Proof. Since automorphisms in $\operatorname {Aut}_0(K)$ deform with $K$, their fixed loci deform as well. Therefore, it suffices to prove the lemma for the generalized Kummer variety $K=K^3(A)$ on an abelian surface $A$. In this case we show that

\[ G=A_2\times \langle -1\rangle \cong ({\mathbb{Z}}/2{\mathbb{Z}})^5. \]

We have $\operatorname {Aut}_0(K^3(A))= A_4\rtimes \langle -1\rangle$. If $h\in \operatorname {Aut}_0(K^3(A))$ is induced by translation by $0\neq \epsilon \in A_4$, then, by [Reference OguisoOgu20, Lemma 3.5], the fixed locus $(K^3(A))^h$ is a union of K3 surfaces if $2\epsilon =0$ and consists of isolated point if $2\epsilon \neq 0$. Consider now $h=(\epsilon, -1)$. The fixed locus of $h=(\epsilon, -1)$ on $A^{(4)}$ consists of

\begin{align*} W_{\epsilon} = \{ (a, -a -\epsilon , b , - b - \epsilon), \ \text{for } a,b \in A \} \subset A^{(4)}, \end{align*}

and subvarieties of lower dimension. If $\epsilon \in A_2$, then $W_{\epsilon }$ is contained in $A^{(4)}_0$ and, hence, ${(\epsilon, -1)\in G}$. If $\epsilon \notin A_2$, then $W_{\epsilon }$ has empty intersection with $A^{(4)}_0$; the action of $h$ on $A^{(4)}_0$ fixes isolated points and the surfaces $\{ (\alpha, b, -b-\epsilon, - \alpha + \epsilon ),\ b\in A\}$, where $\alpha \in A$ satisfies $\alpha =-\alpha -\epsilon$. The general point of such a surface is supported on four distinct points. Therefore, the fixed locus $(K^3(A))^h$ consists of surfaces and isolated points, because it is a union of symplectic varieties and the fibres of $\nu$ have dimension at most $3$. We conclude that $G$ is generated by the $16$ involutions $(\tau,-1)$ with $\tau \in A_2$ and, hence, $G=A_2\times \langle -1\rangle$.

Proof. We prove part (i). As translations are fixed point free, for any $\alpha \in A/\langle \tau, (\theta, -1)\rangle$ the support of $\psi _{\tau, \theta } (\alpha )\in V_{\tau, \theta }$ consists of either two or four distinct points, according to whether $\alpha$ is a node or not. If $\alpha$ is not a node $q^{-1}(\alpha )$ is a single point at which $\phi _{\tau, \theta }$ is well-defined. Otherwise, $\psi _{\tau, \theta } (\alpha ) =(a, a+\tau, a, a+\tau )$ for some $a$ in $A_{2, \theta }$ or $A_{2,\tau +\theta }$. Then, there is a canonical identification $\nu ^{-1}(\psi _{\tau, \theta } (\alpha ))= {\mathbb {P}}(T_aA) \times {\mathbb {P}}(T_{a+\tau } A)$. The exceptional divisor in $\operatorname {\mathrm {Km}}^{\theta }(A/\langle \tau \rangle )$ corresponding to the node $\alpha$ is identified with $ {\mathbb {P}}(T_a A)$. Translation by $\tau$ gives an isomorphism $ {\mathbb {P}}(T_aA)\xrightarrow {\ \sim \ } {\mathbb {P}}(T_{a+\tau}A)$, and the morphism $\phi _{\tau, \theta }$ is extended via $\phi _{\tau, \theta } (t):= (t, \tau (t))$ for $t\in {\mathbb {P}}(T_aA)$.

The proof of part (ii) is similar. If $\psi _{\tau } (\alpha, \beta )$ consists of four distinct points, then $q^{-1}(\alpha, \beta )$ is a single point at which $\phi _{\tau }$ is well-defined. If the support of $\psi _{\tau }(\alpha, \beta )$ consists of three distinct points, then exactly one between $\alpha$ and $\beta$ is a node of $A/\langle (\tau, -1) \rangle$, and $\psi _{\tau }(\alpha, \beta )=(a, \epsilon, -a+\tau, \epsilon )$ for some $a\in A\setminus A_{2,\tau }$ and $\epsilon \in A_{2,\tau }$; we extend $\phi _{\tau }$ via the canonical identifications $q^{-1}(\alpha, \beta ) = {\mathbb {P}}(T_{\epsilon }A)=\nu ^{-1}(\psi _{\tau }(\alpha,\beta ))$. If the support of $\psi _{\tau }(\alpha, \beta )$ consists of two distinct points then either $\alpha =\beta$ for some smooth point or $\alpha \neq \beta$ with both $\alpha$ and $\beta$ nodes of $A/\langle (\tau,-1)\rangle$. In the first case there exists $a\in A\setminus A_{2,\tau }$ such that $\psi _{\tau }(\alpha,\beta )=(a, a, -a+\tau, -a+\tau )$. Then $q^{-1}(\alpha, \beta ) = \mathbb {P}(T_a A)$ and $\nu ^{-1}(\psi _{\tau } (\alpha, \beta ))= {\mathbb {P}}(T_{a}A)\times {\mathbb {P}}(T_{-a+\tau } A)$, and we define $\phi _{\tau }$ by $t\mapsto (t, (\tau, -1)(t))$. In the second case $\psi _{\tau }(\alpha, \beta )= (\epsilon _1, \epsilon _2, \epsilon _1, \epsilon _2)$ for some $\epsilon _1\neq \epsilon _2$ both in $A_{2,\tau }$, and we extend $\phi _{\tau }$ via the canonical isomorphisms $q^{-1}(\alpha, \beta ) = {\mathbb {P}}(T_{\epsilon _1}A)\times {\mathbb {P}}(T_{\epsilon _2}A) = \nu ^{-1}(\psi _{\tau }(\alpha,\beta ))$. Finally, consider the case when $\alpha =\beta$ and $\alpha$ is a node. Locally analytically around the node $\alpha$ of $A/\langle (\tau, -1)\rangle$, the morphism $q$ is isomorphic to the composition

\[ (\mathrm{Bl}_0({\mathbb{C}}^2/\iota))^{[2]}\to (\mathrm{Bl}_0({\mathbb{C}}^2/\iota))^{(2)} \to ({\mathbb{C}}^2/\iota)^{(2)}, \]

where $\iota (x,y)=(-x,-y)$ is the restriction of the involution $(\tau, -1)$. The rational map $\phi _{\tau }$ is identified with $\Phi \colon (\mathrm {Bl}_0( {\mathbb {C}}^2/\iota ))^{[2]} \dashrightarrow ( {\mathbb {C}}^2)^{[4]}$ which maps a general $\zeta =(\alpha,\beta )$ to the subscheme $\xi =(\alpha,\iota (\alpha ), \beta, \iota (\beta ))$. Then Proposition 2.8 implies that $\phi _\tau$ extends to a closed embedding $(\mathrm {Bl}_0( {\mathbb {C}}^2/\iota ))^{[2]}\hookrightarrow ( {\mathbb {C}}^2)^{[4]}$, completing the proof.

Proof. (i) A point $\xi \in A_0^{(4)}$ belongs to ${W_{\tau _1}}\cap {W_{\tau _2}}$ if and only if it can be written as $\xi =(a,b,-a+\tau _1, -b+\tau _1)$ as well as $\xi =(a', b' , -a'+\tau _2, -b'+\tau _2)$, for some $a,b,a',b'$ in $A$. Then, we may assume that $a'=a$; since $\tau _1\neq \tau _2$, it follows that $-a+\tau _1 \neq -a'+\tau _2$. This forces either $b' =-a+\tau _1$ or $-b'+\tau _2= -a +\tau _1$; in both cases, we can write

\[ \xi = (a, -a+\tau_1, -a+\tau_2, a+\tau_1+\tau_2), \]

i.e. $\xi \in V_{\tau _1+\tau _2, \tau _1}$. Thus, ${W_{\tau _1}}\cap {W_{\tau _2}}=V_{\tau _1+\tau _2, \tau _1}$ and, hence, for their strict transforms we have $\overline {W_{\tau _1}}\cap \overline {W_{\tau _2}}=\overline {V_{\tau _1+\tau _2, \tau _1}}$.

(ii) By part (i), a point $\xi$ lies in the intersection $W_{\tau _1}\cap W_{\tau _2}\cap W_{\tau _2}$ if and only if it can be written as $\xi = (a, a+\tau _1+\tau _2, -a+\tau _1, -a+\tau _2)$ as well as $\xi =(a'',b'', -a''+\tau _3, -b''+\tau _3)$. Again, we may assume $a''= a$. Then we must have $-a+\tau _3= a+\tau _1+\tau _2$; equivalently, $a\in A_{2,\tau _1+\tau _2+\tau _3}$. Moreover, either we have $b''=-a+\tau _1$ or $b''=-a+\tau _2$. Note that, since $a\in A_{2, \tau _1+\tau _2+\tau _3}$, in the first case $-b''+\tau _3= -a+\tau _2$, while in the second case $-b''+\tau _3= - a+\tau _1$. This shows that

\[ W_{\tau_1}\cap W_{\tau_2}\cap W_{\tau_3} = \{ (a, a+\tau_1+\tau_2, -a+\tau_1, -a+\tau_2) \in A_0^{(4)}, \ a\in A_{2, \tau_1+\tau_2+\tau_3}\}. \]

This set consists of four distinct points. Indeed, $G$ acts transitively on it and the subgroup of $G$ generated by the $(\tau _i, -1)$, $i=1,2,3$ acts trivially. Moreover, it is easy to see that if $g\in G$ is not an element of this subgroup, then it has no fixed points in the above set. Hence, the intersection $W_{\tau _1}\cap W_{\tau _2}\cap W_{\tau _3}$ is in bijection with $( {\mathbb {Z}}/2 {\mathbb {Z}})^2$. No point of this set belongs to the exceptional locus of the Hilbert–Chow morphism, so that $\overline {W_{\tau _1}}\cap \overline {W_{\tau _2}}\cap \overline {W_{\tau _3}}$ consists of four distinct points.

(iii) Note that $W_{\tau _1}\cap W_{\tau _2}\cap W_{\tau _3}\cap W_{\tau _1+\tau _2+\tau _3}$ is empty. We have seen above that if $\theta \in A_2\setminus \{\tau _1, \tau _2, \tau _3, \tau _1+\tau _2+\tau _3\}$, the action of $(\theta, -1)$ on $W_{\tau _1}\cap W_{\tau _2}\cap W_{\tau _3}$ is free; as ${W_{\theta }}$ is clearly fixed by $(\theta, -1)$, it follows that $W_{\tau _1}\cap W_{\tau _2}\cap W_{\tau _3}\cap W_{\theta } =\emptyset$.

Proof. Let $g=(\tau, 1)$ with $\tau \neq 0\in A_2$. The action of $g$ on $\xi =(a,b,c,d)\in A^{(4)}$ is given by $\xi \mapsto (a+\tau, b+\tau, c+\tau, d+\tau )$. We see immediately that the surfaces $V_{\tau, \theta }$ are fixed by $g$. Conversely, if $\xi$ is fixed and $a$ belongs to its support, then also $a+\tau$ must be in the support. Therefore, $\xi =(a, a+\tau, b, b+\tau )$; if $\xi \in A^{(4)}_0$, then $2a+2b=0$, so that $\theta := a+b$ belongs to $A_2$ and $\xi \in V_{\tau, \theta }$. By the description of $\overline {V_{\tau, \theta }}$ given in Lemma 2.9(i), the fixed locus of $g$ in $\nu ^{-1}(V_{\tau, \theta })$ is precisely $\overline {V_{\tau, \theta }}$. Since the fixed locus is smooth, $K^g$ is the disjoint union of the eight K$3$ surfaces $\overline {V_{\tau, \theta }}$, for $\theta \in A_2$.

Assume now that $g=(\tau, -1)$. If $\xi =(a,b,c,d) \in A^{(4)}$, the action of $g$ is given by $\xi \mapsto (-a+\tau, -b+\tau, -c+\tau, -d+\tau )$. Note that $W_{\tau }$ is contained in the fixed locus of $g$ on $A^{(4)}_0$. Moreover any $\xi$ entirely supported at points of $A_{2,\tau }$ is fixed by $g$. Conversely, let $\xi \in A_0^{(4)}$ be fixed. Then we have the following possibilities. If the support of $\xi$ does not intersect $A_{2, \tau }$, we have $\xi = (a, -a+\tau, b, -b+\tau )$ for some $a,b\in A\setminus A_{2, \tau }$ and $\xi$ belongs to $W_{\tau }$. If the support of $\xi$ contains some $\epsilon \in A_{2,\tau }$ and some $a\notin A_{2,\tau }$, then $\xi = (a, -a+\tau, \epsilon, b)$ for some $b\in A$; as $\xi \in A_0^{(4)}$, we must have $b=\epsilon$, and hence $\xi$ belongs to $W_{\tau }$. Finally, assume that $\xi =( \epsilon _1, \epsilon _2, \epsilon _3, \epsilon _4)$ is entirely supported at $A_{2,\tau }$. As $\xi$ belongs to $A^{(4)}_0$ we have $\sum _i \epsilon _i =0$ in $A$. If the support of $\xi$ consists of four distinct points, then $\xi \notin W_{\tau }$. Otherwise, $\xi =(\epsilon _1, \epsilon _1, \epsilon _2, \epsilon _2)$ lies in $W_{\tau }$.

We conclude that the fixed locus of $g$ acting on $A^{(4)}_0$ consists of $W_{\tau }$ and the isolated points in the set

\[ \biggl\{(\epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4) , \ \epsilon_i \in A_{2,\tau} \text{ pairwise distinct and such that } \sum_{i=1}^4 \epsilon_i =0 \in A \biggr\}. \]

There are $140$ isolated fixed points. Indeed, the total number of ordered sequences $(\epsilon _1, \epsilon _2, \epsilon _3, \epsilon _4)$ such that $\sum _i \epsilon _i = 0$ is $16^3$. There are $16$ such sequences with support a single point, and $6\binom{16}{2}=16 \cdot 45$ with support two distinct points. The number of ordered sequences as above supported on four distinct points is $16^3-16-16\cdot 45=16\cdot 210$; under the symmetric group, each of them has an orbit of cardinality $24$. Hence, the number of isolated fixed points is $\tfrac {210\cdot 16}{24} = 140$.

Therefore, $K^g$ consists of $140$ isolated fixed points and the fixed locus of $g$ acting on $\nu ^{-1}(W_\tau )$. To conclude we have to check that the latter coincides with $\overline {W_{\tau }}$. Using the description of $\overline {W_{\tau }}$ given in the proof of Lemma 2.9(ii) and the second assertion of Proposition 2.8 it is readily seen that $\nu ^{-1}(\xi ) \cap K^g = \nu ^{-1}(\xi ) \cap \overline {W_{\tau }}$ for any $\xi \in W_{\tau }$.

Proof of Theorem 2.2 Let us first assume that $K$ is the generalized Kummer variety associated to an abelian surface $A$. Let $g= (\tau, 1)\in G$, with $\tau \neq 0\in A_2$. Consider a component $\overline {V_{\tau, \theta }}$ of $K^g$. The surface $V_{\tau, \theta }$ is clearly contained in $W_{\theta }$ and, hence, the same holds for their strict transforms. Now let $g= (\tau, -1)\in G$. Let $P$ be an isolated fixed point in $K^{g}$. Then we have $P=(\epsilon _1, \epsilon _2, \epsilon _3, \epsilon _4)$ for some pairwise distinct $\epsilon _i \in A_{2,\tau }$ summing up to $0$ in $A$. The last condition implies that we can write $P=(\epsilon _1, \epsilon _2, \epsilon _1+\theta, \epsilon _2+\theta )$ for some $0\neq \theta \in A_{2}$. This is the same as

\[ P = (\epsilon_1, \epsilon_2, -\epsilon_1+\theta+\tau, -\epsilon_2+\theta+\tau), \]

which lies in $\overline {W_{\tau +\theta }}$. By Lemma 2.12 we conclude that for a generalized Kummer variety the union of the fixed loci $K^g$ consists of the $16$ varieties $\overline {W_{\tau }}$ of $\mathrm {K}3^{[2]}$ type. The intersections of these components were computed in Lemma 2.10.

Now let $f\colon \mathcal {K}\to B$ be a smooth and proper family of $\mathrm {Kum}^3$ manifolds over a connected base $B$, with a fibre $\mathcal {K}_0$ isomorphic to the generalized Kummer variety on an abelian surface $A$. By the result of Hassett and Tschinkel [Reference Hassett and TschinkelHT13, Theorem 2.1], up to a finite étale base change, there is a fibrewise action of $G=A_2\times \langle -1 \rangle$ on $\mathcal {K}$. By [Reference FujikiFuj83, Lemma 3.10], this action is locally trivial with respect to $f$, i.e. any $x\in \mathcal {K}$ has a neighborhood $U_x=(f^{-1}f(U_x)\cap \mathcal {K}_{f(x)} )\times f(U_x)$ such that the action on $U_x$ of the stabilizer subgroup of $x$ is induced by that on the fibre $\mathcal {K}_{f(x)}$. Therefore, for any $\tau \in A_{2}$ the fixed locus $\mathcal {K}^g$ of $g=(\tau, -1)$ contains a smooth family $\overline {\mathcal {W}_{\tau }}\to B$ of manifolds of $\mathrm {K}3^{[2]}$ type as the unique component of $\mathcal {K}^g$ with positive-dimensional intersection with the fibres of $f$. We conclude that for any $b\in B$ we have

\[ \bigcup_{1\neq g\in G} (\mathcal{K}_b)^g = \bigcup_{\tau\in A_2} \overline{\mathcal{W}_{\tau}}_{\lvert_{\mathcal{K}_b}}, \]

and that the components $\overline {\mathcal {W}_{\tau }}_{\lvert _{\mathcal {K}_b}}$ intersect as claimed.

Proof. For each $k$, the $k$th power map $[k]\colon \overline {\operatorname {\mathrm {Pic}}}^d(\tilde {\mathcal {C}}/ {\mathbb {P}}^3) \dashrightarrow \overline {\operatorname {\mathrm {Pic}}}^{kd}(\tilde {\mathcal {C}}/ {\mathbb {P}}^3)$ restricts to a rational map $[k]\colon K_J(v_d) \dashrightarrow K_J(v_{kd})$. This can be easily checked at smooth curves $\tilde {\mathcal {C}_b}$ in the family using the description of the Albanese map $L\mapsto \sum c_2(L)$.

Consider the following commutative diagram.

Both the horizontal maps are dominant and generically finite of the same degree $2^6$. Indeed, they preserve the fibres of the respective support morphisms, which, generically, are torsors under abelian threefolds. Hence, the degree of $[2]$ is the number of points of order $2$ on an abelian threefold.

It thus suffices to prove the lemma for $\mathrm {Nm}_\pi \colon K_J(v_{2d}) \dashrightarrow \overline {\operatorname {\mathrm {Pic}}}^{2d}({\mathcal {D}}/ {\mathbb {P}}^3)$. Since $K_J(v_{2d})$ is the closure of the image of $\pi ^*$, the involution $\sigma ^*$ is the identity on $K_J(v_{2d})$. Hence, the rational map

coincides with multiplication by $[2]$; this composition is therefore dominant and generically finite of degree $2^6$. Since $\pi ^*\colon \overline {\operatorname {\mathrm {Pic}}}^{2d}({\mathcal {D}}/ {\mathbb {P}}^3)\dashrightarrow K_J(v_{4d})$ is generically finite of degree $2$, it follows that $\mathrm {Nm}_\pi$ is dominant and it has degree $2^5$.

Proof. We first show that $\operatorname {\mathrm {Pic}}^0(J)_2\times \langle -1\rangle$ acts on $K_J(v_d)$ via automorphisms trivial on the second cohomology and which preserve the Lagrangian fibration given by the support morphism.

Since $-1\colon J\to J$ acts trivially on the cohomology of $J$ in even degrees, the pull-back of sheaves defines an automorphism $(-1)^*$ of $K_J(v_d)$; this automorphism is trivial for $d$ even but not for $d$ odd. By [Reference Mongardi and WandelMW15, Lemma 2.34], the action of $(-1)^*$ is symplectic, i.e. the induced action on the transcendental cohomology $H^2_{\mathrm {tr}}(K_J(v_d), {\mathbb {Z}})$ is the identity. Since all curves in $\lvert 2\Theta \rvert$ are stable under $-1$, the automorphism $(-1)^*$ preserves all fibres of the support fibration. The Picard rank of $K_J(v_d)$ is $2$, because $J$ is a general abelian surface by assumption (see [Reference YoshiokaYos01]). Hence, by Lemma 3.10, the induced action of $(-1)^*$ on $\operatorname {\mathrm {NS}}(K_J(v_d))$ is also the identity. Since $H^2(K_J(v_d), {\mathbb {Z}})$ is torsion free, we conclude that $(-1)^*$ acts trivially on the second cohomology.

Now let $L\in \operatorname {\mathrm {Pic}}^0(J)_2$. Let $B\subset \lvert 2\Theta \rvert$ be the locus of smooth curves, and consider the universal curve $j\colon \tilde {\mathcal {C}}_B\hookrightarrow J\times B$. By [Reference WieneckWie18, Lemma 6.9], the corresponding pull-back $j^*\colon \operatorname {\mathrm {Pic}}^0(J)\times B \to \overline {\operatorname {\mathrm {Pic}}}^0(\tilde {\mathcal {C}}/ {\mathbb {P}}^3)_B$ is an injection of abelian schemes. Note that the covering involution $\sigma$ for the cover $\pi \colon \tilde {\mathcal {C}}_B\to {\mathcal {C}}_B$ is the restriction of $-1\times \mathrm {id}_B$. Hence, the induced involution $\sigma ^*$ on $\overline {\operatorname {\mathrm {Pic}}}^0(\tilde {\mathcal {C}}/ {\mathbb {P}}^3)_B$ is the inverse map on $j^*_b(\operatorname {\mathrm {Pic}}^0(J))$, for each $b\in B$. As $K_J(v_0)$ is the fixed locus of this involution, the $2$-torsion line bundle $L$ on $J$ corresponds uniquely to a $2$-torsion section $s_L$ of $K_J(v_0)_B\to B$. For any $d$, the variety $K_J(v_d)_B$ is a torsor under $K_J(v_0)_B$ and, hence, $F\mapsto F\otimes L$ induces a birational automorphism $g_L$ of $K_J(v_d)$, which restricts to a translation on the smooth fibres of the support morphism. This implies that $g_L$ is symplectic. We deduce from Lemma 3.10 that $g_L$ extends to a regular automorphism of $K_J(v_d)$, whose induced action is trivial on $\operatorname {\mathrm {NS}}(K_J(v_d))$ as well.

According to [Reference KimKim21, Theorem 5.1] we have described all automorphisms in $\operatorname {Aut}_0(K_J(v_d))$ preserving the fibration $\mathrm {supp}\colon K_J(v_d)\to {\mathbb {P}}^3$. We claim now that an automorphism in $\operatorname {Aut}_0(K_J(v_d))$ which does not preserve the fibration cannot fix a component of dimension $4$ on $K_J(v_d)$; this will show that $G=\operatorname {\mathrm {Pic}}^0(J)_2\times \langle -1\rangle$.

Let $D = \mathrm {supp}^* (\mathcal {O}_{ {\mathbb {P}}^n}(1)) \in \operatorname {\mathrm {Pic}}(K_J(v_d))$ be the line bundle inducing the fibration. Since $\operatorname {Aut}_0(K_J(v_d))$ acts trivially on $H^2(K_J(v_d), {\mathbb {Z}})$, it acts on $|D|= {\mathbb {P}}^3=|2\Theta |$. We obtain an action of $\operatorname {Aut}_0(K_J(v_d))/(\operatorname {\mathrm {Pic}}^0(J)_2\times \langle -1\rangle ) \cong ( {\mathbb {Z}}/2 {\mathbb {Z}})^4$ on $|2\Theta |$, which, up to conjugation by an automorphism of $ {\mathbb {P}}^3$, is identified with the action generated by

\begin{align*} (x,y,z,w) \mapsto (z,w,x,y),&\quad (x,y,z,w) \mapsto (y,x,w,z), \\ (x,y,z,w) \mapsto (x,y,-z,-w),&\quad (x,y,z,w) \mapsto (x, -y, z, -w); \end{align*}

see [Reference Gonzalez-DorregoGon94, Lemma 1.52, Note 1.4]. Thus, $h\in \operatorname {Aut}_0(K_J(v_d))$ either acts trivially on $|2\Theta |$ or it fixes a pair of skew lines. Assume by contradiction that $h\in \operatorname {Aut}_0(K_J(v_d))$ fixes a fourfold but does not preserve the support fibration. Then $(K_J(v_d))^h$ contains one of the varieties ${Z_{j}}$ of $\mathrm {K}3^{[2]}$ type from Theorem 2.2, and the image of $Z_j$ via the fibration must be a line $R \subset |2\Theta |$ fixed by $h$. But, by a theorem of Matsushita [Reference MatsushitaMat99], the $\mathrm {K}3^{[2]}$ variety $Z_j$ does not admit any non-constant morphism to $ {\mathbb {P}}^1$.

Proof. The Néron–Severi group $\operatorname {\mathrm {NS}}(X)$ is a rank $2$ lattice of signature $(1,1)$. Any birational automorphism $g$ induces an isometry $g^*$ of $H^2(X, {\mathbb {Z}})$ which restricts to an isometry of $\operatorname {\mathrm {NS}}(X)$. Denote by $D\in \operatorname {\mathrm {NS}}(X)$ the class of the line bundle $f^*(\mathcal {O}_{ {\mathbb {P}}^n}(1))$, and pick $E\in \operatorname {\mathrm {NS}}(X)$ such that $D$ and $E$ generate $\operatorname {\mathrm {NS}}(X)\otimes {\mathbb {Q}}$. Since $f\circ g=f$, we have $g^*(D)=D$. Using that $D$ is isotropic one sees that this forces $g^*(E)=E$ as well, so that $g^*$ is the identity on $\operatorname {\mathrm {NS}}(X)$; by [Reference FujikiFuj81], $g$ extends to an automorphism of $X$.

Proof of Theorem 3.1 Using the identification of $G$ given in Proposition 3.8 we will show that the norm map descends to a birational map $K_J(v_d)/G\dashrightarrow \overline {\operatorname {\mathrm {Pic}}}^d(\mathcal {D}/ {\mathbb {P}}^3)$, for any odd $d$. The varieties $\overline {\operatorname {\mathrm {Pic}}}^d(\mathcal {D}/ {\mathbb {P}}^3)$ are of $\mathrm {K}3^{[3]}$ type, and $\overline {\operatorname {\mathrm {Pic}}}^3(\mathcal {D}/ {\mathbb {P}}^3)$ is birational to $\operatorname {\mathrm {Km}}(J)^{[3]}$, see [Reference BeauvilleBea99, Proposition 1.3].

For a smooth curve $\tilde {\mathcal {C}}_b \in \lvert 2\Theta \rvert$ the norm map is induced by the map of divisors which sends $\sum _i a_i[P_i]$ to $\sum _i a_i [\pi (P_i)]$. It is then clear that $\mathrm {Nm}_\pi ((-1)^*(F))= \mathrm {Nm}_\pi (F)$ for any $F\in K_J(v_d)_B$. Instead let $L\in \operatorname {\mathrm {Pic}}^0(J)_2$, and let $s_L\colon {\mathbb {P}}^3 \dashrightarrow K_J(v_0)$ be the rational section defined by $s_L(b)= j_b^*(L)$, where $j\colon \tilde {\mathcal {C}}_B\hookrightarrow J\times B$ denotes the natural inclusion. By Remark 3.9, the section $s_L$ is contained in the relative Prym variety; in particular, the composition $\mathrm {Nm}_\pi \circ s_L$ gives the zero section of $\overline {\operatorname {\mathrm {Pic}}}^0(\mathcal {D}/ {\mathbb {P}}^3)_B\to {\mathbb {P}}^3$.

Therefore, for any $g\in G$ and $y\in K_J(v_d)_B$ we have $\mathrm {Nm}_\pi (g(y))= \mathrm {Nm}_\pi (y)$, which means that $\mathrm {Nm}_\pi \colon K_J(v_d)\dashrightarrow \overline {\operatorname {\mathrm {Pic}}}^d(\mathcal {D}/ {\mathbb {P}}^3)$ descends to a rational map

\[ \overline{\mathrm{Nm}}_\pi\colon K_J(v_d)/G \dashrightarrow \overline{\operatorname{\mathrm{Pic}}}^d(\mathcal{D}/{\mathbb{P}}^3). \]

This map is, in fact, birational, because $\mathrm {Nm}_\pi \colon K_J(v_d)\dashrightarrow \overline {\operatorname {\mathrm {Pic}}}^d(\mathcal {D}/ {\mathbb {P}}^3)$ is generically finite of degree $2^5$ by Lemma 3.6, and $G$ has also order $2^5$.

Proof. Theorem 2.2 implies immediately that $X^j$ is empty for $j\geqslant 4$. The group $G$ acts freely on $q^{-1}(X^0\setminus X^1)$ and, hence, $X^0\setminus X^1$ is smooth.

If $\mathcal {K}\to B$ is a smooth proper family of $\mathrm {Kum}^3$ manifolds, the quotient $\mathcal {K}/G \to B$ is a locally trivial family (see [Reference FujikiFuj83, Lemma 3.10]): any point $x\in \mathcal {K}/G$ has a neighborhood $U_x$ of the form to $f(U_x)\times (f^{-1}(f(x))\cap U_x)$. Therefore, to prove the proposition we may assume that $K$ is the generalized Kummer sixfold on an abelian surface $A$.

In this case the components $Z_{i}$ are the explicit $\overline {W_{\tau }}$, for $\tau \in A_2$ (see Definition 2.6). Recall that $\overline {W_{\tau }}\subset K$ is the unique positive-dimensional component of the fixed locus of $(\tau, -1)\in G$, and that the induced action of $G/\langle (\tau,-1) \rangle$ on $\overline {W_{\tau }}$ is faithful. For any $z\in \overline {W_{\tau }}$ there is a decomposition $T_z(K)=N_{\overline {W_{\tau }}\vert K, z}\oplus T_z(\overline {W_{\tau }})$, where the first factor is the normal space. The action of $(\tau, -1)$ on $T_z(K)$ is $(-1, 1)$. Now let $x\in X^1 \setminus X^2$, and let $z\in K$ be a preimage of $x$. By definition, $z$ belongs to exactly one component $\overline {W_{\tau }}$. The stabilizer is $G_z=\langle (\tau,-1) \rangle$. By the above, there exists a neighborhood $V_z$ of $z\in K$ such that $g(V_z)\cap V_z$ is empty for any $g \notin G_z$, and $V_z=( {\mathbb {C}}^2)^3$ with $(\tau, -1)_{\vert _{V_z}} =(\iota, \mathrm {id}, \mathrm {id})$. The image of $V_z$ under the quotient map is thus a neighborhood $U_x=( {\mathbb {C}}^2/\iota )\times ( {\mathbb {C}}^2)^2$ of $x$ in $K/G$.

In the other cases we proceed similarly. Let $x\in X^2\setminus X^3$, and let $z\in K$ be one of its preimages. Then $z$ belongs to two distinct components $\overline {W_{\tau _1}}$ and $\overline {W_{\tau _2}}$. The stabilizer $G_z\cong ( {\mathbb {Z}}/2 {\mathbb {Z}})^2$ is the subgroup $\langle (\tau _1,-1), (\tau _2,-1)\rangle$ of $G$. There is a decomposition $T_z(K) = N_{\overline {W_{\tau _1}}\vert K, z} \oplus N_{\overline {W_{\tau _2}}\vert K, z} \oplus T_z(\overline {W_{\tau _1}}\cap \overline {W_{\tau _2}})$, and the action of $G_z$ is generated by $(-1, 1, 1)$ and $(1, -1, 1)$. This implies that the image in $K/G$ of a sufficiently small neighborhood of $z$ in $K$ is isomorphic to $( {\mathbb {C}}^2/\iota )^2\times ( {\mathbb {C}}^2)$.

Finally, let $x\in X^3$ and $z\in K$ a preimage of it. Then $z$ belongs to exactly three components $\overline {W_{\tau _i}}$, for $i=1,2,3$. The stabilizer $G_z$ is the subgroup generated by the involutions $(\tau _i, -1)$, for $i=1,2,3$, and there is a decomposition $T_z(K)=\bigoplus _{i=1}^3 N_{\overline {W_{\tau _i}}\vert K,z}$ of the tangent space. The action of $(\tau _i,-1)$ on $T_{z}(K)$ is $-1$ on the $i$th summand and the identity on the complement. We conclude that there exists a neighborhood $V_z$ of $z$ in $K$ whose image in $K/G$ is isomorphic to $( {\mathbb {C}}^2/\iota )^3$.

Proof. Given noetherian schemes $T_1$ and $T_2$ and closed subschemes $V_1\subsetneq T_1$ and $V_2\subsetneq T_2$, we have

\[ \mathrm{Bl}_{(V_1\times T_2) \cup ( T_1\times V_2)} (T_1\times T_2) = \mathrm{Bl}_{V_1}(T_1) \times \mathrm{Bl}_{V_2}(T_2). \]

It is easy to see this when $T_1$ and $T_2$ are affine schemes, using the definition of blowing-up via the Proj construction [Reference HartshorneHar77, II, § 7]. The singular locus of $( {\mathbb {C}}^2/\iota )^j\times ( {\mathbb {C}}^2)^k$ is the union of $\operatorname {pr}_i^{-1}(\{0\})$ for $i$ from $1$ to $j$, where $\operatorname {pr}_i\colon ( {\mathbb {C}}^2/\iota )^j\times ( {\mathbb {C}}^2)^k \to ( {\mathbb {C}}^2/\iota )$ is the projection onto the $i$th factor. Via the above observation, the statement is reduced to the case $j=1$, $k=0$, which is the well-known minimal resolution of an isolated $A_1$-singularity of a surface.

Proof. The singular space $K/G$ is a primitively symplectic V-manifold in the sense of Fujiki. According to his [Reference FujikiFuj83, Proposition 2.9], to show that $Y_K$ admits a symplectic form it suffices to check that the following two conditions are satisfied:

1. (i) for each component $X_j$ of the singular locus of $K/G$, a general point $x\in X_j$ has a neighborhood $U_x = A \times V_x$ in $K/G$, where $A$ is a surface and $V_x=U_x\cap X_j$ is a smooth neighborhood of $x$ in $X_j$;

2. (ii) the restriction of $p\colon Y_K\to K/G$ to the preimage of $U_x$ is the product

   \[ p'\times \mathrm{id}\colon \tilde{A}\times V_x \to A\times V_x, \]
   where $p'$ is the minimal resolution of $A$.

We have already shown in the course of proof of Propositions 4.2 and 4.4 that these conditions hold, with $A$ a neighborhood of the ordinary node $0\in {\mathbb {C}}^2/\iota$, whose minimal resolution is the blow-up of the singular point.

Hence, $Y_K$ is symplectic. From its description as the quotient $\mathrm {Bl}_Z(K)/G$ we obtain ${h^{2,0}(Y_K)=1}$ (see also § 4.7). Finally, $Y_K$ is simply connected by [Reference FujikiFuj83, Lemma 1.2].

Proof of Theorem 1.1 Let $\mathcal {K}\to B$ be a smooth proper family of manifolds of $\mathrm {Kum}^3$ type. Up to a finite étale base-change, $G$ acts fibrewise on $\mathcal {K}$, and the fixed loci of automorphisms in $G$ are smooth families over $B$. By Theorem 2.2, the union $\mathcal {Z}$ of $\mathcal {K}^g$ for $1\neq g\in G$ has $16$ components $\mathcal {Z}_i$, each of which is a smooth family of manifolds of $\mathrm {K}3^{[2]}$ type over $B$. The blow-up of $\mathcal {K}$ along $\mathcal {Z}$ is a smooth family $\widetilde {\mathcal {K}}$ over $B$, and the action of $G$ extends to a fibrewise action on $\widetilde {\mathcal {K}}$. By Propositions 4.4 and 4.6, the quotient $\mathcal {Y}=\widetilde {\mathcal {K}}/G$ is a smooth proper family of hyper-Kähler manifolds over $B$, with fibre over the point $b\in B$ the manifold $Y_{\mathcal {K}_b}$. It thus suffices to find a single $K$ of $\mathrm {Kum}^3$ type such that $Y_K$ is hyper-Kähler of $\mathrm {K}3^{[3]}$ type.

Consider the sixfold $K_J(v_3)$ introduced in § 3, which is constructed from a Beauville–Mukai system on a principally polarized abelian surface $J$. By Theorem 3.1, in this case $Y_{K_J(v_3)}$ is birational to the Hilbert scheme $\operatorname {\mathrm {Km}}(J)^{[3]}$ on the Kummer K3 surface associated to $J$. By [Reference HuybrechtsHuy99, Theorem 4.6], birational hyper-Kähler manifolds are deformation equivalent, and therefore $Y_{K_J(v_3)}$ is of $\mathrm {K}3^{[3]}$ type.

Proof. Let $c_K$ and $c_Y$ be the Fujiki constants of $K$ and $Y$, respectively [Reference FujikiFuj87]. This means that we have $\int _K x^6 = c_K q_K(x,x)^3$ for any $x\in H^2(K, {\mathbb {Z}})$, and similarly for $Y_K$. Let $x\in H_{\mathrm {tr}}^2(K, {\mathbb {Z}})$; since $x$ is $G$-invariant, we have $r^*(r_* x) = 2^5 x$. We compute

\begin{align*} q_{Y_K}(r_*x, r_*x)^3 & = \frac{1}{c_{Y_K}}\int_{Y_K} (r_*x)^6 \\ &= \frac{1}{2^5 c_{Y_K}} \int_K (r^*r_* x)^6\\ & = \frac{1}{2^5 c_{Y_K}} \int_K (2^5 x)^6 \\ & =\frac{2^{25}c_K }{c_{Y_K}} q_K(x,x)^3. \end{align*}

By [Reference RapagnettaRap08], we have $c_{Y_K}= 15$ and $c_K=60$. Hence, $q_{Y_K}(r_*x, r_*x)=2^{9} q_K(x,x)$, so $r_*/2^4$ multiplies the form by $2$ and yields the claimed rational Hodge isometry.

Proof of Theorem 1.2 Let $K$ be a projective variety of $\mathrm {Kum}^3$ type and let $Y_K$ be the crepant resolution of $K/G$. By the above lemma, the transcendental lattice $H^2_{\mathrm {tr}}(Y_K, {\mathbb {Z}})$ is an even lattice of signature $(2, k)$, and rank at most $6$. By [Reference MorrisonMor84, Corollary 2.10], there exists a $\mathrm {K}3$ surface $S_K$ such that $H^2_{\mathrm {tr}}(S_K, {\mathbb {Z}})$ is Hodge isometric to $H^2_{\mathrm {tr}}(Y_K, {\mathbb {Z}})$. A criterion independently due to Mongardi and Wandel [Reference Mongardi and WandelMW15] and Addington [Reference AddingtonAdd16, Proposition 4] ensures that $Y_K$ is birational to $M_{S_{K},H}(v)$, for some primitive Mukai vector $v$ and a $v$-generic polarization $H$ on $S_K$. The surface $S_K$ is uniquely determined up to isomorphism, because two $\mathrm {K}3$ surfaces of Picard rank at least $12$ with Hodge isometric transcendental lattices are isomorphic, see [Reference HuybrechtsHuy16, Chapter 16, Corollary 3.8]

Proof. The second assertion implies the first thanks to Lemma 4.8. Conversely, let $\Phi \colon H^2_{\mathrm {tr}}(S, {\mathbb {Q}}) \hookrightarrow \Lambda _{\mathrm {Kum}^3}(2)\otimes _{ {\mathbb {Z}}} {\mathbb {Q}}$ be an isometric embedding. Choose a primitive sublattice $T\subset \Lambda _{\mathrm {Kum}^3}$ such that $T(2)\otimes _{ {\mathbb {Z}}} {\mathbb {Q}}$ coincides with the image of $\Phi$. Equip $T$ with the Hodge structure induced by that on $H^2_{\mathrm {tr}}(S, {\mathbb {Q}})$ via $\Phi$. By the surjectivity of the period map [Reference HuybrechtsHuy99], there exists a manifold $K$ of $\mathrm {Kum}^3$ type such that $H^2_{\mathrm {tr}}(K, {\mathbb {Z}})$ is Hodge isometric to $T$, and Lemma 4.8 gives a Hodge isometry $H^2_{\mathrm {tr}}(S, {\mathbb {Q}})\xrightarrow {\ \sim \ } H^2_{\mathrm {tr}}(S_K, {\mathbb {Q}})$. Since the signature of $T$ is necessarily $(2, k)$ with $k\leqslant 4$, Huybrechts’ projectivity criterion [Reference HuybrechtsHuy99] implies that $K$ is projective.

Proof. By Remark 5.3 there exists an isogeny $\phi \colon \mathrm {KS}(X)\to \mathrm {KS}(Z)$ of Kuga–Satake varieties. It induces an isomorphism

\[ \phi_*\colon H^2(\mathrm{KS}(X)\times \mathrm{KS}(X)) \xrightarrow{\ \sim\ } H^2(\mathrm{KS}(Z)\times \mathrm{KS}(Z)) \]

of rational Hodge structures. If Conjecture 5.4 holds for $X$, there exists an algebraic cycle $\zeta$ on $X\times \mathrm {KS}(X) \times \mathrm {KS}(X)$ giving an embedding of Hodge structures

\[ \zeta_*\colon H^2_{\mathrm{tr}}(X) \hookrightarrow H^2(\mathrm{KS}(X)\times \mathrm{KS}(X)). \]

It follows that the embedding of Hodge structure given by the composition

\[ \phi_* \circ \zeta_* \circ \gamma_* \colon H^2_{\mathrm{tr}}(Z) \hookrightarrow H^2(\mathrm{KS}(Z)\times \mathrm{KS}(Z)) \]

is induced by an algebraic cycle on $Z\times \mathrm {KS}(Z)\times \mathrm {KS}(Z)$.

Proof. By Lemma 4.10, there exists a projective variety $K$ of $\mathrm {Kum}^3$ type with associated K3 surface $S_K$ and a rational Hodge isometry $t_0\colon H^2_{\mathrm {tr}}(S)\xrightarrow {\ \sim \ } H^2_{\mathrm {tr}}(S_K)$. Denote by $Y_K$ the crepant resolution of $K/G$ given by Theorem 1.1 and let $v, H$ be given by Theorem 1.2, so that $Y_K$ is birational to the moduli space $M_{S_K,H}(v)$. Denote by $r\colon K\dashrightarrow Y_K$ the natural rational map of degree $2^5$.

By [Reference BuskinBus19, Reference HuybrechtsHuy19], the isometry $t_0$ is algebraic. Next, by [Reference MukaiMuk87], there exists a quasi-tautological sheaf $U$ over $S_K \times M_{S_K,H}(v)$, which means that there exists an integer $\rho$ such that for any $F\in M_{S_K,H}(v)$ the restriction of $U$ to $S_K\times \{F\}$ is $F^{\oplus \rho }$. Consider the algebraic class

\[ \gamma:= \frac{1}{\rho}\mathrm{ch}(U)\cdot \mathrm{pr}^* \sqrt{\mathrm{td}_{S_K}}\in H^{\bullet}(S_K\times M_{S_K, H}(v)), \]

where $\mathrm {pr}\colon S_K\times M_{S_K, H}(v) \to S_K$ is the projection. By [Reference O'GradyO'Gr97], its Künneth component $\gamma _3\in H^6 (S_K\times M_{S_K, H}(v))$ induces a Hodge isometry

\[ t_1\colon H^2_{\mathrm{tr}}(S_K) \xrightarrow{\ \sim\ } H^2_{\mathrm{tr}}(M_{S_K, H}(v)). \]

By [Reference Charles and MarkmanCM13], the standard conjectures holds for $M_{S_K, H}(v)$ (this also follows from [Reference BüllesBül20] via the arguments of [Reference ArapuraAra06]). Hence, all Künneth components of $\gamma$ are algebraic. Now let $f\colon M_{S_K, H}(v)\dashrightarrow Y_K$ be a birational map. Then the push-forward $f_*\colon H^2_{\mathrm {tr}}(M_{S_K, H}(v))\xrightarrow {\ \sim \ } H^2_{\mathrm {tr}}(Y_K)$ is a Hodge isometry. By Lemma 4.8, a multiple of the composition $r^* \circ f_*$ gives a Hodge isometry $t_2\colon H^2_{\mathrm {tr}}(M_{S_K, H}(v)) \xrightarrow {\ \sim \ } H^2_{\mathrm {tr}}(K)(2)$.

It follows that the Hodge isometry

\[ t_2\circ t_1\circ t_0 \colon H^2_{\mathrm{tr}}(S)\xrightarrow{\ \sim\ } H^2_{\mathrm{tr}}(K)(2) \]

is induced by an algebraic cycle on $S\times K$. The Kuga–Satake Hodge conjecture holds for $K$ by [Reference VoisinVoi22]. By Lemma 5.6, Conjecture 5.4 holds for the K3 surface $S$ as well.

Proof. By Remark 4.9, there exists a $4$-dimensional family $\mathcal {S}\to B$ of projective K3 surfaces of general Picard rank $16$ such that: for each $b\in B$ the fibre $\mathcal {S}_b$ is of the form $S_{K_b}$ for some $K_b$ of $\mathrm {Kum}^3$ type and for some $0\in B$ there exists a rational Hodge isometry $H^2_{\mathrm {tr}}(S) \xrightarrow {\ \sim \ } H^2_{\mathrm {tr}}(\mathcal {S}_0)$. By [Reference HuybrechtsHuy19], it is sufficient to prove the Hodge conjecture for all powers of $\mathcal {S}_0$.

To this end, it will be enough to check that $\mathcal {S}\to B$ satisfies the two assumptions of [Reference VarescoVar22, Theorem 0.2]. As explained in [Reference VarescoVar22, Theorem 4.1], the first of these assumptions is that the Kuga–Satake variety of a general K3 surface in the family is isogenous to $A^4$ for an abelian fourfold $A$ of Weil type with trivial discriminant. This holds by O'Grady's theorem in [Reference O'GradyO'Gr21] as already mentioned. The other assumption is that the Kuga–Satake Hodge conjecture holds for the surfaces $\mathcal {S}_b$ for all $b\in B$, which is the content of Theorem 5.7.