No cohomologically trivial non-trivial automorphism of generalized Kummer manifolds
Keiji Oguiso
We prove that the natural representaion, on the total Betti cohomolgy group, of the automorphism group of a hyperk\"ahler manifold deformation equivalent to a generalized Kummer manifold is faithful. This is a sort of generalization of an earlier work of Beauville and a recent work of Boiss\'ere, Nieper-Wisskirchan and Sarti, concerning the representation on the second cohomology group.
1208.3750.tex

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\title [Generalized Kummer manifolds]{No cohomologically trivial 
non-trivial automorphism of generalized Kummer manifolds}

\author{Keiji Oguiso}

\address{Keiji Oguiso, Department of Mathematics, Osaka University\\
Toyonaka 560-0043 Osaka, Japan and  Korea Institute for Advanced Study, Hoegiro 87, Seoul, 130-722, Korea} \email{oguiso@math.sci.osaka-u.ac.jp}

%\subjclass[2010]{14J28, 14J50}


\thanks{supported by JSPS Gran-in-Aid (B) No 22340009, JSPS Grant-in-Aid (S), No 22224001, and by KIAS Scholar Program}

\begin{document}

\maketitle

\begin{abstract}
We prove that the natural representaion, on the total Betti cohomolgy group, of the automorphism group of a hyperk\"ahler manifold deformation equivalent to a generalized Kummer manifold is faithful. This is a sort of 
generalization of an earlier work of Beauville and 
a recent work of Boiss\'ere, Nieper-Wisskirchan and Sarti, concerning the representation on the second cohomology group. 
\end{abstract}

\section{Introduction}

Our main result is Theorem (\ref{main}). Thoughout this note, we work over 
${\mathbf C}$. 

Global Torelli theorem for K3 surfaces (\cite{PS71}, \cite{BR75}, 
see also \cite{BHPV04}) says that the natural contravariant representation 
$\rho_2 : {\rm Aut}\,(S) \to {\rm GL}\, (H^2(S, {\mathbf Z}))$ is faithful for any K3 surface $S$. On the other hand, Dolgachev (\cite[4.4]{Do84}) and Mukai and Namikawa (\cite{MN84}, \cite{Mu10}) show that there are Enriques surfaces $E$ such that $\rho_2 : {\rm Aut}\,(E) \to {\rm GL}\, (H^2(E, {\mathbf Z}))$ are not faithful. Here and hereafter ${\rm GL}\, (L) := {\rm Aut}_{{\rm group}}\,(L)$ for a finitely generated abelian group $L$, possibly with non-trivial torsion.

Beauville (\cite[Propositions 9, 10]{Be83-2}) 
considered a similar question for hyperk\"ahler manifolds, by which we mean a simply-connected compact K\"ahler manifold $M$ admitting everywhere non-degenarate global $2$-form $\omega_M$ such that $H^0(M, \Omega_M^2) = 
{\mathbf C} \omega_M$. Most standard examples are the Hilbert scheme 
${\rm Hilb}^n (S)$ of dimension $2n$ of $0$-dimensional closed subschemes 
of length $n$ on a K3 surface $S$, the generalized Kummer manifold 
$K_{n-1}\, (A)$ of dimension $2(n-1) \ge 4$ associated to a 
$2$-dimensional complex torus $A$ (see eg. Section 2 for definition) 
and their deformations (\cite[Sections 6,7]{Be83}). For them, he found the following two opposing phenomena:

\begin{theorem}\label{beauville}

\begin{enumerate}

\item 
$\rho_2 : {\rm Aut}\,({\rm Hilb}^n (S)) \to {\rm GL}\, (H^2({\rm Hilb}^n (S), {\mathbf Z}))$ is faithful. 

\item $\rho_2 : {\rm Aut}\,(K_{n-1}(A)) \to 
{\rm GL}\, (H^2(K_{n-1}(A), {\mathbf Z}))$ 
is not faithful. More precisely, $T(n) \subset {\rm Ker}\, \rho_2$. Here $T(n) \simeq ({\mathbf Z}/n)^{\oplus 4}$ is the group of automorphisms naturally induced from the group of $n$-torsion points $T[n] := \{a \in A \vert na =0\}$ of $A = {\rm Aut}^0(A)$. 

\end{enumerate}
\end{theorem}

Recently, Boissi\`ere, Nieper-Wisskirchen and Sarti (\cite[Theorem 3, Corollary 5 (2)]{BNS11}) generalized the second statement completely as follows:

\begin{theorem}\label{sarti}
${\rm Ker}\, (\rho_2 : {\rm Aut}\,(K_{n-1}(A)) \to 
{\rm GL}\, (H^2(K_{n-1}(A), {\mathbf Z}))) = T(n)\cdot 
\langle \iota \rangle$. 
Here $\iota$ is the automorphism naturally induced from the inversion $-1$ of $A$ and $\cdot$ is the semi-direct product for which $T(n)$ is normal. 
\end{theorem}

Now, a natural question is how about the representation on the total cohomology group 
$H^*(K_{n-1}(A), {\mathbf Z}) := \oplus_{k=0}^{4(n-1)} H^k(K_{n-1}(A), {\mathbf Z})$. Our main result is the following:

\begin{theorem}\label{main}
Let $Y$ be any hyperk\"ahler manifold deformation equivalent to $K_{n-1}(A)$. Then, the natural representation 
$\rho : {\rm Aut}\,(Y) \to 
{\rm GL}\, (H^*(Y, {\mathbf Z}))$ is faithful.
\end{theorem}

First we prove Theorem (\ref{main}) for $K_{n-1}(A)$. By Theorem (\ref{sarti}), it suffices to show that $g^* \vert H^*(K_{n-1}(A), {\mathbf Z}) \not= id$ for each $g \in T(n) \cdot \langle \iota \rangle \setminus \{id\}$. This will be done in Sections 3. We then prove Theorem (\ref{main}) for any $Y$ by using the density result due to Markman and Mehrotra (\cite[Theor extends em 4.1, see also Theorem 1.1]{MM12}) in Section 4. In Remark (\ref{hilb}), we point out a similar 
result for deformation of ${\rm Hilb}^n(S)$. After posting this note on ArXiv, Professor Yuri Tschinkel kindly informed that in his joint work with Hassett (\cite[Theorem 2.1, Proposition 3.1]{HT10}), they show that the natural action 
$T(n) \cdot \langle \iota \rangle$ on $K_{n-1}(A)$ above extends to a faithful action on any deformation $Y$ of $K_{n-1}(A)$, being trivial on $H^2(Y, {\mathbf Z})$. In particular, we have ${\rm Aut}\, (Y) \not= \{id_Y\}$ in Theorem (\ref{main}) even if $Y$ is generic.

Besides its own theoretical interest, Theorem (\ref{main}) is much inspired by the following question asked by Professor Dusa McDuff to me at the conference in Banff (2012, July):

\begin{question}\label{McDuff}
Is there an example of a compact K\"ahler manifold $M$ such that the biholomorphic automorphism group is discrete, i.e., ${\rm Aut}^0\, (M) = \{id_M\}$, but with a biholomorphic automorphism $g \not= id_M$ being homotopic to $id_M$ in the group of diffeomorphisms?
\end{question}

Note that any translation automorphism of a complex torus $T$ of $\dim\, T >0$ is holomorphically homotopic to $id_T$, but ${\rm Aut}^0\, (T) = T$. If such $g \in {\rm Aut}\, (M)$ exists, then $g$ is necessarily cohomologically trivial, i.e., 
$g \in {\rm Ker}\, (\rho : {\rm Aut}\, (M) \to {\rm GL}(H^*(M, {\mathbf Z}))$. 

Cohomologically trivial automorphisms on Enriques surfaces are not affirmative examples of Question (\ref{McDuff}). In fact, any automorphism $g$ on an Enriques surface lifts to automorphisms $\tilde{g}$ of the covering K3 surface in two ways. As pointed out by her, one of the lifted automorphisms is homotopic to $id$ if so is the orginal one, in the groups of diffeomorphisms. But none of the two lifted automorphisms is cohomologically trivial unless $\tilde{g} = id$ by the global Torelli theorem for K3 surfaces. As other candidates, I wonder if Beauville's automorphisms $T(n)$ of $K_{n-1}(A)$ in Theorem (\ref{beauville}) provide affirmative examples or not, as it is derived from translations of $A$. This is my starting point of this work. Unfortunately, Theorem (\ref{main}) says that they are not even cohomologically trivial. 
So, {\it Question (\ref{McDuff}) itself is still completely open and also widely open even for cohomologically trivial level}. 

{\bf Ackowledgement.} I would like to express my thanks to 
Professor Dusa McDuff for inspiring question, Professor Nessim Sibony for invitation to the conference at Banff. I also would like to express my thanks to Professor Yuri Tschinkel for his interest in this work and a valuable information 
about a paper \cite{HT10} and to Professor Igor Dolgachev for valuable comments. 

\section{Preliminaries.}
In this section, following \cite[Section 7]{Be83}, 
we mainly fix notations on a generalized Kummer manifold. 
For more details on generalized Kummer manifolds 
and basic propoerties on hyperk\"ahler manifolds, 
see \cite[Section 7]{Be83} and \cite[Part III]{GHJ03}. 
In order to avoid any notational confusion, we also notice here that $K_n(A)$ in \cite{BNS11} is $K_{n-1}(A)$ in this note and \cite{Be83}, \cite{Be83-2}.

Let $A$ be a $2$-dimensional complex torus and let $n$ be an integer such that 
$n \ge 3$. Let ${\rm Hilb}^n(A)$ be the Hilbert scheme of $0$-dimensional closed subschemes of $A$ of length $n$. Then ${\rm Hilb}^n(A)$ is a smooth K\"ahler manifold of dimension $2n$. Let 
$$\nu = \nu_A : {\rm Hilb}^n(A) \to {\rm Sym}^n(A) = A^n/S_n$$ be the Hilbert-Chow morphism. We denote the sum as $0$-cycles by $\oplus$ and the sum 
in $A$ by $+$. Then an element of ${\rm Sym}^n(A)$ is of the form 
$$\oplus_{i=1}^{k} x_i^{\oplus m_i}\,\, ,$$
where $x_i$ are distinct points on $A$ and $m_i$ are positive integers 
such that $\sum_{i=1}^{k} m_i = n$. Then, we have a natural surjective morphism $$s := s_A : {\rm Sym}^n(A) \to A\,\, ;\,\, \oplus_{i=1}^{k} x_i^{\oplus m_i} \mapsto \sum_{i=1}^{k} m_ix_i\,\, .$$
Hence we have a surjective morphism
$$s \circ \nu = s_A \circ \nu_A : {\rm Hilb}^n(A) \to A\,\, .$$
This is a smooth morphism such that all fibers are isomorphic. Any fiber of 
$s_A \circ \nu_A$ is the generalized Kummer manifold $K_{n-1}(A)$. 
This is in fact a $2(n-1)$-dimensional hyperk\"ahler manifold. When $n=1$, it is a point and when $n=2$, it is the Kummer K3 surface ${\rm Km}\,(A)$ associated to $A$. So, we usually consider only the cases where $2(n-1) \ge 4$, i.e., $n \ge 3$. 

{\it We fix $K_{n-1}(A) = (s \circ \nu)^{-1}(0)$, the fiber over $0$ from now on until the end of this note.} We can also describe $K_{n-1}(A)$ in a slightly different way. Let 
$$A(n-1) := \{(P_1, P_2, \cdots , P_n) \in A^n\, \vert\, \sum_{i=1}^{n} P_i = 0\}\,\,.$$
$A(n-1)$ is a closed submanifold of $A^n$, isomorphic to $A^{n-1}$. 
$A(n-1)$ is stable under the action of $S_n$ on $A^n$, and we have:
$${\rm Sym}^n(A) \supset A^{(n-1)} := s^{-1}(0) = 
A(n-1)/S_n\,\, .$$
Our $K_{n-1}(A)$ is then: 
$$K_{n-1}(A) = \nu^{-1}(A^{(n-1)}) = {\rm Hilb}^n(A) \times_{{\rm Sym}^n(A)} A^{(n-1)}\,\, .$$
It is known that $\dim\, {\rm Def}\, (A) = 4$, while 
$\dim\, {\rm Def}\, (K_{n-1}(A)) = 5$ for $n \ge 3$ and any local deformation 
of hyperk\"ahler manifold is again a hyperk\"ahler manifold. So, there are 
hyperk\"ahler manifolds $Y$ which are deformation equivalent to $K_{n-1}(A)$ 
but not isomorphic to any generalized Kummer manifold.

\section{Proof of Theorem (\ref{main}) for $K_{n-1}(A)$.}

From now until the end of this note, we put $X := K_{n-1}(A)$ and $K := T(n)\cdot \langle \iota \rangle$ (see Introduction for definition), and employ the same notations in Section 2.

In this section, we prove Theorem (\ref{main}) for $K_{n-1}(A)$ with 
$2(n-1) \ge 4$. By Theorem (\ref{sarti}), it suffices to prove the 
following two Claims:

\begin{claim}\label{iota}
Let $g \in K \setminus T(n)$. Then $g^* \vert H^*(X, {\mathbf C}) \not= id$.
\end{claim}

\begin{claim}\label{translation}
Let $a \in T(n) \setminus \{id\}$. 
Then $a^* \vert H^*(X, {\mathbf C}) \not= id$. 
\end{claim}

Let $(z^1_i, z^2_i)$ ($1 \le i \le n$) 
be the standard global coordinates of the universal cover 
${\mathbf C}^2_i$ of the $i$-th factor $A_i = A$ of the product $A^n$. Then the universal cover ${\mathbf C}^{2(n-1)}$ of $A(n-1) \simeq A^{n-1}$ is a closed submanifold of ${\mathbf C}^{2n}$, defined by the two equations:
$$z^1_1 + z^1_2 + \cdots + z^1_{n-1} + z^1_{n} = 0\,\, ,$$
$$z^2_1 + z^2_2 + \cdots + z^2_{n-1} + z^2_{n} = 0\,\, .$$
In particular, $(z^1_i, z^2_i)$ ($1 \le i \le n-1$) give the global linear coordinates of the universal cover ${\mathbf C}^{2(n-1)}$ of $A(n-1)$. 
Note that 
$dz^1_i$ and $dz^2_i$ ($1 \le i \le n$) can be regarded as global 
$1$-forms on $A(n-1)$. They satisfy
$$dz^1_1 + dz^1_2 + \cdots + dz^1_{n-1} + dz^1_{n} = 0\,\, ,$$
$$dz^2_1 + dz^2_2 + \cdots + dz^2_{n-1} + dz^2_{n} = 0\,\, .$$
So, $dz^1_i$ and $dz^2_i$ ($1 \le i \le n-1$) form the basis of the space of global holomorphic $1$-forms on $A(n-1) \simeq A^{n-1}$.

\begin{lemma}\label{form} Consider the following global 
$(2,1)$-form $\tilde{\tau}$ on $A(n-1)$:
$$\tilde{\tau} = dz^1_1 \wedge dz^2_1 \wedge d\overline{z}^2_1 + 
dz^1_2 \wedge dz^2_2 \wedge d\overline{z}^2_2 + \cdots + 
dz^1_{n-1} \wedge dz^2_{n-1} \wedge d\overline{z}^2_{n-1} + 
dz^1_n \wedge dz^2_n \wedge d\overline{z}^2_n\,\, .$$
 Then $\tilde{\tau}$ naturally 
decends to a non-zero element $\tau$ of $H^{2,1}(X)$.
\end{lemma}
\begin{proof} Since $\tilde{\tau}$ is $S_n$-invariant, it descends to 
a global $(2,1)$-form, say $\overline{\tau}$, on the compact K\"ahler 
V-manifold 
$A^{(n-1)}$. For compact K\"ahler $V$-manifolds, the Hodge decompsition is pure and the Hodge theory works in the same way as smooth compact case (\cite{St77}).Consider $\nu \vert X : X \to A^{(n-1)}$. Then $\tau = (\nu \vert X)^* 
\overline{\tau} \in H^{2,1}(X)$. 
It remains to show that $\tau \not= 0$. 
Since $(\nu \vert X)^*$ is injective, it suffices to show that $\overline{\tau} \not= 0$ in $H^{2,1}(A^{(n-1)})$. For this, it suffices to show that $\tilde{\tau} \not= 0$ in $H^{2,1}(A(n-1))$, again by the injectivity applied for 
the quotient map $A(n-1) \to A^{(n-1)}$. 
By the two equalities above, we have
$$\tilde{\tau} = dz^1_1 \wedge dz^2_1 \wedge d\overline{z}^2_1 + 
dz^1_2 \wedge dz^2_2 \wedge d\overline{z}^2_2 + \cdots + 
dz^1_{n-1} \wedge dz^2_{n-1} \wedge d\overline{z}^2_{n-1} 
- (\sum_{k=1}^{n-1} dz^1_k) \wedge (\sum_{k=1}^{n-1} dz^2_k) \wedge 
(\sum_{k=1}^{n-1} d\overline{z}^2_k)\,\, .$$
This gives an expression of $\tilde{\tau}$ in terms of the standard basis of $H^{2,1}(A(n-1))$. By $n-1 \ge 2$, the term $dz^1_1 \wedge dz^2_2 \wedge d\overline{z}^2_2$ appears with coefficient $-1$ and we are done. 
\end{proof}

\begin{lemma}\label{form2} 
Let $g \in K \setminus T(n)$ and $\tau \in H^{2,1}(X)$ be as in Lemma (\ref{form}). Then $g^* \tau = -\tau$. In particular, $g$ is not 
cohomologically trivial. 
\end{lemma}

\begin{proof} $g$ acts equivariantly for 
$A(n-1) \rightarrow A^{(n-1)} \leftarrow X$. Since $g \in K \setminus T(n)$, 
it follows that $g^*dz_i^q = -dz_i^q$ ($1 \le i \le n$, $q=1$, $2$). Hence $g^*\tilde{\tau} = -\tilde{\tau}$ by the shape of $\tilde{\tau}$. Thus $g^*\tau = -\tau$. Since $\tau \not= 0$ in $H^{2,1}(X)$, this implies 
$g^* \vert H^3(X, {\mathbf C}) \not= id$. 
\end{proof}

Lemma (\ref{form2}) completes the proof of Claim (\ref{iota}). 

From now, we prove Claim (\ref{translation}). 

The next proposition is a formal generalization of a result of 
Camere (\cite[Proposition 3]{Ca10}), which states and proves for symplectic 
involutions. Logically, this proposition is not needed in our proof but 
this helps to guess the result of Proposition (\ref{fixed}) 
and may be applicable for other studies. So we state and prove here:

\begin{proposition}\label{symplectic} 
Let $(M, \omega_M)$ be a holomorphic symplectic manifold of dimension 
$2m$, i.e., $M$ is a compact K\"ahler manifold and $\omega_M$ is an everywhere non-degenerate holomorphic $2$-form on $M$. Let $h \in {\rm Aut}\, (M)$ such that 
$h^*\omega_M = \omega_M$ and $h$ is of finite order, say $m$. Let $F$ 
be a connected component of the fixed locus 
$M^h = \{P \in M \vert h(P) = P \}$. Then $(F, \omega_M \vert F)$ is a holomorphic symplectic manifold, in particular, of even dimension. Here we also consider that a point is a symplectic manifold, as it is harmless at all. 
\end{proposition}

\begin{proof} $F$ is isomorphic to the intersection of the graph of $h$ and the diagonal in $M \times M$. So it is compact and K\"ahler. Let $P \in F$. Since $h$ is of finite order, $h$ is locally linearizable at $P$. That is, there are local coordinates $(y_1, y_2, \cdots , y_{2m})$ 
such that $h^*y_i = y_i$ for all $1 \le i \le r$ and $h^*y_j = c_jy_j$ for all $r+1 \le j \le 2m$. Here $c_j \not= 1$ and satisfies $c_j^m = 1$. Then $F$ is locally defined by $x_j = 0$ ($r+1 \le i \le 2d$), in $M$. Hence $F$ 
is smooth. Since $h\vert V = id_V$, the automorphism $h$ induces the linear automorphism $h_{*,P} : T_PM \to T_PM$ of the tangent space of $M$ at $P$. By the description above, we have the decomposition $T_PM = T_PV \oplus N$ 
where $N = \oplus_{j=r+1}^{2m} {\mathbf C}v_j$ with 
$h_{*, P}(v_j) = c_j^{-1}v_j$, and the tangent space $T_PF$ of $F$ is exactly the invariant subspace $(T_PM)^{h_{*, P}}$. Then for $v \in T_PF$ and $v_j$, we have by $h^*\omega_M = \omega_M$:
$$\omega_{M, P}(v, v_j) = (h^*_P(\omega_{M, P}))(h_{*,P}(v), h_{*, P}(v_j)) = 
\omega_{M, P}(v, c_j^{-1}v_j) = c_j^{-1}\omega_{M, P}(v, v_j)\,\, .$$
Since $c_j \not= 1$, it follows that $\omega_{M, P}(v, v_j) = 0$ for all 
$r+1 \le j \le 2m$, that is, the decomposition $T_PM = T_PV \oplus N$ is orthogonal with respect to $\omega_{M, P}$. Hence $\omega_{M, P} = 
\omega_{M, P} \vert T_PV \oplus 
\omega_{M, P} \vert N$. Since $\omega_{M, P}$ is non-degenerate, 
it follows that 
$\omega_{M, P} \vert T_PV$ (indeed, also $\omega_{M, P} \vert N$) is non-degenerate, and in particular $\dim\, F = r$ is even (possibly $0$). This completes 
the proof. 
\end{proof}

From now on, let $a \in T(n) \simeq ({\mathbf Z}/n)^{\oplus 4}$ be an element of order $p \not= 1$ ($p$ is {\it not} assumed to be prime). 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 from the translation automorphism $x \mapsto x +a$ on $A$. 

\begin{proposition}\label{fixed}
The fixed locus $X^a$ consists of $p^3$ connected components $F_i$ 
($1 \le i \le p^3$), each of which is isomorphic to the generalized 
Kummer manifold $K_{d-1}(A/\langle a \rangle)$ associated to the 
$2$-dimensional complex torus $A/\langle a \rangle$. 
\end{proposition}

\begin{proof} Let ${\mathcal S} \subset A$ be a $0$-dimensional closed subscheme of length $n$. Since $\langle a \rangle$ acts freely on $A$, or in other words, the quotient map $\pi : A \to A/\langle a \rangle$ is \'etale of degree $p$, 
it follows that $a_*{\mathcal S} = {\mathcal S}$ 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 ${\mathcal S} = \pi^*{\mathcal T}$. This ${\mathcal T}$ is unique. Thus we have an isomorphism 
${\rm Hilb}^d(A/\langle a \rangle) \simeq ({\rm Hilb}^n(A))^a$ by $\pi^*$. 

Let ${\mathcal S} \in  ({\rm Hilb}^n(A))^a$. Then, by $a_*{\mathcal S} = {\mathcal S}$, it follows that $\nu({\mathcal S}) \in ({\rm Sym}^n(A))^a$ 
and it is of the form:
$$\nu({\mathcal S}) = \oplus_{i=1}^{k} 
\oplus_{j=0}^{p-1} (x_i + ja)^{\oplus m_i}\,\, ,$$ 
where $\sum_{i=1}^{k}m_i = d$ and all points $x_i + ja$ are distinct. Note that $K_{n-1}(A)^a = ({\rm Hilb}^n(A))^a \cap K_{n-1}(A)$. Hence, by definition of $K_{n-1}(A)$, we have ${\mathcal S} \in K_{n-1}(A)^a$ 
if and only if ${\mathcal S} \in K_{n-1}(A)$, whence, if and only if 
$$p(m_1x_1 + m_2x_2 + \cdots + m_kx_k + \alpha) = 0\,\, ---\,\, (*)$$
in $A$. Here $\alpha \in A$ is an element 
such that $p\alpha = (n(p-1)/2) a$ in 
$A$. (We choose and fix such $\alpha$. We also note that $n(p-1)/2 \in {\mathbf Z}$.) The last equation (*) is equivalent to 
$$m_1x_1 + m_2x_2 + \cdots + m_kx_k + \alpha \in A[p]\,\, --- (**)\,\, ,$$ 
where $A[p]$ is the group of $p$-torsion points of $A$. Since $a$ is also a $p$-torsion point, this condition (**) is also equivalent to
$$m_1\pi(x_1) + m_2\pi(x_2) + \cdots + m_k\pi(x_k) + \pi(\alpha) \in \pi(A[p]) = A[p]/\langle a \rangle\,\, --- (***)\,\, .$$ 
Write ${\mathcal S} = \pi^*({\mathcal T})$. Then, the last condition (***) holds if and only if ${\mathcal T}$ is in the fibers of 
$$s_{A/\langle a \rangle} \circ \nu_{A/\langle a \rangle} : {\rm Hilb}^d(A/\langle a \rangle) \to A/\langle a \rangle$$ 
over $\pi(A[p])$. We have $\vert \pi(A[p]) \vert = p^3$, 
as $a$ is also $p$-torsion. Hence, by $\pi^*$, the fixed locus $K_{n-1}(A)^a$ is isomorphic to the union of $p^3$ fibers of $s_{A/\langle a \rangle} 
\circ \nu_{A/\langle a \rangle}$, 
each of which is, by definition, isomorphic to 
$K_{d-1}(A/\langle a \rangle)$. This completes the proof.  
\end{proof}

We recall the following fundamental result due to G\"ottche and Soergel 
(\cite[Corollary 1]{GS93}, see also \cite{Go94}, \cite{De10}):

\begin{theorem}\label{goettche}
The topological Euler number $\chi_{\rm top}(K_{n-1}(A))$ of $K_{n-1}(A)$ is 
$n^3 \sigma(n)$, where 
$\sigma(n) = \sum_{1 \le b \vert n} b$, the sum of all positive 
divisors of $n$. (This is valid also for $n=1$, $2$.)
\end{theorem}

Consider now the Lefschetz number of $h \in {\rm Aut}\, (X)$:
$$L(h) := \sum_{k=0}^{4(n-1)} (-1)^k\,\, {\rm tr}\,\, (h^{*} \vert H^{k}(X, {\mathbf C}))\,\, .$$

\begin{proposition}\label{lefschetz}

(1) If $h \in {\rm Aut}\, (X)$ is cohomologically trivial, then $L(h) = n^3\sigma(n)$. 

(2) $L(a) = n^3\sigma(d)$, where $d = n/p$. 
\end{proposition}

\begin{proof} If $h$ is cohomologically trivial, then ${\rm tr}\, (h^{*} \vert H^{k}(X, {\mathbf C})) = b_k(X)$. This implies (1). By the topological Lefschetz fixed point formula, Proposition (\ref{fixed}) and Theorem (\ref{goettche}), we have
$$L(a) = \chi_{\rm top}(X^a) = p^3 \chi_{\rm top}(K_{d-1}(A/\langle a \rangle) 
= p^3\cdot d^3\sigma(d) = n^3\sigma(d)\,\, .$$ 
This is nothing but the assertion (2). 
\end{proof}

Since $d \vert n$ and $d \not= n$, it follows that $\sigma(d) \le \sigma(n) -n < \sigma(n)$. Hence by Proposition (\ref{lefschetz})(1), (2), 
$a$ is not cohomologically trivial. 
This shows Claim (\ref{translation}). Now, the proof 
of Theorem (\ref{main}) for $K_{n-1}(A)$ is completed.

\section{Proof of Theorem (\ref{main}).}

In this section, we shall prove Theorem (\ref{main}). 

Let $\Lambda = (\Lambda, (*, **))$ be a fixed abstract lattice 
isometric to 
$(H^2(K_{n-1}(A), {\mathbf Z}), b)$, $b$ being the Beauville-Bogomolov form.
Let $Y$ be a hyperk\"ahler manifold deformation equivalent to a generalized 
Kummer manifold $X = K_{n-1}(A)$. Let $g \in {\rm Aut}\, (Y)$ such that 
$g^* \vert H^*(Y, {\mathbf Z}) = id$. We shall show that $g = id_Y$. 

Let ${\mathcal M}^0$ be the connected component of 
the marked moduli space of ${\mathcal M}_{\Lambda}$, containing $(Y, \eta)$. 
Here $\eta : H^2(Y, {\mathbf Z}) \to \Lambda$ is a marking. 
The marked moduli space ${\mathcal M}_{\Lambda}$ is constructed by Huybrechts 
(\cite[1.18]{Hu99}) by patching Kuranishi spaces via local Torelli. 
By construction, ${\mathcal M}_{\Lambda}$ is smooth, but highly non-Hausdorff 
as explained briefly below. He also showed that 
the period map 
$$p : {\mathcal M}^0 \to {\mathcal D} = \{[\omega] \in {\mathbf P}(\Lambda \otimes {\mathbf C}\, \vert (\omega, \omega) = 0 \, ,\, (\omega, \overline{\omega}) > 0 \}$$ is a surjective holomorphic map of degree $1$ (\cite[Theorem 8.1]{Hu99}, see also \cite{Ve09}, \cite{Hu11} for degree and futher development). For $[\omega] \in {\mathcal D}$, if $p^{-1}([\omega])$ ($\subset {\mathcal M}^0$) is not a single point, then  $p^{-1}([\omega])$ consists of points, being mutually inseparable, corresponding to birational hyperk\"ahler manifolds. Markman and Mehrotra (\cite[Theorem 4.1]{MM12}) show that marked generalized Kummer manifolds are dense in ${\mathcal M}^0$. Actually they proved a strong density (see also an argument of Theorem 1.1 in \cite{MM12}), based on the Hodge theoretic Torelli type Theorem (\cite{Ma11}): 

\begin{theorem}\label{markman}
There is a dense subset ${\mathcal D}' \subset {\mathcal D}$ such that each point of $p^{-1}([\omega])$ corresponds to a generalized Kummer manifold for $[\omega] \in {\mathcal D}'$. 
\end{theorem}

Consider the Kuranishi family $u : {\mathcal U} \to {\mathcal K}$ of $Y$. Here and hereafter we freely shrink ${\mathcal K}$ around $0 = [Y]$. 
Since the Kuranishi family is universal, $g \in {\rm Aut}\, (Y)$ induces automorphisms 
$\tilde{g} \in {\rm Aut}\, ({\mathcal U})$ and $\overline{g} \in {\rm Aut}\, ({\mathcal K})$ such that $u \circ \tilde{g} = \overline{g} \circ u$ 
and 
$\tilde{g}\vert Y = g$. Since ${\mathcal K}$ is locally isomorphic to ${\mathcal D}$ by the local Torelli theorem (\cite[Theorem 5]{Be83}), the locus ${\mathcal K}' \subset {\mathcal K}$, 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 (\ref{markman}) and 
the construction of ${\mathcal M}^0$ briefly explained above. Here we note that the density in ${\mathcal M}^0$ is not sufficient. Now we follow Beauville's argument (\cite[Proof of Proposition 10]{Be83-2}) 
to conclude. 

We can take ${\mathcal K}$ as a small polydisk in 
$H^1(Y, TY)$ with center $0$. 
Recall that $H^1(Y, TY) \simeq H^1(X, \Omega_Y^1)$ by $\omega_Y$. Then 
$\overline{g} = id_{\mathcal K}$. Indeed, since $g$ is cohomologically trivial 
and $H^{2,0}(Y) = H^0(Y, \Omega_Y^2) = {\mathbf C}\omega_Y$, we have $g^*\omega_Y = \omega_Y$ and $g^* \vert H^1(X, \Omega_Y^1) = id$. 
Let $t \in {\mathcal K}$ be any point of ${\mathcal K}$. Then  
$\tilde{g}$ preserves the fiber $Y_t = u^{-1}(t)$, 
i.e., $\tilde{g}\vert Y_t \in {\rm Aut}\, (Y_t)$. Put $g_t := \tilde{g}\vert Y_t$. Then $g_t$ is also cohomologically trivial, because $g_t^*$ 
is derived from the action of $\tilde{g}$ on the constant system $\oplus_{k=0}^{4(n-1)} R^ku_*{\mathbf Z}$. But then $g_t = id_{Y_t}$ over the dense subset ${\mathcal K}'$ by 
Theorem (\ref{main}) for $K_{n-1}(A)$. Since ${\mathcal U}$ is Hausdorff 
and $\tilde{g}$ is continuous, this implies that 
$\tilde{g} = id_{\mathcal U}$. Hence 
$g = g_0 = id_Y$ as well. This completes the proof of Theorem (\ref{main}). 

\begin{remark}\label{hilb}
Markman and Mehrotra (\cite[Theorem 1.1]{MM12}) also show the strong density result for ${\rm Hilb}^n (S)$ of K3 surfaces $S$. So, the same argument here together with Beauville's result (Theorem (\ref{beauville})(1)) 
implies the following:
Let $W$ be a hyperk\"ahler manifold deformation equivalent to 
${\rm Hilb}^n (S)$. Let $g \in {\rm Aut}\, (W)$ such that $g^* \vert H^2(W, 
{\mathbf Z}) = id$. Then $g = id_W$. 
\end{remark}

\begin{thebibliography}{BHPV04}

\bibitem[BHPV04]{BHPV04} Barth, W. P., Hulek, K.,  Peters, C. A. M., 
Van de Ven, A., : {\em Compact complex surfaces}, Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 4. Springer-Verlag, Berlin, 2004. xii+436 pp.

\bibitem[Be83]{Be83} Beauville, A., : {\em Vari\'et\'es K\"ahleriennes dont la premi\'ere classe de Chern est nulle}, J. Differential Geom. 
{\bf 18} (1983) 755--782. 

\bibitem[Be83-2]{Be83-2} Beauville, A., : {\em  Some remarks on K\"ahler manifolds with $c_1 =0$}, Classification of algebraic and analytic manifolds (Katata, 1982), 1--26, Progr. Math., 39, Birkh\"auser Boston, Boston, MA, 1983. 

\bibitem[BNS11]{BNS11} Boissi\`ere, S., Nieper-Wisskirchen, M., 
Sarti, A., : {\em Higher dimensional Enriques varieties and automorphisms of generalized Kummer varieties}, J. Math. Pures Appl. {\bf 95} (2011) 553--563.

\bibitem[BR75]{BR75} Burns, D., Jr., Rapoport, M. : {\em On the Torelli problem for k\"ahlerian K-3 surfaces}, Ann. Sci. \'Ecole Norm. Sup. {\bf 8} 
(1975) 235--273. 

\bibitem[Ca10]{Ca10} 
Camere C., : {\em Symplectic involutions of holomorphic symplectic
fourfolds}, arXiv:1001.2607.

\bibitem[De10]{De10}  Debarre, O., : {\em On the Euler characteristic of generalized Kummer varieties}, Amer. J. Math. {\bf 121} (1999) 577--586.

\bibitem[Do84]{Do84} Dolgachev, I., : {\em On automorphisms of Enriques surfaces}, Invent. Math. {\bf 76} (1984) 163--177. 

\bibitem[Go94]{Go94} G\"ottsche, L., : {\em Hilbert schemes of zero-dimensional subschemes of smooth varieties}, Lecture Notes in Mathematics, 1572. Springer-Verlag, Berlin, 1994. x+196 pp. 

\bibitem[GS93]{GS93} G\"ottsche, L., Soergel, W., : {\em Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces}, 
Math. Ann. {\bf 296} (1993) 235--245.


\bibitem[GHJ03]{GHJ03} Gross M., Huybrechts D., Joyce D., : {\em Calabi-Yau manifolds and related geometries. Lectures from the Summer School held in Nordfjordeid, June 2001}, Universitext. Springer-Verlag, Berlin, 2003.


\bibitem[Hu99]{Hu99} Huybrechts, D., : {\em Compact hyper-K\"ahler manifolds: basic results}, Invent. Math. {\bf 135} (1999) 63--113,  {\em Erratum: "Compact hyper-K\"ahler manifolds: basic results''}, Invent. Math.{\bf 152} (2003) 209--212.

\bibitem[HT10]{HT10} Hassett, B., Tschinkel, Y., : {\em Hodge theory and Lagrangian planes on generalized Kummer fourfolds}, arXiv:1004.0046, to appear in Moscow Math. J.

\bibitem[Hu11]{Hu11} Huybrechts, D., : {\em A global Torelli theorem for hyperkaehler manifolds (after Verbitsky)}, arXiv:1106.5573.

\bibitem[Ma11]{Ma11} Markman, E., : {\em A survey of Torelli and monodromy results for holomorphic-symplectic varieties}, arXiv:1101.4606. 

\bibitem[MM12]{MM12} Markman, E., Mehrotra, S., : {\em Hilbert schemes of K3 surfaces are dense in moduli}, arXiv:1201.0031 

\bibitem[Mu10]{Mu10} Mukai, S., : {\em Numerically trivial involutions of Kummer type of an Enriques surface}, Kyoto J. Math. {\bf 50} (2010) 889--902.

\bibitem[MN84]{MN84} Mukai, S., Namikawa, Y., : {\em Automorphisms of Enriques surfaces which act trivially on the cohomology groups}, Invent. Math. 
{\bf 77} (1984) 383--397. 


\bibitem[PS71]{PS71} Pjateckii--Shapiro, I. I., Shafarevich, I. R., : {\em  Torelli's theorem for algebraic surfaces of type K3}, Izv. Akad. Nauk SSSR Ser. Mat. {\bf 35} (1971) 530--572.

\bibitem[St77]{St77} Steenbrink, J. H. M., : {\em Mixed Hodge structure on the vanishing cohomology}, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) 525--563. Sijthoff and Noordhoff, Alphen aan den Rijn, 1977.

\bibitem[Ve09]{Ve09} Verbitsky, M., : {\em A global Torelli theorem for hyperk\"ahler manifolds}, arXiv:0908.4121.

\end{thebibliography}
\end{document}

1208.3750.extracted_bib
\begin{thebibliography}{BHPV04}

\bibitem[BHPV04]{BHPV04} Barth, W. P., Hulek, K.,  Peters, C. A. M., 
Van de Ven, A., : {\em Compact complex surfaces}, Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 4. Springer-Verlag, Berlin, 2004. xii+436 pp.

\bibitem[Be83]{Be83} Beauville, A., : {\em Vari\'et\'es K\"ahleriennes dont la premi\'ere classe de Chern est nulle}, J. Differential Geom. 
{\bf 18} (1983) 755--782. 

\bibitem[Be83-2]{Be83-2} Beauville, A., : {\em  Some remarks on K\"ahler manifolds with $c_1 =0$}, Classification of algebraic and analytic manifolds (Katata, 1982), 1--26, Progr. Math., 39, Birkh\"auser Boston, Boston, MA, 1983. 

\bibitem[BNS11]{BNS11} Boissi\`ere, S., Nieper-Wisskirchen, M., 
Sarti, A., : {\em Higher dimensional Enriques varieties and automorphisms of generalized Kummer varieties}, J. Math. Pures Appl. {\bf 95} (2011) 553--563.

\bibitem[BR75]{BR75} Burns, D., Jr., Rapoport, M. : {\em On the Torelli problem for k\"ahlerian K-3 surfaces}, Ann. Sci. \'Ecole Norm. Sup. {\bf 8} 
(1975) 235--273. 

\bibitem[Ca10]{Ca10} 
Camere C., : {\em Symplectic involutions of holomorphic symplectic
fourfolds}, arXiv:1001.2607.

\bibitem[De10]{De10}  Debarre, O., : {\em On the Euler characteristic of generalized Kummer varieties}, Amer. J. Math. {\bf 121} (1999) 577--586.

\bibitem[Do84]{Do84} Dolgachev, I., : {\em On automorphisms of Enriques surfaces}, Invent. Math. {\bf 76} (1984) 163--177. 

\bibitem[Go94]{Go94} G\"ottsche, L., : {\em Hilbert schemes of zero-dimensional subschemes of smooth varieties}, Lecture Notes in Mathematics, 1572. Springer-Verlag, Berlin, 1994. x+196 pp. 

\bibitem[GS93]{GS93} G\"ottsche, L., Soergel, W., : {\em Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces}, 
Math. Ann. {\bf 296} (1993) 235--245.


\bibitem[GHJ03]{GHJ03} Gross M., Huybrechts D., Joyce D., : {\em Calabi-Yau manifolds and related geometries. Lectures from the Summer School held in Nordfjordeid, June 2001}, Universitext. Springer-Verlag, Berlin, 2003.


\bibitem[Hu99]{Hu99} Huybrechts, D., : {\em Compact hyper-K\"ahler manifolds: basic results}, Invent. Math. {\bf 135} (1999) 63--113,  {\em Erratum: "Compact hyper-K\"ahler manifolds: basic results''}, Invent. Math.{\bf 152} (2003) 209--212.

\bibitem[HT10]{HT10} Hassett, B., Tschinkel, Y., : {\em Hodge theory and Lagrangian planes on generalized Kummer fourfolds}, arXiv:1004.0046, to appear in Moscow Math. J.

\bibitem[Hu11]{Hu11} Huybrechts, D., : {\em A global Torelli theorem for hyperkaehler manifolds (after Verbitsky)}, arXiv:1106.5573.

\bibitem[Ma11]{Ma11} Markman, E., : {\em A survey of Torelli and monodromy results for holomorphic-symplectic varieties}, arXiv:1101.4606. 

\bibitem[MM12]{MM12} Markman, E., Mehrotra, S., : {\em Hilbert schemes of K3 surfaces are dense in moduli}, arXiv:1201.0031 

\bibitem[Mu10]{Mu10} Mukai, S., : {\em Numerically trivial involutions of Kummer type of an Enriques surface}, Kyoto J. Math. {\bf 50} (2010) 889--902.

\bibitem[MN84]{MN84} Mukai, S., Namikawa, Y., : {\em Automorphisms of Enriques surfaces which act trivially on the cohomology groups}, Invent. Math. 
{\bf 77} (1984) 383--397. 


\bibitem[PS71]{PS71} Pjateckii--Shapiro, I. I., Shafarevich, I. R., : {\em  Torelli's theorem for algebraic surfaces of type K3}, Izv. Akad. Nauk SSSR Ser. Mat. {\bf 35} (1971) 530--572.

\bibitem[St77]{St77} Steenbrink, J. H. M., : {\em Mixed Hodge structure on the vanishing cohomology}, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) 525--563. Sijthoff and Noordhoff, Alphen aan den Rijn, 1977.

\bibitem[Ve09]{Ve09} Verbitsky, M., : {\em A global Torelli theorem for hyperk\"ahler manifolds}, arXiv:0908.4121.

\end{thebibliography}