname{\mathrm{Pic}}(S) \simeq H^2(S,\mathbb{Z}) \cap H^{1,1}(S) , \text{ then } \, \sigma \in (v^\perp)_{\mathbb{C}} \,. \end{align} $$

Indeed, due to Remark 2.5 and since the integral BBF form of $S$ coincides with the intersection product on $H^2(S,\mathbb {Z})$ , we infer that $\sigma \in H^{2,0}(S)$ is orthogonal – with respect to $\langle \cdot , \cdot \rangle _{\mathbb {C}}$ and under its identification with the vector $(0,\sigma ,0) \in \widetilde {H}(S,\mathbb {C})$ – to $\ell \in \operatorname {\mathrm {Pic}}(S)$ , and hence to the given Mukai vector $v = (r, \ell , s) \in \widetilde {H}(S,\mathbb {Z})$ .

Given a coherent sheaf $\mathcal {F}$ on a K3 surface $S$ , we can naturally attach to it a Mukai vector by setting

$$\begin{align*}v(\mathcal{F}) := \operatorname{ch}(\mathcal{F}) \sqrt{\operatorname{td}(S)} = \big( \operatorname{{\mathrm{rk}}} (\mathcal{F}), \, c_1(\mathcal{F}), \, \frac{1}{2}c_1(\mathcal{F})^2 -c_2(\mathcal{F}) + \operatorname{{\mathrm{rk}}}(\mathcal{F}) \big) \in \widetilde{H}(S,\mathbb{Z}) , \end{align*}$$

where $\frac {1}{2}c_1(\mathcal {F})^2 -c_2(\mathcal {F}) = \operatorname {ch}_2(\mathcal {F})$ . By the Hirzebruch–Riemann–Roch formula, we have $\chi (\mathcal {F}) = \operatorname {ch}_2(\mathcal {F}) + 2\operatorname {{\mathrm {rk}}}(\mathcal {F})$ and hence the third component of $v(\mathcal {F})$ is often written as $\chi (\mathcal {F}) - \operatorname {{\mathrm {rk}}} (\mathcal {F})$ . By construction, the associated Mukai vector $v(\mathcal {F})=(r, \ell , s)$ is of type $(1,1)$ and satisfies one of the following conditions:

1. (1) $r>0$ ;

2. (2) $r=0$ and $\ell \in H^{1,1}(S,\mathbb {Z})$ is the first Chern class of a strictly effective divisor; or

3. (3) $r = \ell = 0$ and $s>0$ .

In view of the previous observations, a nonzero element $v = (r, \ell , s) \in \widetilde {H}(S,\mathbb {Z})$ with $\ell \in \operatorname {\mathrm {NS}}(S)$ satisfying $v^2 \geq 2$ and any of the above three conditions is called a positive Mukai vector.

Finally, for the definition and the comparison of the notions of $v$ -generic and $v$ -general polarizations $H \in \operatorname {{\mathrm {Amp}}}(S)$ with respect to a positive Mukai vector $v \in \widetilde {H}(S,\mathbb {Z})$ , which will appear in the following paragraph as well as in Subsection 5.3, we refer to [Reference Perego and Rapagnetta54, Subsection 2.1]. In particular, due to [Reference Perego and Rapagnetta54, Lemmas 2.9, 2.10 and 2.12], we will only consider $v$ -generic polarizations in this paper; see also [Reference Perego and Rapagnetta54, Remark 2.13].

Moduli spaces of semistable sheaves on K3 surfaces: Let $S$ be a projective K3 surface. According to [Reference Perego and Rapagnetta54, Theorem 1.10], if $v \in \widetilde {H}(S,\mathbb {Z})$ is a positive Mukai vector and if $H \in \operatorname {{\mathrm {Amp}}}(S)$ is a $v$ -generic polarization, then the moduli space $M_S(v,H)$ of Gieseker $H$ -semistable sheaves on $S$ with Mukai vector $v$ is a projective irreducible symplectic variety of dimension $v^2 +2 \geq 4$ whose smooth locus $M^s_S(v,H)$ is the moduli space of Gieseker $H$ -stable sheaves on $S$ with Mukai vector $v$ . In particular, if $v$ is primitive, then by works of Mukai and Yoshioka we know that $M^s_S(v,H) = M_S(v,H)$ is a projective IHS manifold that is deformation equivalent to the Hilbert scheme $\operatorname {\mathrm {Hilb}}^n(S)$ of $n = \frac {1}{2} (v^2 +2)$ points on the K3 surface $S$ . Finally, O’Grady [Reference O’Grady49, Main Theorem] and Perego–Rapagnetta [Reference Perego and Rapagnetta55, Proposition 5.12] showed that there also exists a Hodge isometry

(2.12) $$ \begin{align} \theta \colon \big( v^\perp , \langle \cdot,\cdot\rangle |_{v^\perp} \big) \to \Big( H^2 \big( M_S(v,H) , \mathbb{Z} \big) , \, q_{M_S(v,H)} \Big). \end{align} $$

In particular, the symplectic form on $M_S(v,H)$ arises from the symplectic form $\sigma $ on $S$ ; namely, $\theta _{\mathbb {C}} (\sigma ) \in H^{2,0}\big ( M_S(v,H) \big ) $ .

2.4 Primitive Enriques varieties

In this subsection we first give the definition and establish some basic properties of the main objects of study in this paper.

A primitive Enriques variety $Y \simeq X/G$ is a normal, compact, Kähler variety with klt singularities and torsion canonical class; see [Reference Popp56, Lecture 3], [Reference Varouchas63, Section II, Corollary 3.2.1], and [Reference Denisi, Ríos Ortiz, Tsakanikas and Xie17, Remark 2.7(i)]. At first glance, Definition 2.9 seems to be the Kähler version of our earlier definition of a projective primitive Enriques variety, namely [Reference Denisi, Ríos Ortiz, Tsakanikas and Xie17, Definition 5.11]. However, we will now show that the two definitions coincide; that is, primitive Enriques varieties and the (a priori Kähler) primitive symplectic varieties determining them are automatically projective.

Proof. Recall that $Y \simeq X/G$ is a normal, compact, Kähler variety with klt, and hence rational, singularities. By Proposition 2.3 and [Reference Denisi, Ríos Ortiz, Tsakanikas and Xie17, Remark 2.6(iv)] we obtain

$$\begin{align*}h^2(Y, \mathcal{O}_Y) = h^0 \big(Y, \Omega_Y^{[2]} \big) = 0. \end{align*}$$

Therefore, [Reference Graf26, Proposition 4.10] implies now that $Y$ is projective, so it admits an ample line bundle $\mathcal {L}$ . Observe that the quotient map $\gamma \colon X \to Y$ is finite. Thus, using the Finite Mapping Theorem [Reference Grauert and Remmert29, pp. 64-65], the projection formula [Reference Hartshorne34, Chapter III, Exercise 8.3] and the relative ampleness criterion [Reference Nakayama47, p. 25, (1) $ \iff $ (4)] for the line bundle $\gamma ^* \mathcal {L}$ and the composite map $X \to Y \to \operatorname {\mathrm {Spec}} (\mathbb {C})$ , we deduce that $\gamma ^* \mathcal {L}$ is an ample line bundle on $X$ , so $X$ is also projective, as claimed.

2.4.1 Markings and period domains for PEVs

We now define a notion of marking for primitive Enriques varieties. To this end, let $(L,q)$ be an abstract nondegenerate lattice with signature of type $(3,*)$ , endowed with an orthogonal $\mu _d(\mathbb {C})$ -action. We further impose the condition that the induced Hermitian form on the weight space $L_{\mathbb {C},1} := (L\otimes _{\mathbb {Z}}\mathbb {C})_1$ is nondegenerate with signature of type $(2,*)$ for $d=2$ and $(1,*)$ for $d\geq 3$ .

Consider a primitive Enriques variety $Y \simeq X/G$ , where $X$ is a primitive symplectic variety and $G \subseteq \operatorname {\mathrm {Aut}}(X)$ is a finite group of order $d \geq 2$ , identified with $\mu _d(\mathbb {C})$ due to Remark 2.11(1) and (2.7), which acts nonsymplectically and freely in codimension one on $X$ . In view of Lemma 2.7 and the fact that the BBF lattice $\big ( H^2(X,\mathbb {Z})_{\mathrm{tf}} \, , q_X \big )$ of $X$ is endowed with an orthogonal $\mu _d(\mathbb {C})$ -action by (2.10), an $L$ -marking of $Y$ is a $\mu _d(\mathbb {C})$ -equivariant lattice isomorphism

$$\begin{align*}\phi \colon \big( H^2(X,\mathbb{Z})_{\mathrm{tf}} \, , q_X \big) \to (L, q) , \end{align*}$$

and the $L$ -marked primitive Enriques variety is denoted by $(Y,\phi )$ .

Definition 2.12. In the above setting, we also give the following definitions:

1. (1) The period domain for $L$ -marked primitive Enriques varieties is defined to be

   $$\begin{align*}\mathcal{D}_L := \big\{ [\omega]\in \mathbb{P}(L_{\mathbb{C},1}) \mid q(\omega) = 0, \ q(\omega, \overline{\omega})> 0 \big \}. \end{align*}$$

2. (2) The period point of an $L$ -marked primitive Enriques variety $(Y,\phi )$ is defined to be the point $[\phi (\sigma )] \in \mathcal {D}_L$ .

We comment briefly on the previous definition. In the first item above, $\overline {\omega }$ denotes the complex conjugate of $\omega $ inside the complexification $L_{\mathbb {C}} = L \otimes _{\mathbb {Z}} \mathbb {C}$ . In the second item above, slightly abusing notation, $\phi $ also denotes the complexification of the given lattice isomorphism and $\sigma \in H^{2,0}(X) \subseteq H^2(X,\mathbb {C})_1$ is the cohomology class of the (unique up to scalar) symplectic form on $X$ . Since $q_X(\sigma ) = 0$ and $q_X(\sigma ,\overline {\sigma })> 0$ by (2.3) and (2.5), we indeed have that $[\phi (\sigma )] \in \mathcal {D}_L$ , as asserted.

Finally, note that the period domains $\mathcal {D}_L$ inside $\mathbb {P}(L_{\mathbb {C},1})$ are locally closed with respect to the classical topology, and hence inherit the structure of a complex manifold. In fact, as we will see in the proof of Theorem 1.2, $\mathcal {D}_L$ is an open subset of a smooth quadric in $\mathbb {P}(L_{\mathbb {C},1})$ for $d=2$ , and an open subset of $\mathbb {P}(L_{\mathbb {C},1})$ for $d \geq 3$ . Incidentally, the precise description of $\mathcal {D}_L$ as bounded symmetric domains (for $d=2$ and $d \geq 3$ , respectively) is given in [Reference Oguiso and Schröer51, Proposition 2.2], which remains valid also in our setting with an identical proof.

2.5 Irreducible Enriques varieties

We recall here the definition of a more special kind of primitive Enriques variety and we refer to [Reference Denisi, Ríos Ortiz, Tsakanikas and Xie17, Subsections 5.1 and 5.3] for a more detailed discussion, including several examples.

For other characterizations of irreducible Enriques varieties we refer to [Reference Denisi, Ríos Ortiz, Tsakanikas and Xie17, Proposition 5.6]. Note also that irreducible Enriques varieties are automatically primitive Enriques varieties, but the converse does not hold; see [Reference Denisi, Ríos Ortiz, Tsakanikas and Xie17, Example 5.24(b)] for a counterexample.

2.6 Notions from deformation theory

We refer to [Reference Bakker and Lehn4, Section 4] for the basic notions of the deformation theory of compact complex spaces, see also [Reference Grauert28, Reference Flenner and Kosarew24, Reference Sernesi61, Reference Grivaux32]. Here, we only recall the following definitions.

Definition 2.15. A deformation $(\mathscr {X}\to S, \, \alpha \colon \mathscr {X}_0\to X)$ of a compact complex space $X$ is called versal if for every deformation $(\mathscr {X}'\to S', \, \beta \colon \mathscr {X}^{\prime }_0\to X)$ of $X$ there exist a map $\varphi \colon (S',0)\to (S,0)$ of germs of complex spaces and an isomorphism $\mathscr {X}'\to \mathscr {X}\times _S S'$ such that the following diagrams commute:

A versal deformation is called miniversal if the differential $d\varphi _0 \colon T_0 S' \to T_0 S$ of $\varphi $ is uniquely determined, and universal if moreover the map $\varphi $ is unique.

The different notions of versality are defined analogously for locally trivial deformations of $X$ .

Any compact complex space $X$ admits a miniversal deformation $\mathscr {X} \to \operatorname {\mathrm {Def}}(X)$ by [Reference Grauert28, Hauptsatz, p. 140], called the Kuranishi family of $X$ , as well as a locally trivial miniversal deformation $\mathscr {X} \to {{\mathrm {Def}}^{\mathrm {lt}}}(X)$ by [Reference Flenner and Kosarew24, Corollary 0.3], called the locally trivial Kuranishi family of $X$ . Note that ${{\mathrm {Def}}^{\mathrm {lt}}}(X)$ is a closed complex subspace of $\operatorname {\mathrm {Def}}(X)$ by the proof of [Reference Flenner and Kosarew24, Corollary 0.3].

2.6.1 Auxiliary results

The content of the following result should be well known to the experts.

Proposition 2.17. Let $\pi \colon \mathscr {X} \to S$ be a flat and proper morphism of complex spaces with reduced fibers $\mathscr {X}_s := \pi ^{-1}(s)$ , $s \in S$ , where $\mathscr {X}$ (and hence $S$ ) is also reduced. Assume that $\pi $ is topologically locally trivial, that is, the underlying continuous map is a topological fiber bundle, and that $R^2 \pi _* \mathcal {O}_{\mathscr {X}} = 0 $ . Given $t \in S$ and after sufficiently shrinking $S$ around $t$ , the following statements hold:

1. (i) If $L_t$ is a holomorphic line bundle on $\mathscr {X}_t$ , then there exists a holomorphic line bundle $\mathcal {L}$ on $\mathscr {X}$ such that $c_1 \big ( \mathcal {L} |_{\mathscr {X}_t} \big ) = c_1(L_t)$ .

2. (ii) If, additionally, $R^1\pi _*\mathcal {O}_{\mathscr {X}}=0$ , then the lift in (i) is unique (up to isomorphism).

Proof. (i) By shrinking $S$ , we can assume that it is Stein. First, we show that

(2.13) $$ \begin{align} H^2(\mathscr{X},\mathcal{O}_{\mathscr{X}}) = 0. \end{align} $$

Indeed, the Leray spectral sequence applied to the morphism $\pi $ and the sheaf $\mathcal {O}_{\mathscr {X}}$ yields the following exact sequence

$$\begin{align*}H^2 \big( S,\pi_*\mathcal{O}_{\mathscr{X}}) \to \mathrm{ker} \left(H^2(\mathscr{X},\mathcal{O}_{\mathscr{X}}) \to H^0(S,R^2\pi_*\mathcal{O}_{\mathscr{X}})\right) \to H^1(S,R^1\pi_*\mathcal{O}_{\mathscr{X}}). \end{align*}$$

Since the analytic sheaves $R^i\pi _* \mathcal {O}_{\mathscr {X}}$ , $i \geq 0$ , are coherent by the Direct Image Theorem [Reference Grauert and Remmert29, p. 207, Theorem] and since the space $S$ is Stein, we have

$$\begin{align*}H^1(S, R^1\pi_*\mathcal{O}_{\mathscr{X}}) = 0 = H^2(S,\pi_*\mathcal{O}_{\mathscr{X}}). \end{align*}$$

Thus, (2.13) follows immediately from the above and the hypothesis that $R^2 \pi _* \mathcal {O}_{\mathscr {X}} = 0 $ .

Next, we consider the exponential sequence of the reduced complex space $\mathscr {X}$ , which gives, together with (2.13), the following exact sequence in cohomology:

(2.14) $$ \begin{align} H^1(\mathscr{X}, \mathcal{O}_{\mathscr{X}}) \to H^1(\mathscr{X}, \mathcal{O}^*_{\mathscr{X}}) \xrightarrow{c_1} H^2(\mathscr{X}, \mathbb{Z}) \to H^2(\mathscr{X}, \mathcal{O}_{\mathscr{X}}) = 0. \end{align} $$

Since $\pi \colon \mathscr {X} \to S$ is topologically locally trivial, after shrinking $S$ even further so that it becomes contractible, we infer that for any $s \in S$ there is an isomorphism

$$\begin{align*}H^2(\mathscr{X}, \mathbb{Z}) \simeq H^2(\mathscr{X}_s,\mathbb{Z}). \end{align*}$$

Therefore, the class $c_1(L_t) \in H^2 ( \mathscr {X}_t, \mathbb {Z})$ of the given holomorphic line bundle $L_t$ on $\mathscr {X}_t$ lifts to a class $\alpha \in H^2(\mathscr {X}, \mathbb {Z})$ on the total space $\mathscr {X}$ , which in turn satisfies $\alpha =c_1(\mathcal {L})$ for some holomorphic line bundle $\mathcal {L}$ on $\mathscr {X}$ due to (2.14).

(ii) Since $R^1\pi _*\mathcal {O}_{\mathscr {X}}=0$ by assumption, the five-term exact sequence induced by the Leray spectral sequence now gives $H^1(\mathscr {X},\mathcal {O}_{\mathscr {X}})=0$ . Hence, $H^1(\mathscr {X},\mathcal {O}^*_{\mathscr {X}})\cong H^2(\mathscr {X},\mathbb {Z})$ by (2.14), which implies that any holomorphic line bundle $L_t$ on $\mathscr {X}_t$ uniquely lifts to $\mathscr {X}$ , up to isomorphism. This completes the proof.

Remark 2.18. Let $\pi \colon \mathscr {X} \to S$ be a flat and proper morphism of complex spaces. If

$$\begin{align*}H^i (\mathscr{X}_s, \mathcal{O}_{\mathscr{X}_s}) = 0 \, \text{ for all } \, s \in S , \end{align*}$$

then

$$\begin{align*}R^i \pi_* \mathcal{O}_{\mathscr{X}} = 0 \end{align*}$$

by [Reference Bănică and Stănăşilă5, Chapter III, Corollary 3.5].

Recall that a proper morphism $X \to Y$ between complex spaces is called projective if there exists a relatively ample line bundle on $X$ over $Y$ ; see [Reference Nakayama47, Definition II.1.10]. We now derive the following application of Proposition 2.17 and Remark 2.18, which will play a key role in Subsection 5.3.

Corollary 2.19. Let $\pi \colon \mathscr {X} \to (S,0)$ be a smooth family, that is, a smooth and proper morphism between connected complex spaces whose fibers are compact, connected, complex manifolds. Assume that $\mathscr {X}$ (and hence $S$ ) is reduced and that $\mathscr {X}_0 = \pi ^{-1}(0)$ is Kähler with $H^2(\mathscr {X}_0, \mathcal {O}_{\mathscr {X}_0}) = 0$ . Then there exists a sufficiently small neighborhood $V$ of $0$ in $S$ such that the restricted morphism $\pi |_{\pi ^{-1}(V)} \colon \pi ^{-1}(V) \to V$ is projective.

Proof. Note first that the given smooth family is topologically locally trivial by [Reference Gross, Huybrechts and Joyce33, Theorem 14.5]. Moreover, since the central fiber $\mathscr {X}_0$ is by assumption a compact Kähler manifold such that $h^{0,2}(\mathscr {X}_0) = h^{2,0}(\mathscr {X}_0) = 0$ , it is actually projective by Kodaira’s embedding theorem, so it admits an ample line bundle $H_0$ . On the other hand, by the upper semi-continuity of the function $t \mapsto H^2(\mathscr {X}_s,\mathcal {O}_{\mathscr {X}_s})$ , see [Reference Bănică and Stănăşilă5, Chapter III, Theorem 4.12(i)], and after shrinking $S$ , we infer that $H^2 (\mathscr {X}_s, \mathcal {O}_{\mathscr {X}_s}) = 0$ for all $s \in S$ . It follows now from Proposition 2.17 and Remark 2.18 that, after possibly further shrinking $S$ , $H_0$ lifts to a line bundle $\mathcal {H}$ on $\mathscr {X}$ such that $c_1 \big ( \mathcal {H} |_{\mathscr {X}_0} \big ) = c_1(H_0)$ . But ampleness is an open condition in smooth families. Indeed, being Kähler is an open condition in such families; see [Reference Gross, Huybrechts and Joyce33, Proposition 22.2(i)]. In particular, small perturbations of integral Kähler classes are represented by ample line bundles due to Kodaira’s embedding theorem. Therefore, up to shrinking $S$ , the line bundle $\mathcal {H}$ on $\mathscr {X}$ is relatively ample over $S$ .

The following observation will only be used for the special case of an Enriques surface in Subsection 5.3, where a relative moduli space construction will be discussed.

Remark 2.20. If $\big ( \pi _Y \colon \mathscr {Y} \to \operatorname {\mathrm {Def}}(Y) , \, \mathscr {Y}_0 \simeq Y \big )$ is the universal Kuranishi family of an Enriques manifold $Y$ , then Proposition 2.19 applies to $\pi _Y$ , showing that it restricts to a smooth projective morphism over some small open neighborhood of $0 \in \operatorname {\mathrm {Def}}(Y)$ . Indeed, the morphism $\pi _Y \colon \mathscr {Y} \to \operatorname {\mathrm {Def}}(Y)$ is smooth and proper by Kuranishi’s theorem, see [Reference Grauert28, Einleitung, 1.]. Since $\operatorname {\mathrm {Def}}(Y)$ is smooth by [Reference Oguiso and Schröer51, Proposition 1.2], $\pi _Y$ is also locally trivial in the sense of Definition 2.14, which implies that $\pi _Y$ is in particular topologically locally trivial and that $\mathscr {Y}$ is reduced as well. The remaining conditions in Proposition 2.19 are satisfied, after sufficiently shrinking the Kuranishi space $\operatorname {\mathrm {Def}}(Y)$ , due to [Reference Oguiso and Schröer50, Proposition 2.6 and Corollary 2.7] and [Reference Oguiso and Schröer51, Proposition 1.2].

2.6.2 Deformations and group actions

Given a compact complex space $X$ endowed with a holomorphic group action $G$ , we consider the $G$ -equivariant deformations of $X$ , namely, a usual deformation $(\mathscr {X}\to S, \mathscr {X}_0 \xrightarrow {\sim } X )$ of $X$ equipped with a holomorphic group action on $\mathscr {X}$ over $S$ , which extends the given $G$ -action on $X$ .

When $X$ is an algebraic variety, Rim showed in [Reference Rim58, Corollary, p. 225] the existence of an equivariant $G$ -structure on the miniversal formal deformation of $X$ , extending the given $G$ -action on $X$ , under the hypotheses that $X$ is complete or affine with only a finite number of singularities, and $G$ is a linearly reductive group.

In the analytic setting, Cathelineau established in [Reference Cathelineau16, Théorème 1] the existence of a miniversal $G$ -equivariant deformation of a given compact complex space $X$ endowed with an action of a group $G$ satisfying further properties, see [Reference Cathelineau16, Définition, p. 392]. When $X$ is in particular a compact complex manifold, Doan proved the following analog of Rim’s result:

Theorem 2.21 [Reference Doan18, Theorem 5.2].

Let $\mathscr {X}\to S$ be the Kuranishi family of a compact complex manifold $X$ endowed with a holomorphic action of a complex reductive Lie group $G$ . Then there exist local equivariant $G$ -actions on $\mathscr {X}$ over $S$ extending the given $G$ -action on $X$ .

One of the main problems of extending the given $G$ -action on $X$ to the whole family $\mathscr {X}\to S$ is the following: when $X$ admits nonzero holomorphic vector fields, there is no canonical choice of such an extension. However, in the opposite case, we have a unique extension of the $G$ -action to the universal deformation of $X$ :

Lemma 2.22. Let $X$ be a compact complex space, let $G$ be a subgroup of $\operatorname {\mathrm {Aut}} (X)$ , and let $\mathscr {D} = (\pi \colon \mathscr {X}\to S, \, \alpha \colon \mathscr {X}_0\to X)$ be a versal deformation of $X$ . Assume that $\mathscr {D}$ is universal. Then there exists a group homomorphism

$$\begin{align*}\Phi\colon G\to \operatorname{\mathrm{Aut}}\big((S,0)\big), \ g \mapsto \varphi_g, \end{align*}$$

where $\operatorname {\mathrm {Aut}}\big ((S,0)\big )$ denotes the group of germs of biholomorphisms that fix the marked point $0 \in S$ . Assume moreover that $H^0(X, \mathscr {T}_X) = \{ 0 \}$ . Then there exists a group homomorphism

$$\begin{align*}\Psi \colon G \to \operatorname{\mathrm{Aut}}_S (\mathscr{X}), \ g \mapsto \psi_g, \end{align*}$$

where $\operatorname {\mathrm {Aut}}_S (\mathscr {X})$ denotes the group of $S$ -automorphisms of the complex space $\mathscr {X}$ .

Proof. Let $g,h \in G$ . Setting $\beta := g\circ \alpha $ and $\gamma := h\circ \alpha $ , we obtain two deformations of $X$ :

$$\begin{align*}\mathscr{D}_g = (\pi\colon\mathscr{X}\to S, \, \beta \colon \mathscr{X}_0\to X) \quad \text{ and } \quad \mathscr{D}_h = (\pi\colon\mathscr{X}\to S, \, \gamma \colon \mathscr{X}_0\to X). \end{align*}$$

By the versality of $\mathscr {D}$ there exist two maps $\varphi _g, \varphi _h \colon (S,0) \to (S,0)$ and two morphisms $\psi _g, \psi _h \colon \mathscr {X} \to \mathscr {X}$ such that the following diagrams commute:

and

Since $\psi _g|_{\mathscr {X}_0} = \alpha ^{-1}\circ g\circ \alpha $ and $\psi _h|_{\mathscr {X}_0} = \alpha ^{-1}\circ h\circ \alpha $ , we have

(2.15) $$ \begin{align} (\psi_g|_{\mathscr{X}_0}) \circ (\psi_h|_{\mathscr{X}_0}) = \alpha^{-1} \circ (g\circ h) \circ \alpha = \psi_{g\circ h}|_{\mathscr{X}_0}. \end{align} $$

Now, by the universality of $\mathscr {D}$ , the map

$$\begin{align*}\Phi\colon G\to \operatorname{\mathrm{Aut}}\big((S,0)\big), \ g \mapsto \varphi_g. \end{align*}$$

is well defined and is a group homomorphism.

If, additionally, $H^0(X, \mathscr {T}_X) = \{ 0 \}$ , then $\operatorname {\mathrm {Aut}}(\mathscr {D}) = \{ \operatorname {\mathrm {Id}}_{\mathscr {D}} \}$ ; see, for example, [Reference Grivaux32, Proposition A.1] and observe that smoothness does not play any role in the proof of this result. Hence, it follows from (2.15) that the map $\Psi $ is well defined and is a group homomorphism.

4.1 A local Torelli theorem for PEVs

Proof of Theorem 1.2.

We proceed in four steps, following the strategy of the proof of [Reference Oguiso and Schröer51, Theorem 2.4]. The notion of marking of a primitive Enriques variety, which will be used in the proof below, can be found in Subsection 2.4.

Step 1: Construction of the local period map.

By shrinking the smooth base of the locally trivial Kuranishi family $\mathscr {Y}\to {{\mathrm {Def}}^{\mathrm {lt}}}(Y)$ of the given primitive Enriques variety $Y = X / G$ , we obtain a locally trivial deformation $f_Y \colon \mathscr {Y} \to S$ of $Y$ over a simply connected and smooth base $S \subseteq {{\mathrm {Def}}^{\mathrm {lt}}}(Y)$ . We denote by $f_X \colon \mathscr {X} \times _{{{\mathrm {Def}}^{\mathrm {lt}}}(X)} S \to S$ the corresponding locally trivial deformation of $X$ , where $\mathscr {X} \to {{\mathrm {Def}}^{\mathrm {lt}}}(X)$ is the locally trivial Kuranishi family of the primitive symplectic variety $X$ . Since $S \subseteq {{\mathrm {Def}}^{\mathrm {lt}}}(Y)$ is by construction a sufficiently small neighborhood of $0 \in {{\mathrm {Def}}^{\mathrm {lt}}}(X)$ , for each $s \in S$ we have $\mathscr {Y}_s \simeq \mathscr {X}_s / G$ , so that the BBF lattice $\big ( H^2( \mathscr {X}_s ,\mathbb {Z})_{\mathrm{tf}} \, , q_{\mathscr {X}_s} \big )$ of $\mathscr {X}_s$ is endowed with an orthogonal $\mu _d(\mathbb {C})$ -action under the identification $G = \langle g \rangle \leftrightarrow \mu _d(\mathbb {C})$ ; see Subsection 2.4.

Fix now an $L$ -marking $\phi \colon \big ( H^2(X,\mathbb {Z})_{\mathrm{tf}} \, , q_X \big ) \to (L,q)$ of $Y$ . Since the locally trivial family $f_X \colon \mathscr {X} \to S$ is topologically locally trivial by [Reference Amerik and Verbitsky1, Proposition 5.1], the local system $R^2 (f_X)_* \mathbb {Z}_{\mathscr {X}}$ on $S$ is actually a constant sheaf on $S$ by [Reference El Zein and Snoussi21, Proposition 1.5]. Note that its stalk at a point $s \in S$ is canonically isomorphic to the singular cohomology group $H^2(\mathscr {X}_s,\mathbb {Z})$ , where $\mathscr {X}_s = f_X^{-1}(s)$ . Hence, all these cohomology groups (for varying $s\in S$ ), and thus their torsion-free parts $H^2(\mathscr {X}_s,\mathbb {Z})_{\mathrm{tf}}$ , are isomorphic. In view of the invariance of the BBF form under locally trivial deformations due to [Reference Bakker and Lehn4, Lemma 5.7], it follows that the $L$ -marking of $Y$ can be uniquely extended to an $L$ -marking of the restricted family $f_Y \colon \mathscr {Y}\to S$ ; the fiberwise marking is denoted again by $\phi $ with a slight abuse of notation.

Consequently, we obtain a period map

$$\begin{align*}\mathscr{P}_Y \colon S \to \Omega(Y), \ s \mapsto \big[ \phi(\sigma_{\mathscr{X}_s}) \big] , \end{align*}$$

which is holomorphic. Indeed, it is the restriction of the period map $\mathscr {P}_X$ , which is itself holomorphic, since $(f_X)_* \, \Omega ^{[2]}_{\mathscr {X}/S}$ is a holomorphic subbundle of $H^2(X,\mathbb {C}) \otimes \mathcal {O}_S$ ; see the proof of [Reference Bakker and Lehn4, Proposition 5.5].

Step 2: Reduction step.

Since both $S$ and $\Omega (Y)$ are smooth, to prove the statement, it suffices to show that

1. (a) the differential $d (\mathscr {P}_Y)_0$ of $\mathscr {P}_Y$ at $0\in S$ is injective, and

2. (b) $\dim T_0 S = \dim T_{[\phi (\sigma )]} \, \Omega (Y)$ .

First, as far as item (a) is concerned, since the differential $d (\mathscr {P}_X)_0$ is bijective by [Reference Bakker and Lehn4, Proposition 5.5], taking into account the horizontal inclusions and the vertical isomorphisms from the diagram (3.3) we conclude that the differential $d (\mathscr {P}_Y)_0$ is injective, as desired. Next, to prove item (b), we first note that

(4.1) $$ \begin{align} \dim T_0 S = h^1(Y,\mathscr{T}_Y) = h^{1,1}(X)_1 \end{align} $$

according to Lemma 3.1(iii). The remainder of the proof is therefore devoted to the computation of $\dim T_{[\phi (\sigma )]} \, \Omega (Y)$ . We proceed by distinguishing two cases for $d$ and we use the following observation: as $\phi $ induces an isomorphism $H^2(X,\mathbb {C})_1\simeq L_{\mathbb {C},1}$ , by [Reference Voisin64, Proposition 10.1] we know that

(4.2) $$ \begin{align} T_{[\phi(\sigma)]} \, \mathbb{P}(L_{\mathbb{C},1}) \simeq \operatorname{\mathrm{Hom}} \big( \mathbb{C} \phi(\sigma) , L_{\mathbb{C},1}/ \mathbb{C} \phi(\sigma) \big) \simeq \operatorname{\mathrm{Hom}} \big( \mathbb{C} \sigma , H^2(X,\mathbb{C})_1/ \mathbb{C} \sigma \big). \end{align} $$

Step 3: Case $d\geq 3$ .

For any $\omega \in H^2(X, \mathbb {C})_1$ we have $g^{[*]} \omega = \xi \omega $ , and thus

$$\begin{align*}q_X(\omega) \overset{(\ast)}{=\!=} q_X \big( g^{[*]} \omega \big) = q_X ( \xi \omega) = \xi^2 q_X (\omega), \end{align*}$$

where $(\ast )$ holds due to (2.10). Since $d>2$ , we infer that the weight space $L_{\mathbb {C},1}$ is isotropic in $L_{\mathbb {C}}$ . Therefore,

$$\begin{align*}\Omega(Y) = \big\{ [\omega] \in \mathbb{P}(L_{\mathbb{C},1}) \mid q_X(\omega,\overline{\omega})>0 \big\} \end{align*}$$

is an open subset of $\mathbb {P}(L_{\mathbb {C},1})$ . It follows now from (4.2) that

(4.3) $$ \begin{align} \dim T_{[\phi(\sigma)]} \, \Omega(Y)= \dim \operatorname{\mathrm{Hom}} \big( \mathbb{C} \sigma , H^2(X,\mathbb{C})_1 / \mathbb{C} \sigma \big). \end{align} $$

Now, since the Hodge decomposition (2.1) is preserved by automorphisms of $X$ , we have

$$\begin{align*}H^2(X,\mathbb{C})_1 = H^{2,0}(X)_1 \oplus H^{1,1}(X)_1 \oplus H^{0,2}(X)_1 , \end{align*}$$

where $H^{0,2}(X)_1 = 0$ , because $g^{[*]}$ acts via multiplication by $\overline {\xi }$ on $H^{0,2}(X)$ and $d>2$ . Hence, by (4.1) and (4.3) we deduce that

$$\begin{align*}\dim T_0 S = h^{1,1}(X)_1 = \dim T_{[\phi(\sigma)]} \, \Omega(Y) , \end{align*}$$

which finishes the proof in the case $d \geq 3$ .

Step 4: Case $d=2$ .

Since $d=2$ , we have $\xi =\overline {\xi } = -1$ , and thus $\overline {\sigma } \in H^{0,2}(X)_{-1} = H^{0,2}(X)_{1}$ . The Hodge decomposition (2.1) yields

$$ \begin{align*} H^2(X,\mathbb{C})_1 & = H^{2,0}(X)_1 \oplus H^{1,1}(X)_1 \oplus H^{0,2}(X)_1 \\ & = \mathbb{C} \sigma \oplus H^{1,1}(X)_1 \oplus \mathbb{C} \overline{\sigma}. \end{align*} $$

By (2.3), (2.4) and (2.5) we deduce that the orthogonal complement of $\sigma \in H^2(X,\mathbb {C})_1$ with respect to $q_X(\cdot , \cdot )$ is the subspace $V = \mathbb {C} \sigma \oplus H^{1,1}(X)_1$ . Therefore,

(4.4) $$ \begin{align} \dim \operatorname{\mathrm{Hom}} (\mathbb{C} \sigma, V/ \mathbb{C} \sigma) = h^{1,1}(X)_1. \end{align} $$

Now, since $\Omega (Y)$ is an open subset of the quadric in $\mathbb {P}(L_{\mathbb {C},1})$ defined by the equation $q_X(\cdot ) = 0 $ , the tangent space $T_{[\phi (\sigma )]} \, \Omega (Y)$ is precisely the tangent space of the quadric at the point $[\phi (\sigma )]$ . Since the quadratic form $q_X$ is nondegenerate, the quadric $\big \{ q(\cdot )= 0 \big \}$ is smooth. Using this, together with (4.2), we deduce that the tangent space of the quadric at the point $[\phi (\sigma )]$ is precisely $\operatorname {\mathrm {Hom}} (\mathbb {C} \sigma , V/\mathbb {C} \sigma )$ . It follows now from (4.1) and (4.4) that

$$\begin{align*}\dim T_0 S = h^{1,1}(X)_1 = \dim T_{[\phi(\sigma)]} \, \Omega(Y) , \end{align*}$$

which completes the proof in the case $d = 2$ .

4.2 A local Torelli theorem for bielliptic surfaces

In this subsection we describe the Kuranishi family of a bielliptic surface (see Proposition 4.1) and we provide subsequently a local Torelli theorem for bielliptic surfaces (see Theorem 4.2).

First of all, we recall some basic facts about abelian surfaces. Specifically, if $A$ is an abelian surface, then $H^2(A,\mathbb {Z})$ is a free $\mathbb {Z}$ -module of rank $ 6$ by [Reference Birkenhake and Lange10, Corollary 1.3.3], while by [Reference Birkenhake and Lange10, Lemma 1.3.1] the cup product induces an isomorphism

$$\begin{align*}\bigwedge^2 H^2(A,\mathbb{Z}) \simeq H^4(A,\mathbb{Z}) \simeq \mathbb{Z}. \end{align*}$$

Therefore, $H^2(A,\mathbb {Z})$ has a lattice structure and the resulting nondegenerate, integral, quadratic form on $H^2(A,\mathbb {Z})$ is denoted by $q_A$ . Note that $\big ( H^2(A,\mathbb {Z}) , q_A \big )$ is an even, unimodular, nondegenerate lattice with signature of type $(3,\ast )$ , which is isomorphic to $U^{\oplus 3}$ , where $U$ is the hyperbolic plane over $\mathbb {Z}$ ; see [Reference Huybrechts36, Chapter 3, Subsections 1.3, 2.2 and 2.3]. The direct summands of the Hodge decomposition of $H^2(A,\mathbb {C})$ have dimension

$$\begin{align*}h^{2,0}(A) = h^{0,2}(A) = 1 \quad \text{ and } \quad h^{1,1}(A) = 4 , \end{align*}$$

and the symplectic form $\sigma _A$ on $A$ induces an isomorphism

(4.5) $$ \begin{align} \mathscr{T}_A \simeq \Omega^1_A , \end{align} $$

which yields

$$\begin{align*}H^1(A,\mathscr{T}_A) \simeq H^1(A,\Omega^1_A) \simeq H^{1,1}(A). \end{align*}$$

Furthermore, the Kuranishi space $\operatorname {\mathrm {Def}}(A)$ of $A$ is smooth of dimension

$$\begin{align*}\dim \operatorname{\mathrm{Def}}(A) = \dim_{\mathbb{C}} H^1(A,\mathscr{T}_A) = h^{1,1}(A) = 4 \end{align*}$$

by the Bogomolov–Tian–Todorov unobstructedness theorem, and the Kuranishi family $\pi _A \colon \mathscr {A} \to \operatorname {\mathrm {Def}}(A)$ of $A$ is universal; see, for example, [Reference Wavrik65, Theorems 1.1 and 4.2]. In particular, up to shrinking $\operatorname {\mathrm {Def}}(A)$ , each fiber of $\pi _A$ is an abelian surface, and also the following local Torelli statement holds: the local period map

$$\begin{align*}\mathscr{P}_A \colon \operatorname{\mathrm{Def}}(A) \to \Omega(A) , \ t \mapsto \mathbb{P} \big( H^{2,0}( \mathscr{A}_t ) \big), \end{align*}$$

where

$$\begin{align*}\Omega(A) := \big\{ [\omega] \in \mathbb{P} \big( H^2(A,\mathbb{C}) \big) \mid q_A(\omega)=0 , \ q_A(\omega,\overline{\omega})>0 \big\}, \end{align*}$$

is a local isomorphism; see [Reference Voisin64, Subsection 10.3.2].

We now turn our attention to bielliptic surfaces in order to derive an analogous statement to the above. Given a bielliptic surface $B$ , recall that there exist an abelian surface $A$ and a finite cyclic group $G$ of order $d \in \{ 2, 3, 4, 6 \}$ that acts freely on $A$ such that $B \simeq A/G$ ; see [Reference Oguiso and Schröer51, Section 6] and [Reference Ferrari, Tirabassi, Vodrup and Bergström23, Subsections 2.1 and 2.2]. The quotient map $A \to B$ is the global index-one cover of $B$ (also referred to as the canonical cover of $B$ ); see [Reference Kollár and Mori41, Definition 5.19]. Since $H^0(A,\omega _A)^G \simeq H^0(B, \omega _B) = \{ 0 \} $ , the group $G \subseteq \operatorname {\mathrm {Aut}}(A)$ must contain at least one nonsymplectic automorphism. In fact, by inspecting the classification table of bielliptic surfaces due to Bagnera–De Franchis, one can readily check that

(4.6) $$ \begin{align} G \text{ acts on }A\text{ nonsymplectically} , \end{align} $$

that is, the generator of $G$ is a purely nonsymplectic automorphism of order $d$ ; see [Reference Oguiso and Schröer51, Section 6] or [Reference Suwa62, Theorem, pp. 473-474]. Taking (4.5) and (4.6) into account and arguing as in the proof of Lemma 3.1(iii), we infer that

(4.7) $$ \begin{align} H^1(B,\mathscr{T}_{B}) \simeq H^1(A,\mathscr{T}_A)^G = H^1(A, \mathscr{T}_A)_0 \simeq H^1(A,\Omega^1_A)_1 \simeq H^{1,1}(A)_1 \,. \end{align} $$

We can now describe the Kuranishi family of $B \simeq A/G$ .

Proof. To show that $\big ( \mathscr {A} \times _{\operatorname {\mathrm {Def}}(A)} F \big ) / G \to F$ is a miniversal deformation of $B \simeq \mathscr {B}_0$ and that $F$ is smooth, we closely follow the proof of Theorem 1.1(i), but with the following differences: in the second paragraph, we apply Theorem 2.21 instead of both [Reference Bakker and Lehn4, Lemma 4.6] and Lemma 2.22 to extend the $G$ -action on $A$ equivariantly to its universal family $\mathscr {A} \to \operatorname {\mathrm {Def}}(A)$ , while in the third paragraph, we replace Lemma 3.1(iii) by (4.7).

Since, according to [Reference Grauert28, p. 107, 3rd paragraph], miniversal deformations of a given compact complex manifold are uniquely determined (up to isomorphism), we conclude that the above deformation is actually the Kuranishi family $\mathscr {B} \to \operatorname {\mathrm {Def}}(B) \simeq F$ of $B$ ; in particular, the germ $\big ( \operatorname {\mathrm {Def}}(B), 0 \big )$ is reduced. Hence, we have

$$\begin{align*}\dim \operatorname{\mathrm{Def}}(B) = h^1(B, \mathscr{T}_B) = \begin{cases} 2 , & \text{if } \ d = 2, \\ 1 , & \text{if } \ d \in \{3,4,6\} , \end{cases} \end{align*}$$

where the second equality holds by [Reference Lang42, Theorem 4.9].

Next, to show that $\mathscr {B} \to \operatorname {\mathrm {Def}}(B)$ is universal for $B$ (and thus for all of its fibers due to the openness of versality), by [Reference Wavrik65, Theorems 1.1 and 4.2] it suffices to show that $h^0(\mathscr {B}_t, \mathscr {T}_{\mathscr {B}_t})$ is independent of $t$ in a sufficiently small neighborhood of $0 \in \operatorname {\mathrm {Def}}(B)$ . However, granting (ii) for a moment, this follows immediately from [Reference Lang42, Theorem 4.9], which shows that bielliptic surfaces carry a unique (up to scalar) holomorphic vector field.

Consequently, to complete the proof of the theorem, it remains to show (ii). To do so, we proceed in two steps.

Step 1: We will show that $G$ acts freely on any fiber of $\pi _B$ over a sufficiently small neighborhood of $0 \in \operatorname {\mathrm {Def}}(B) \subset \operatorname {\mathrm {Def}}(A)$ .

To this end, consider the nonfree locus $\mathscr {A}^{\mathrm{nf}}$ of the induced $G$ -action on $\mathscr {A}$ , that is,

$$\begin{align*}\mathscr{A}^{\text{nf}} := \big\{ x \in \mathscr{A} \mid G_x \neq \{ \operatorname{\mathrm{Id}}_G \} \big\} , \end{align*}$$

which is a closed subset of $\mathscr {A}$ and does not contain any point of $\mathscr {A}_0 \simeq A$ by assumption. Since the morphism $\pi _A \colon \mathscr {A} \to \operatorname {\mathrm {Def}}(A)$ is smooth and proper, the image $\pi _A \big ( \mathscr {A}^{\mathrm{nf}} \big )$ is a closed subset of $\operatorname {\mathrm {Def}}(A)$ that does not contain $0 \in \operatorname {\mathrm {Def}}(A)$ and the restricted morphism $\pi _A|_{\mathscr {A} \setminus \mathscr {A}^{\mathrm{nf}}} \colon \mathscr {A} \setminus \mathscr {A}^{\mathrm{nf}} \to \operatorname {\mathrm {Def}}(A) \setminus \pi \big ( \mathscr {A}^{\mathrm{nf}} \big )$ is also smooth and proper. This yields the assertion; cf. the proof of [Reference Oguiso and Schröer51, Proposition 1.2].

Step 2: We will demonstrate that $G$ acts nonsymplectically on any fiber of $\pi _B$ over a sufficiently small neighborhood of $0 \in \operatorname {\mathrm {Def}}(B) \subset \operatorname {\mathrm {Def}}(A)$ .

For $t$ in a sufficiently small neighborhood of $0 \in \operatorname {\mathrm {Def}}(B) \subset \operatorname {\mathrm {Def}}(A)$ as in Step 1, the quotient map $\mathscr {A}_t \to \mathscr {B}_t \simeq \mathscr {A}_t/G$ is a finite étale cover. Therefore, $\mathscr {B}_t$ has torsion canonical class. But since Hodge numbers are constant in the smooth family $\mathscr {B} \to \operatorname {\mathrm {Def}}(B)$ whose central fiber $\mathscr {B}_0$ is the given bielliptic surface $B$ , we conclude that $\mathscr {B}_t \simeq \mathscr {A}_t/G$ is also a bielliptic surface. In particular, the induced $G$ -action on $\mathscr {A}_t$ is nonsymplectic, as explained earlier in this subsection; see (4.6). This finishes the proof.

As a short interlude, we explain here a notion of marking for bielliptic surfaces, which will be tacitly used both in the statement and in the proof of Theorem 4.2. Specifically, the discussion that precedes Lemma 2.7 remains valid also in the present framework, and consequently an identical version of Lemma 2.7 with respect to the Hermitian form associated with $q_A$ can be established for the canonical abelian cover $A$ of a bielliptic surface $B \simeq A/G$ . This is due to the following two observations: (2.10) can be replaced by the invariance of the cup product under homeomorphisms, while (2.3) can be replaced by the equalities $[\sigma _A] \smile [\sigma _A] = 0 = [\overline {\sigma _A}] \smile [\overline {\sigma _A}]$ and the Hodge–Riemann bilinear relations; see [Reference Voisin64, Theorem 6.32]. Therefore, an $L$ -marking of a bielliptic surface can be defined in the same way as it was previously done for a primitive Enriques variety; see Subsection 2.4.

Finally, we establish the desired statement for bielliptic surfaces, which will only be used in the proof of Proposition 5.2.

Theorem 4.2 (Local Torelli theorem for bielliptic surfaces).

Let $B \simeq A/G$ be a bielliptic surface, where $A$ is an abelian surface and $G \subseteq \operatorname {\mathrm {Aut}}(A)$ is generated by a purely nonsymplectic automorphism $g \in \operatorname {\mathrm {Aut}}(A)$ of order $d \in \{2, 3, 4, 6\}$ and acts freely on $A$ . Let

$$\begin{align*}\Omega(B) := \big\{ [\omega] \in \mathbb{P} \big( H^2(A,\mathbb{C})_1 \big) \mid q_A(\omega)=0 , \ q_A(\omega,\overline{\omega})>0 \big\} \end{align*}$$

be the period domain for $B$ inside $\mathbb {P} \big ( H^2(A,\mathbb {C})_1 \big )$ . If $\pi _B \colon \mathscr {B} \to \operatorname {\mathrm {Def}}(B)$ is the universal Kuranishi family of $B$ and if $\mathscr {A}_t := \pi _A^{-1}(t)$ is the fiber over $t \in \operatorname {\mathrm {Def}}(A)$ of the universal Kuranishi family $\pi _A \colon \mathscr {A} \to \operatorname {\mathrm {Def}}(A)$ of $A$ , then the local period map

$$\begin{align*}\mathscr{P}_B \colon \operatorname{\mathrm{Def}}(B) \to \Omega(B) , \ s \mapsto \mathbb{P} \big( H^{2,0}(\mathscr{A}_s) \big) \end{align*}$$

is a local isomorphism at $0 \in \operatorname {\mathrm {Def}}(B)$ .

Proof. First, arguing as in Step 1 of the proof of Theorem 1.2, we obtain a holomorphic local period map

$$\begin{align*}\mathscr{P}_B \colon \operatorname{\mathrm{Def}}(B) \to \Omega(B) , \ s \mapsto \big[ \phi(\sigma_{\mathscr{A}_s}) \big] , \end{align*}$$

which is the restriction of the local period map $\mathscr {P}_A \colon \operatorname {\mathrm {Def}}(A) \to \Omega (A)$ for $A$ . Here, $\phi $ denotes the fiberwise marking (involved in the proof) of the deformation over a simply connected (up to shrinking) and smooth base $\operatorname {\mathrm {Def}}(B)$ of the bielliptic surface $B \simeq A/G$ , and $\sigma _{\mathscr {A}_s}$ denotes the symplectic form on the fiber $\mathscr {A}_s$ over $s \in \operatorname {\mathrm {Def}}(B) \hookrightarrow \operatorname {\mathrm {Def}}(A)$ of the corresponding deformation of the abelian surface $A$ .

Next, to conclude that $\mathscr {P}_B$ is a local isomorphism at $0 \in \operatorname {\mathrm {Def}}(B)$ , arguing as in Step 2 of the proof of Theorem 1.2, except that we now invoke the local Torelli theorem for abelian surfaces and (4.7) instead of [Reference Bakker and Lehn4, Proposition 5.5] and Lemma 3.1(iii), respectively, we see that it remains to compute $\dim _{\mathbb {C}} T_{[\phi (\sigma _A)]} \, \Omega (B)$ and to verify that it is equal to $\dim _{\mathbb {C}} T_0 \operatorname {\mathrm {Def}}(B)$ . This will be done below by distinguishing cases for $d$ and by noting that we now work with the lattice $\big ( H^2(A,\mathbb {Z}) , q_A \big )$ , where the nondegenerate symmetric bilinear form $q_A(\cdot , \cdot )$ associated to $q_A(\cdot )$ is the cup product in cohomology.

If $d \in \{3,4,6\}$ , then the same argument as in Step 3 of the proof of Theorem 1.2 works also in our context due to the following observation: For any $\omega \in H^2(A, \mathbb {C})_1$ we have $g^* \omega = \xi \omega $ , where $\xi $ is a primitive $d$ -th root of unity, and thus

$$\begin{align*}\omega \smile \omega \overset{(\ast \ast)}{=\!=\!=} g^* \omega \smile g^* \omega = \xi^2 (\omega \smile \omega) , \end{align*}$$

where $(\ast \ast )$ holds because the cup product is preserved under homeomorphisms.

If, finally, $d=2$ , then we repeat verbatim the argument from Step 4 of the proof of Theorem 1.2, except that we now determine the orthogonal complement of the symplectic form $\sigma _A \in H^{2,0}(A) \subseteq H^2(A,\mathbb {C})_1$ with respect to $q_A(\cdot , \cdot )$ using the Hodge–Riemann bilinear relations [Reference Voisin64, Theorem 6.32].

5.1 Locally trivial deformations of $\operatorname {\mathrm {Hilb}}^n(S)/G$

Consider a K3 surface $S$ with a purely nonsymplectic automorphism $g \in \operatorname {\mathrm {Aut}}(S)$ of finite order $d \geq 2$ such that $G := \langle g \rangle $ acts freely in codimension one on $S$ . Note that $S$ is projective by [Reference Beauville7, Proposition 6(i)]. Pick $n \geq 1$ and let $G^{[n]} := \langle g^{[n]} \rangle \subseteq \operatorname {\mathrm {Aut}} \big ( S^{[n]} \big )$ be the induced cyclic group of automorphisms of the Hilbert scheme $S^{[n]}$ of $n$ points on $S$ . Since $G^{[n]}$ acts on $S^{[n]}$ nonsymplectically by construction and freely in codimension one either by construction when $n=1$ or by [Reference Denisi, Ríos Ortiz, Tsakanikas and Xie17, Proposition 4.14] when $n\geq 2$ , the quotient

$$\begin{align*}Y := S^{[n]} / G^{[n]} = \operatorname{\mathrm{Hilb}}^n(S) / G^{[n]} \end{align*}$$

is a primitive Enriques variety of dimension $2n \geq 2$ , which is also $\mathbb {Q}$ -factorial by [Reference Kollár and Mori41, Proposition 5.15].Footnote ¹ We will show in Proposition 5.1 that any small locally trivial deformation of $Y = S^{[n]} / G^{[n]}$ is of the same form.

To this end, we first construct a family of primitive Enriques varieties with central fiber $Y$ as follows. Consider the locally trivial Kuranishi family $\mathscr {T} \to {{\mathrm {Def}}^{\mathrm {lt}}}(T)$ of the primitive Enriques surface $T := S / G$ . According to Theorem 1.1, there is a composition of morphisms

$$\begin{align*}\mathscr{S} \to \mathscr{T} \simeq \mathscr{S}/G \to {{\mathrm{Def}}^{\mathrm{lt}}}(T) \end{align*}$$

which determines a family of K3 surfaces with central fiber $\mathscr {S}_0 \simeq S$ and, up to shrinking ${{\mathrm {Def}}^{\mathrm {lt}}}(T)$ , we may also assume that the induced action of $G$ on $\mathscr {S}$ is fiberwise nonsymplectic and free in codimension one. The relative Douady space of points yields then a family

$$\begin{align*}\big( \mathscr{S} / {{\mathrm{Def}}^{\mathrm{lt}}}(T) \big)^{[n]} \to {{\mathrm{Def}}^{\mathrm{lt}}}(T) \end{align*}$$

of IHS manifolds, which are Hilbert schemes $\mathscr {S}_t^{[n]}$ of $n$ points on the K3 surfaces $\mathscr {S}_t$ , with central fiber $\mathscr {S}_0^{[n]} \simeq \operatorname {\mathrm {Hilb}}^n (\mathscr {S}_0) \simeq S^{[n]}$ , see [Reference Beauville6, p. 778]. Note that the total space

$$\begin{align*}\big( \mathscr{S}/{{\mathrm{Def}}^{\mathrm{lt}}}(T) \big)^{[n]} \end{align*}$$

is naturally endowed with a $G^{[n]}$ -action, which is compatible with the induced fiberwise $G^{[n]}$ -action on $\mathscr {S}_t^{[n]}$ . The latter is in turn nonsymplectic and free in codimension one by construction. It follows from the proof of [Reference Graf26, Proposition 6.2] that

$$\begin{align*}\mathscr{Y}' := \big( \mathscr{S} / {{\mathrm{Def}}^{\mathrm{lt}}}(T) \big)^{[n]} / G^{[n]} \to {{\mathrm{Def}}^{\mathrm{lt}}}(T) \end{align*}$$

is a locally trivial deformation of $Y = S^{[n]} / G^{[n]} = \operatorname {\mathrm {Hilb}}^n(S) / G^{[n]}$ . Moreover, up to further shrinking ${{\mathrm {Def}}^{\mathrm {lt}}}(T)$ , any member $\mathscr {Y}^{\prime }_t$ of this family is a $\mathbb {Q}$ -factorial primitive Enriques variety of the form

$$\begin{align*}\mathscr{Y}^{\prime}_t \simeq \mathscr{S}_t^{[n]} / G^{[n]} = \operatorname{\mathrm{Hilb}}^n (\mathscr{S}_t) / G^{[n]} \end{align*}$$

by construction.

Now, by the universality of the locally trivial Kuranishi family $\mathscr {Y}\to {{\mathrm {Def}}^{\mathrm {lt}}}(Y)$ of the primitive Enriques variety $Y$ , we obtain a unique morphism ${{\mathrm {Def}}^{\mathrm {lt}}}(T) \to {{\mathrm {Def}}^{\mathrm {lt}}}(Y)$ . We may also assume that these two complex manifolds are simply connected, possibly after further shrinking them. The next result indicates how small locally trivial deformations of $Y$ look and generalizes [Reference Oguiso and Schröer51, Proposition 3.1]; see also [Reference Oguiso and Schröer51, Remark 3.2].

Proof. The proof is similar to the one of [Reference Oguiso and Schröer51, Proposition 3.1], but we provide the details for the reader’s convenience. We identify the groups $G$ and $G^{[n]}$ with $\mu _d(\mathbb {C})$ below.

Let $(L,q) := \big ( H^2(S^{[n]},\mathbb {Z}) \, , q_{S^{[n]}} \big )$ be the Beauville–Bogomolov–Fujiki lattice of $S^{[n]}$ , which is endowed with the canonical $\mu _d(\mathbb {C})$ -action. According to [Reference Beauville6, Proposition 6 and Remark on p. 768], there is a $\mu _d(\mathbb {C})$ -equivariant primitive embedding of lattices

$$\begin{align*}\iota \colon H^2(S,\mathbb{Z}) \hookrightarrow H^2(S^{[n]},\mathbb{Z}) , \end{align*}$$

compatible with Hodge structures, such that

$$\begin{align*}H^2(S^{[n]},\mathbb{Z}) \simeq \iota \left( H^2(S,\mathbb{Z}) \right) \oplus \mathbb{Z} \delta , \end{align*}$$

where $2\delta =[E]$ and $E$ is the prime exceptional divisor of the $\mu _d(\mathbb {C})$ -equivariant Hilbert–Chow morphism $ S^{[n]} = \operatorname {\mathrm {Hilb}}^n(S) \to S^{(n)} = \operatorname {\mathrm {Sym}}^n(S) $ . Since $E$ is $\mu _d(\mathbb {C})$ -invariant by [Reference Boissière and Sarti13, Theorem 1], we obtain an identification of weight spaces:

(5.1) $$ \begin{align} H^2(S,\mathbb{C})_1 \simeq H^2(S^{[n]},\mathbb{C})_1 \,. \end{align} $$

Set $L' := \iota (H^2(S,\mathbb {Z}))$ . Since $\iota $ is $\mu _d(\mathbb {C})$ -equivariant, we obtain an $L$ -marking for the primitive Enriques variety $Y$ and an $L'$ -marking for the primitive Enriques surface $T$ . Due to the hypothesis that ${{\mathrm {Def}}^{\mathrm {lt}}}(T)$ and ${{\mathrm {Def}}^{\mathrm {lt}}}(Y)$ are simply connected, these markings uniquely extend to markings of the considered families $\mathscr {Y} \to {{\mathrm {Def}}^{\mathrm {lt}}}(Y)$ and $\mathscr {T} \to {{\mathrm {Def}}^{\mathrm {lt}}}(T)$ , respectively. Taking now Definition 2.12(1) and (5.1) into account, we observe that the period domain $\mathcal {D}_L$ for the $L$ -marked primitive Enriques varieties coincides with the period domain $\mathcal {D}_{L'}$ for the $L'$ -marked primitive Enriques surfaces.

Finally, consider the diagram

where $\mathscr {P}'$ and $\mathscr {P}$ are the local period maps for the marked families of primitive Enriques surfaces and primitive Enriques varieties, respectively. By Theorem 1.2, both $\mathscr {P}'$ and $\mathscr {P}$ are local isomorphisms at the reference point $0$ . Thus, to complete the proof, it only remains to check that the above diagram is commutative. But this follows from the commutativity of the diagram in [Reference Beauville6, p. 780].

5.2 Locally trivial deformations of $\operatorname {\mathrm {Kum}}^n(A)/G$

Consider a bielliptic surface $B$ . As in Subsection 4.2, write $B \simeq A/G$ , where $A$ is an abelian surface and $G \subseteq \operatorname {\mathrm {Aut}}(A)$ is generated by a purely nonsymplectic automorphism $g \in \operatorname {\mathrm {Aut}}(A)$ of order $d \in \{2,3,4,6\}$ acting freely on $A$ . There exists a finite étale cover $\widetilde {A} \to B$ factoring through $A$ such that $\widetilde {A} \simeq E \times F$ , where $E$ and $F$ are elliptic curves, and also $\widetilde {A} / \widetilde {K} \simeq A$ , where $\widetilde {K} := \ker (\widetilde {A} \to A)$ is a finite group. We refer to [Reference Oguiso and Schröer51, Section 6] and [Reference Ferrari, Tirabassi, Vodrup and Bergström23, Subsections 2.1 and 2.2] for more details.

Next, pick $n \geq 2$ and let $G^{[n+1]} := \langle g^{[n+1]} \rangle \subseteq \operatorname {\mathrm {Aut}} \big ( \operatorname {\mathrm {Hilb}}^{n+1}(A) \big )$ be the induced cyclic group of automorphisms of the Hilbert scheme of $n+1$ points on $A$ . Note that $G^{[n+1]}$ acts on $\operatorname {\mathrm {Hilb}}^{n+1}(A)$ nonsymplectically. According to [Reference Oguiso and Schröer50, Proposition 6.1], if $d \mid n+1$ and $z \in F$ is an $(n+1)$ -torsion point, then the generalized Kummer variety $\operatorname {\mathrm {Kum}}^n(A)$ is $G^{[n+1]}$ -invariant. Since $G^n := \langle g^{[n+1]} |_{\operatorname {\mathrm {Kum}}^n(A)} \rangle \subseteq \operatorname {\mathrm {Aut}} \big ( \operatorname {\mathrm {Kum}}^n(A) \big )$ acts on $\operatorname {\mathrm {Kum}}^n(A)$ nonsymplectically by construction, and hence freely in codimension one by [Reference Denisi, Ríos Ortiz, Tsakanikas and Xie17, Proposition 4.14], the quotient

$$\begin{align*}Y := \operatorname{\mathrm{Kum}}^n(A) / G^n \end{align*}$$

is a $\mathbb {Q}$ -factorial primitive Enriques variety of dimension $2n$ .Footnote ²

We will demonstrate in Proposition 5.2 that any small locally trivial deformation of $\operatorname {\mathrm {Kum}}^n(A)/G^n$ is of the same form. To this end, consider the Kuranishi family

$$\begin{align*}\widetilde{\mathscr{A}} \to \operatorname{\mathrm{Def}}(E \times F) \end{align*}$$

of $\widetilde {A} \simeq E \times F$ . Since the latter is endowed with a holomorphic $(G \times \widetilde {K})$ -action (see the bottom of [Reference Oguiso and Schröer50, p. 230], the table given in [Reference Oguiso and Schröer50, p. 231], and [Reference Ferrari, Tirabassi, Vodrup and Bergström23, Section 2] for further details), so is the Kuranishi family of $\widetilde {A}$ by Theorem 2.21. Let $\widetilde {\operatorname {\mathrm {Def}}(B)}$ be the connected component of the fixed locus of the $(G \times \widetilde {K})$ -action on $\operatorname {\mathrm {Def}}(E \times F)$ containing the reference point, that is, the one over which the fiber is isomorphic to $E \times F$ , and set

$$\begin{align*}\widetilde{\mathscr{A}}_{|\operatorname{\mathrm{Def}}(B)} := \widetilde{\mathscr{A}}\times_{\operatorname{\mathrm{Def}}(E\times F)} \widetilde{\operatorname{\mathrm{Def}}(B)}. \end{align*}$$

It follows from [Reference Graf26, Proposition 6.2] that the Kuranishi family of the bielliptic surface $B \simeq (E \times F) / (G \times \widetilde {K})$ is

$$\begin{align*}\mathscr{B} = \widetilde{\mathscr{A}}_{|\operatorname{\mathrm{Def}}(B)}/(G \times \widetilde{K}) \to \operatorname{\mathrm{Def}}(B) = \widetilde{\operatorname{\mathrm{Def}}(B)}. \end{align*}$$

Observe now that the morphism $\mathscr {A} \to \mathscr {B}$ constructed in Proposition 4.1 is the relative canonical cover of $\mathscr {B}$ ,Footnote ³ that is, the index-one cover associated with any lift of the line bundle $\mathcal {O}_{B_t}(K_{B_t})$ to $\mathscr {B}$ . Such an extension exists by Proposition 2.17 and Remark 2.18. Then $\widetilde {\mathscr {A}}_{|\operatorname {\mathrm {Def}}(B)} \to \operatorname {\mathrm {Def}}(B)$ factors through $\mathscr {A} \to \operatorname {\mathrm {Def}}(B)$ , because the group $\widetilde {K} \subset G \times \widetilde {K}$ acts on $\widetilde {\mathscr {A}}_{|\operatorname {\mathrm {Def}}(B)}$ . Thus,

$$\begin{align*}\widetilde{\mathscr{A}}_{|\operatorname{\mathrm{Def}}(B)} / \widetilde{K} \cong \mathscr{A} \quad \text{and} \quad \mathscr{A} / G \cong \mathscr{B}; \end{align*}$$

see [Reference Ferrari, Tirabassi, Vodrup and Bergström23, Section 2].

At this point, after possibly shrinking $\operatorname {\mathrm {Def}}(B)$ , we can perform the relative Kummer construction for the family $\mathscr {A} \to \operatorname {\mathrm {Def}}(B)$ of abelian surfaces with central fiber $\mathscr {A}_0 \simeq A$ and obtain a family

(5.2) $$ \begin{align} \operatorname{\mathrm{Kum}}^n \big( \mathscr{A} / \operatorname{\mathrm{Def}}(B) \big) \to \operatorname{\mathrm{Def}}(B) \end{align} $$

of IHS manifolds, which are generalized Kummer varieties $\operatorname {\mathrm {Kum}}^n(\mathscr {A}_t)$ associated with the abelian surfaces $\mathscr {A}_t$ , with central fiber $\operatorname {\mathrm {Kum}}^n(A)$ . By our choice of the $(G \times \widetilde {K})$ -action on $E\times F$ it follows that the $(G \times \widetilde {K})$ -action on the nearby fibers of $\widetilde {\mathscr {A}}_{|\operatorname {\mathrm {Def}}(B)}\to \operatorname {\mathrm {Def}}(B)$ stays of the same type. Then, according to [Reference Oguiso and Schröer50, Proposition 6.1], the family (5.2) comes equipped with a $G^n$ -action that is fiberwise nonsymplectic and free in codimension one. Moreover, this $G^n$ -action is induced by the natural $G^{[n+1]}$ -action on the relative Douady space of points

$$\begin{align*}\big( \mathscr{A} / \operatorname{\mathrm{Def}}(B) \big)^{[n+1]}\to \operatorname{\mathrm{Def}}(B). \end{align*}$$

Thus, arguing as in Step 2 of the proof of [Reference Graf26, Proposition 6.2], we obtain a locally trivial deformation

$$\begin{align*}\mathscr{Y}' := \operatorname{\mathrm{Kum}}^n \big( \mathscr{A} / \operatorname{\mathrm{Def}}(B) \big) / G^n \to \operatorname{\mathrm{Def}}(B) \end{align*}$$

of $Y = \operatorname {\mathrm {Kum}}^n(A) / G^n$ such that any member $\mathscr {Y}^{\prime }_t$ of this family is a $\mathbb {Q}$ -factorial primitive Enriques variety of the form

$$\begin{align*}\mathscr{Y}_t' \simeq \operatorname{\mathrm{Kum}}^n(\mathscr{A}_t) / G^n. \end{align*}$$

Now, by the universality of the locally trivial Kuranishi family $\mathscr {Y} \to {{\mathrm {Def}}^{\mathrm {lt}}}(Y)$ of $Y$ , we obtain a unique morphism $\operatorname {\mathrm {Def}}(B) \to {{\mathrm {Def}}^{\mathrm {lt}}}(Y)$ , and we may also assume that $\operatorname {\mathrm {Def}}(B)$ and ${{\mathrm {Def}}^{\mathrm {lt}}}(Y)$ are simply connected. The next result describes the small locally trivial deformations of $Y$ and generalizes [Reference Oguiso and Schröer51, Proposition 3.5].