\section{Introduction}

The {\it Kummer variety} of an abelian variety $A$ is the quotient
${\rm K}(A):=A/\langle\iota\rangle$, where $\iota(a)=-a$ for any $a\in
A$. The singular variety ${\rm K}(A)$ has an orbifold structure and
it admits a minimal crepant resolution ${\rm Km}(A)$
if and only if the dimension of $A$ is 2. In this case ${\rm Km}(A)$ is a
K3 surface (i.e.\ it is simply connected and its dualizing sheaf is
trivial) and it is called the {\it Kummer surface of $A$}. More generally, we can associate to the global quotient ${\rm K}(A)$ the smooth quotient stack
$[A/\langle\iota\rangle]$.

In \cite{HLOY} Hosono, Lian, Oguiso and Yau proved that,
\begin{itemize}
\item[(A)] {\it given two abelian surfaces $A$ and $B$, ${\rm D}^{\rm b}(A)\cong{\rm D}^{\rm b}(B)$ if
and only if $${\rm D}^{\rm b}({\rm Km}(A))\cong{\rm D}^{\rm b}({\rm Km}(B)).$$}
\end{itemize}
Their argument runs as follows: They notice that, due to the
geometric construction of the Kummer surfaces ${\rm Km}(A)$ and ${\rm Km}(B)$,
the transcendental lattices of $A$ and $B$ are Hodge-isometric if
and only if the transcendental lattices of ${\rm Km}(A)$ and ${\rm Km}(B)$ are
Hodge-isometric. Then, they apply a deep result of Orlov which says
that two abelian or K3 surfaces have equivalent derived categories
if and only if their transcendental lattices are Hodge-isometric
(see Theorem \ref{thm:orlov1}). From this it is evident that (A) can
be reformulated in the following way:
\begin{itemize}
\item[(B)] {\it given two abelian surfaces $A$ and $B$,
${\rm D}^{\rm b}({\rm Km}(A))\cong{\rm D}^{\rm b}({\rm Km}(B))$ if and only if there exists a Hodge
isometry between the transcendental lattices of $A$ and $B$.}
\end{itemize}
Since Mukai proved in \cite{Mu} that two K3 surfaces with Picard
number greater than 11 and with Hodge-isometric transcendental
lattices are isomorphic, (A) and (B) are equivalent to the
following statement:
\begin{itemize}
\item[(C)] {\it given two abelian surfaces $A$ and $B$,
${\rm D}^{\rm b}(A)\cong{\rm D}^{\rm b}(B)$ if and only if ${\rm Km}(A)\cong{\rm Km}(B).$}
\end{itemize}

The aim of this paper is to address (A), (B) and (C) in two more
general contexts. Our first result shows that if $A_1$ and $A_2$
are abelian varieties with equivalent derived categories, then the
derived categories of the stacks $[A_1/\langle\iota\rangle]$ and
$[A_2/\langle\iota\rangle]$ are equivalent as well. In fact we
will prove the following:

\begin{thm}\label{cor:equiv} Let $A_1$ and $A_2$ be abelian varieties.
If ${\rm D}^{\rm b}(A_1)\cong{\rm D}^{\rm b}(A_2)$, then there exists a Fourier-Mukai
equivalence
${\rm D}^{\rm b}([A_1/\langle\iota\rangle])\cong{\rm D}^{\rm b}([A_2/\langle\iota\rangle])$.

Conversely, if ${\rm D}^{\rm b}([A_1/\langle\iota\rangle])$ and
${\rm D}^{\rm b}([A_2/\langle\iota\rangle])$ are equivalent, then there is an
isomorphism of Hodge structures $\widetilde
H(A_1,\mathbb{Q})\cong\widetilde H(A_2,\mathbb{Q})$.\end{thm}

The Hodge structures mentioned in the second part of the previous
statement will be defined in Section \ref{subsec:Hodge} and the
proof of this result will occupy almost all Section
\ref{sec:equivav}. As we will show in Section
\ref{subsec:geomappl1}, when we deal with abelian surfaces this
result leads to a direct proof of one implication in (A). An
application to the number of birational generalized Kummer varieties
is given in Proposition \ref{cor:genKum}.

Our second main result treats the two-dimensional twisted case.
Indeed, according to (B), we prove that the twisted derived
categories of two Kummer surfaces are equivalent if and only if the
generalized transcendental lattices of the corresponding abelian
surfaces are Hodge isometric. More precisely the result (proved in
Sections \ref{subsec:Br} and \ref{subsec:proofsec}) is as follows:

\begin{thm}\label{thm:main2} Let $A_1$ and $A_2$ be abelian
surfaces. Then the following two conditions are equivalent:
\begin{itemize}
\item[(i)] there exist $\alpha_i$ in the Brauer group of ${\rm Km}(A_i)$
and an equivalence between the derived categories
${\rm D}^{\rm b}({\rm Km}(A_1),\alpha_1)$ and ${\rm D}^{\rm b}({\rm Km}(A_2),\alpha_2)$;
\item[(ii)] there exist $\beta_i$ in the Brauer group of $A_i$ such
that the twisted abelian surfaces $(A_1,\beta_1)$ and
$(A_2,\beta_2)$ have Hodge-isometric generalized transcendental
lattices.
\end{itemize}
Furthermore, if one of these two equivalent conditions holds true,
then $A_1$ and $A_2$ are isogenous.\end{thm}

The notations an definitions involved in the formulation of the
previous result will be explained in Sections \ref{subsec:Hodge} and
\ref{subsec:Br}. We will observe that the analogues of (A) and (C)
in the twisted setting are no longer true (see Remark
\ref{rmk:commenti}). Nevertheless we completely generalize the
results in \cite{HLOY} about the number of Kummer structures on K3
surfaces in the twisted context (Proposition \ref{prop:twnum}). A
geometric example involving abelian surfaces with Picard number two
is discussed.

\section{Derived categories of abelian varieties and K3 surfaces}\label{sec:prelim}

In this section we recall some facts and definitions concerning the
derived categories of coherent sheaves on abelian varieties and K3
surfaces. In the following pages ${\rm D}^{\rm b}(X)$ will always mean the
bounded derived category of coherent sheaves on the smooth
projective variety $X$ (we will also use the same notation for the
bounded derived category of coherent sheaves on a smooth quotient
stack according to \cite{Ka}).

Suppose that $X_1$ and $X_2$ are smooth projective varieties. Let
${\rm D}^{\rm b}(X_1)$ and ${\rm D}^{\rm b}(X_2)$ be the bounded derived categories of
coherent sheaves on $X_1$ and $X_2$. Orlov proved in \cite{Or1} that
any equivalence $\Phi:{\rm D}^{\rm b}(X_1)\rightarrow{\rm D}^{\rm b}(X_2)$ is a {\it
Fourier-Mukai equivalence}, i.e. there exists ${\cal E}\in{\rm D}^{\rm b}(X_1\times
X_2)$ and an isomorphism of functors
\begin{eqnarray}\label{eqn:FMtr}
\Phi\cong{\bf R}p_{2*}({\cal E}\stackrel{\bf L}{\otimes}p_1^*(-)),
\end{eqnarray}
where $p_i:X_1\times X_2\rightarrow X_i$ is the projection and
$i\in\{1,2\}$ (see also \cite{CS} for a more general statement). The
complex ${\cal E}$ is the {\it kernel} of $\Phi$ and it is uniquely (up
to isomorphism) determined. We write $\Phi_{\cal E}$ for a Fourier-Mukai
equivalence whose kernel is ${\cal E}$. In general, given
${\cal E}\in{\rm D}^{\rm b}(X_1\times X_2)$, we write $\Phi_{\cal E}$ for a functor
defined as in \eqref{eqn:FMtr} (notice that $\Phi_{\cal E}$ is not
necessarily an equivalence).

\subsection{Derived categories of abelian varieties} Assume that $A_1$ and $A_2$ are abelian varieties of dimension
$d$. For $i\in\{1,2\}$, let ${\cal P}_i$ be the Poicar{\'e} line bundle
on $A_i\times\widehat{A_i}$, let $\mu_i:A_i\times A_i\rightarrow
A_i\times A_i$ be the isomorphism such that $(a,b)\mapsto (a+b,b)$
and let $\Phi_i:=\mu_{i*}\circ(\mathrm{id}\times\Phi_{{\cal P}_i})$. If
$\Phi_{\cal E}:{\rm D}^{\rm b}(A_1)\rightarrow{\rm D}^{\rm b}(A_2)$ is a Fourier-Mukai
equivalence with kernel ${\cal E}$, we get the following commutative
diagram:
\begin{eqnarray}\label{eqn:der}
\xymatrix{{\rm D}^{\rm b}(A_1\times\widehat{A_1})\ar[dr]^{\mathrm{id}\times\Phi_{{\cal P}_1}}\ar[rrr]^{F_{\cal E}}\ar[dd]_{\Phi_1}&&&
{\rm D}^{\rm b}(A_2\times\widehat{A_2})\ar[dl]_{\mathrm{id}\times\Phi_{{\cal P}_2}}\ar[dd]_{\Phi_2}\\
&{\rm D}^{\rm b}(A_1\times A_1)\ar[dl]_{\mu_{1*}}& {\rm D}^{\rm b}(A_2\times A_2)\ar[dr]^{\mu_{2*}}&\\
{\rm D}^{\rm b}(A_1\times
A_1)\ar[rrr]^{\Phi_{\cal E}\times\Phi_{{\cal E}_R}}&&&{\rm D}^{\rm b}(A_2\times A_2),}
\end{eqnarray}
where $F_{\cal E}$ is the functor completing the diagram,
${\cal E}_R={\cal E}^\vee[d]$ and $\Phi_{\cal E}\times\Phi_{{\cal E}_R}$ is the
Fourier-Mukai equivalence whose kernel is ${\cal E}\boxtimes{\cal E}_R$.
Observe that since $\Phi_{\cal E}$, $\Phi_1$ and $\Phi_2$ are
equivalences, $\Phi_{\cal E}\times\Phi_{{\cal E}_R}$ and $F_{\cal E}$ are
equivalences as well.

For $i\in\{1,2\}$, the K\"{u}nneth formula yields a decomposition
$$H_1(A_i\times\widehat{A_i},\mathbb{Z})\cong H_1(A_i,\mathbb{Z})\oplus
H_1(\widehat{A_i},\mathbb{Z}).$$ Since $H_1(\widehat{A_i},\mathbb{Z})\cong
H_1(A_i,\mathbb{Z})^\vee$, the group $H_1(A_i\times\widehat{A_i},\mathbb{Z})$ is
endowed with a natural quadratic form. Indeed, if
$(a_1,\alpha_1),(a_2,\alpha_2)\in H_1(A_i\times\widehat{A_i},\mathbb{Z})$,
we define
\[
\langle(a_1,\alpha_1),(a_2,\alpha_2)\rangle_i:=\alpha_1(a_2)+\alpha_2(a_1),
\]
where $i\in\{1,2\}$. Consider the set of isomorphisms
\[
{\rm U}(A_1,A_2):=\{f\in\mathrm{Isom}(A_1\times\widehat{A_1},A_2\times\widehat{A_2}):\langle
f_*(a_1,\alpha_1),f_*(a_2,\alpha_2)\rangle_2=\langle
(a_1,\alpha_1),(a_2,\alpha_2)\rangle_1\}.
\]

\begin{thm}\label{thm:orlov} {\bf(\cite{Or}, Theorem 2.19 and Proposition 4.12.)}  Let $A_1$ and $A_2$ be abelian varieties. If $\Phi_{\cal E}:{\rm D}^{\rm b}(A_1)\rightarrow{\rm D}^{\rm b}(A_2)$ is an
equivalence, then, for any ${\cal F}\in{\rm D}^{\rm b}(A_1)$,
$$F_{\cal E}({\cal F})=f_{{\cal E}\,*}({\cal F})\otimes N_{\cal E},$$ where $F_{\cal E}$ is the
equivalence in \eqref{eqn:der}, $f_{\cal E}\in{\rm U}(A_1,A_2)$ and
$N_{\cal E}\in{\rm Pic}(A_2\times\widehat{A_2})$. Moreover, there exists a
surjective map
\begin{eqnarray}\label{eqn:gamma}
\gamma:{\rm Eq}({\rm D}^{\rm b}(A_1),{\rm D}^{\rm b}(A_2))\longrightarrow{\rm U}(A_1,A_2)
\end{eqnarray}
such that $\gamma(\Phi_{\cal E})=f_{\cal E}$, where ${\rm Eq}({\rm D}^{\rm b}(A_1),{\rm D}^{\rm b}(A_2))$
is the set of equivalences between ${\rm D}^{\rm b}(A_1)$ and
${\rm D}^{\rm b}(A_2)$.\end{thm}

\subsection{Hodge structures and derived categories}\label{subsec:Hodge} If $X$ is a smooth
projective variety of dimension $d$, we denote by $\widetilde
H(X,\mathbb{Q})$ the even cohomology group $H^{2*}(X,\mathbb{Q})$ with the
weight-$d$ Hodge structure defined as follows:
\begin{eqnarray}\label{eqn:HSnuova}
\widetilde{H}^{p,q}(X)=\bigoplus_{p-q=r-s} H^{r,s}(X),
\end{eqnarray}
where $H^{r,s}(X)$ is the $(r,s)$-part of the usual Hodge
decomposition of $H^{r+s}(X,\mathbb{C})\subset H^{2*}(X,\mathbb{C})$. An
equivalent way to put on $H^*(X,\mathbb{C})$ such a Hodge structure could
be obtained considering the natural grading on the Hochschild
homology of $X$ (see, for example, \cite{C3}).

Suppose now that $X$ is either an abelian or a K3 surface,
$H^{2,0}(X)=\langle\sigma_X\rangle$ and $B$ is any class in
$H^2(X,\mathbb{Q})$. Then
$$\varphi:=\exp(B)(\sigma_X)=\sigma_X+B\wedge\sigma_X\in
H^2(X,\mathbb{C})\oplus H^4(X,\mathbb{C})$$ is a {\it generalized Calabi-Yau
structure on $X$} (for a complete picture see \cite{Hu}). Let
$T(X,B)$ be the minimal primitive sublattice of $H^2(X,\mathbb{Z})\oplus
H^4(X,\mathbb{Z})$ such that $\varphi\in T(X,B)\otimes\mathbb{C}$. The lattice
$T(X,B)$ is the {\it generalized transcendental lattice of
$\varphi$} (see \cite{Hu} and \cite{HS1}). Let $\widetilde H(X,\mathbb{Z})$
be the $\mathbb{Z}$-module $H^0(X,\mathbb{Z})\oplus H^2(X,\mathbb{Z})\oplus H^4(X,\mathbb{Z})$
endowed with the \emph{Mukai pairing}
\[
\langle(a_0,a_2,a_4),(b_0,b_2,b_4)\rangle=a_2\cdot b_2-a_0\cdot
b_4-a_4\cdot b_0,
\]
where $(a_0,a_2,a_4),(b_0,b_2,b_4)\in H^{2*}(X,\mathbb{Z})$ and
``$\cdot$'' is the cup-product. We write $\widetilde H(X,B,\mathbb{Z})$
for the lattice $\widetilde H(X,\mathbb{Z})$ with the weight-two Hodge
structure such that
\[
\widetilde H^{2,0}(X,B):=\exp(B)(\widetilde H^{2,0}(X))
\]
and $\widetilde H^{1,1}(X,B)$ is its orthogonal complement in
$H^2(X,\mathbb{C})$. It is clear that $T(X,B)$ inherits from $\widetilde
H(X,B,\mathbb{Z})$ a weight-two Hodge structure. By definition,
$T(X)=T(X,0)$ is the {\it transcendental lattice of $X$} and
$${\rm NS}(X):=T(X)^\perp\subset H^2(X,\mathbb{Z})$$ is the {\it
N{\'e}ron-Severi group of $X$}. The number
$\rho(X):=\mathrm{rk}{\rm NS}(X)$ is the {\it Picard number} of $X$. If
$L_1$ and $L_2$ are lattices endowed with a weight-$k$ Hodge
structure, then an isometry $f:L_1\rightarrow L_2$ is a {\it Hodge
isometry} if it preserves the Hodge structures.

For abelian and K3 surfaces, Orlov proved in \cite{Or1} (using
results in \cite{Mu}) the following theorem:

\begin{thm}\label{thm:orlov1} {\bf(\cite{Or1}, Theorem 3.3.)} Let $X_1$ and $X_2$ be either
abelian or K3 surfaces. Then the following two conditions are
equivalent:

{\rm (i)} there exists an equivalence ${\rm D}^{\rm b}(X_1)\cong{\rm D}^{\rm b}(X_2)$;

{\rm (ii)} there exists a Hodge isometry $T(X_1)\cong
T(X_2)$.\end{thm}

\section{Derived categories of the smooth stacks}\label{sec:equivav}

This section is mainly devoted to the proof of Theorem
\ref{cor:equiv}. As it will turn out, such a proof, which will be
given in Section \ref{subsec:proof}, relies on some results about
the equivariant derived categories of coherent sheaves on abelian
varieties proved in Section \ref{subsec:equivcat}. In Section
\ref{subsec:geomappl1} some geometric applications are discussed.

\subsection{Equivariant derived categories and abelian varieties}\label{subsec:equivcat}
Consider the simple case of an abelian variety $A$ with the action
of $G:=\mathbb{Z}/2\mathbb{Z}$ induced by the automorphism $\iota:A\rightarrow A$
such that $\iota(a)=-a$, for any $a\in A$. A $G$-linearization for a
coherent sheaf ${\cal E}\in\cat{Coh}(A)$ is an isomorphism
$\lambda:{\cal E}\rightarrow\iota^*{\cal E}$ such that
$\iota^*(\lambda)=\lambda$ and
$\iota^*(\lambda)\circ\lambda=\lambda\circ\lambda=\mathrm{id}$.

$\cat{Coh}^G(A)$ is the abelian category whose objects are the pairs
$({\cal E},\lambda)$, where ${\cal E}\in\cat{Coh}(A)$ admits a $G$-linearization
and $\lambda$ is a $G$-linearization for ${\cal E}$. The morphisms in
$\cat{Coh}^G(A)$ are the morphisms in $\cat{Coh}(A)$ compatible with the
$G$-linearizations. We define ${\rm D}^{\rm b}_G(A):={\rm D}^{\rm b}(\cat{Coh}^G(A))$ to be the
bounded derived category of $\cat{Coh}^G(A)$.  A complete discussion
about the general case when $G$ is any finite group acting on a
smooth projective variety can be found in \cite{BKR}.

If $A_1$ and $A_2$ are abelian varieties and $G_\Delta\cong\mathbb{Z}/2\mathbb{Z}$
acts on $A_1\times A_2$ via the automorphism $\iota\times\iota$, the
set of $G_\Delta$-invariant equivalences has the following
description:
\[
{\rm Eq}({\rm D}^{\rm b}(A_1),{\rm D}^{\rm b}(A_2))^{G_{\Delta}}=\{\Phi_{\cal G}\in{\rm Eq}({\rm D}^{\rm b}(A_1),{\rm D}^{\rm b}(A_2)):{\cal G}\in{\rm D}^{\rm b}(A_1\times
A_2)\mbox{ is $G_\Delta$-invariant}\}.\\
\]
An equivalence $\Phi:{\rm D}^{\rm b}_G(A_1)\cong{\rm D}^{\rm b}_G(A_2)$ is a Fourier-Mukai
equivalence if there is an isomorphism as in \eqref{eqn:FMtr}, where
the kernel ${\cal E}$ is in ${\rm D}^{\rm b}_{G\times G}(A_1\times A_2)$.
${\rm Eq}({\rm D}^{\rm b}_G(A_1),{\rm D}^{\rm b}_G(A_2))$ is the set whose elements are the
equivalences of this type.

\begin{prop}\label{thm:main1} Let $A_1$ and $A_2$ be abelian varieties
and let $G=\mathbb{Z}/2\mathbb{Z}$ act on $A_1$ and $A_2$ as above. Then the
restriction
\[
\gamma:{\rm Eq}({\rm D}^{\rm b}(A_1),{\rm D}^{\rm b}(A_2))^{G_\Delta}\longrightarrow{\rm U}(A_1,A_2)
\]
of the map in \eqref{eqn:gamma} is surjective and
${\rm Eq}({\rm D}^{\rm b}_G(A_1),{\rm D}^{\rm b}_G(A_2))$ is non-empty if ${\rm U}(A_1,A_2)$ is
non-empty.\end{prop}

\begin{proof}  By definition, we can think of any $f\in{\rm U}(A_1,A_2)$ as represented by a matrix
\[
\left(\begin{array}{cc} x_f & y_f\\ z_f & w_f\end{array}\right).
\]
Define $S(A_1,A_2):=\{f\in{\rm U}(A_1,A_2): y_f\mbox{ is an isogeny}\}$
and let $f\in S(A_1,A_2)$. Using results from \cite{Mu1}, Orlov
proved in \cite{Or} (see, in particular, \cite[Prop.\ 4.12]{Or})
that there exists a vector bundle ${\cal E}$ on $A_1\times A_2$ with the
following properties:
\begin{itemize}
\item[(a)] ${\cal E}$ is simple and $\Phi_{\cal E}$ is an equivalence;
\item[(b)] for any $(a,b)\in A_1\times A_2$, if $T_{(a,b)}$ is the translation with respect to the point $(a,b)$,
then $T_{(a,b)*}{\cal E}\cong{\cal E}\otimes P$ for some
$P\in\mathrm{Pic}^0(A_1\times A_2)$;
\item[(c)] $\gamma(\Phi_{\cal E})=f$.
\end{itemize}
Consider now the sheaf ${\cal F}:=(\iota\times\iota)^*{\cal E}$. It is clear
that $\gamma(\Phi_{\cal F})=\gamma(\Phi_{\cal E})=f$.

For a brief proof of this fact, consider the maps $\Phi_i$,
$\Phi_{{\cal P}_i}$ and $\mu_i$ in \eqref{eqn:der}. A straightforward
calculation shows that $(\iota\times\iota)^*{\cal P}_i\cong{\cal P}_i$.
Moreover, $\mu_i$ is a morphism of abelian varieties. Hence
$(\iota\times\iota)^*\circ(\mathrm{id}\times\Phi_{{\cal P}_i})\circ(\iota\times\iota)^*=\mathrm{id}\times\Phi_{{\cal P}_i}$
and
$(\iota\times\iota)^*\circ\mu_{i*}\circ(\iota\times\iota)^*=\mu_{i*}$.
This implies that
$(\iota\times\iota)^*\circ\Phi_i\circ(\iota\times\iota)^*=\Phi_i$,
for $i\in\{1,2\}$. Since $\Phi_{\cal F}=\iota^*\circ\Phi_{\cal E}\circ\iota^*$
and
$\Phi_{\cal F}\times\Phi_{{\cal F}_R}=(\iota\times\iota)^*\circ(\Phi_{\cal E}\times\Phi_{{\cal E}_R})\circ(\iota\times\iota)^*$,
we rewrite the commutative diagram \eqref{eqn:der} in the following
way:
\begin{eqnarray}\label{eqn:dia1}
\xymatrix{{\rm D}^{\rm b}(A_1\times\widehat{A_1})\ar[d]_{\Phi_1}\ar[r]_{(\iota\times\iota)^*}
\ar@/^2pc/[rrr]^{F_{\cal F}} &
{\rm D}^{\rm b}(A_1\times\widehat{A_1})\ar[d]_{\Phi_1}\ar[r]^{F_{\cal E}} &
{\rm D}^{\rm b}(A_2\times
\widehat{A_2})\ar[d]_{\Phi_2}\ar[r]_{(\iota\times\iota)^*} &
{\rm D}^{\rm b}(A_2\times\widehat{A_2})\ar[d]_{\Phi_2}\\
{\rm D}^{\rm b}(A_1\times A_1)\ar[r]^{(\iota\times\iota)^*}
\ar@/_2pc/[rrr]^{\Phi_{\cal F}\times\Phi_{{\cal F}_R}}
 & {\rm D}^{\rm b}(A_1\times A_1)\ar[r]^{\Phi_{\cal E}\times\Phi_{{\cal E}_R}} & {\rm D}^{\rm b}(A_2\times
 A_2)\ar[r]^{(\iota\times\iota)^*} & {\rm D}^{\rm b}(A_2\times A_2).}
\end{eqnarray}
By Theorem \ref{thm:orlov}, for any
${\cal G}\in{\rm D}^{\rm b}(A_1\times\widehat{A_1})$,
$F_{\cal F}({\cal G})=f_{{\cal F}*}({\cal G})\otimes N_{\cal F}$, for some
$f_{\cal F}\in{\rm U}(A_1,A_2)$ and $N_{\cal F}\in{\rm Pic}(A_2\times\widehat{A_2})$.
Hence, from \eqref{eqn:dia1} we deduce that
{\setlength\arraycolsep{2pt}
\begin{eqnarray*}
F_{\cal F}({\cal G})&=&((\iota\times\iota)^*\circ
F_{\cal E}\circ(\iota\times\iota)^*)({\cal G})\\&=&((\iota\times\iota)^*\circ
f_{{\cal E}*}\circ(\iota\times\iota)^*)({\cal G})\otimes
M\\&=&f_{{\cal E}*}({\cal G})\otimes M,
\end{eqnarray*}}for some $M\in{\rm Pic}(A_2\times\widehat{A_2})$. Observe that the last
equality holds true because $f_{\cal E}$ is a morphism of abelian
varieties. This proves that
$\gamma(\Phi_{\cal F})=\gamma(\iota^*\circ\Phi_{\cal E}\circ\iota^*)=\gamma(\Phi_{\cal E})$
which is what we claimed.

Due to this last remark and to \cite[Cor.\ 3.4]{Or}, there exist
$a\in A_1$ and $\alpha\in\widehat{A_1}$ such that
\begin{eqnarray}\label{eqn:orlsh}
{\cal F}=T_{(a,0)*}{\cal E}\otimes p^*P_\alpha[i],
\end{eqnarray}
where $p:A_1\times A_2\rightarrow A_1$ is the projection, $i$ is an
integer and $P_\alpha$ is the degree zero line bundle on $A_1$
corresponding to $\alpha$. In the following arguments, without loss
of generality, we will forget about the shift $[i]$ in
\eqref{eqn:orlsh}.

Since ${\cal E}$ satisfies (b), from \eqref{eqn:orlsh} we get
${\cal F}={\cal E}\otimes Q$, where $Q$ is a degree zero line bundle on
$A_1\times A_2$. Let $N\in\mathrm{Pic}^0(A_1\times A_2)$ be such
that $N^2=Q$ and consider the sheaf ${\cal E}_f:={\cal E}\otimes N$. It is
easy to see that
{\setlength\arraycolsep{2pt}
\begin{eqnarray*}
(\iota\times\iota)^*({\cal E}_f) & =&(\iota\times\iota)^*({\cal E}\otimes
N)\nonumber \\
 & \cong&{\cal E}\otimes Q\otimes N^\vee\\
 & \cong&{\cal E}\otimes N\\
 & =&{\cal E}_f.\nonumber
\end{eqnarray*}}Due to \cite[Prop.\ 3.3]{Or} and to (c),
$\gamma(\Phi_{{\cal E}_f})=\gamma(\Phi_{\cal E})=f$.

Let $f\in{\rm U}(A_1,A_2)$. Orlov observed in \cite[Sect.\ 4]{Or} that
there exist $g_1\in S(A_1,A_2)$ and $g_2\in S(A_2,A_2)$ such that
$f=g_2\circ g_1$. From its very definition, the map $\gamma$ in
Theorem \ref{thm:orlov} preserves the compositions. Hence $\gamma$
restricts to a surjective map
$\gamma:{\rm Eq}({\rm D}^{\rm b}(A_1),{\rm D}^{\rm b}(A_2))^{G_\Delta}\rightarrow{\rm U}(A_1,A_2)$.

To prove the second claim in Proposition \ref{thm:main1}, consider
the set
\[
{\rm Ker}(A_1,A_2,G_\Delta):=\{({\cal G},\lambda)\in{\rm D}^{\rm b}_{G_\Delta}(A_1\times
A_2):\Phi_{\cal G}\in{\rm Eq}({\rm D}^{\rm b}(A_1),{\rm D}^{\rm b}(A_2))\}.
\]
Since the group cohomology
$H^2(\mathbb{Z}/2\mathbb{Z},\mathbb{C}^*)$ is trivial, \cite[Thm.\
6]{Pl} shows the existence of two maps
\[
\begin{array}{l}
\psi_1:{\rm Ker}(A_1,A_2,G_\Delta)\longrightarrow{\rm Eq}({\rm D}^{\rm b}(A_1),{\rm D}^{\rm b}(A_2))^{G_{\Delta}}\\
\psi_2:{\rm Ker}(A_1,A_2,G_\Delta)\longrightarrow{\rm Eq}({\rm D}^{\rm b}_G(A_1),{\rm D}^{\rm b}_G(A_2))
\end{array}
\]
such that, for any $({\cal G},\lambda)\in{\rm Ker}(A_1,A_2,G_\Delta)$,
$\psi_1(({\cal G},\lambda))=\Phi_{\cal G}$ and
$\psi_2(({\cal G},\lambda))=\Phi_{\cal H}$, where
${\cal H}:=({\cal G}\oplus(\iota,\mathrm{id})^*{\cal G},\lambda')$ and $\lambda'$
is the natural $(G\times G)$-linearization induced by $\lambda$.

We previously proved that for any $f\in S(A_1,A_2)$, there exists
$\Phi_{{\cal E}_f}\in{\rm Eq}({\rm D}^{\rm b}(A_1),{\rm D}^{\rm b}(A_2))^{G_{\Delta}}$ such that
$\gamma(\Phi_{{\cal E}_f})=f$. From \cite{Pl} it follows that $\psi_1$
is surjective and that the set ${\rm Ker}(A_1,A_2,G_\Delta)$ is
non-empty if ${\rm Eq}({\rm D}^{\rm b}(A_1),{\rm D}^{\rm b}(A_2))^{G_{\Delta}}$ is non-empty.
Hence, there exists $\Psi_f\in{\rm Ker}(A_1,A_2,G_\Delta)$ such that
$\psi_1(\Psi_f)=\Phi_{{\cal E}_f}$. The functor $\psi_2(\Psi_f)$ is in
${\rm Eq}({\rm D}^{\rm b}_G(A_1),{\rm D}^{\rm b}_G(A_2))$.\end{proof}

The special case $A_1=A_2$ is also treated in \cite{Pl}.

\subsection{Proof of Theorem \ref{cor:equiv}}\label{subsec:proof} Let $A_1$ and $A_2$
be abelian varieties and suppose that ${\rm D}^{\rm b}(A_1)\cong{\rm D}^{\rm b}(A_2)$. Due
to Theorem \ref{thm:orlov}, the set ${\rm U}(A_1,A_2)$ is non-empty.
Therefore, if $G=\mathbb{Z}/2\mathbb{Z}$ acts on $A_1$ and $A_2$ as prescribed
at the beginning of Section \ref{subsec:equivcat}, then
Proposition \ref{thm:main1} yields an equivalence
$\Psi:{\rm D}^{\rm b}_{G}(A_1)\mor[\sim]{\rm D}^{\rm b}_{G}(A_2)$.

Consider the stacks $[A_1/G]$ and $[A_2/G]$ (see \cite{Go} and
\cite{Ka}). For any $i\in\{1,2\}$, let ${\rm D}^{\rm b}([A_i/G])$ be the bounded
derived category of the abelian category $\cat{Coh}([A_i/G])$ of coherent
sheaves on $[A_i/G]$ (see \cite{Ka}). Obviously
${\rm D}^{\rm b}([A_i/G])\cong{\rm D}^{\rm b}_G(A_i)$, because
$\cat{Coh}([A_i/G])\cong\cat{Coh}^G(A_i)$, for any $i\in\{1,2\}$. This implies
that $\Psi$ can be rewritten as
$\Phi:{\rm D}^{\rm b}([A_1/G])\mor[\sim]{\rm D}^{\rm b}([A_2/G])$. Due to \cite{Ka}, $\Phi$ is
of Fourier-Mukai type (i.e.\ there is an isomorphism as in
\eqref{eqn:FMtr}). Hence, the first part of Theorem \ref{cor:equiv}
is proved.

Assume that an equivalence $\Phi:{\rm D}^{\rm b}([A_1/G])\mor[\sim]{\rm D}^{\rm b}([A_2/G])$
is given. As before, the results in \cite{Ka} imply that we can
think of $\Phi$ as a Fourier-Mukai equivalence whose kernel is a
$(G\times G)$-linearized complex $({\cal E},\lambda)$. Obviously, the
inverse $\Phi^{-1}$ is a Fourier-Mukai equivalence as well.
Suppose that its kernel is $({\cal F},\lambda')$. It is an easy
exercise to show that the kernel of the identity functor
$\mathrm{id}=\Phi\circ\Phi^{-1}:{\rm D}^{\rm b}_G(A_i)\rightarrow{\rm D}^{\rm b}_G(A_i)$
is the $(G\times G)$-linearized sheaf $({\cal O}_\Delta\oplus
(\iota,\mathrm{id})^*{\cal O}_\Delta,\mu)$, where $\mu$ is the natural
linearization and $\Delta\hookrightarrow A_i\times A_i$ is the
diagonal embedding.

Consider the functors $\Phi_{\cal E}$, $\Phi_{\cal F}$ and $\Phi_{{\cal O}_\Delta\oplus
(\iota,\mathrm{id})^*{\cal O}_\Delta}$. Although they are
no longer equivalences, they induce the commutative diagram
\begin{eqnarray}\label{eqn:import}
\xymatrix{
{\rm D}^{\rm b}(A_1)\ar[d]_{\mathrm{ch}}\ar[r]^{\Phi_{\cal E}}\ar@/^2pc/[rr]^{\Phi_{{\cal O}_\Delta\oplus
(\iota,\mathrm{id})^*{\cal O}_\Delta}} &
{\rm D}^{\rm b}(A_2)\ar[d]_{\mathrm{ch}}\ar[r]^{\Phi_{\cal F}} &
{\rm D}^{\rm b}(A_1)\ar[d]_{\mathrm{ch}}\\
H^{2*}(A_1,\mathbb{Q})\ar[r]^{\Phi_{\cal E}^H}
\ar@/_2pc/[rr]_{\Phi_{{\cal O}_\Delta\oplus
(\iota,\mathrm{id})^*{\cal O}_\Delta}^H}
 & H^{2*}(A_2,\mathbb{Q})\ar[r]^{\Phi_{\cal F}^H} & H^{2*}(A_1,\mathbb{Q}),}
\end{eqnarray}
where $\Phi_{\cal E}^H:H^{2*}(A_1,\mathbb{Q})\rightarrow H^{2*}(A_2,\mathbb{Q})$ is
such that $\Phi^H_{\cal E}(a)=p_{2*}(\mathrm{ch}({\cal E})\cdot p_1^*(a))$ and
$p_i:A_1\times A_2\rightarrow A_i$ is the projection. Take analogous
definitions for $\Phi_{\cal F}^H$ and $\Phi_{{\cal O}_\Delta\oplus
(\iota,\mathrm{id})^*{\cal O}_\Delta}^H$.

Observe that $(\iota,\mathrm{id})^*{\cal O}_\Delta$ is the kernel of the
Fourier-Mukai equivalence $\iota^*:{\rm D}^{\rm b}(A_i)\mor[\sim]{\rm D}^{\rm b}(A_i)$. Since
$\iota^*$ acts as the identity on the cohomology lattice $\widetilde
H(A_i,\mathbb{Z})$, from \eqref{eqn:import} we deduce
$$\Phi_{\cal F}^H\circ\Phi_{\cal E}^H=(\Phi_{{\cal O}_\Delta\oplus
(\iota,\mathrm{id})^*{\cal O}_\Delta})^H=2\mathrm{id}.$$ Hence
$\Phi_{\cal E}^H$ is injective. Exchanging the roles of $\Phi_{\cal E}$ and
$\Phi_{\cal F}$ in \eqref{eqn:import}, we see that $\Phi_{\cal E}^H$ is an
isomorphism of $\mathbb{Q}$-vector spaces. In particular,
$\mathrm{dim}(A_1)=\mathrm{dim}(A_2)=n$.

The fact that the Hodge structures defined in \eqref{eqn:HSnuova}
are preserved follows from the standard argument for Fourier-Mukai
equivalences (see \cite[Prop.\ 5.39]{Hu1}). Indeed, one just needs
to observe that $\mathrm{ch}({\cal E})\in\widetilde H^{2n,2n}(A_1\times
A_2)$. This concludes the proof of Theorem \ref{cor:equiv}.

\begin{remark} Of course, in general, $\Phi_{\cal E}^H$ does not preserve the Mukai
pairing naturally defined on $H^{2*}(A_i,\mathbb{Q})$ by means of the cup
product (\cite[Chapter 5]{Hu1}). Indeed, it is easy to see that the
Mukai pairing is preserved up to a factor 2.\end{remark}

\subsection{Geometric applications}\label{subsec:geomappl1} Assume that $A_1$ and $A_2$
are abelian surfaces. The main result in \cite{BKR} yields an
equivalence
$\Psi_i:{\rm D}^{\rm b}([A_i/\langle\iota\rangle])\mor[\sim]{\rm D}^{\rm b}({\rm Km}(A_i))$, for any
$i\in\{1,2\}$. Thus, if $\Upsilon:{\rm D}^{\rm b}(A_1)\mor[\sim]{\rm D}^{\rm b}(A_2)$ is a
Fourier-Mukai equivalence, we immediately get a second Fourier-Mukai
equivalence
\begin{eqnarray}\label{eqn:BKR1}
\Phi:{\rm D}^{\rm b}({\rm Km}(A_1))\stackrel{\Psi_1^{-1}}{\longrightarrow}{\rm D}^{\rm b}([A_1/\langle\iota\rangle])\mor[\sim]{\rm D}^{\rm b}([A_2/\langle\iota\rangle])\stackrel{\Psi_2}{\longrightarrow}{\rm D}^{\rm b}({\rm Km}(A_2)),
\end{eqnarray}
where the middle equivalence is produced by Theorem \ref{cor:equiv}
and the kernel of $\Phi$ can be easily computed using \cite{BKR}.
This leads to a different and explicit proof of the ``only if''
implication in (A) without using the lattice theoretical description
of the transcendental lattices of an abelian surface and of the
associated Kummer surface.

Let us discuss a second geometric application. Assume that $A$ is
an abelian surface. We denote by $\mathrm{K}^n(A)$ the {\it $n$-th
generalized Kummer variety of $A$}. Recalling the construction in
\cite{Be}, we see that $\mathrm{K}^n(A)$ is the fiber over $0$
with respect to the map $\Psi$ which is the composition of the
morphisms in the following diagram:
\[
\Psi:\mathrm{Hilb}^{n+1}(A)\stackrel{\rho}{\longrightarrow}
\mathrm{Sym}^{n+1}(A)\stackrel{\sigma}{\longrightarrow} A,
\]
where $\rho$ is the Hilbert-Chow morphism and
$\sigma(a_1,\ldots,a_{n+1})=a_1+\ldots+a_{n+1}$. It is easy to
see that ${\rm K}^n(A)$ is smooth and that $\mathrm{K}^1(A)={\rm Km}(A)$.
Furthermore, in \cite{Be} Beauville proved that these varieties are
examples of irreducible symplectic manifolds.

\begin{prop}\label{cor:genKum} Let $A$ be an abelian surface and let
$n\geq 2$ be an integer. The number of generalized Kummer
varieties ${\rm K}^n(B)$ birational to ${\rm K}^n(A)$ is finite up to
isomorphisms. Moreover if ${\rm K}^n(A)$ and ${\rm K}^n(B)$ are birational,
then ${\rm Km}(A)\cong{\rm Km}(B)$ and $A$ and $B$ are isogenous.\end{prop}

\begin{proof} Let $A_1$ and $A_2$ be abelian surfaces and let $\varphi$ be
a birational morphism between $\mathrm{K}^n(A_1)$ and
$\mathrm{K}^n(A_2)$. Obviously, $\varphi$ induces an isomorphism
$g:H^2(\mathrm{K}^n(A_1),\mathbb{Z})\mor[\sim]
H^2(\mathrm{K}^n(A_2),\mathbb{Z})$. Furthermore, there exists an
isometry of lattices $H^2(\mathrm{K}^n(A_i),\mathbb{Z})\mor[\sim]
H^2(A_i,\mathbb{Z})\oplus\mathbb{Z}[E_i]$, where $E_i$ is the
restriction to ${\rm K}^n(A)$ of the exceptional locus of
$\mathrm{Hilb}^{n+1}(A_i)$. The left hand side of the isomorphism is
endowed with the Beauville-Bogomolov form  while the quadratic form
on $H^2(A_i,\mathbb{Z})$ is the cup-product (see \cite[Lemma
4.10]{Be} and \cite[Prop.\ 4.11]{Y}).

Since $E_1$ and $E_2$ are algebraic, $g$ yields an isomorphism
$T(A_1)\cong T(A_2)$. Using Theorem \ref{thm:orlov1}, we get an
equivalence ${\rm D}^{\rm b}(A_1)\cong{\rm D}^{\rm b}(A_2)$. To prove that $A_1$ is
isogenous to $A_2$ observe that if ${\rm D}^{\rm b}(A_1)\cong{\rm D}^{\rm b}(A_2)$, then
$A_1\times\widehat{A_1}\cong A_2\times\widehat{A_2}$ (Theorem
\ref{thm:orlov}). Hence $A_1\times A_1$ and $A_2\times A_2$ are
isogenous and $A_1$ and $A_2$ are isogenous as well. On the other
hand, as there are only finitely many isomorphism classes of abelian
surfaces $A$ such that ${\rm D}^{\rm b}(A)\cong{\rm D}^{\rm b}(A_1)$ (see \cite[Prop.\
5.3]{BM}), the number of generalized Kummer varieties ${\rm K}^n(A_2)$
birational to $\mathrm{K}^n(A_1)$ is finite up to isomorphism.
Moreover, Theorem \ref{cor:equiv} yields an equivalence
${\rm D}^{\rm b}([A_1/\langle\iota\rangle])\cong{\rm D}^{\rm b}([A_2/\langle\iota\rangle])$.
Due to \eqref{eqn:BKR1} and Theorem \ref{cor:equiv},
${\rm D}^{\rm b}({\rm Km}(A))\cong{\rm D}^{\rm b}({\rm Km}(B))$ and then ${\rm Km}(A)\cong{\rm Km}(B)$ (see
\cite{Mu}).\end{proof}

An analogous result for Hilbert schemes of points on K3 surfaces was proved in \cite{Pl}.

\begin{remark}\label{rmk:cont} Observe that, in general, if $A$ and $B$ are abelian
surfaces such that ${\rm Km}(A)\cong{\rm Km}(B)$, then ${\rm K}^n(A)$ and
${\rm K}^n(B)$ are not necessarily birational. Indeed, consider an
abelian surface $A$ such that $A\not\cong\widehat A$ and
${\rm NS}(A)=\langle H\rangle$ with $H^2=6$. Obviously
${\rm D}^{\rm b}(A)\cong{\rm D}^{\rm b}(\widehat A)$. Due to Theorem \ref{cor:equiv},
${\rm D}^{\rm b}({\rm Km}(A))\cong{\rm D}^{\rm b}({\rm Km}(\widehat A))$ and
${\rm Km}(A)\cong{\rm Km}(\widehat A)$ (\cite{Mu}). On the other hand,
Namikawa (\cite[Sect.\ 5]{Na}) proved that ${\rm K}^2(A)$ and
${\rm K}^2(\widehat A)$ are not birational.

Furthermore, Example \ref{ex:geom} yields very explicit examples of
isogenous abelian surfaces $A$ and $B$ such that
${\rm D}^{\rm b}(A)\not\cong{\rm D}^{\rm b}(B)$. In particular ${\rm Km}(A)\not\cong{\rm Km}(B)$ and
${\rm K}^n(A)$ is not birational to ${\rm K}^n(B)$ for any positive integer
$n$.\end{remark}

\section{Derived categories of twisted Kummer surfaces}\label{sec:brauer}

In this section we prove Theorem \ref{thm:main2} which relates the
existence of equivalences between the twisted derived categories of
two Kummer surfaces and the existence of Hodge isometries between
the generalized transcendental lattices of the corresponding abelian
surfaces. More precisely, in Section \ref{subsec:Br} we introduce
the Brauer group of smooth projective varieties and we prove a
preliminary result (Lemma \ref{lemma:brK3ab}) which will be used in
the proof of Theorem \ref{thm:main2} contained in Section
\ref{subsec:proofsec}. We also discuss a geometric example and an
application to the problem of determining the number of possible
twisted Kummer structures on a twisted K3 surface (respectively in
Sections \ref{ex:geom} and \ref{subsec:num}).

\subsection{Brauer groups and twisted sheaves}\label{subsec:Br} Recall that the {\it Brauer group} ${\rm Br}(X)$ of a smooth projective
variety $X$ is the torsion part of $H^2(X,\mathcal{O}_X^*)$ in the
analytic topology (or, equivalently,
$H^2_{\mathrm{\acute{e}t}}(X,{\cal O}_X^*)$ in the \'{e}tale topology)
(see \cite{C1,dJ}).

Assume that $X$ is either a K3 or an abelian surface. It is known
that any $\alpha\in{\rm Br}(X)$ is determined (not uniquely) by some
$B\in H^2(X,\mathbb{Q})$ (see Chapter 1 of \cite{C1} for the case of K3
surfaces and use a similar argument to deal with abelian surfaces).
This follows from the fact that $H^2(X,\mathbb{Z})$ is unimodular and
$H_1(X,\mathbb{Z})$ is torsion free. More precisely, we deduce the
existence of natural isomorphisms ${\rm Br}(X)\cong
T(X)^\vee\otimes\mathbb{Q}/\mathbb{Z}\cong{\rm
Hom}(T(X),\mathbb{Q}/\mathbb{Z})$ and for any $t\in T(X)$,
$\alpha:t\longmapsto t\cdot B\pmod{\mathbb{Z}}$, where ``$\cdot$'' is the
cup-product. From this we get a surjective map
\[
\kappa_X:H^2(X,\mathbb{Q})\longrightarrow{\rm Br}(X).
\]

\begin{lem}\label{lemma:brK3ab}  If $A$ is an abelian surface, there exists an
isomorphism $\Theta_A:{\rm Br}(A)\rightarrow{\rm Br}({\rm Km}(A))$.\end{lem}

\begin{proof} The K3 surface ${\rm Km}(A)$ is the crepant resolution of
${\rm K}(A)=A/\langle\iota\rangle$. Hence there exists a rational map
$\pi:A\dashrightarrow{\rm Km}(A)$. Furthermore, as it was observed in
Remark 2 of \cite{Ni1} (see also \cite[Sect.\ 4]{Mo}), the
homomorphism $\pi_{*}$ induces a Hodge isometry
\begin{eqnarray}\label{eqn:trasc}
\pi_{*}:T(A)(2)\longrightarrow T({\rm Km}(A)).
\end{eqnarray}
(Recall that, given a lattice $L$ with quadratic form $b_L$, the
lattice $L(m)$, with $m\in\mathbb{Z}$, coincides with $L$ as a group but
its quadratic form $b_{L(m)}$ is such that
$b_{L(m)}(l_1,l_2)=mb_L(l_1,l_2)$, for any $l_1,l_2\in L$.)

In particular, we get a natural morphism
$\Xi:H^2(A,\mathbb{Q})\rightarrow T({\rm Km}(A))\otimes\mathbb{Q}$ defined by
\begin{eqnarray}\label{eqn:Theta}
\Xi:B\longmapsto\frac{\pi_{*}(p(B))}{2},
\end{eqnarray}
where $p:H^2(A,\mathbb{Q})\rightarrow T(A)\otimes\mathbb{Q}$ is the orthogonal
projection. This yields a morphism
$\Theta_A:{\rm Br}(A)\longrightarrow{\rm Br}({\rm Km}(A))$ of Brauer groups defined
by the commutative diagram
\begin{eqnarray}\label{eqn:diaTheta}
\xymatrix{H^2(A,\mathbb{Q})\ar[r]^{\Xi}\ar@{>>}[d]_{\kappa_A}&
T({\rm Km}(A))\otimes\mathbb{Q}\ar[d]^{\kappa_{{\rm Km}(A)}}\\
{\rm Br}(A)\ar[r]^{\Theta_A}&{\rm Br}({\rm Km}(A)).}
\end{eqnarray}
Observe that $\Theta_A$ is well-defined because, obviously, the
restriction $\kappa_{{\rm Km}(A)}|_{T({\rm Km}(A))\otimes\mathbb{Q}}$ is still
surjective. An easy check then shows that $\Theta_A$ is an
isomorphism.\end{proof}

Any $\alpha\in{\rm Br}(X)$ can be represented by a \v{C}ech 2-cocycle on
an analytic cover $\{U_i\}_{i\in I}$ of $X$ using sections
$\alpha_{ijk}\in\Gamma(U_i\cap U_j\cap U_k,{\cal O}^*_X)$. An
\emph{$\alpha$-twisted coherent sheaf} ${\cal F}$ is a pair
$(\{{\cal F}_i\}_{i\in I},\{\varphi_{ij}\}_{i,j\in I})$, where ${\cal F}_i$ is
a coherent sheaf on $U_i$ and $\varphi_{ij}:{\cal F}_j|_{U_i\cap
U_j}\to{\cal F}_i|_{U_i\cap U_j}$ is an isomorphism such that
$\varphi_{ii}=\mathrm{id}$, $\varphi_{ji}=\varphi_{ij}^{-1}$ and
$\varphi_{ij}\circ\varphi_{jk}\circ\varphi_{ki}=\alpha_{ijk}\mathrm{id}$.
Given $\alpha\in{\rm Br}(X)$, we denote by $\cat{Coh}(X,\alpha)$ the abelian
category of $\alpha$-twisted coherent sheaves on $X$ while
${\rm D}^{\rm b}(X,\alpha):={\rm D}^{\rm b}(\cat{Coh}(X,\alpha))$ is the bounded derived category
of $\cat{Coh}(X,\alpha)$ (see \cite{C1} and \cite{HS2} for details).

If $X$ and $Y$ are smooth projective varieties and $\alpha\in{\rm Br}(X)$
while $\beta\in{\rm Br}(Y)$, an equivalence
$\Phi:{\rm D}^{\rm b}(X,\alpha)\rightarrow{\rm D}^{\rm b}(Y,\beta)$ is a {\it twisted
Fourier-Mukai equivalence} if and only if it there is an isomorphism
as in \eqref{eqn:FMtr} whose kernel ${\cal E}$ is in ${\rm D}^{\rm b}(X\times Y,
\alpha^{-1}\boxtimes\beta)$ (see also \cite{CS}).

As in \cite{HS1}, a \emph{twisted variety} is a pair $(X,\alpha)$,
where $X$ is a smooth projective variety and $\alpha\in{\rm Br}(X)$. An
isomorphism $f:(X,\alpha)\rightarrow (Y,\beta)$ of the twisted
varieties $(X,\alpha)$ and $(Y,\beta)$ is an isomorphism
$f:X\rightarrow Y$ such that $f^*\beta=\alpha$.

\subsection{Proof of Theorem \ref{thm:main2}}\label{subsec:proofsec} First
of all, observe that, if $X$ is either a K3 or an abelian surface
and $\alpha\in{\rm Br}(X)$, the lattice $T(X,\alpha):=\ker(\alpha)$
inherits from $T(X)$ a weight-two Hodge structure. Secondly, if
$\Theta_{A_i}:{\rm Br}(A_i)\rightarrow{\rm Br}({\rm Km}(A_i))$ is the isomorphism
in Lemma \ref{lemma:brK3ab}, the isometry
$\pi_{i*}:T(A_i)(2)\rightarrow T({\rm Km}(A_i))$ defined in
\eqref{eqn:trasc} yields a Hodge isometry
$$f_i:T(A_i,\alpha)(2)\longrightarrow
T({\rm Km}(A_i),\Theta_{A_i}(\alpha)),$$ for any $\alpha\in{\rm Br}(A_i)$
and $i\in\{1,2\}$.

Proposition 4.7 in \cite{Hu}, originally proved for K3 surfaces,
works perfectly in the case of abelian surfaces as well. Therefore
if $X$ is either a K3 or an abelian surface, $\alpha\in{\rm Br}(X)$ and
$B\in H^2(X,\mathbb{Q})$ is such that $\alpha=\kappa_X(B)$, then there
exists a Hodge isometry
\begin{eqnarray}\label{eqn:Huy}
\begin{array}{rcl}
\exp(B):T(X,\alpha)(k)&\longrightarrow&T(X,B)(k)\\
\gamma&\longmapsto&(\gamma,B\wedge\gamma),
\end{array}
\end{eqnarray}
for any $k\in\{1,2\}$. Given $B_i\in H^2(A_i,\mathbb{Q})$, let $\widetilde
B_i\in H^2({\rm Km}(A_i),\mathbb{Q})$ be such that
$$\Theta_{A_i}(\kappa_{A_i}(B_i))=\kappa_{{\rm Km}(A_i)}(\widetilde B_i).$$ Define
$\widetilde\alpha_i:=\Theta_{A_i}(\kappa_{A_i}(B_i))$. If $g:
T(A_1,B_1)\rightarrow T(A_2,B_2)$ is a Hodge isometry, the diagram
\[
\xymatrix{T(A_1,B_1)(2)\ar[rrr]^{g}\ar[ddd]\ar[dr]_{\exp(-B_1)}&&&
T(A_2,B_2)(2)\ar[ddd]\ar[dl]^{\exp(-B_2)}\\
& T(A_1,\kappa_{A_1}(B_1))(2)\ar[d]_{f_1}\ar[r]& T(A_2,\kappa_{A_2}(B_2))(2)\ar[d]^{f_2}&\\
& T({\rm Km}(A_1),\widetilde\alpha_1)\ar[r]& T({\rm Km}(A_2),\widetilde\alpha_2)&\\
T({\rm Km}(A_1),\widetilde B_1)\ar[rrr]^{f}\ar[ur]^{\exp(\widetilde
B_1)}&&& T({\rm Km}(A_2),\widetilde B_2)\ar[ul]_{\exp(\widetilde B_2)}}
\]
commutes and yields a Hodge isometry $f: T({\rm Km}(A_1),\widetilde
B_1)\rightarrow T({\rm Km}(A_2),\widetilde B_2)$. Conversely, since
$\Theta_i$ is an isomorphism (Lemma \ref{lemma:brK3ab}), the same
diagram and remarks show that any Hodge isometry between the
generalized transcendental lattices of ${\rm Km}(A_1)$ and ${\rm Km}(A_2)$
determined by some $\widetilde B_i\in H^2({\rm Km}(A_i),\mathbb{Q})$ induces a
Hodge isometry of the generalized transcendental lattices of $A_1$
and $A_2$ corresponding to $B_i\in H^2(A_i,\mathbb{Q})$ such that
$$\kappa_{A_i}(B_i)=\Theta_{A_i}^{-1}(\kappa_{{\rm Km}(A_i)}(\widetilde
B_i))\in{\rm Br}(A_i).$$

Since the Picard number of ${\rm Km}(A_i)$ is greater than 11, the
equivalence between item (i) and item (ii) of Theorem
\ref{thm:main2} follows from \cite[Thm.\ 0.4]{HS1}. Indeed such a
result proves that, for any $B_i\in H^2({\rm Km}(A_i),\mathbb{Q})$, there
exists a twisted Fourier-Mukai equivalence
\[
{\rm D}^{\rm b}({\rm Km}(A_1),\kappa_{{\rm Km}(A_1)}(B_1))\cong{\rm D}^{\rm b}({\rm Km}(A_2),\kappa_{{\rm Km}(A_2)}(B_2))
\]
if and only if there exists a Hodge isometry
$T({\rm Km}(A_1),B_1)\cong T({\rm Km}(A_2),B_2)$.

Due to what we have just proved, any twisted Fourier-Mukai
equivalence ${\rm D}^{\rm b}({\rm Km}(A_1),\alpha_1)\cong{\rm D}^{\rm b}({\rm Km}(A_2),\alpha_2)$
induces a Hodge isometry $T({\rm Km}(A_1))\otimes\mathbb{Q}\cong
T({\rm Km}(A_2))\otimes\mathbb{Q}$ which yields a Hodge isometry
$T(A_1)\otimes\mathbb{Q}\cong T(A_2)\otimes\mathbb{Q}$. Consider the Kuga-Satake
varieties ${\rm KS}(A_1)$ and ${\rm KS}(A_2)$ associated to the weight-two
Hodge structures on $T(A_1)\otimes\mathbb{Q}$ and $T(A_2)\otimes\mathbb{Q}$ (see
Section 4 in \cite{Mo1} for the definition). Theorem 4.3 and Lemma
4.4 in \cite{Mo1} show that, for any $i\in\{1,2\}$,
$$\underbrace{{\rm KS}(A_i)\times\cdots\times{\rm KS}(A_i)}_{2^{\rho(A_i)}\mbox{ times}}\sim\underbrace{A_i\times\cdots\times A_i}_{8\mbox{ times}},$$ where ``$\sim$'' denotes
an isogeny of abelian varieties. By construction
${\rm KS}(A_1)\sim{\rm KS}(A_2)$ and then $A_1^8\sim A_2^8$. In particular,
$A_1$ and $A_2$ are isogenous and this concludes the proof of
Theorem \ref{thm:main2}.

To shorten the notation and according to \cite{HS1}, we introduce
two equivalence relations:

\begin{definition}\label{def:DTequiv} Let $(X_1,\alpha_1)$ and $(X_2,\alpha_2)$ be twisted K3
or abelian surfaces.
\begin{itemize}
\item[(i)] They are {\it $D$-equivalent} if there exists a twisted
Fourier-Mukai equivalence
$$\Phi:{\rm D}^{\rm b}(X_1,\alpha_1)\rightarrow{\rm D}^{\rm b}(X_2,\alpha_2).$$

\item[(ii)] They are {\it $T$-equivalent} if there exist $B_i\in
H^2(X_i,\mathbb{Q})$ such that $\alpha_i=\kappa_{A_i}(B_i)$ and a Hodge
isometry
$$\varphi:T(X_1,B_1)\rightarrow
T(X_2,B_2).$$\end{itemize}\end{definition}

We can now prove the following easy corollary of Theorem
\ref{thm:main2}:

\begin{cor}\label{cor:cons} {\rm (i)} $({\rm Km}(A_1),1)$ is
D-equivalent to $({\rm Km}(A_2),1)$ if and only if $(A_1,1)$ and
$(A_2,1)$ are T-equivalent.

{\rm (ii)} If $(A_1,\alpha_1)$ and $(A_2,\alpha_2)$ are D-equivalent
twisted abelian surfaces, then $({\rm Km}(A_1),\Theta_{A_1}(\alpha_1))$
and $({\rm Km}(A_2),\Theta_{A_2}(\alpha_2))$ are D-equivalent.\end{cor}

\begin{proof} Due to the isomorphism in Lemma \ref{lemma:brK3ab}, (i) follows trivially from Theorem \ref{thm:main2}. The machinery in \cite{HS1} applied to the case of abelian surfaces shows that if $(A_1,\alpha_1)$ and $(A_2,\alpha_2)$ are D-equivalent, then they are T-equivalent as well. Then use Theorem \ref{thm:main2}.\end{proof}

Notice that part (i) of Corollary \ref{cor:cons} is exactly the
analogue of (B) in the introduction.

\begin{remark}\label{rmk:commenti} (i) Due to \cite[Prop.\ 8.1]{HS1},
if $\alpha_j\in{\rm Br}({\rm Km}(A_j))$ is non-trivial for any
$j\in\{1,2\}$, then the existence of an equivalence
${\rm D}^{\rm b}({\rm Km}(A_1),\alpha_1)\cong{\rm D}^{\rm b}({\rm Km}(A_2),\alpha_2)$ does not imply
that ${\rm Km}(A_1)\cong{\rm Km}(A_2)$ (see also Example \ref{ex:geom}). This is one of the main differences
with the untwisted case treated by Hosono, Lian, Oguiso and Yau in
\cite{HLOY} (see (A) and (C) in the introduction).

(ii) As suggested by Corollary \ref{cor:cons}, we would expect (ii)
in Theorem \ref{thm:main2} to be equivalent to the existence of a
twisted Fourier-Mukai equivalence
${\rm D}^{\rm b}(A_1,\beta_1)\cong{\rm D}^{\rm b}(A_2,\beta_2)$, where $\beta_i\in{\rm Br}(A_i)$.
This would lead to a twisted version of (A). Actually this is not
the case. Indeed, since the period map is surjective for abelian
surfaces (\cite{Sh}), one can produce a counterexample to this
expectation by adapting Example 4.11 in \cite{HS1}.

(iii) Let $A_1$ and $A_2$ be two abelian surfaces with
${\rm NS}(A_1)=\langle H_1\rangle$ and ${\rm NS}(A_2)=\langle H_2\rangle$. If
there exist $\alpha_1\in{\rm Br}({\rm Km}(A_1))$ and
$\alpha_2\in{\rm Br}({\rm Km}(A_2))$ such that
${\rm D}^{\rm b}({\rm Km}(A_1),\alpha_1)\cong{\rm D}^{\rm b}({\rm Km}(A_2),\alpha_2)$ then
$H_1^2/H_2^2$ is a square in $\mathbb{Q}$. Indeed, by Theorem
\ref{thm:main2} (and by \cite[Sect.\ 7]{HS1}), if
${\rm D}^{\rm b}({\rm Km}(A_1),\alpha_1)\cong{\rm D}^{\rm b}({\rm Km}(A_2),\alpha_2)$ then there
exists an isogeny $\varphi:A_1\rightarrow A_2$ inducing a Hodge
isometry $\varphi^*:H^2(A_2,\mathbb{Q})\rightarrow H^2(A_1,\mathbb{Q})$ such that
$\varphi^*(H_2)=qH_1$, for some $q\in\mathbb{Q}$. In particular
$H_2^2=q^2H_1^2$.\end{remark}

\subsection{An explicit example}\label{ex:geom} In this example, we use Theorem
\ref{thm:main2} to establish a connection between the twisted
derived categories of some nice Kummer surfaces with Picard number
2. Recall that the lattices $U$ and $U(n)$ are the free abelian
group $\mathbb{Z}\oplus\mathbb{Z}$ endowed respectively with the quadratic forms
represented by the matrices
\[
\left(\begin{array}{cc} 0 & 1\\ 1 &
0\end{array}\right)\;\;\;\mbox{and}\;\;\;\left(\begin{array}{cc} 0 & n\\
n & 0\end{array}\right).
\]

Let $A$ be an abelian surface such that ${\rm NS}(A)\cong U(n)$, for
some positive integer $n$. We first show that there exist two
elliptic curves $E$ and $F$ and a subgroup $C_n\cong\mathbb{Z}/n\mathbb{Z}$ of
$E\times F$ such that either $A\cong(E\times F)/C_n$ or
$\widehat{A}\cong(E\times F)/C_n$.

To see this, let us first observe that, since ${\rm NS}(A)\cong U(n)$,
the transcendental lattice $T(A)$ is isometric to $U(n)\oplus U$.
Indeed for any abelian surface $A$, $H^2(A,\mathbb{Z})$, endowed with the
cup-product, is isometric to the lattice $U\oplus U\oplus U$ (see
\cite{Mo} for more details).

We choose a basis $\langle e_1,e_2,f_1,f_2\rangle=U\oplus
U(n)\hookrightarrow U^3$, an isometry $\varphi:H^2(A,\mathbb{Z})\rightarrow
U^3$ and $c\in\mathbb{C}$ such that
\begin{eqnarray}\label{eqn:period}
\varphi(c\sigma_A)=e_1-n\omega_1\omega_2e_2+\omega_1f_1+\omega_2f_2,
\end{eqnarray}
where $H^{2,0}(A)=\langle\sigma_A\rangle$. We define in $\mathbb{C}$ the
lattices $\Gamma_1=\mathbb{Z}+\omega_1\mathbb{Z}$ and $\Gamma_2=\mathbb{Z}+\omega_2\mathbb{Z}$
and the elliptic curves $E:=\mathbb{C}/\Gamma_1$ and $F:=\mathbb{C}/\Gamma_2$.
Notice that, since $T(A)\cong U(n)\oplus U$ and $\sigma_A^2=0$,
the numbers $1$, $\omega_1$, $\omega_2$ and $\omega_1\omega_2$ are
linearly independent over $\mathbb{Q}$. So, in particular, $E$ and $F$
are not isogenous. If $H_1(E\times
F,\mathbb{Z})=\langle\gamma_1,\gamma_2,\delta_1,\delta_2\rangle$, then we
consider the subgroup $C_n$ of $E\times F$ such that
\[
H_1((E\times
F)/C_n,\mathbb{Z})=\left\langle\frac{\gamma_1+\delta_1}{n},\gamma_2,\delta_1,\delta_2\right\rangle.
\]
Let $S:=(E\times F)/C_n$. In terms of the dual bases of the bases of
$H_1(E\times F,\mathbb{Z})$ and  $H_1(S,\mathbb{Z})$ just described, we write
$H^1(S,\mathbb{Z})=\langle{\rm d} z_1,{\rm d} z_2,{\rm d} w_1,{\rm d} w_2\rangle$ and
$H^1(E\times F,\mathbb{Z})=\langle{\rm d} x_1,{\rm d} x_2,{\rm d} y_1,{\rm d} y_2\rangle$.
If $\pi:E\times F\rightarrow S$ is the natural surjection, the map
$\theta:=\pi^*:H^1(S,\mathbb{Z})\rightarrow H^1(E\times F,\mathbb{Z})$ is such
that:
\begin{eqnarray}\label{eqn:theta}
\begin{array}{l}
\theta({\rm d} z_1)=n{\rm d} x_1,\\
\theta({\rm d} z_2)={\rm d} x_2,\end{array}\;\;\;\;\;\;\;\;\;\;\begin{array}{l}\theta({\rm d} w_1)=-{\rm d} x_1+{\rm d} y_1,\\
\theta({\rm d} w_2)={\rm d} y_2.
\end{array}
\end{eqnarray}
Observe that ${\rm NS}(E\times F)=\langle{\rm d} x_1\wedge{\rm d} x_2,{\rm d}
y_1\wedge{\rm d} y_2\rangle$. Furthermore, due to the properties in
\eqref{eqn:theta} which characterize the morphism
$\stackrel{2}{\wedge}\theta:H^2(S,\mathbb{Z})\rightarrow H^2(E\times
F,\mathbb{Z})$ and due to the fact that $\stackrel{2}{\wedge}\theta$
preserves the Hodge structures on $H^2(S,\mathbb{Z})$ and $H^2(E\times
F,\mathbb{Z})$,
{\setlength\arraycolsep{2pt}
\begin{eqnarray*}
{\rm NS}(S)&=&\langle{\rm d} z_1\wedge{\rm d} z_2,n{\rm d} w_1\wedge{\rm d} w_2+{\rm d}
z_1\wedge{\rm d} w_2\rangle,\\
T(S)&=&\langle{\rm d} z_1\wedge{\rm d} w_1,{\rm d} z_2\wedge{\rm d} w_2,{\rm d}
z_1\wedge{\rm d} w_2,-n{\rm d} w_1\wedge{\rm d} z_2+{\rm d} z_1\wedge{\rm d} w_2\rangle.
\end{eqnarray*}}In particular, ${\rm NS}(S)\cong U(n)$ and $T(S)\cong U\oplus U(n)$.

Consider the two cohomology classes
{\setlength\arraycolsep{2pt}
\begin{eqnarray*}
\sigma_{E\times F}&=&{\rm d} x_1\wedge{\rm d} y_1+\omega_2{\rm d} x_1\wedge{\rm d}
y_2+\omega_1{\rm d} x_2\wedge{\rm d} y_1+\omega_1\omega_2{\rm d} x_2\wedge{\rm d}
y_2;\\
\sigma_S&=&{\rm d} z_1\wedge{\rm d} w_1+\omega_2{\rm d} z_1\wedge{\rm d}
w_2+\omega_1(n{\rm d} w_1\wedge{\rm d} z_2-{\rm d} z_1\wedge{\rm d}
w_2)+n\omega_1\omega_2{\rm d} z_2\wedge{\rm d} w_2.
\end{eqnarray*}}Obviously, $\sigma_{E\times F}\in T(E\times F)\otimes\mathbb{C}$ and
$\sigma_S\in T(S)\otimes\mathbb{C}$. Since $\langle\sigma_{E\times
F}\rangle=H^{2,0}(E\times F)$ and since an easy calculation shows
that $\stackrel{2}{\wedge}\theta(\sigma_S)=n\sigma_{E\times F}$,
$\langle\sigma_S\rangle=H^{2,0}(S)$. This implies that, due to
\eqref{eqn:period}, there exists an isometry
$\eta:H^2(S,\mathbb{Z})\rightarrow U^3$ such that
$\eta^{-1}\circ\varphi:H^2(A,\mathbb{Z})\rightarrow H^2(S,\mathbb{Z})$ is a Hodge
isometry (see \cite{Sh}). The Torelli Theorem for abelian surfaces
shows that either $A\cong(E\times F)/C_n$ or
$\widehat{A}\cong(E\times F)/C_n$.

Observe that, since ${\rm NS}(A)\cong U(n)$, the abelian surface $A$ is
principally polarized if and only if $n=1$. This means that, if
$n\neq 1$, $(E\times F)/C_n$ and its dual are not isomorphic.
Furthermore, $A$ and $E\times F$ are isogenous but $T(A)\not\cong
T(E\times F)$. Therefore, due to Theorem \ref{thm:orlov1},
${\rm D}^{\rm b}(A)\not\cong{\rm D}^{\rm b}(E\times F)$. This proves that there are
isogenous abelian surfaces whose derived categories of (untwisted)
coherent sheaves are not equivalent (see Remark \ref{rmk:cont}).

Choose the standard basis $\{g_1,g_2,k_1,k_2\}$ for $U\oplus U$. Due
to the explicit description of $T(A)$ that we have previously given,
it is straightforward to see that there exists an inclusion
$i_1:T(A)\rightarrow U\oplus U$ where $i_1(e_j)=g_j$
($j\in\{1,2\}$), $i_1(f_1)=nk_1$ and $i_1(f_2)=nk_2$. Let
$\sigma:=i(\sigma_A)\in U^2\otimes\mathbb{C}$. Due to \eqref{eqn:period},
we can write
$\sigma=g_1-n\omega_1\omega_2g_2+n\omega_1h_1+\omega_2h_2$.

Consider in $\mathbb{C}$ the lattice $\Gamma_3=\mathbb{Z}+n\omega_1\mathbb{Z}$ and the
elliptic curve $E_1:=\mathbb{C}/\Gamma_3$. Of course, $E$ and $E_1$ are
isogenous. Reasoning as before and using the surjectivity of the
period map and the Torelli Theorem for abelian surfaces (\cite{Sh}),
we get an isometry $\varphi_1:T(E_1\times F)\rightarrow U^2$ fitting
in the following commutative diagram:
\[
\xymatrix{0\ar[dr] & & T(E_1\times
F)\ar[dd]^{\varphi_1}\ar[rd]^{\alpha} & & 0 & \\  &
T(A)\ar[dr]^{i_1}\ar[ru]^{i} &  & \mathbb{Z}/n\mathbb{Z}\ar[rd]\ar[ru] &  \\
0\ar[ru] &
 & U\oplus U\ar[ru]^{\alpha_1} &  & 0.}
\]
Of course, $i$ preserves the Hodge structures and
$\alpha\in{\rm Br}(E_1\times F)$. Proposition 4.7 in \cite{Hu} yields
$B\in H^2(E\times F,\mathbb{Q})$ such that $(A,1)$ and $(E\times
F,\kappa_{E\times F}(B))$ are T-equivalent. By Theorem
\ref{thm:main2}, there exist $\beta\in{\rm Br}({\rm Km}(E_1\times F))$ of
order $n$ and a twisted Fourier-Mukai equivalence
${\rm D}^{\rm b}({\rm Km}(A))\cong{\rm D}^{\rm b}({\rm Km}(E_1\times F),\beta)$.

\subsection{The number of twisted Kummer structures}\label{subsec:num} As an easy corollary of Lemma \ref{lemma:brK3ab}, we get a
surjective map
\[
\Psi:\{\mbox{Twisted abelian
surfaces}\}/\mathrm{isom}\longrightarrow\{\mbox{Twisted Kummer
surfaces}\}/\mathrm{isom}
\]
which sends the isomorphism class $[(A,\alpha)]$ to the isomorphism
class $[({\rm Km}(A),\Theta_A(\alpha))]$. The main result in \cite{HLOY}
proves that the preimage of $[({\rm Km}(A),1)]$ is finite, for any
abelian surface $A$ and $1\in{\rm Br}(A)$ the trivial class (see
\cite[Thm.\ 0.1]{HLOY}). On the other hand \cite{HLOY} shows that
the cardinality of the preimages of $\Psi$ can be arbitrarily large.
This answered an old question by Shioda. Namely, there can be many
non-isomorphic (untwisted) abelian surfaces giving rise to
isomorphic (untwisted) Kummer surfaces (a partial result in this
direction is also contained in \cite{GH}). This is usually rephrased
saying that on a K3 surface one can put many non-isomorphic
(untwisted) Kummer structures.

Using Theorem \ref{thm:main2}, the picture in \cite{HLOY} can be
completely generalized to the twisted case.

\begin{prop}\label{prop:twnum} {\rm (i)} For any twisted Kummer surface
$({\rm Km}(A),\alpha)$, the preimage $\Psi^{-1}([({\rm Km}(A),\alpha)])$ is
finite.

{\rm (ii)} For positive integers $N$ and $n$, there exists a twisted
Kummer surface $({\rm Km}(A),\alpha)$ with $\alpha$ of order $n$ in
${\rm Br}({\rm Km}(A))$ and such that $|\Psi^{-1}([({\rm Km}(A),\alpha)])|\geq
N$.\end{prop}

\begin{proof} Suppose that
$\Psi([(A_1,\alpha_1)])=\Psi([(A_2,\alpha_2)])=[({\rm Km}(A),\alpha)]$,
i.e.\ suppose that there exists an isomorphism
$f:{\rm Km}(A)\mor[\sim]{\rm Km}(A_i)$ such that
$f^*\Theta_{A_i}(\alpha_i)=\alpha$. In particular,
$${\rm D}^{\rm b}({\rm Km}(A_1),\Theta_{A_1}(\alpha_1))\cong{\rm D}^{\rm b}({\rm Km}(A_2),\Theta_{A_2}(\alpha_2)).$$

Due to Theorem \ref{thm:main2}, the proof of (i) amounts to show
that, up to isomorphisms, there are finitely many T-equivalent
twisted abelian surfaces $(A',\beta)$ such that
$\Psi([(A',\beta)])=[({\rm Km}(A),\alpha)]$. Since, up to isomorphisms,
there are just finitely many abelian surfaces $A'$ with
${\rm D}^{\rm b}(A')\cong{\rm D}^{\rm b}(A)$ (\cite[Prop.\ 5.3]{BM}), we can just fix $A'$
with such a property and show that, up to isomorphisms, there exists
a finite number of $\beta'\in{\rm Br}(A')$ such that $(A',\beta)$ and
$(A',\beta')$ are T-equivalent. But this is the content of
\cite[Prop.\ 3.4]{HS1} for the case of abelian surfaces.

Applying the results in \cite{Og} and \cite{St} to abelian surfaces,
we see that, for any positive integer $N$, there exist $N$
non-isomorphic abelian surfaces $A_1,\ldots,A_N$ such that
${\rm D}^{\rm b}(A_i)\cong{\rm D}^{\rm b}(A_j)$ ($i,j\in\{1,\ldots,N\}$). Due to Theorem
\ref{thm:orlov1}, for any $i\in\{2,\ldots,N\}$, there is a Hodge
isometry $$g_i:T(A_1)\rightarrow T(A_i).$$ Take $B_1\in
T(A_1)\otimes\mathbb{Q}$ such that $\alpha_1:=\kappa_{A_1}(B_1)$ and
$\Theta_{A_1}(\alpha_1)$ are not trivial in ${\rm Br}(A_1)$ and
${\rm Br}({\rm Km}(A_1))$ respectively. We can also choose $\alpha_1$ such
that the order of $\Theta_{A_1}(\alpha_1)$ is $n$ in
${\rm Br}({\rm Km}(A_1))$. Then, for any $i\in\{2,\ldots,N\}$, define
$\alpha_i:=\kappa_{A_i}(g_i(B_1))$. Obviously, $(A_i,\alpha_i)$ and
$(A_j,\alpha_j)$ are T-equivalent when $i,j\in\{1,\ldots,N\}$.
Theorem \ref{thm:main2} immediately implies that
$({\rm Km}(A_i),\Theta_{A_i}(\alpha_i))$ and
$({\rm Km}(A_j),\Theta_{A_j}(\alpha_j))$ are D-equivalent.

For any $i\in\{2,\ldots,N\}$, the isometry $g_i$ induces a Hodge
isometry $f_i:T({\rm Km}(A_1))\rightarrow T({\rm Km}(A_i))$ which (due to
\cite[Thm.\ 1.14.4]{Ni}) extends to a Hodge isometry
$h_i:H^2({\rm Km}(A_1),\mathbb{Z})\rightarrow H^2({\rm Km}(A_i),\mathbb{Z})$. The Torelli
Theorem yields an isomorphism $\varphi_i:{\rm Km}(A_1)\rightarrow
{\rm Km}(A_i)$ such that
$\varphi_i^*(\Theta_{A_i}(\alpha_i))=\Theta_{A_1}(\alpha_1)$
(possibly changing $\alpha_i$ with $\alpha_i^{-1}$), for any
$i\in\{2,\ldots,N\}$. This concludes the proof of (ii).\end{proof}

In other words, Proposition \ref{prop:twnum} shows that on a twisted
K3 surface we can put just a finite number of non-isomorphic
\emph{twisted Kummer structures}. Nevertheless, such a number can be
arbitrarily large even when the twist is non-trivial and has any
possible order.

\medskip

{\small\noindent{\bf Acknowledgements.}  The final version of this
paper was written during the author stay at the Max-Planck-Institut
f\"{u}r Mathematik (Bonn). The results in this paper benefit of many
interesting discussions with Daniel Huybrechts and Bert van Geemen.
It is a great pleasure to thank them for their help and
suggestions.}


\section{Introduction}
Black holes are perhaps the most curious objects described by physics. Their construction requires
only our concepts of space and time~\cite{Chandrasekhar:1985kt}, and they are completely described
by only a few parameters, such as their mass, charge and spin. In this, black holes are exactly like
elementary particles. Another property that black holes share with fundamental particles is our
complete lack of knowledge about their internal structure, including whether any such structure
exists. But the source of this ignorance appears to be different for the two kinds of objects.
Elementary particles are point objects which cannot be probed further, since that would require
infinite energy of the probe. A black hole on the other hand presents to the universe a closed surface
of finite size, but it is impossible to observe anything about its internal structure, as no information
passes from the inside of this surface to the outside, at least classically.

The startling discovery by Hawking that stationary black holes radiate like a black body
with a finite surface temperature~\cite{Hawking:1974sw}, following Bekenstein's suggestion
that standard laws of thermodynamics applied to a black hole provided we assume that its
entropy is proportional to the surface area of its horizon~\cite{Bekenstein:1973ur}, implies
the possibility that a black hole has associated with it a very large number of microscopic
states. It is natural to think that these states are in some way related to the degrees of
freedom of the horizon. In the membrane paradigm, one replaces the black hole by a
membrane with certain classical properties at the stretched horizon, i.e. a small distance
outside the event horizon (an excellent overview is provided by the collection of articles
in~\cite{Thorne:1986iy}). This is a sensible description from the perspective of an external
stationary observer, who finds that particles cannot classically leave the interior of the
black hole or reach the horizon from the outside in finite time.

Thus it seems that the classical or semi-classical dynamics of fields, or gravity, on a
spacetime which includes a horizon, may be studied by looking at the bulk and the horizon,
and completely ignoring what happens in the interior of horizon. It has been suggested that
in this view it should be sufficient to consider fields on a manifold with boundary. For
gravity, this approach leads to a quantum description in which an infinite set of
observables are localized on the boundary~\cite{Balachandran:1994up, Balachandran:1995qa, Carlip:1998wz, Carlip:1999cy}. Recently there has been a resurgence of interest in
studying the behaviour of quantum fields near black hole horizons, motivated by various
paradoxes and puzzles~\cite{Almheiri:2012rt, Braunstein:2009my}. The near horizon behaviour
of fields have also been investigated in the space time of isolated
horizons~\cite{Ashtekar:1998sp, Ashtekar:1999yj, Ashtekar:2000sz, Ashtekar:2001is,
Booth:2000iq, Booth:2001gx}.

Boundary conditions on the classical fields play a crucial role in all these investigations.
In most of these papers, though not in all of them, Dirichlet (or Neumann) conditions are imposed
on the boundary, i.e. fields (or their derivatives) are set to vanish on the (stretched) horizon.
This is a
convenient choice for most calculations, but somewhat of an overkill, since the (stretched)
horizon is not a physical boundary of spacetime. In particular, fields need not vanish on
the horizon -- only invariants constructed out of the stress energy tensor need to remain
finite. For that it is sufficient for invariants made out of the physical fields to remain
finite on the horizon. Gauge theories are even more special in this regard, since components
of gauge fields are not physical, but defined up to gauge transformations. So strictly speaking
it is not necessary to impose finiteness on components of gauge fields on the horizon.

Gauge theories are characterized by the presence of redundant degrees of
freedom, which leads to the presence of constraints, usually relations among
the corresponding momenta. The formalism for studying
the dynamics of constrained systems was discovered by Dirac~\cite{Dirac:1950pj}
and independently by Bergmann et al.~\cite{Bergmann:1949zz,Anderson:1951ta}, and has
been applied
to numerous theories of interest over the years~\cite{Dirac-lect-1964,Henneaux:1992ig}.
In this paper, we will be concerned with the
classical dynamics of gauge theories defined on spherically symmetric
curved backgrounds, with horizons for boundaries.

While the formalism for constrained field theories set up by Dirac generalizes to curved backgrounds~\cite{Dirac:1951zz},  the more general formulation in terms of shift and lapse variables was introduced by Arnowitt, Deser and Misner~\cite{Arnowitt:1962hi}. Apart from the gravitational field itself, the ADM decomposition has been used to determine aspects of the Maxwell field on curved backgrounds, which includes the electromagnetic self-energy problem of a point charge~\cite{Arnowitt:1960zza}, the behaviour of the fields near the horizons of stationary black hole spacetimes~\cite{MacDonald:1982zz}, and its quantization~\cite{Sorkin:1979ja}. Through the works of Isenberg and Nester, the ADM decomposition has been subsequently used to better understand the initial value problem of fields theories~\cite{Isenberg:1977ja} and the description of derivative coupled theories ~\cite{Isenberg:1977fs}. While the relevance of boundary terms in the description of the gravitational field has also been considered in ~\cite{Isenberg:1981fa}, the general formulation of constrained theories with boundaries has not been provided in these works.

Boundaries lead to the inclusion of surface terms which are essential in the formulation of the action in gravity~\cite{Regge:1974zd, Gibbons:1978ac, York:1986} and other field theories~\cite{Gervais:1976ec, Anco:2001gm, Anco:2001gk,Romero:2002vg, G.:2013zca, G.:2015uda}. The generalization to cases where the boundaries are null~\cite{Parattu:2015gga, Booth:2001gx}
as well as non-orthogonal~\cite{Hawking:1996ww}, have also been considered in the literature.
There is also much recent activity regarding the asymptotic symmetries of gauge theories and
gravity~\cite{Balachandran:2013wsa, Campiglia:2014yka, Strominger:2013lka, He:2014cra,
	Strominger:2013jfa, He:2014laa}, which can be formulated in terms of fields localized
on the null boundary of a conformally compactified asymptotically flat
spacetime~\cite{Mohd:2014oja}.
While surface terms in these contexts provide an important topic of investigation in its own
right~\cite{Donnelly:2016auv}, it is not what our paper seeks to address.

We will be largely concerned in the role
boundaries play in the classical
and quantum descriptions of constrained field theories. There has been some consideration of
these in the literature. The modifications of constraints through the presence of boundaries,
close to the spirit in which we will carry out our work,  has been investigated in~\cite{SheikhJabbari:1999xd,Zabzine:2000ds}. In~\cite{Balachandran:1993tm,Balachandran:1992qg}, by studying the quantization of the the Chern-Simons theory on a disk, the role of boundaries on the vacuum structure of the theory has been covered in detail. While these works have made some ground in addressing how boundaries affect constraints,  many questions still remain open. As far as we are aware of, there appears to be no general prescription on how boundaries are to be considered in the case of constrained theories, and a formulation on curved backgrounds with horizons is completely lacking. The present work is an attempt to address this issue.

The organization of our paper is as follows. In Sec.~\ref{geom}, we describe the foliation
which will be implemented to carry out the 3+1 decomposition of the spherically
symmetric spacetime, as well as the form of the matter action defined on it. In
Sec.~\ref{Max}, we consider the specific example of Maxwell's electrodynamics
as a constrained theory, for which the concrete manifestation of horizons in the
description of the dynamics of the fields are pointed out as they arise. Finally, in
Sec.~\ref{Disc} we discuss some unexpected results and possible applications of
our findings.

\section{General Algorithm} \label{geom}

We will work on a static, spherically symmetric spacetime endowed with at least one horizon.
In other words, we assume that there is a timelike Killing vector field $\xi^a\,$
with norm given by $\xi^a\xi_a = -\lambda^2\,,$ satisfying
\begin{align}
\xi_{[a}\nabla_b \xi_{c ]} & = 0 \,. \label{Frobenius}
\end{align}
The horizon is defined by $\xi^a$ becoming null, $\lambda=0\,.$
It follows that there is a spacelike hypersurface $\Sigma$ which is
everywhere orthogonal to $\xi^a\,.$ The situation we have in mind is that
of fields living on the background of a
static black hole. For an asymptotically flat or anti-de Sitter
space, $\Sigma$ is the region `outside the horizon', while for a positive
cosmological constant, we may have a static de Sitter black hole spacetime,
in which case $\Sigma$ is the region `between the horizons'.

The induced metric on $\Sigma$ is given by
\begin{equation}
h_{ab} = g_{ab} + {\lambda}^{-2} \xi_{a} \xi_{b} \,,
\label{gen.met}
\end{equation}
leading to the following expression for the determinant of spacetime metric
\begin{equation}
\sqrt{-g} = \lambda \, \sqrt{h} \, . \label{gen.det}
\end{equation}

The action functional for $N$ fields $\Phi_A\,,\, A = 1, \cdots, N\,,$ is given by the time
integral of the Lagrangian $L$, or equivalently the integral of
the Lagrangian density ${\cal L}$ over the four volume,
\begin{equation}
S[\Phi_A] = \int dt\, L
= \int dt \int \limits_{\Sigma} \lambda dV_x ~ {\cal L}(\Phi_A(x) , \nabla_a \Phi_A(x)) \, ,
\label{gen.act}
\end{equation}
where $dV_x$ is the volume element on $\Sigma\,,$
and ${\cal L}(\Phi_A(x) , \nabla_a \Phi_A(x))$ is the Lagrangian density.  The Lagrangian density
can be written in terms of the `spatial' and `temporal' derivatives of the fields,
\begin{equation}
{\cal L} \equiv {\cal L}(\Phi_A(x),  \mathcal{D}_a \Phi_A(x),  \dot{\Phi}_A(x) )\,,
\label{gen.den}
\end{equation}
where $\mathcal{D}_a \Phi_A = h_a^b \nabla_b \Phi_A$ are the $\Sigma$-projected derivatives
of the fields $\Phi_A\,,$ and $\dot{\Phi}_A$ are their time derivatives, defined as their Lie derivatives
with respect to $\xi\,,$
\begin{equation}
\dot{\Phi}_A := \pounds_{\xi} \Phi_A \,.
\label{gen.dot}
\end{equation}

The momenta $\Pi^A$ canonically conjugate to the fields $\Phi_A\,$ are defined as
\begin{equation}
\Pi^A  = \frac{\delta L}{\delta \dot{\Phi}_A}  =  -\lambda^{-1}~ \xi_a~ \frac{\partial {\cal L}}{\partial (\nabla_a \Phi_A)} \, ,
\label{gen.mom}
\end{equation}
where the functional derivative in this definition is taken on the hypersurface $\Sigma\,,$ i.e.
it is an `equal-time' functional derivative, defined as
\begin{equation}
\frac{\delta\Phi_A(\vec{x}, t)}{\delta\Phi_B(\vec{y}, t)} = \delta^B_A\, \delta(x, y) =
\frac{\delta\dot\Phi_A(\vec{x}, t)}{\delta\dot\Phi_B(\vec{y}, t)}\, .
\label{gen.var}
\end{equation}
The $\delta(x,y)$ in Eq.~(\ref{gen.var}) is the three-dimensional covariant delta function defined
on $\Sigma\,,$
\begin{equation}
\int\limits_\Sigma dV_y \delta(x,y) f(\vec{y}, t) = f(\vec{x}, t)\,.
\label{gen.del}
\end{equation}

Given a Lagrangian $L$ we can construct the canonical Hamiltonian through the Legendre transform
\begin{equation}
H_C = \int \limits_{\Sigma} dV_x ~(\Pi^A \dot{\Phi}_A) -  L \,.
\label{gen.Ham}
\end{equation}
Dynamics in the Hamiltonian formalism is determined using the Poisson bracket, which for two
functionals $F(\Phi_A(x), \Pi^A(x))$ and $G(\Phi_A(x), \Pi^A(x))$ of the fields and their momenta
is defined as
\begin{equation}
\left[F ,G\right]_P = \int dV_z \left[\frac{\delta F}{\delta \Phi_A(z)} \frac{\delta G}{\delta \Pi^A(z)}
- \frac{\delta G}{\delta \Phi_A(z)} \frac{\delta F}{\delta \Pi^A(z)} \right]\,. \label{gen.PB}
\end{equation}
The canonical Poisson brackets between the fields and their momenta follows from setting
$F(x) = \Phi_A(\vec{x}, t)$ and $G(y) = \Pi^B(\vec{y}, t)$
\begin{equation}
\left[\Phi_A(\vec{x}, t), \Pi^B(\vec{y}, t) \right]_P = \delta_{A}^{B} \delta(x,y)\,.
\label{gen.can}
\end{equation}
The time evolution of any functional of the fields and momenta is determined
from its Poisson bracket with the Hamiltonian
\begin{equation}
\dot{F}(x) = \left[F(x), H_C \right]_P\,.
 \end{equation}

The Hamiltonian, obtained by a Legendre transform from the Lagrangian,
provides a complete description of the dynamics of the system only if
all velocities of the theory uniquely map into momenta by
Eq.~(\ref{gen.mom}). In the case of constrained theories,
such a mapping is not possible due to the presence of constraints. In these theories, the Hamiltonian
must be constructed by determining all the constraints of the theory via the Dirac-Bergmann
algorithm. The usual Poisson brackets of these theories are modified in the presence of
constraints, and as we will argue below, the constraints of the theory are
modified in the presence of horizons.

\section{The Maxwell field}\label{Max}
For the sake of concreteness, we will consider the specific example of electromagnetism
as a constrained theory on spherically symmetric spacetimes with horizon(s). The action is
\begin{equation}
S_{EM} =  \int dV_4 \left(-\tfrac{1}{4} F_{a b} F_{c d} g^{a c} g^{b d}\right) \, ,
\end{equation}
where $dV_4 = \lambda dV_x$ is the four dimensional volume form on the manifold $\Sigma\times\mathbb{R}$, and
$F_{a b} = 2 \partial_{[a}
A_{b]}$. Defining $e_a = - \lambda^{-1} \xi^c F_{c d}$ and $f_{a b} = F_{c d} h^c_a h^d_b\,,$ we
can rewrite this action as
\begin{equation}
S_{EM} = - \int dt \int \limits_{\Sigma} dV_x \frac{\lambda}{4} \left[f_{a b} f^{a b} - 2 e_{a} e^{a} \right] \, ,
\label{H.Lag}
\end{equation}
Recalling Eq.~(\ref{gen.dot}), we write
\begin{align}
\dot A_b \equiv \pounds_{\xi} A_b  &= \xi^{a}\nabla_a A_b + A_a \nabla_a \xi^a \, \notag \\
&=\xi^a F_{a b} + \nabla_a (A_b \xi^b) \,,
\end{align}
and defining
	{$\phi = A_a \xi^{a}$},
	we have for the projection $a_b\,,$
\begin{equation}
\dot a_b = -\lambda e_b + \mathcal{D}_b \phi \,.
\label{H.elf}
\end{equation}
Since the velocity term $\pounds_{\xi} \phi$ does not appear in the electromagnetic Lagrangian Eq.~(\ref{H.Lag}), it implies that the momentum conjugate to  $\phi$ vanishes,
\begin{equation}
\frac{\partial L_{E M}}{\partial \dot{\phi}} = \pi^{\phi} = 0 \,,
\label{H.con}
\end{equation}
thus producing the only constraint following from the Lagrangian. The momenta
corresponding to the $a_b$ are given by
\begin{equation}
\pi^b = \frac{\partial L_{E M}}{\partial \dot{a}_b} = - e^b \, .
\label{H.mom}
\end{equation}

The canonical Hamiltonian is then
		\begin{align}
H_C &= \int \limits_{\Sigma} dV_x \, \left(\pi^b \dot{a}_b\right) -  L \notag \\
&= \int \limits_{\Sigma} dV_x \, \left[\lambda\left(\frac{1}{2} \pi^b \pi_b + \frac{1}{4} f_{a b} f^{a b}\right) + \pi^b \mathcal{D}_b \phi \right]\,.
\end{align}
The constraint of Eq.~(\ref{H.con}) is now added to this and a new Hamiltonian is defined,
		\begin{equation}
H_0  = \int \limits_{\Sigma} dV_x \, \left[\lambda \left(\frac{1}{2} \pi^b \pi_b + \frac{1}{4} f_{a b} f^{a b}\right) + \pi^b \mathcal{D}_b \phi + v_{\phi} \pi^{\phi} \right] \, ,
\label{H.pri}
\end{equation}
where $v_{\phi}$ is an undetermined multiplier. The canonical Poisson brackets of
Eq.~(\ref{gen.can}) are in this case
\begin{align}
\left[\phi(x), \pi^{\phi}(y) \right]_P & = \delta(x,y) \notag\\
\left[ a_a(x) , \pi^b(y) \right]_P & = \delta^{b}_{a}\delta(x,y) \,.
\label{H.can}
\end{align}
\subsection{The Dirac-Bergmann algorithm} \label{MDBA}
We will now apply the Dirac-Bergman algorithm to determine all the constraints of the theory
and construct the unconstrained Hamiltonian.
For that, we need to check that the constraint is obeyed at all times, or in other words,
$\dot \pi^{\phi} \approx 0 \,.$ The Poisson bracket between $\pi^\phi$ and the Hamiltonian
is calculated with the help of a smearing function $\epsilon$ as follows,
\begin{align}
\int \limits_{\Sigma} dV_y \epsilon(y) \dot\pi^\phi(y) &= \int \limits_{\Sigma} dV_y
\epsilon (y) \left[\pi^{\phi}(y) , H_0 \right]_P  \,\notag \\
&= \int \limits_{\Sigma} dV_y
\epsilon (y) \left[\pi^{\phi}(y) , \int \limits_{\Sigma} dV_x \pi^b(x)
\mathcal{D}^x_b \phi(x) \right]_P \notag \\
&=  - \oint \limits_{\partial \Sigma} da_y \, \epsilon(y)  n^y_b \pi^b(y) +
\int  \limits_{\Sigma} dV_y \, \epsilon(y) \left( \mathcal{D}^y_b \pi^b(y)\right)\,.
\label{U.PB1}
\end{align}
Here we have used the canonical Poisson brackets given in Eq.~(\ref{H.can})
and an integration by parts. The smearing function $\epsilon$ is assumed
to be well behaved, but we make no further assumption regarding its properties.
In particular we do not assume that $\epsilon$ vanishes on the
horizon (or horizons, if $\Sigma$ is the region between the horizons in a de Sitter
black hole spacetime).

The surface integral is finite, since using Schwarz inequality we get
\begin{equation}
\left| n_b \pi^b \right|  \leq \, \sqrt{\left|n_b n^b\right| \,
	\left|\pi_b \pi^b\right|}  \,.
\label{U.SI}
\end{equation}
In this, $n_bn^b = 1$ by definition since $n$ is the `unit normal' to the horizon,
and $\pi_b\pi^b = e_b e^b$ appears in the energy momentum tensor (more precisely
in invariant scalars such as $T^{ab}T_{ab}$), and therefore may not
diverge at the horizon.
So the integral over $\partial\Sigma$ is finite and in general different from zero.
Thus the boundary integral is finite at the horizon and we have a non-vanishing
contribution from $\partial\Sigma$\,, which was one of the things we wanted to show.

We note that since the smearing function $\epsilon$ is present in the integrand,
specific assumptions about the class of allowed smearing functions may be required to
produce physically sensible results. For the Maxwell case, the assumption
that $\epsilon$ is regular at the horizon (with no dependence on $\lambda$)
is sufficient.

The right hand side of Eq.~(\ref{U.PB1}), comprising of a bulk and a surface term,
must vanish weakly, giving a constraint. Since we are working on a spherically symmetric
background, we can use a radial delta function to convert the surface integral to
a volume integral,
\begin{equation}
\oint \limits_{\partial \Sigma} da_y  K(y) = \int  \limits_{\Sigma} dV_y \lambda(y) K(y)
\delta(r(y) - r_H) \,,
\label{U.area}
\end{equation}
where $K$ is any well-behaved function, $r_H$ is the radius of the sphere corresponding
to $\partial\Sigma\,,$ $\delta$
is the usual Dirac delta, defined by $\int\, dr\, \delta(r - R) f(r) = f(R)$ for
any well-behaved function $f(r)\,,$ and we have assumed that $h^{rr} = \lambda$ for the
spherically symmetric metrics that we consider. (This is solely for notational convenience,
and $h^{rr}$ should replace $\lambda$ in this formula if there is any confusion.) If we have
a de Sitter black hole spacetime, we will need to consider a sum over two spheres,
corresponding to inner and outer horizons.
Thus we can rewrite the last equality in Eq.~(\ref{U.PB1}) as
\begin{equation}
\int \limits_{\Sigma} dV_y \epsilon(y) \dot\pi^\phi(y)
= \int  \limits_{\Sigma} dV_y \, \epsilon(y) \left[ - \lambda(y)  n^y_b \pi^b(y)
\delta (r(y) - r_H) + \mathcal{D}^y_b \pi^b(y) \right]\,,
\label{U.PB11}
\end{equation}
which produces the constraint
\begin{equation}
\Omega_2 =  - \lambda n_b  \pi^b\delta (r - r_H) + \mathcal{D}_b \pi^b \approx 0 \, .
\label{U.con2}
\end{equation}

We now need to check if there are any further constraints resulting from $\dot \Omega_2 \approx 0$. We first include the new constraint with a multiplier into the existing Hamiltonian given in Eq.~(\ref{H.pri}), which gives us
\begin{equation}
H_T = H_0 + \int \limits_{\Sigma} dV_x v_1 \left[\mathcal{D}_b  \pi^b - \lambda n_b \pi^b\delta (r - r_H) \right] \,,
\label{U.sham}
\end{equation}
and consider the time evolution to be governed by this Hamiltonian,
		\begin{align}
\int \limits_{\Sigma} dV_y\, \epsilon(y) \dot \Omega_2(y) &= \int \limits_{\Sigma} dV_y\,
\epsilon(y)  \left[ \Omega_2(y) , H_T \right]_P \notag \\
&= - \int \limits_{\Sigma} dV_y  \mathcal{D}^y_b\left( \epsilon (y)\right) \int \limits_{\Sigma} dV_x \left[ \pi^b(y) , \mathcal{D}^x_a a_c(x) \right]_P f^{a c}(x) \notag \\
&= \int \limits_{\Sigma} dV_y  \mathcal{D}^y_b \left( \epsilon(y)\right) \int \limits_{\Sigma} dV_x \mathcal{D}^x_a \left(\delta(x,y)\right) f^{a b}(x)  \notag\\
&= \int \limits_{\Sigma} dV_x \mathcal{D}^x_a \left(\int \limits_{\Sigma} dV_y  \mathcal{D}^y_b \left( \epsilon(y)\right)\delta(x,y)\right) f^{a b}(x)  \notag\\
& = \int \limits_{\Sigma} dV_y   \mathcal{D}^y_a \mathcal{D}^y_b \epsilon(y) f^{a b}(y) \notag\\
& = 0\,.
\label{U.PB2}
\end{align}
The last equality follows from the antisymmetry of $f^{ac}$ in its indices, and we have used the fact that
$\mathcal{D}$ is torsion-free. Since these constraints commute with one another, they are also first class.
Thus the full Hamiltonian is the $H_T$ defined earlier,
\begin{equation}
H_T = \int\limits_{\Sigma} dV_x \left[ \lambda \left( \frac{1}{4} f_{a b} f^{a b} + \frac{1}{2} \pi_a \pi^a\right) + v_1 \left( \mathcal{D}_b \pi^b -  \lambda n_b \pi^b \delta (r - r_H) \right) + \pi^b \mathcal{D}_b \phi + v_{\phi} \pi^{\phi} \right]
\label{U.Htot1}
\end{equation}
The multipliers $v_1$ and $v_\phi$ may be determined by examining the equations of motion.
The evolution of $\phi$ is given by
\begin{align}
\int \limits_{\Sigma} dV_y \epsilon(y) \dot \phi(y) &= \int \limits_{\Sigma} dV_y
\epsilon(y)\left[  \phi(y) , H_T \right]_P \notag \\
&=  \int \limits_{\Sigma} dV_y \epsilon(y) \int \limits_{\Sigma} dV_x v_{\phi}(x)
\left[ \phi(y) , \pi^{\phi}(x) \right]_P \notag \\
&=  \int \limits_{\Sigma} dV_y \epsilon (y) v_{\phi}(y) \, ,
\end{align}
which tells us that we can set $\dot \phi = v_{\phi}$. The evolution of $a_b$ can also
be determined in the same manner,
 %
\begin{align}
\int \limits_{\Sigma} & dV_y\, \epsilon(y) \dot{a}_b(y) = \int \limits_{\Sigma} dV_y\, \left[ \epsilon(y) a_b(y) , H_T \right]_P \notag \\
&= \int \limits_{\Sigma} dV_y\, \epsilon(y)  \int \limits_{\Sigma} dV_x \left[a_b(y) , \pi^c(x) \right]_P \left(\lambda(x) \pi_c(x) + \mathcal{D}^x_c\phi(x) - \mathcal{D}^x_c v_1(x)\right)  \notag \\
&=  \int \limits_{\Sigma} dV_y \epsilon(y) \left[ \lambda(y) \pi_b(y) + \mathcal{D}^y_b \phi(y) - \mathcal{D}^y_b v_1(y) \right] \, .
\label{U.vpt}
\end{align}
Comparing this with Eq.~(\ref{H.elf}), we deduce that we can set
$\mathcal{D}_b v_1 =  0$. There may be many ways in which this
condition could be satisfied,
but for simplicity, we will simply assume that $v_1 =0$.
Then Eq.~(\ref{U.vpt}) produces
\begin{equation}
\dot a_b = \lambda\pi_b + \mathcal{D}_b \phi \, ,
\end{equation}
and we thus find that the total Hamiltonian takes the form
	\begin{equation}
H_T = \int\limits_{\Sigma} dV_x \left[\lambda \left(\frac{1}{4} f_{a b} f^{a b} + \frac{1}{2} \pi_a \pi^a\right) + \pi^b \mathcal{D}_b \phi + \dot{\phi} \pi^{\phi} \right] \,.
\end{equation}

\subsection{Gauge transformations and Gauge fixing}

We have found two constraints, both first class, which depend on the momenta in the theory. These on account of being first class will generate gauge transformations, i.e. they transform the fields while not transforming the Hamiltonian (or the Lagrangian).

For the constraint $\Omega_1 = \pi^{\phi}$ , the only non-vanishing Poisson Bracket with
the fields is
\begin{align}
	\delta_1 \phi(y) &= \left[ \phi(y) , \int\limits_{\Sigma} dV_x \epsilon_1 (x) \pi^{\phi}(x) \right]_P \notag \\
	&= \epsilon_1 (y) \,.
	\label{GT1}
\end{align}
For the other first class constraint $\Omega_2$ of the theory, we have the following
non-vanishing Poisson bracket with the fields
\begin{align}
	\delta_2 a_b(y) &= \left[ a_b(y) , \int\limits_{\Sigma} dV_x \epsilon_2 (x) \left(\mathcal{D}^x_c \pi^c(x) -  \lambda(x) n^x_c \pi^c(x) \delta(r(x) - r_H) \right) \right]_P \notag \\
	&= \left[ a_b(y) ,  \int\limits_{\Sigma} dV_x \pi^c(x) \mathcal{D}^x_c \epsilon_2(x) \right]_P \notag \\
	&= \mathcal{D}^y_b \epsilon_2(y)\, .
	\label{GT2}
\end{align}
These transformations can be identified with the usual gauge transformations $A_\mu \to A_\mu + \partial_\mu \psi$ if we write
$\epsilon_2(y) = \psi$ and
$\epsilon_1(y) = \pounds_{\xi} \psi$. This can be seen by
simply projecting the gauge transformation one finds from the Lagrangian
\begin{align}
	A_a(x) + \nabla^x_a \psi(x) & = \delta^b_a \left( A_b(x) + \nabla^x_b \psi(x) \right) \notag\\
	&=a_a + \mathcal{D}^x_a \psi(x) -\lambda^{-2}(x) \xi_{a} \left(\phi(x) + \pounds_{\xi} \psi(x) \right) \, .
\label{gf.GTL}
\end{align}
We note that the gauge transformations for this background are the same as in the
absence of horizons. The boundary terms which arise in the constraints are such that the
gauge transformations remain unaltered.

To proceed further, we take the approach of converting the gauge constraints into
second class ones by fixing the gauge. Let us choose a `radiation-like' gauge, in
which the gauge-fixing condition or constraint is analogous to the usual radiation
gauge, with an additional boundary contribution motivated by the
surface term in Eq.~(\ref{U.con2}). This choice considers how the horizon could
affect the dynamics of the theory, which was our original motivation for this work.
Thus we now have a total of four constraints given by
\begin{align}
	\Omega_1 & = \pi^{\phi} \notag \\
	\Omega_2 & =  \mathcal{D}_a \pi^a
	- \lambda n_a \pi^a \delta (r - r_H) \notag\\
	\Omega_3 & = \phi \notag \\
	\Omega_4 & = \mathcal{D}^b a_b
	- \lambda n^b a_b \delta (r - r_H)\,.
	\label{gf.con}
\end{align}
We note that $a_b$ is not a physical observable since it changes under a gauge transformation.
In particular, the near horizon behaviour of the coefficient of the $\delta$-function in the
last term of the gauge-fixing constraint\, $\Omega_4$ cannot be fixed from any physical
consideration. The Poisson brackets of these constraints are easily calculated,
\begin{align}
	\left[\Omega_1 (x), \Omega_3 (y) \right]_P & = - \delta(x,y) \, , \notag \\
	\left[\Omega_2 (x), \Omega_4 (y) \right]_P & =
	 \mathcal{D}_{a} \mathcal{D}^{a}\delta(x,y)  \, ,
	\label{gf.pb}
\end{align}
with all other Poisson brackets vanishing. The first Poisson bracket in
Eq.~(\ref{gf.pb}) follows directly from the canonical relations. The
second Poisson bracket gives

\begin{align}
	& \left[ \int dV_x \eta(x) \Omega_2 (x) ,

	\int dV_y \epsilon(y) \Omega_4(y) \right]_P \notag\\
	 & \qquad \qquad  = \left[ \int dV_x  \left(\mathcal{D}^x_a
	 \eta(x) \right)  \pi^a(x)\,, \int dV_y \left(\mathcal{D}_y^b\epsilon(y) \right) a_b(y) \right]_P \notag \\
	& \qquad \qquad  = - \int dV_y  \left(\mathcal{D}^y_a
	\eta(y)\right) \left(\mathcal{D}_y^a\epsilon(y) \right) \notag \\
	& \qquad \qquad  = -\oint da_y \epsilon(y) n_y^a  \left(\mathcal{D}^y_a\eta(y)\right) + \int dV_y \epsilon(y) \mathcal{D}_y^a \mathcal{D}^y_a\eta(y)\,,
	\label{gf.pb2}
\end{align}
where we have used integration by parts in deriving the equalities.
The surface integral in the last equality of Eq.~(\ref{gf.pb2}) vanishes,
which can be seen by using Schwarz's inequality
\begin{align}
	\left| n^a D_a \left(\eta\right) \right|^2 & \leq \left|n^a n_a \right| \left|h^{ab} \left(D_a\eta\right) \left(D_b\eta\right)\right| \notag\\
	& = h^{a b} \left(D_a\eta\right) \left(D_b\eta\right) \,.
	\label{gf.srf}
\end{align}
The smearing function and its derivatives are regular on the horizon, while
$h^{rr} \sim \lambda^2$ on spherically symmetric backgrounds. Hence the surface integral
in the last equality of Eq.~(\ref{gf.pb2}) vanishes, and only the
volume term contributes.
Thus the Poisson brackets between the constraints are those given in Eq.~(\ref{gf.pb}).
The matrix of the Poisson brackets between these constraints have
a non-vanishing determinant and is invertible. This matrix,
$C_{\alpha\beta}\left(x,y \right) = \left[\Omega_\alpha (x),
\Omega_\beta(y)\right]_P\,,$ is given by
\begin{equation}
	C (x,y) =
	\begin{pmatrix}
		0 & 0 & -\delta(x,y)\ & 0 \\
		0 & 0 & 0 & \mathcal{D}_a \mathcal{D}^a\delta(x,y) \\
		\delta(x,y) & 0 & 0 & 0 \\
		0 & - \mathcal{D}_a \mathcal{D}^a\delta(x,y) & 0 & 0
	\end{pmatrix} \, .
	\label{gf.dirm}
\end{equation}
The Dirac bracket for any two dynamical entities $F$ and $G$ (which may be
functions on the phase space, or functionals, the duals of functions) is defined as
\begin{equation}
	\left[F\,, \,G \right]_{D} = \left[F\,, \,G \right]_{P} - \int dV_w \int dV_z
	\left[F\,, \,\Omega_\alpha(w)\right]_P C^{-1}_{\alpha\beta}\left(w, z\right) \left[\Omega_\beta(z)\,, \,G\right]_P\,.
	\label{gf.dir}
\end{equation}
Thus we need to find the inverse of the operator
$\mathcal{D}_a
		\mathcal{D}^a$.
Let us formally write the inverse as $\tilde{G}(x, y)\,,$ i.e.
\begin{equation}
	\mathcal{D}_a\mathcal{D}^a \tilde{G}\left(x, y\right) = \delta\left(x,y\right) \,.
	\label{U.gfs}
\end{equation}
This $\tilde{G}(x, y)\,$ is the Green's function
for the {\em spatial} Laplacian operator $\mathcal{D}_a\mathcal{D}^a $\,, but not of the wave operator,
which is the {\em spacetime} Laplacian. With the help of this, the inverse matrix
$C^{-1}_{\alpha\beta}(x,y)$ can be written as

\begin{equation}
	C^{-1}(x,y) =
	\begin{pmatrix}
		0 & 0 & \delta(x,y) & 0\\
		0 & 0 & 0 & -\tilde{G}\left(x,y\right) \\
		-\delta(x,y) & 0 & 0 & 0 \\
		0 & \tilde{G}\left(x,y \right) & 0 & 0
	\end{pmatrix} \, .
	\label{gf.cin}
\end{equation}
We can now substitute Eq.~(\ref{gf.cin}) in Eq.~(\ref{gf.dir}) to find the following Dirac brackets for the fields,
\begin{align}
	\left[a_a(x),\pi^b(y)\right]_{D} & = \delta(x,y)\delta_a^b - \mathcal{D}_a^x \mathcal{D}^b_x \tilde{G}\left(x,y \right)\,,
\label{gf.cdir}
\end{align}
all other Dirac brackets being zero.

We could choose to fix the gauge so that the resulting Dirac brackets would involve
the (static) Green's function for the spacetime Laplacian. The corresponding gauge-fixing
constraints are
\begin{align}
	\Omega_3 & = \phi \, ,\notag \\
	\Omega_4 & = \mathcal{D}^b\left(\lambda a_b\right) \, .
	\label{gf.con2}
\end{align}
In this gauge the Dirac brackets are given by
\begin{align}
	\left[a_a(x),\pi^b(y)\right]_{D} & = \delta(x,y)\,\delta_a^b -
	\mathcal{D}_a^x\left(\lambda(x)\mathcal{D}^b_x G\left(x,y \right)\right)\,,
\label{gf.cdir2}
\end{align}
all other Dirac brackets being zero. Here $G(x, y)$ is the time-independent Green's function
for the spacetime Laplacian,
\begin{equation}
	\mathcal{D}_a^x\left(\lambda(x)\mathcal{D}^a_x G\left(x,y \right)\right) = \delta(x,y)\,.
\end{equation}

Although the choice of gauge-fixing functions determine the form of Dirac brackets, we know
that physical observables and measurable quantities must be independent of that choice.
However, the choice of Green's function is determined by the boundary conditions
we wish to impose on the fields at the horizon. We have mentioned earlier that
the horizon is not a boundary of the spacetime, in particular it is not necessary to
impose boundary conditions which force the respective fields to vanish on the
horizon, or even remain finite on the horizon if we are considering gauge fields.
We expect that the utility of our choice of gauge-fixing in Eq.~(\ref{gf.con}) could lie
in its allowance for more general boundary conditions for gauge fields at the horizon,
and in its ability to describe the dynamics of fields at the horizon. We will leave a
detailed investigation for later work, but discuss one interesting result, as well
as some open questions, in the next section.

We note in passing that explicit forms of both kinds of Green's functions considered in this section are known for the Schwarzschild
black hole as well as for the static de Sitter backgrounds~\cite{Cop:1928, J.Math.Phys.12.1845, Hanni:1973fn, Linet:1976sq, Bunch:1977sq, Chernikov:1968zm, Tagirov:1972vv, Dowker:1977, Fernandes:2016sue}.

\section{Discussion}\label{Disc}
In this work, we have argued that horizons affect the constraints of gauge theories, and
will in turn affect their dynamics in ways that spatial boundaries cannot. In the previous
section, we demonstrated that the Gauss law constraint of the Maxwell field receives a surface
contribution on spherically symmetric backgrounds with horizons. This however will not
typically happen for backgrounds with spatial boundaries. While the behaviour of gauge
fields at spatial boundaries cannot be determined by physical considerations alone, they must
nevertheless be continuous across them.  When surface terms occur, they are present on either
side of the spatial boundary and cancel out. When the boundary corresponds to the physical end
of the manifold, regularity of the fields require that they vanish there. The surface terms of
the previous section exist because the horizon hides the `other' side of the horizon from
observations. The only requirement are that physical fields, more precisely gauge-invariant
(and local Lorentz-invariant) scalars constructed out of the fields, must be finite at the
horizon. Horizons thus lead to a richer set of possibilities for field theories,
and in particular for gauge theories.

We have also considered the gauge fixing of the Maxwell field as a specific example to test
our claim. We can get some insight into the role played by the horizon in the choice of
boundary conditions by comparing the two gauge fixing constraints of Eq.~(\ref{gf.con}) and
Eq.~(\ref{gf.con2}). These gauge choices led to Dirac brackets which involve the
Green's functions for the spatial Laplacian and the spacetime Laplacian, respectively. The
difference in the actions of the two Laplacian operators on a time-independent
scalar field $F$ is seen in the following identity,
\begin{equation}
\nabla_a\nabla^a F - \mathcal{D}_a \mathcal{D}^a F =  \left(\lambda^{-1} \mathcal{D}^a \lambda\right) D_{a} F \, .
\label{App.Lap}
\end{equation}
When $\lambda=\lambda(r)$, as is the case for spherically symmetric backgrounds, the limit of Eq.~(\ref{App.Lap}) as $r \to r_H$ is
\begin{equation}
\left[\nabla_a\nabla^a F - \mathcal{D}_a \mathcal{D}^a F\right]_{r=r_H} = \kappa_H \left(\partial_r F\right)_{r=r_H} \, .
\label{App.diff}
\end{equation}
Thus the action of the spacetime and spatial Laplacians differ at the horizon by a term which depends on the surface
gravity $\kappa_H$ of the background. Eq.~(\ref{App.diff}) also indicates how boundary conditions affect the behaviour
of the operators on the left hand side. For instance,
the operators disagree on the horizon when either Dirichlet or Robin boundary conditions are assumed.
This suggests that the eigenvalues, and thus the description of the horizon states may be different for the two gauge choices,
however that is an investigation we leave for another occasion.

The modified Gauss law constraint in Eq.~(\ref{U.con2}) has further implications on the quantization of this theory, including horizon states and the charge. By integrating this constraint over a bounded volume whose radius is greater than the event horizon of the black hole (and less than the cosmological horizon, should it exist), we can derive the expression for the charge contained within it. In considering the Reissner-Nordstr\"om solution for simplicity, the electric field given in Eq.~(\ref{H.elf}) reduces to $\lambda \pi_b = - \mathcal{D}_b \phi$. Integrating up to a spatial boundary of radius $r_B$, where $r_B > r_H$, we have
\begin{align}
Q &=  \int \limits_{\Sigma_B} \Omega_2 \notag\\
&= - \oint \limits_{\partial_{\Sigma}} \left[r^2\partial_r \phi(r) \right]_{r = r_B} +  \oint \limits_{\partial_{\Sigma}} \left[r^2\partial_r \phi(r) \right]_{r = r_H} -  \oint \limits_{\partial_{\Sigma}} \left[r^2\partial_r \phi(r) \right]_{r = r_H} \notag\\
&= - \oint \limits_{\partial_{\Sigma}} \left[r^2\partial_r \phi(r) \right]_{r = r_B} \, .
\label{App.charge}
\end{align}
In Eq.~(\ref{App.charge}), `$\Sigma_B$' indicates that the volume integral on the hypersurface is evaluated from the horizon up to a sphere of constant radius $r_B$. The usual spherically symmetric solution, $\phi = \frac{Q}{4 \pi r}$, satisfies this equation. A crucial difference occurs if the above integral is performed \emph{exactly} at the black hole horizon, in which case we have
\begin{align}
Q_H &= \int \limits_{\Sigma_H} \left[\mathcal{D}_a \pi^{a} - \lambda n_a \pi^a \delta(r - r_H) \right] \notag\\
&=0\,.
\label{App.nocharge}
\end{align}
Note that Eq.~(\ref{App.nocharge}) is the charge at the horizon(s) and holds regardless of the solution for the charge outside the horizon. Equations (\ref{App.charge}) and (\ref{App.nocharge}) suggests that the horizon may be viewed as a dipole layer, with the charge on one side of the horizon being screened from observation. For an observer outside the horizon, the black hole is a charged body which follows from the bulk contribution to the constraint. When the observer is at the horizon, the cancellation of ``positive" and ``negative" charges leads to the result in Eq.~(\ref{App.nocharge}).

The constraint in Eq.~(\ref{U.con2}) will also necessarily affect the quantization of the Maxwell field on such backgrounds. Quantization of gauge fields is most effectively carried out within the BRST formalism, where the first class constraints of the theory and the inclusion of additional ghost fields leads to the construction of the BRST charge operator. Within the Hamiltonian BRST formalism, this operator leads to the derivation of the gauge fixing and ghost actions at tree level, both of which are BRST invariant \cite{Henneaux:1985kr,Henneaux:1992ig}.  On backgrounds with boundaries, the requirement of BRST invariance imposes certain restrictions on the allowed boundary values of the fields involved. Such considerations have been made on curved backgrounds with spatial boundaries
in~\cite{Moss:1989wu,Moss:1990yq,Moss:1996ip,Moss:2013vh}, and more recently in~\cite{Huang:2014pfa,Donnelly:2014fua,Donnelly:2015hxa} in relation to edge state entanglement
entropy calculations. These investigations however did not consider the modification of the Gauss law constraint. That the constraint derived in Eq.~(\ref{U.con2}) does contain a non-vanishing contribution at the horizon should lead to the derivation of ``horizon terms" for the action, which are guaranteed to be BRST invariant. Further, it is clear that when Eq.~(\ref{U.con2}) is imposed as an operator on physical states, it will relate the bulk and horizon states in a non-trivial way. To our knowledge, such considerations have not been made in the literature, and could be particularly relevant in describing the behaviour of horizon states of gauge fields.

To summarize, we have presented a covariant formalism for describing constrained field theories in
the presence of an event horizon of spherically symmetric black hole spacetimes, which are
either asymptotically flat or have an outer cosmological horizon. In the process we also
determined that the presence of horizons lead to non-trivial surface contributions to the
constraints in general, and demonstrated this explicitly in the Maxwell case. We have argued that surface contributions to the constraints will modify both the quantization of these theories and the description of states at the horizon. We leave further investigation of these topics to future work.

\section*{Acknowledgments}

KF thanks Steve Carlip, Daniel Kabat, Aruna Kesavan, George Paily and Robert Wald for stimulating discussions.