2 Preliminaries

In this section, we mainly fix notations we shall use. We follow [Reference BeauvilleBe83] and [Reference BeauvilleBe83-2]. So, $K_{n}(A)$ in [Reference Boissière, Nieper-Wisskirchen and SartiBNS11] is $K_{n-1}(A)$ in this note.

We refer to [Reference BeauvilleBe83, Section 7] and [Reference Gross, Huybrechts and JoyceGHJ03, Part III] for more details on generalized Kummer manifolds and basic properties on hyper-Kähler manifolds.

Let $A$ be a $2$-dimensional complex torus and let $n$ be an integer such that $n\geqslant 3$. Let $\text{Hilb}^{n}(A)$ be the Hilbert scheme of $0$-dimensional closed subschemes of $A$ of length $n$. Then $\text{Hilb}^{n}(A)$ is a smooth Kähler manifold of dimension $2n$. Let

$$\begin{eqnarray}\unicode[STIX]{x1D708}=\unicode[STIX]{x1D708}_{A}:\text{Hilb}^{n}(A)\rightarrow \text{Sym}^{n}(A)=A^{n}/S_{n}\end{eqnarray}$$

be the Hilbert–Chow morphism. We denote the sum as $0$-cycles by $\oplus$ and the sum in $A$ by $+$. Then each element of $\text{Sym}^{n}(A)$ is of the form

$$\begin{eqnarray}\bigoplus _{i=1}^{k}x_{i}^{\oplus m_{i}}.\end{eqnarray}$$

Here, $x_{i}$ are distinct points on $A$ and $m_{i}$ are positive integers such that $\sum _{i=1}^{k}m_{i}=n$. We have the following surjective morphism

$$\begin{eqnarray}s:=s_{A}:\text{Sym}^{n}(A)\rightarrow A;\qquad \bigoplus _{i=1}^{k}x_{i}^{\oplus m_{i}}\mapsto \mathop{\sum }_{i=1}^{k}m_{i}x_{i}.\end{eqnarray}$$

The generalized Kummer manifold $K_{n-1}(A)$ is defined by

$$\begin{eqnarray}K_{n-1}(A):=(s\circ \unicode[STIX]{x1D708})^{-1}(0).\end{eqnarray}$$

Note that the morphism

$$\begin{eqnarray}s\circ \unicode[STIX]{x1D708}=s_{A}\circ \unicode[STIX]{x1D708}_{A}:\text{Hilb}^{n}(A)\rightarrow A\end{eqnarray}$$

is a smooth surjective morphism such that all fibers are isomorphic [Reference BeauvilleBe83, Section 7]. So, $K_{n-1}(A)$ is isomorphic to any fiber of $s_{A}\circ \unicode[STIX]{x1D708}_{A}$. One can also describe $K_{n-1}(A)$ in a slightly different way, as follows. Let

$$\begin{eqnarray}A(n-1):=\left\{(P_{1},P_{2},\ldots ,P_{n})\in A^{n}\,\bigg|\,\mathop{\sum }_{i=1}^{n}P_{i}=0\right\}.\end{eqnarray}$$

Then $A(n-1)$ is a closed submanifold of $A^{n}$ and $A(n-1)\simeq A^{n-1}$. Moreover, $A(n-1)$ is stable under the action of $S_{n}$ on $A^{n}$ and

$$\begin{eqnarray}\text{Sym}^{n}(A)\supset A^{(n-1)}:=s^{-1}(0)=A(n-1)/S_{n}.\end{eqnarray}$$

From this, we deduce that

$$\begin{eqnarray}K_{n-1}(A)=\unicode[STIX]{x1D708}^{-1}(A^{(n-1)})=\text{Hilb}^{n}(A)\times _{\text{Sym}^{n}(A)}A^{(n-1)}.\end{eqnarray}$$

Recall that $\dim \text{Def}(A)=4$, while $\dim \text{Def}(K_{n-1}(A))=5$ for $n\geqslant 3$ and any local deformation of a hyper-Kähler manifold is a hyper-Kähler manifold [Reference BeauvilleBe83, Section 7]. So, there are hyper-Kähler manifolds which are deformation equivalent to $K_{n-1}(A)$ but are not isomorphic to any generalized Kummer manifold.

From now until the end of this note, we denote by $X:=K_{n-1}(A)$ ($n\geqslant 3$) the generalized Kummer manifold, of dimension $2(n-1)$, associated to a $2$-dimensional complex torus $A$ and by $K:=T(n){\vartriangleleft}\langle \unicode[STIX]{x1D704}\rangle$ the subgroup of $\text{Aut}(K_{n-1}(A))$ defined in the Introduction. We also use the notations introduced in this section freely in the remaining sections.

3 Proof of Theorem 1.3 for $K_{n-1}(A)$

In this section, we prove Theorem 1.3 for $K_{n-1}(A)$.

First we prove the following.

Proof. Let $(z_{i}^{1},z_{i}^{2})$ ($1\leqslant i\leqslant n$) be the standard global coordinates of the universal cover $\mathbb{C}_{i}^{2}$ of the $i$th factor $A_{i}=A$ of $A^{n}$. Then the universal cover $\mathbb{C}^{2(n-1)}$ of $A(n-1)\simeq A^{n-1}$ is a closed submanifold of $\mathbb{C}^{2n}$ defined by

(3.1)$$\begin{eqnarray}z_{1}^{1}+z_{2}^{1}+\cdots +z_{n-1}^{1}+z_{n}^{1}=0,\qquad z_{1}^{2}+z_{2}^{2}+\cdots +z_{n-1}^{2}+z_{n}^{2}=0.\end{eqnarray}$$

In particular, $(z_{i}^{1},z_{i}^{2})$ ($1\leqslant i\leqslant n-1$) give the global coordinates of the universal cover $\mathbb{C}^{2(n-1)}$ of $A(n-1)$. Note that $1$-forms $dz_{i}^{1}$ and $dz_{i}^{2}$ ($1\leqslant i\leqslant n$) can be regarded as global $1$-forms on $A(n-1)$. They satisfy

(3.2)$$\begin{eqnarray}dz_{1}^{1}+dz_{2}^{1}+\cdots +dz_{n-1}^{1}+dz_{n}^{1}=0,\qquad dz_{1}^{2}+dz_{2}^{2}+\cdots +dz_{n-1}^{2}+dz_{n}^{2}=0,\end{eqnarray}$$

and $\{dz_{i}^{1},dz_{i}^{2}(1\leqslant i\leqslant n-1)\}$ forms a basis of the space of global holomorphic $1$-forms on $A(n-1)\simeq A^{n-1}$. Consider the following global $(2,1)$-form $\tilde{\unicode[STIX]{x1D70F}}$ on $A(n-1)$:

$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D70F}}=dz_{1}^{1}\wedge dz_{1}^{2}\wedge d\overline{z}_{1}^{2}+\cdots +dz_{n-1}^{1}\wedge dz_{n-1}^{2}\wedge d\overline{z}_{n-1}^{2}+dz_{n}^{1}\wedge dz_{n}^{2}\wedge d\overline{z}_{n}^{2}.\end{eqnarray}$$

Lemma 3.2. $\tilde{\unicode[STIX]{x1D70F}}$ descends to a nonzero element $\unicode[STIX]{x1D70F}$ of $H^{2,1}(X)$.

Proof. Recall that, for compact Kähler orbifolds, the Hodge decomposition is pure and the Hodge theory works in the same way as smooth compact manifolds [Reference SteenbrinkSt77].

Since $\tilde{\unicode[STIX]{x1D70F}}$ is $S_{n}$-invariant, it descends to a global $(2,1)$-form, say $\overline{\unicode[STIX]{x1D70F}}$, on the compact Kähler orbifold $A^{(n-1)}$. Then $\unicode[STIX]{x1D70F}=(\unicode[STIX]{x1D708}|_{X})^{\ast }\overline{\unicode[STIX]{x1D70F}}\in H^{2,1}(X)$ under $\unicode[STIX]{x1D708}|_{X}:X\rightarrow A^{(n-1)}$. It remains to show that $\unicode[STIX]{x1D70F}\not =0$. Since $(\unicode[STIX]{x1D708}|_{X})^{\ast }$ is injective, it suffices to show that $\overline{\unicode[STIX]{x1D70F}}\not =0$ in $H^{2,1}(A^{(n-1)})$. For this, it suffices to show that $\tilde{\unicode[STIX]{x1D70F}}\not =0$ in $H^{2,1}(A(n-1))$, as $q^{\ast }$ is also injective for the quotient map $q:A(n-1)\rightarrow A^{(n-1)}$. By Equation (3.2), we have

$$\begin{eqnarray}\displaystyle \tilde{\unicode[STIX]{x1D70F}} & = & \displaystyle dz_{1}^{1}\wedge dz_{1}^{2}\wedge d\overline{z}_{1}^{2}+\cdots +dz_{n-1}^{1}\wedge dz_{n-1}^{2}\wedge d\overline{z}_{n-1}^{2}\nonumber\\ \displaystyle & & \displaystyle -\,\left(\mathop{\sum }_{k=1}^{n-1}dz_{k}^{1}\right)\wedge \left(\mathop{\sum }_{k=1}^{n-1}dz_{k}^{2}\right)\wedge \left(\mathop{\sum }_{k=1}^{n-1}d\overline{z}_{k}^{2}\right).\nonumber\end{eqnarray}$$

This is the expression of $\tilde{\unicode[STIX]{x1D70F}}$ in terms of the standard basis of $H^{2,1}(A(n-1))$. As $n-1\geqslant 2$, the term

$$\begin{eqnarray}dz_{1}^{1}\wedge dz_{2}^{2}\wedge d\overline{z}_{2}^{2}\end{eqnarray}$$

appears with coefficient $-1$ in this expression. Hence $\tilde{\unicode[STIX]{x1D70F}}\not =0$ in $H^{2,1}(A(n-1))$. This proves Lemma 3.2.◻

Lemma 3.3. Let $g\in K\setminus T(n)$ and $\unicode[STIX]{x1D70F}\in H^{2,1}(X)$ be as in Lemma 3.2. Then $g^{\ast }\unicode[STIX]{x1D70F}=-\unicode[STIX]{x1D70F}$. In particular, $g^{\ast }|_{H^{3}(X,\mathbb{C})}\not =\text{id}$.

Proof. The automorphism $g$ acts equivariantly on $A(n-1)\rightarrow \,\,A^{(n-1)}\leftarrow X$. For $\unicode[STIX]{x1D704}$, hence for $g\in K\setminus T(n)$, we have

$$\begin{eqnarray}\unicode[STIX]{x1D704}^{\ast }dz_{i}^{q}=-dz_{i}^{q},\qquad g^{\ast }dz_{i}^{q}=-dz_{i}^{q}\;(1\leqslant i\leqslant n,q=1,2).\end{eqnarray}$$

Hence $g^{\ast }\tilde{\unicode[STIX]{x1D70F}}=-\tilde{\unicode[STIX]{x1D70F}}$ by the shape of $\tilde{\unicode[STIX]{x1D70F}}$. Thus $g^{\ast }\unicode[STIX]{x1D70F}=-\unicode[STIX]{x1D70F}$. By Lemma 3.2, $\unicode[STIX]{x1D70F}\not =0$ in $H^{2,1}(X)$. Hence $g^{\ast }|_{H^{3}(X,\mathbb{C})}\not =\text{id}$ as claimed.◻

Lemma 3.3 completes the proof of Proposition 3.1. ◻

Next we prove the following.

Proof. Let $a\in T(n)\simeq (\mathbb{Z}/n\mathbb{Z})^{\oplus 4}$ be an element of order $p\not =1$ ($p$ is not necessarily a prime number). Set $d=n/p$. Then $d$ is a positive integer such that $d<n$. We freely regard $a$ also as a torsion element of order $p$ in $A$ and automorphisms of various spaces which are naturally and equivariantly induced by the translation automorphism $x\mapsto x+a$ of $A$.

We will show first Lemma 3.5, Theorem 3.6 and Lemma 3.7, and then we will conclude the proof of Proposition 3.4.

Lemma 3.5. The fixed locus $X^{a}$ consists of $p^{3}$ connected components $F_{i}\;(1\leqslant i\leqslant p^{3})$. Moreover, each $F_{i}$ is isomorphic to the generalized Kummer manifold $K_{d-1}(A/\langle a\rangle )$ associated to the $2$-dimensional complex torus $A/\langle a\rangle$.

Proof. Let ${\mathcal{S}}\subset A$ be a $0$-dimensional closed subscheme of length $n$. As $\langle a\rangle$ acts freely on $A$, the quotient map $\unicode[STIX]{x1D70B}:A\rightarrow A/\langle a\rangle$ is étale of degree $p$. It follows that $a_{\ast }{\mathcal{S}}={\mathcal{S}}$ if and only if there is a $0$-dimensional closed subscheme ${\mathcal{T}}\subset A/\langle a\rangle$ of length $d=n/p$ such that ${\mathcal{S}}=\unicode[STIX]{x1D70B}^{\ast }{\mathcal{T}}$. This ${\mathcal{T}}$ is clearly unique and we obtain an isomorphism

(3.3)$$\begin{eqnarray}\text{Hilb}^{d}(A/\langle a\rangle )\simeq (\text{Hilb}^{n}(A))^{a};\qquad {\mathcal{T}}\mapsto \unicode[STIX]{x1D70B}^{\ast }{\mathcal{T}}.\end{eqnarray}$$

Let ${\mathcal{S}}\in (\text{Hilb}^{n}(A))^{a}$. Then $\unicode[STIX]{x1D708}({\mathcal{S}})\in (\text{Sym}^{n}(A))^{a}$ as well, and $\unicode[STIX]{x1D708}({\mathcal{S}})$ is then of the form

$$\begin{eqnarray}\unicode[STIX]{x1D708}({\mathcal{S}})=\bigoplus _{i=1}^{k}\hspace{0.0pt}\bigoplus _{j=0}^{p-1}(x_{i}+ja)^{\oplus m_{i}}\end{eqnarray}$$

(and vice versa). Here $\sum _{i=1}^{k}m_{i}=d$ and all points $x_{i}+ja$ are distinct. Observe also that

$$\begin{eqnarray}K_{n-1}(A)^{a}=(\text{Hilb}^{n}(A))^{a}\cap K_{n-1}(A).\end{eqnarray}$$

As ${\mathcal{S}}\in (\text{Hilb}^{n}(A))^{a}$ by our choice of ${\mathcal{S}}$, it follows from the equality above that ${\mathcal{S}}\in K_{n-1}(A)^{a}$ if and only if ${\mathcal{S}}\in K_{n-1}(A)$, that is, ${\mathcal{S}}\in (\text{Hilb}^{n}(A))^{a}$ satisfies (by the definition of $K_{n-1}(A)$ and by the shape of $\unicode[STIX]{x1D708}({\mathcal{S}})$) that

(3.4)$$\begin{eqnarray}p(m_{1}x_{1}+m_{2}x_{2}+\cdots +m_{k}x_{k}+\unicode[STIX]{x1D6FC})=0\end{eqnarray}$$

in $A$. Here $n(p-1)/2\in \mathbb{Z}$ and $\unicode[STIX]{x1D6FC}\in A$ is an element such that

$$\begin{eqnarray}p\unicode[STIX]{x1D6FC}=(n(p-1)/2)a\end{eqnarray}$$

in $A$. We choose and fix such $\unicode[STIX]{x1D6FC}$.

Let $A[p]$ be the group of $p$-torsion points of $A$. Then, Equation (3.4) is equivalent to

(3.5)$$\begin{eqnarray}m_{1}x_{1}+m_{2}x_{2}+\cdots +m_{k}x_{k}+\unicode[STIX]{x1D6FC}\in A[p].\end{eqnarray}$$

Since $a$ is also a $p$-torsion point, Equation (3.5) is also equivalent to

(3.6)$$\begin{eqnarray}m_{1}\unicode[STIX]{x1D70B}(x_{1})+m_{2}\unicode[STIX]{x1D70B}(x_{2})+\cdots +m_{k}\unicode[STIX]{x1D70B}(x_{k})+\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6FC})\in \unicode[STIX]{x1D70B}(A[p])=A[p]/\langle a\rangle .\end{eqnarray}$$

Write ${\mathcal{S}}=\unicode[STIX]{x1D70B}^{\ast }{\mathcal{T}}$. Then, Equation (3.6) holds if and only if ${\mathcal{T}}$ is in the fibers of

$$\begin{eqnarray}s_{A/\langle a\rangle }\circ \unicode[STIX]{x1D708}_{A/\langle a\rangle }:\text{Hilb}^{d}(A/\langle a\rangle )\rightarrow A/\langle a\rangle\end{eqnarray}$$

over $\unicode[STIX]{x1D70B}(A[p])$. We have $|\unicode[STIX]{x1D70B}(A[p])|=p^{3}$, as $a$ is also $p$-torsion. Hence, by the isomorphism (3.3), the fixed locus $K_{n-1}(A)^{a}$ is isomorphic to the union of $p^{3}$ fibers of $s_{A/\langle a\rangle }\circ \unicode[STIX]{x1D708}_{A/\langle a\rangle }$ and each fiber is isomorphic to $K_{d-1}(A/\langle a\rangle )$ as remarked in Section 2. This completes the proof of Lemma 3.5.◻

Set

$$\begin{eqnarray}\unicode[STIX]{x1D70E}(n)=\mathop{\sum }_{1\leqslant b|n}b,\end{eqnarray}$$

the sum of all positive divisors of a positive integer $n$. The following fundamental result due to Göttche and Soergel ([Reference Göttsche and SoergelGS93, Corollary 1], see also [Reference GöttscheGo94, Reference DebarreDe10]) is crucial in our proof.

Theorem 3.6. The topological Euler number $\unicode[STIX]{x1D712}_{\text{top}}(K_{n-1}(A))$ of $K_{n-1}(A)$ is $n^{3}\unicode[STIX]{x1D70E}(n)$. (This is also valid for $n=1,2$.)

Now we consider the Lefschetz number of $h\in \text{Aut}(X)$:

$$\begin{eqnarray}L(h):=\mathop{\sum }_{k=0}^{4(n-1)}(-1)^{k}\,\text{tr}\,h^{\ast }|_{H^{k}(X,\mathbb{C})}.\end{eqnarray}$$

Lemma 3.7.

1. (1) If $h\in \text{Aut}(X)$ is cohomologically trivial, then $L(h)=n^{3}\unicode[STIX]{x1D70E}(n)$.

2. (2) $L(a)=n^{3}\unicode[STIX]{x1D70E}(d)$ for any element $a$ of order $p$ in $T(n)\setminus \{\text{id}\}$ with $d=n/p$.

Proof. If $h$ is cohomologically trivial, then $\text{tr}\,h^{\ast }|_{H^{k}(X,\mathbb{C})}=b_{k}(X)$. This implies (1). By the topological Lefschetz fixed point formula, Lemma 3.5 and Theorem 3.6, we obtain

$$\begin{eqnarray}L(a)=\unicode[STIX]{x1D712}_{\text{top}}(X^{a})=p^{3}\unicode[STIX]{x1D712}_{\text{top}}(K_{d-1}(A/\langle a\rangle ))=p^{3}\cdot d^{3}\unicode[STIX]{x1D70E}(d)=n^{3}\unicode[STIX]{x1D70E}(d).\end{eqnarray}$$

This is nothing but the assertion (2). This proves Lemma 3.7. ◻

Since $d|n$ and $d\not =n$, it follows that

$$\begin{eqnarray}\unicode[STIX]{x1D70E}(d)\leqslant \unicode[STIX]{x1D70E}(n)-n<\unicode[STIX]{x1D70E}(n).\end{eqnarray}$$

Hence $a\in T(n)\setminus \{\text{id}\}$ is not cohomologically trivial by Lemma 3.7. This proves Proposition 3.4.◻

Theorem 1.3 for $K_{n-1}(A)$ now follows from Theorem 1.2, Propositions 3.1 and 3.4. This completes the proof of Theorem 1.3 for $K_{n-1}(A)$.

4 Proof of Theorem 1.3

In this section, we shall prove Theorem 1.3 for any $Y$.

Let $\unicode[STIX]{x1D6EC}=(\unicode[STIX]{x1D6EC},(\ast ,\ast \ast ))$ be a fixed abstract lattice isometric to $(H^{2}(K_{n-1}(A),\mathbb{Z}),b)$. Here $b$ is the Beauville–Bogomolov form of $K_{n-1}(A)$ (see e.g. [Reference Gross, Huybrechts and JoyceGHJ03, Example 23.20]).

Let $Y$ be a hyper-Kähler manifold deformation equivalent to a generalized Kummer manifold $X=K_{n-1}(A)$. Let $g\in \text{Aut}(Y)$ such that $g^{\ast }|_{H^{\ast }(Y,\mathbb{Z})}=\text{id}$. We are going to show that $g=\text{id}_{Y}$.

Let ${\mathcal{M}}^{0}$ be the connected component of the marked moduli space of ${\mathcal{M}}_{\unicode[STIX]{x1D6EC}}$, containing $(Y,\unicode[STIX]{x1D702})$. Here $\unicode[STIX]{x1D702}:H^{2}(Y,\mathbb{Z})\rightarrow \unicode[STIX]{x1D6EC}$ is a marking. Huybrechts constructed the marked moduli space ${\mathcal{M}}_{\unicode[STIX]{x1D6EC}}$ [Reference HuybrechtsHu99, 1.18] by patching Kuranishi spaces via local Torelli theorem for hyper-Kähler manifolds ([Reference BeauvilleBe83, Theorem 5], [Reference Gross, Huybrechts and JoyceGHJ03, 25.2]). By construction, ${\mathcal{M}}_{\unicode[STIX]{x1D6EC}}$ is smooth, but highly non-Hausdorff. He also showed that the period map

$$\begin{eqnarray}p:{\mathcal{M}}^{0}\rightarrow {\mathcal{D}}=\{[\unicode[STIX]{x1D714}]\in \mathbb{P}(\unicode[STIX]{x1D6EC}\otimes \mathbb{C})|(\unicode[STIX]{x1D714},\unicode[STIX]{x1D714})=0,(\unicode[STIX]{x1D714},\overline{\unicode[STIX]{x1D714}})>0\}\end{eqnarray}$$

is a surjective holomorphic map of degree $1$ ([Reference HuybrechtsHu99, Theorem 8.1], see also [Reference VerbitskyVe13, Reference HuybrechtsHu12] for degree and further development). Let $[\unicode[STIX]{x1D714}]\in {\mathcal{D}}$. If $p^{-1}([\unicode[STIX]{x1D714}])$ ($\subset {\mathcal{M}}^{0}$) is not a single point, then $p^{-1}([\unicode[STIX]{x1D714}])$ consists of points, being mutually inseparable, corresponding to birational hyper-Kähler manifolds [Reference HuybrechtsHu99, Theorem 8.1].

By using the Hodge theoretic Torelli type theorem [Reference MarkmanMa11], Markman and Mehrotra [Reference Markman and MehrotraMM17, Theorem 4.1] proved that the marked generalized Kummer manifolds are dense in ${\mathcal{M}}^{0}$. Actually they proved the following stronger density result:

Consider the Kuranishi family $u:{\mathcal{U}}\rightarrow {\mathcal{K}}$ of $Y$. Here and hereafter we freely shrink ${\mathcal{K}}$ around $0=[Y]$. Since the Kuranishi family is universal, $g\in \text{Aut}(Y)$ induces automorphisms $\tilde{g}\in \text{Aut}({\mathcal{U}})$ and $\overline{g}\in \text{Aut}({\mathcal{K}})$ such that $u\circ \tilde{g}=\overline{g}\circ u$ and $\tilde{g}|Y=g$. Since ${\mathcal{K}}$ is locally isomorphic to ${\mathcal{D}}$ by the local Torelli theorem, the locus

$$\begin{eqnarray}{\mathcal{K}}^{\prime }\subset {\mathcal{K}},\end{eqnarray}$$

consisting of the point $t$ such that $u^{-1}(t)$ is a generalized Kummer manifold, is dense in ${\mathcal{K}}$. This is a direct consequence of Theorem 4.1 and the construction of ${\mathcal{M}}^{0}$ explained above. Here we also emphasize that the density in ${\mathcal{M}}^{0}$is not sufficient to conclude this.

From now, we follow Beauville’s argument [Reference BeauvilleBe83-2, proof of Proposition 10].

Let $T_{Y}$ be the tangent bundle of $Y$. Then, one can take ${\mathcal{K}}$ as a small polydisk in $H^{1}(Y,T_{Y})$ with center $0$. As $\unicode[STIX]{x1D714}_{Y}$ is everywhere nondegenerate, we have an isomorphism

$$\begin{eqnarray}H^{1}(Y,T_{Y})\simeq H^{1}(X,\unicode[STIX]{x1D6FA}_{Y}^{1})\end{eqnarray}$$

induced by the isomorphism

$$\begin{eqnarray}T_{Y}\simeq \unicode[STIX]{x1D6FA}_{Y}^{1}=T_{Y}^{\ast }:v\mapsto \unicode[STIX]{x1D714}_{Y}(v,\ast ).\end{eqnarray}$$

As $g$ is cohomologically trivial and $H^{2,0}(Y)=H^{0}(Y,\unicode[STIX]{x1D6FA}_{Y}^{2})=\mathbb{C}\unicode[STIX]{x1D714}_{Y}$, we have $g^{\ast }\unicode[STIX]{x1D714}_{Y}=\unicode[STIX]{x1D714}_{Y}$ and $g^{\ast }|_{H^{1}(Y,\unicode[STIX]{x1D6FA}_{Y}^{1})}=\text{id}$. Hence by the isomorphism above, we obtain $g^{\ast }|_{H^{1}(Y,T_{Y})}=\text{id}$, and therefore, $\overline{g}=\text{id}_{{\mathcal{K}}}$.

Let $t\in {\mathcal{K}}$ be any point of ${\mathcal{K}}$. Then, by $\overline{g}=\text{id}_{{\mathcal{K}}}$, the morphism $\tilde{g}$ preserves the fiber $Y_{t}=u^{-1}(t)$, that is,

$$\begin{eqnarray}\tilde{g}|_{Y_{t}}\in \text{Aut}(Y_{t}).\end{eqnarray}$$

Put $g_{t}:=\tilde{g}|_{Y_{t}}$. Then $g_{t}$ is also cohomologically trivial, because $g_{t}^{\ast }|_{H^{\ast }(Y_{t},\mathbb{Z})}$ is derived from the action of $\tilde{g}$ on the constant system $\bigoplus _{k=0}^{4(n-1)}R^{k}u_{\ast }\mathbb{Z}$. Then $g_{t}=\text{id}_{Y_{t}}$ for all $t\in {\mathcal{K}}^{\prime }$, as we already proved Theorem 1.3 for $K_{n-1}(A)$ in Section 3. Since ${\mathcal{U}}$ is Hausdorff and $\tilde{g}$ is continuous, it follows that $\tilde{g}=\text{id}_{{\mathcal{U}}}$. Hence $g=g_{0}=\text{id}_{Y}$ as well. This completes the proof of Theorem 1.3.

5 A few concluding remarks

In this section, we remark a few relevant facts, which should be known to some experts.

Our first remark is about an analogue of Theorem 1.3 for a hyper-Kähler manifold deformation equivalent to the Hilbert scheme $\text{Hilb}^{n}(S)$ of a K3 surface $S$.

Markman and Mehrotra [Reference Markman and MehrotraMM17, Theorem 1.1] also proved the strong density result for $\text{Hilb}^{n}(S)$ of K3 surfaces $S$. So, the same argument as in Section 4 together with Beauville’s result (Theorem 1.1(1)) implies the following result due to Mongardi [Reference MongardiMo13, Lemma 1.2]:

Our second remark is about the fixed locus of symplectic automorphism of finite order.

In Lemma 3.5, we described the fixed locus $X^{a}$. Our description shows that $X^{a}$ is a disjoint union of smooth hyper-Kähler manifolds. However, this is not accidental:

Proof. $F$ is isomorphic to the intersection of the graph of $h$ and the diagonal $\unicode[STIX]{x1D6E5}$ in $M\times M$. So it is compact and Kähler, possibly singular. Let $P\in F$. Since $h$ is of finite order, $h$ is locally linearizable at $P$ (see the proof of [Reference KatsuraKa84, Lemma 1.3]). That is, there are local coordinates $(y_{1},y_{2},\ldots ,y_{2d})$ at $P$ such that

(5.1)$$\begin{eqnarray}h^{\ast }y_{i}=y_{i}\;(1\leqslant \forall i\leqslant r),\qquad h^{\ast }y_{j}=c_{j}y_{j}\;(r+1\leqslant \forall j\leqslant 2d).\end{eqnarray}$$

Here $c_{j}\not =1$ and satisfies $c_{j}^{m}=1$. Then $F$ is locally defined by $y_{j}=0$ ($r+1\leqslant j\leqslant 2d$) in $M$. Hence $F$ is smooth. Consider the linear differential map

$$\begin{eqnarray}dh_{P}:T_{M,P}\rightarrow T_{M,P}\end{eqnarray}$$

of the tangent space $T_{M,P}$ of $M$ at $P$. By Equation (5.1), we have the decomposition

(5.2)$$\begin{eqnarray}T_{M,P}=T_{F,P}\oplus N.\end{eqnarray}$$

Here, $N=\bigoplus _{j=r+1}^{2d}\mathbb{C}v_{j}$ for some $v_{j}$ ($r+1\leqslant j\leqslant 2d$) such that $dh_{P}(v_{j})=c_{j}^{-1}v_{j}$ and the tangent space $T_{F,P}$ of $F$ at $P$ is exactly the invariant subspace $(T_{M,P})^{dh_{P}}$. Using $h^{\ast }\unicode[STIX]{x1D714}_{M}=\unicode[STIX]{x1D714}_{M}$, we deduce that

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{M,P}(v,v_{j})=\unicode[STIX]{x1D714}_{M,P}(dh_{P}(v),dh_{P}(v_{j}))=\unicode[STIX]{x1D714}_{M,P}(v,c_{j}^{-1}v_{j})=c_{j}^{-1}\unicode[STIX]{x1D714}_{M,P}(v,v_{j}),\end{eqnarray}$$

for any $v\in T_{F,P}$ and $v_{j}$ ($r+1\leqslant j\leqslant 2d$). As $c_{j}\not =1$, it follows that

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{M,P}(v,v_{j})=0\end{eqnarray}$$

for all $v\in T_{F,P}$ and $v_{j}$ with $r+1\leqslant j\leqslant 2d$. Hence, the decomposition (5.2) is orthogonal with respect to $\unicode[STIX]{x1D714}_{M,P}$. As $\unicode[STIX]{x1D714}_{M,P}$ is nondegenerate, it follows from the orthogonality of the decomposition that $\unicode[STIX]{x1D714}_{M,P}|_{T_{F,P}}$ is also nondegenerate on $T_{F,P}$ (possibly $\{0\}$). Hence $(F,\unicode[STIX]{x1D714}|_{F})$ is a smooth symplectic manifold (possibly a point) as well. This completes the proof of Proposition 5.2.◻