# A note on Fourier-Mukai partners of abelian varieties over positive
characteristic fields

Zhiyuan Li  Shanghai Center for Mathematical Science
2005 Songhu Road 200438, Shanghai<EMAIL_ADDRESS>and  Haitao Zou
Shanghai Center for Mathematical Science
2005 Songhu Road 200438, Shanghai<EMAIL_ADDRESS>
###### Abstract.

Over complex numbers, the Fourier-Mukai partners of abelian varieties are
well-understood. A celebrated result is Orlov’s derived Torelli theorem. In
this note, we study the FM-partners of abelian varieties in positive
characteristic. We notice that, in odd characteristics, two abelian varieties
of odd dimension are derived equivalent if their associated Kummer stacks are
derived equivalent, which is Krug and Sosna’s result over complex numbers. For
abelian surfaces in odd characteristic, we show that two abelian surfaces are
derived equivalent if and only if their associated Kummer surfaces are
isomorphic. This extends the result [7] to odd characteristic fields, which
solved a classical problem originally from Shioda. Furthermore, we establish
the derived Torelli theorem for supersingular abelian varieties and apply it
to characterize the quasi-liftable birational models of supersingular
generalized Kummer varieties.

###### Key words and phrases:

Fourier-Mukai partner, abelian variety, derived Torelli Theorem, Kummer
variety

###### 2010 Mathematics Subject Classification:

Primary 14F05; Secondary 14K05

##  1\. Introduction

Let $A$ be an abelian variety over an algebraically closed field $k$. It is
desirable to have a description of the derived category of $A$ via its
associated Kummer stack $\mathrm{K}(A)\coloneqq[A/\iota]$, where $\iota$ acts
on $A$ by the involution $-1\colon A\to A$. When $A$ is an abelian surface,
the singular Kummer variety $A/\iota$ admits a crepant resolution, denoted by
$\operatorname{\rm Km}(A)$. A classical problem raised by Shioda is

######  Question 1.1 ([23]).

For two abelian surfaces $A_{1}$ and $A_{2}$, if $\operatorname{\rm
Km}(A_{1})\cong\operatorname{\rm Km}(A_{2})$, then can we conclude that
$A_{1}\cong A_{2}$?

Over complex numbers, Question 1.1 was solved in [7, 18]: two abelian surfaces
have isomorphic Kummer surfaces if and only if they are derived equivalent.
Their proof relies on the use of Hodge theory and global Torelli theorem for
abelian surfaces, which are missing in positive characteristic fields. More
generally, Stellari has investigated this problem for abelian varieties of
arbitrary dimension in [26].

In this paper, we are interested in above questions over positive
characteristic fields. In particular, we would like to extend the result of
[7] and [26] to fields with odd characteristic. The following result gives an
answer of Shioda’s question over positive characteristic fields.

######  Theorem 1.2.

Assume $\mathrm{char}(k)\neq 2$. Let $A_{1}$ and $A_{2}$ be abelian varieties
of dimension $n$ over $k$. Then the following holds:

  1. (i)

If $A_{1}$ is derived equivalent to $A_{2}$, then Kummer stack
$\operatorname{\rm K}(A_{1})$ is derived equivalent to $\operatorname{\rm
K}(A_{2})$.

  2. (ii)

If $n$ is odd, then the converse of (i) holds.

  3. (iii)

If $n=2$, then $A_{1}$ and $A_{2}$ are derived equivalent if and only if
$\operatorname{\rm Km}(A_{1})\cong\operatorname{\rm Km}(A_{2})$. If $A_{1}$ is
supersingular, then $A_{2}$ is derived equivalent to $A_{1}$ if and only if
$A_{2}\cong A_{1}$.

The statements (i) and (ii) will be proved with techniques around equivariant
derived categories, which is already known in [9, 20] with $k=\mathbb{C}$. In
the same way, we will generalize them to the case that
$\operatorname{char}(k)>2$ .

The proof of statement (iii) will be divided into two cases.

  * •

For the finite height case, we can prove a lifting theorem for Kummer
structures (see §3.3). Then the specialization argument of derived
equivalences will imply this statement.

  * •

For the supersingular case, it can be concluded by supersingular Torelli
theorem for abelian varieties (see §4.2).

Notice that Theorem 1.2 shows that supersingular abelian surfaces do not have
any non-trivial Fourier-Mukai partners. We expect that there is a similar
characterization for higher dimension supersingular abelian varieties.

As an application, we can characterize the quasi-liftably birational class
(cf. [6, Definition 3.3]) of irreducible symplectic varieties which come from
the moduli space of sheaves on supersingular abelian surfaces. Consider the
moduli space of Gieseker-Maruyama $H$-stable sheaves on $A$ with Mukai vector
$v$ such that $p\nmid v^{2}$, denoted by $M_{H}(A,v)$. Recall that the fiber
$K_{v}(A)$ of the Albanese morphism

$(\det,c_{2})\colon M_{H}(A,v)\to\operatorname{Pic}^{0}(A)\times A,$

is an irreducible symplectic variety of dimension $v^{2}-2$ in the sense of
loc.cit. . A consequence in loc.cit. asserts that the generalized Kummer type
variety $K_{v}(A)$ is quasi-liftably birational to some generalized Kummer
variety $K_{n}(A^{\prime})$, where $A^{\prime}$ is a Fourier-Mukai partner of
$A$. Here we verify that $A\cong A^{\prime}$.

######  Theorem 1.3.

Let $n=\frac{v^{2}}{2}$. Suppose that $p\nmid n$ and $p\nmid n+1$. Let $A$ be
a supersingular abelian surface and let $K_{v}(A)$ be the generalized Kummer
type variety. Then

  1. (i)

$K_{v}(A)$ is quasi-liftably birational equivalent to $K_{n}(A)$.

  2. (ii)

if $K_{v}(A)$ is quasi-liftably birational to $K_{v}(A^{\prime})$ for some
abelian surface $A^{\prime}$ and $v^{\prime}$, then $A\cong A^{\prime}$.

Acknowledgement: We are grateful to Lie Fu for helpful discussions and
comments. We also want to thank the referee for several useful comments. The
authors are supported by NKRD Program of China (No. 2020YFA0713200), NSFC
General Program (No. 11771086) and Shanghai Pilot Program for Basic Research
(No. 21TQ00).

##  2\. Equivariant derived category of schemes

###  2.1. Equivariant quasi-coherent sheaves

Let $G$ be a constant finite group scheme over $k$ and let $X$ be a quasi-
compact and quasi-separated scheme over a field $k$ with a $G$-action

$\mu\colon G\times_{k}X\to X.$

A $G$-equivariant quasi-coherent sheaf on $X$ is a pair
$(\mathcal{F},\lambda)$ such that $\mathcal{F}$ is a quasi-coherent sheaf on
$X$ and $\lambda$ is a family $\left\\{\lambda_{g}\colon\mathcal{F}\to
g^{*}\mathcal{F}\right\\}_{g\in G}$ of isomorphisms satisfying the following
cocycle condition:

$\lambda_{fg}=\lambda_{f}\circ\lambda_{g}\colon\mathcal{F}\to
g^{*}\mathcal{F}\to(fg)^{*}\mathcal{F}.$ (2.1.1)

We call $\lambda$ a _$G$ -linearization_ of $\mathcal{F}$. For instance,
$\mathcal{O}_{X}^{\rm
can}=(\mathcal{O}_{X},\operatorname{id}_{\mathcal{O}_{X\times G}})$ is a
$G$-equivariant quasi-coherent sheaf. In this paper, we denote by
$\operatorname{QCoh}_{G}(X)$ the category of $G$-equivariant quasi-coherent
$\mathcal{O}_{X}$-modules.

######  Remark 2.1.

If the cocyle condition (2.1.1) for $\gamma$ is missing, then $\mathcal{F}$
will be called _$G$ -invariant_.

Let $[X/G]$ be the quotient stack given by the $G$-action on $X$ and denote by
$\operatorname{QCoh}([X/G])$ the category of quasi-coherent sheaves on $[X/G]$
(cf. [15, §9]). There is a well-known stacky description for $G$-equivariant
quasi-coherent sheaves on $X$ as follows.

######  Lemma 2.2.

There is a canonical equivalence of categories

$\operatorname{QCoh}([X/G])\simeq\operatorname{QCoh}_{G}(X).$

Moreover, if $X$ is locally noetherian, then we also have
$\operatorname{Coh}([X/G])\cong\operatorname{Coh}_{G}(X)$.

###### Proof.

Assume that $X$ is locally noetherian. Consider the functor

$\displaystyle\Phi\colon\operatorname{QCoh}_{G}(X)$
$\displaystyle\to\operatorname{QCoh}([X/G])$
$\displaystyle(\mathcal{F},\lambda)$
$\displaystyle\mapsto\widetilde{\mathcal{F}}$

where $\widetilde{\mathcal{F}}$ is the sheaf on $[X/G]$ defined as follows.
For any object

${\mathscr{P}}$${X_{T}}$${T}$$\scriptstyle{\pi}$

lying on a $k$-scheme $T$, the pull-back $\pi^{*}\mathcal{F}_{T}$ is a
$G$-equivariant quasi-coherent sheaf on $\mathscr{P}$ as $\pi$ is
$G$-equivariant. The descent theory along $G$-torsor for the stack
$\operatorname{QCoh}_{k}$ of quasi-coherent sheaves establishes a canonical
equivalence:

$\operatorname{QCoh}(T)\simeq\operatorname{QCoh}_{G}(\mathscr{P}),$ (2.1.2)

see [27, Theorem 4.46] for example. Take $\widetilde{F}(T,\mathscr{P},\pi)$ to
be one in the isomorphism class in $\operatorname{QCoh}(T)$ corresponding to
$\pi^{*}\mathcal{F}_{T}$. It remains to show that $\widetilde{F}$ is quasi-
coherent (resp. coherent).

Consider the smooth covering $(G_{X},\mu)\colon X\to[X/G]$ where $G_{X}$ is
the trivial torsor on $X$ defined by $p_{X}\colon X\times_{k}G\to X$ and $\mu$
is the group action of $G$ on $X$. Since the $\lambda$ is an isomorphism

$p_{X}^{*}\mathcal{F}\xrightarrow{\sim}\mu^{*}\mathcal{F},$

we can see $\widetilde{\mathcal{F}}(X,G_{X},\mu)\cong\mathcal{F}$ by the
previous construction, which is quasi-coherent (resp. coherent). Thus by [15,
Proposition 9.1.15], we can see $\widetilde{\mathcal{F}}$ is quasi-coherent
(resp. coherent).

The converse is similar. We just take $\Phi^{-1}\widetilde{\mathcal{F}}$ to be
the quasi-coherent sheaf (resp. coherent sheaf)
$\widetilde{\mathcal{F}}(X,G_{X},\mu)$. The linearization $\lambda$ is from
the definiton of quasi-coherent sheaves (resp. coherent sheaves) on $[X/G]$. ∎

The Lemma 2.2 implies that the category $\operatorname{QCoh}_{G}(X)$ is a
Grothendieck category (cf. [25, Tag 0781]). This promises a nice homological
algebraic theory on $G$-equivariant quasi-coherent sheaves. In the following
literature, the _$G$ -equivariant derived category_ of $X$ means the bounded
derived category of $\operatorname{Coh}_{G}(X)$, denoted by
$\operatorname{D}^{b}_{G}(X)$. A useful fact for $G$-equivariant derived
category is the derived McKay correspondence established by
Bridgeland–King–Reid [3]. The restriction of the Hilbert–Chow morphism
$X^{[n]}\to X^{(n)}$ gives a morphism

$\tau\colon X\sslash G\to X/G.$

under the natural inclusion $X\sslash G\hookrightarrow X^{[n]}$.

######  Theorem 2.3 (Bridgeland–King–Reid).

Assume that $(|G|,p)=1$. Suppose the following two conditions hold

  1. (BKR1)

$\omega_{X}$ is locally trivial as a $G$-bundle,

  2. (BKR2)

$X\sslash G\times_{\tau}X\sslash G$ has dimension $d\leq\dim X+1$.

Then there is a derived equivalence between
$\operatorname{D}^{b}(X\\!\sslash\\!G)$ and $\operatorname{D}^{b}_{G}(X)$.

###### Proof.

The original proof in loc.cit. is for $k=\mathbb{C}$. However, this also
proceeds for general case that $(|G|,p)=1$ (cf. [4, Theorem 2.4.5]). ∎

The following consequence is well-known over complex numbers (cf. [3, §10.2]
or [20, §3.2]).

######  Corollary 2.4.

Suppose $\operatorname{char}(k)\neq 2$. Let $A$ be an abelian surface over
$k$. Let $\iota$ be the involution on $A$ and $\langle\iota\rangle$ the finite
group scheme over $k$ generated by $\iota$. Then
$\operatorname{D}^{b}([A/\langle\iota\rangle])\simeq\operatorname{D}^{b}(\operatorname{\rm
Km}(A))$.

###### Proof.

We can view $\operatorname{\rm Km}(A)$ as the closure of the subset of reduced
$\langle\iota\rangle$-clusters in the Hilbert scheme
$\operatorname{Hilb}^{2}(A)$ of two points on $A$. In this case (BKR2) is
satisfied and the canonical sheaf is trivial as an
$\langle\iota\rangle$-bundle. Thus Theorem 2.3 implies that there is a
Fourier-Mukai transform

$\Phi^{P}\colon\operatorname{D}^{b}(\operatorname{\rm
Km}(A))\to\operatorname{D}^{b}_{\langle\iota\rangle}(A)\simeq\operatorname{D}^{b}([A/\langle\iota\rangle])$

as $\operatorname{char}(k)\neq 2$. ∎

Let us recall some general duality theorem for the $G$-equivariant derived
category. The $G$-action on $X$ also induces $G$-action on the triangulated
category $\operatorname{D}^{b}(X)$. Thus we can also consider the
$G$-equivariant category $\operatorname{D}^{b}(X)^{G}$ (cf. [5, §2]). Under
the assumption that $(|G|,p)=1$, we have an exact equivalence

$\operatorname{D}^{b}(X)^{G}\simeq\operatorname{D}^{b}_{G}(X)$ (2.1.3)

by loc.cit. Theorem 7.1. Therefore, we will not distinguish
$\operatorname{D}^{b}_{G}(X)$ and $\operatorname{D}^{b}(X)^{G}$ in the rest of
the paper if $G$ acts on $X$ and $(\lvert G\rvert,p)=1$.

Let $\widehat{G}=\operatorname{Hom}(G,k^{*})$ be the character group of $G$.
There exists an induced $\widehat{G}$-action on the $G$-equivariant derived
category $\operatorname{D}^{b}_{G}(X)$ as follows. For each
$\chi\in\widehat{G}$, one can define a line bundle on $[X/G]$ twisted by
$\chi$ as

$\mathcal{L}_{\chi}\coloneqq\mathcal{O}_{X}^{can}\otimes_{k}\chi\cong(\mathcal{O}_{X},\operatorname{id}_{\mathcal{O}_{G\times
X}}\otimes\chi)\quad\text{ for $\chi\in\widehat{G}$ }.$

The action of $\chi$ on $\operatorname{D}^{b}([X/G])$ is given by tensoring
$\mathcal{L}_{\chi}$.

######  Definition 2.5.

For any $G$-equivariant object $(E,\lambda)$ in $\operatorname{D}^{b}_{G}(X)$,
the $\widehat{G}$-action on a $(E,\lambda)$ is given by twisting the
linearization:

$\lambda\otimes_{k}\chi\coloneqq\left\\{E\xrightarrow{\lambda_{g}\cdot\chi(g)}g^{*}E\right\\}_{g\in
G}\quad\text{ for any $\chi\in\widehat{G}$}.$

This gives a $\widehat{G}$-action on the triangulated categories
$\operatorname{D}^{b}_{G}(X)\simeq\operatorname{D}^{b}(X)^{G}$.

Consider the identification
$\operatorname{D}^{b}([X/G])\simeq\operatorname{D}^{b}(X)^{G}$.

######  Proposition 2.6 (A. Elagin).

Assume $X$ is noetherian. There are exact equivalences

$\operatorname{D}^{b}_{G}(X)^{\widehat{G}}\simeq(\operatorname{D}^{b}(X)^{G})^{\widehat{G}}\simeq\operatorname{D}^{b}(X).$

###### Proof.

The first exact equivalence is given by (2.1.3). For the second exact
equivalence, we shall note that $\operatorname{D}^{b}_{G}(X)$ is idempotent
complete for any noetherian scheme or algebraic stack $X$ as
$\operatorname{Coh}(X)$ is a Grothendieck category and admits all products.
Now we can apply the duality theorem [5, Theorem 4.2] to conclude it. ∎

######  Remark 2.7.

The Proposition 2.6 is also referred for idempotent complete
$\mathbb{C}$-linear triangulated categories in [9] as a crucial fact (cf.
Proposition 2.2 loc.cit. ). We use the slightly general form for field $k$
with $(\lvert G\rvert,\operatorname{char}(k))=1$.

With the same notations in Corollary 2.4, we have
$\widehat{\langle\iota\rangle}\cong\langle\iota^{*}\rangle$, where $\iota^{*}$
is the character dual to $\iota$, acting naturally on
$\operatorname{D}^{b}([A/\langle\iota\rangle])$.

######  Corollary 2.8.

If $\operatorname{char}(k)>2$, then
$\operatorname{D}^{b}\left([A/\iota]\right)^{\langle\iota^{*}\rangle}\simeq\operatorname{D}^{b}(A)$.
In particular, if $A$ is an abelian surface, then
$\operatorname{D}^{b}(\operatorname{\rm
Km}(A))^{\langle\iota^{*}\rangle}\simeq\operatorname{D}^{b}(A)$. ∎

##  3\. Lifting of derived equivalences

###  3.1. Equivariant derived equivalences

In this part, we will recollect some preliminary facts on lifting theory and
descent theory of equivariant equivalences in [9, 20] and extend them to all
algebraically closed fields. With the notations as in §2, we will always
assume the order of $\lvert G\rvert$ is coprime to $p$. Let $X_{1}$ and
$X_{2}$ be two projective $k$-schemes or quotient stacks equipped with $G$
actions. Let
$\Phi\colon\operatorname{D}^{b}(X_{1})\to\operatorname{D}^{b}(X_{2})$ be an
exact functor.

######  Definition 3.1.

An exact functor
$\widetilde{\Phi}\colon\operatorname{D}^{b}_{G}(X_{1})\to\operatorname{D}^{b}_{G}(X_{2})$
is called a descent of $\Phi$ if it fits into the following 2-commutative
diagrams

${\operatorname{D}^{b}(X_{1})}$${\operatorname{D}^{b}(X_{2})}$${\operatorname{D}^{b}_{G}(X_{1})}$${\operatorname{D}^{b}_{G}(X_{2})}$$\scriptstyle{\Phi}$$\scriptstyle{\pi_{1,*}}$$\scriptstyle{\pi_{2,*}}$$\scriptstyle{\widetilde{\Phi}}$
${\operatorname{D}^{b}_{G}(X_{1})}$${\operatorname{D}^{b}_{G}(X_{2})}$${\operatorname{D}^{b}(X_{1})}$${\operatorname{D}^{b}(X_{2}),}$$\scriptstyle{\widetilde{\Phi}}$$\scriptstyle{\pi_{1}^{*}}$$\scriptstyle{\pi_{2}^{*}}$$\scriptstyle{\Phi}$

where $\pi_{i}\colon X_{i}\to[X_{i}/G]$ are structure morphisms of quotients.
We may also call $\Phi$ a lift of $\widetilde{\Phi}$.

######  Remark 3.2.

The push-forward and pull-back functors $\pi^{*}_{i},\pi_{i,*}$ are both exact
under the assumption that $(|G|,p)=1$ (cf. [5, Lemma 3.8]). Thus the functors
in Definition 3.1 are all exact.

Consider the $k$-linear triangluated category
$\mathcal{T}=\operatorname{D}^{b}_{G}(X)$. The non-trivial line bundles
$\mathcal{L}_{\chi}$ on $[X/G]$ induce a natural $\widehat{G}$-action on
$\mathcal{T}$ by taking tensor product (cf. Definition 2.5). This leads to the
following definition.

######  Definition 3.3.

Let $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ be two $k$-linear triangulated
categories with $G_{1}$ and $G_{2}$ actions respectively. Let $\gamma\colon
G_{1}\xrightarrow{\cong}G_{2}$ be a group isomorphism. An exact functor
$\Phi\colon\mathcal{T}_{1}\to\mathcal{T}_{2}$ is called $\gamma$-twisted
equivariant if

$\Phi\circ g^{*}\simeq\gamma(g)^{*}\circ\Phi\quad\text{for all }g\in G_{1}.$

When $\gamma=\operatorname{id}_{G}$, the $\Phi$ will be called $G$-equivariant
for simplicity. If a descent $\widetilde{\Phi}$ of $\Phi$ is
$\widehat{G}$-equivariant as an exact functor, then it will be called a
_$\widehat{G}$ -equivariant descent_ of $\Phi$.

The following examples of equivariant functors is the most frequently used
throughout this paper. Let $\gamma\colon G\xrightarrow{\cong}G$ be an abstract
group isomorphism, then there is an action of $G$ on $X_{1}\times_{k}X_{2}$
(or $\operatorname{D}^{b}(X_{1}\times X_{2})$) by

$g\cdot(x_{1},x_{2})=(g\cdot x_{1},\gamma(g)\cdot x_{2}).$

For instance, if $\gamma=\operatorname{id}$, then this action is just the
diagonal action of $G$. The $G$-equivariant derived category of
$X_{1}\times_{k}X_{2}$ under the action given by $\gamma$ is denoted by
$\operatorname{D}^{b}_{G_{\gamma}}(X_{1}\times X_{2})$. If there is an object
$\mathcal{P}=(P,\lambda)$ in $\operatorname{D}^{b}_{G_{\gamma}}(X_{1}\times
X_{2})$ such that

$\Phi\simeq\mathbf{R}\\!p_{2,*}(p_{1}^{*}(-)\otimes\mathcal{P}),$

then $\Phi$ is a $\gamma$-twisted $G$-equivariant exact functor. A
$\gamma$-twisted $G$-equivariant functor of this form is called integral and
will be denoted by $\Phi^{\mathcal{P}}$, where the object $\mathcal{P}$ is
called its _kernel_. In this case, we can forget the linearization of the
kernel $\mathcal{P}$, which will defines an integral functor

$\Phi^{P}\colon\operatorname{D}^{b}(X_{1})\to\operatorname{D}^{b}(X_{2}).$

It is not hard to check that $\Phi^{P}$ is a lift of $\Phi^{\mathcal{P}}$.

A well-known fact is that a $G$-equivariant descent of a Fourier-Mukai
transform is still an exact equivalence (cf. [9, Proposition 3.10] or [20,
Lemma 5]).

######  Proposition 3.4.

Let $\mathcal{P}=(P,\lambda)$ be a $G_{\gamma}$-equivariant object in
$\operatorname{D}^{b}_{G_{\gamma}}(X_{1}\times X_{2})$ for some abstract
isomorphism $\gamma\colon G\xrightarrow{\cong}G$. If $\Phi^{P}_{X_{1}\to
X_{2}}$ is an exact equivalence, then the integral functor

$\Phi\coloneqq\Phi^{\mathcal{P}}\colon\operatorname{D}^{b}_{G}(X_{1})\to\operatorname{D}^{b}_{G}(X_{2})$

will also be an exact equivalence.

The proof in loc.cit. also works for any algebraically closed field $k$ such
that $(|G|,p)=1$. We sketch it here for reader’s convenience.

###### Proof.

Take the kernel of right adjoint inverse of $\Phi$:

$Q=P^{\vee}\otimes p_{X_{1}}^{*}\omega_{X_{1}}[\dim X_{1}].$

It admits a natural $G$-lineaization $\lambda^{\prime}$ induced by $\lambda$.
Denote $(Q,\lambda^{\prime})$ by $\mathcal{Q}$. Note that the pull-back
functor $\pi^{*}$ is just the forgetful functor of linearization. Therefore
$\pi^{*}(\mathcal{Q}\circ\mathcal{P})\cong Q\circ
P\cong\mathcal{O}_{\Delta_{X_{1}}}$. On the other hand, we have
$\operatorname{Aut}(\mathcal{O}_{\Delta_{X_{1}}})=k^{*}$ in
$\operatorname{D}^{b}(X_{1}\times X_{1})$. Via computing the cohomology of
$G$, we can show that the set of linearizations of
$\mathcal{O}_{\Delta_{X_{1}}}$ forms a principal homogeneous space under the
action of $\widehat{G}=\operatorname{Hom}(G,k^{*})$ (cf. [20, Lemma 1]). In
particular, since $\mathcal{O}_{\Delta_{X_{1}}}$ admits the trivial
linearization, the set of its linearizations is equal to $\widehat{G}$. It
means that there is some $\chi\in\widehat{G}$ such that
$\mathcal{Q}\circ\mathcal{P}\cong\Delta_{*}\mathcal{L}_{\chi}$. Since
tensoring line bundles will induce a derived equivalence, we can conclude that
$\Phi$ is an exact equivalence. ∎

Suppose $G$ is a cyclic group. Then
$\mathrm{H}^{2}(G,k^{*})=\mathrm{H}^{2}(\widehat{G},k^{*})=0$ by a direct
computation. This implies that any Fourier-Mukai kernel $P$ (resp.
$\mathcal{P}=(P,\lambda)$) admits a $G$-linearization (resp.
$\widehat{G}$-linearization). Now we can lift a $G$-equivariant derived
equivalence along cyclic Galois coverings, i.e. morphisms of smooth $k$-stacks
$Y_{i}\to X_{i}=Y_{i}/G$111The quotient here is in stack-theoretical sense
(cf. [22])..

######  Corollary 3.5 ([9, Proposition 4.3]).

Let $Y_{1}\to X_{1}$ and $Y_{2}\to X_{2}$ be two cyclic Galois coverings.
Suppose that there is an isomorphism
$\gamma\colon\widehat{G}\xrightarrow{\cong}\widehat{G}$ . Then any
$\gamma$-twisted equivariant Fourier-Mukai transform

$\Phi^{\mathcal{P}}\colon\operatorname{D}^{b}(X_{1})\xrightarrow{\sim}\operatorname{D}^{b}(X_{2})$

lifts to a derived equivalence

$\widetilde{\Phi}\colon\operatorname{D}^{b}(Y_{1})\xrightarrow{\sim}\operatorname{D}^{b}(Y_{2}).$

∎

###  3.2. Proof of Theorem 1.2 (i) and (ii)

######  Proposition 3.6 (Proposition 3.1, [26]).

If there is a Fourier-Mukai transform

$\Phi^{P}\colon\operatorname{D}^{b}(A_{1})\xrightarrow{\sim}\operatorname{D}^{b}(A_{2}),$

then
$\operatorname{D}^{b}([A_{1}/\iota])\simeq\operatorname{D}^{b}([A_{2}/\iota])$.

###### Proof.

We repeat Stellari’s proof for this statement here. Let $\gamma$ be the
abstract isomorphism $[-1]_{A_{1}}\mapsto[-1]_{A_{2}}$. Let $\tau$ be the
generator of $G_{\gamma}$. Let $T_{(x,y)}$ be the translation of a point
$(x,y)$ on $A_{1}\times A_{2}$. As
$\Phi^{\tau^{*}P}\in\text{Eq}(\operatorname{D}^{b}(A_{1}),\operatorname{D}^{b}(A_{2}))$,
by [19, Corollary 3.4], we have

$\tau^{*}P\cong T_{(a,0),*}P\otimes p_{A_{1}}^{*}\mathcal{L}_{\alpha}$

for some $a\in A_{1}(k)$, line bundle
$\mathcal{L}_{\alpha}\in\operatorname{Pic}^{0}(A_{1})\cong\widehat{A}_{1}$.

It is well-known that the Fourier-Mukai kernel $P$ is isomorphic to
$\mathcal{E}[i]$ for some semi-homogeneous sheaf $\mathcal{E}$ on $A_{1}\times
A_{2}$, which implies

$T_{(a,0),*}P\cong T_{(a,0),*}\mathcal{E}[i]\cong P\otimes\mathcal{L}_{\beta}$

for some $\mathcal{L}_{\beta}\in\operatorname{Pic}^{0}(A_{1}\times A_{2})(k)$.
Thus $\tau^{*}P\cong P\otimes\mathcal{L}$, where
$\mathcal{L}=\mathcal{L}_{\beta}\otimes p_{A}^{*}\mathcal{L}_{\alpha}[i]$ . As
$\operatorname{Pic}^{0}(A_{1}\times A_{2})(k)$ is divisible, we can find a
line bundle $\mathcal{N}$ on $A_{1}\times A_{2}$ such that
$\mathcal{N}^{2}=\mathcal{L}$. Let $\widetilde{P}=P\otimes\mathcal{N}$. We can
see

$\tau^{*}\widetilde{P}\cong\tau^{*}P\otimes\tau^{*}N\cong
P\otimes\mathcal{L}\otimes N^{\vee}\cong\widetilde{P}.$ (3.2.1)

On the other hand, since $\widetilde{P}$ is obtained from $P$ by tensoring
line bundles, it also induces a Fourier-Mukai transform from $A_{1}$ to
$A_{2}$. The isomorphism (3.2.1) ensures that $\widetilde{P}$ is
$G_{\Delta}$-invariant, which also implies $\widetilde{P}$ admits a
$G_{\gamma}$ linearization $\lambda$ as $\mathrm{H}^{2}(G_{\gamma},k^{*})=0$.
Therefore
$\operatorname{D}^{b}([A/\iota])\simeq\operatorname{D}^{b}([B/\iota])$ by
Proposition 3.4. ∎

Combining the identification in Corollary 2.4, the statement (i) in Theorem
1.2 can be concluded.

Krug and Sosna proved the converse direction for odd dimensional abelian
varieties (cf. [9, Proposition 5.13]). Their statement there is over
$\mathbb{C}$, while we extend it to algebraically closed fields in positive
characteristic.

######  Proposition 3.7.

Let $A_{1}$ and $A_{2}$ are two abelian varieties in dimension $2n+1$,
$n\in\mathbb{Z}_{>0}$. Any Fourier-Mukai transform between Kummer stacks

$\Phi^{\mathcal{P}}\colon\operatorname{D}^{b}([A_{1}/\iota])\xrightarrow{\sim}\operatorname{D}^{b}([A_{2}/\iota])$

has a lift

$\widetilde{\Phi}\colon\operatorname{D}^{b}(A_{1})\xrightarrow{\sim}\operatorname{D}^{b}(A_{2}).$

###### Proof.

The stack $[A/\iota]$ has non-trivial torsion canonical bundle when $\dim A$
is odd (cf. [9]). Let $X_{i}=[A_{i}/\iota]$. We can take the abstract
isomorphism $\langle K_{X_{1}}\rangle\xrightarrow[\sim]{\gamma}\langle
K_{X_{2}}\rangle$, whose actions are given by tensoring. Notice that
$K_{X_{i}}\otimes(-)$ is just the shift of Serre functor
$S_{i}[-\dim_{X_{i}}]$ on $\operatorname{D}^{b}(X_{i})$. Therefore
$\Phi^{\mathcal{P}}$ is $\gamma$-twisted equivariant as Serre functor commutes
with all Fourier-Mukai transforms. Thus any
$\operatorname{D}^{b}([A_{1}/\iota])\xrightarrow{\sim}\operatorname{D}^{b}([A_{2}/\iota])$
can be lifted to an exact equivalence
$\operatorname{D}^{b}(A_{1})\simeq\operatorname{D}^{b}(A_{2})$ by the
Corollary 3.5. ∎

This proves Theorem 1.2 (ii).

######  Remark 3.8.

We wonder if the Kummer stacks $[A_{1}/\iota]\cong[A_{2}/\iota]$ will imply
$\operatorname{D}^{b}(A_{1})\cong\operatorname{D}^{b}(A_{2})$ or not. The same
question can be asked for two pairs $(X_{1},G)$ and $(X_{2},G)$ satisfying
(BKR1) and (BKR2) such that their quotients have isomorphic crepant
resolution.

###  3.3. Lifting Kummer structures

To prove the Theorem (iii), we need to use the lifting of Kummer surfaces. A
technical lemma is

######  Lemma 3.9.

Let $X$ be a K3 surface over $k$. Suppose $\mathcal{X}$ is a relative K3
surface, lifting $X$ to some finite extension $V$ of the ring of Witt vectors
$W(k)$, such that the specialization map is an isomorphism

$\operatorname{NS}(\mathcal{X}_{F})\xrightarrow{\sim}\operatorname{NS}(X).$
(3.3.1)

For any abelian surface $A$ satisfying $\operatorname{\rm Km}(A)\cong X$,
there is an abelian scheme $\mathcal{A}$ over $V$ such that
$\mathcal{A}_{k}\cong A$ and $\mathcal{X}\cong\operatorname{\rm
Km}(\mathcal{A})$.

###### Proof.

We may follow the idea of [24, Proposition 1.1] to prove the existence of the
lifting. Let $E_{a}$ be the exceptional curve on $X$ with respect to the
$2$-torsion point $a\in A[2]$. It has a unique Cartier divisor extension
$\mathcal{E}_{a}\in\operatorname{Pic}(\mathcal{X})$ by the construction.
Moreover, the divisor

$\mathcal{E}\coloneqq\sum_{a\in
A[2]}\mathcal{E}_{a}\in\operatorname{Pic}(\mathcal{X})$

is $2$-divisible as $\mathcal{E}$ is equal to the extension of
$\frac{1}{2}\sum_{a\in A[2]}E_{a}\in\operatorname{NS}(X)$ via (3.3.1).
Therefore, we have a double covering of $V$-schemes

$\pi\colon\mathcal{Y}\longrightarrow\mathcal{X},$

which is branched along $\mathcal{E}\subset\mathcal{X}$. We can contract the
curves $\mathcal{C}_{a}\subset\mathcal{Y}$ lying over $\mathcal{E}_{a}$ to
points:

$f\colon\mathcal{Y}\longrightarrow\mathcal{A},$

where $\mathcal{A}$ is a smooth proper scheme over $V$. Thus we can see
$\mathcal{A}$ is the required abelian scheme since
$\mathcal{X}\cong\mathcal{Y}/\iota\cong\operatorname{\rm Km}(\mathcal{A})$
where $\iota$ is the involution of $\mathcal{Y}$ induced by the double
covering $\pi$. ∎

###  3.4. Derived Torellli theorem and specialization

Let $A_{1}$ and $A_{2}$ be two abelian varieties. Consider the set of
_symplectic isomorphisms_

$U(A_{1},A_{2})=\left\\{f=\begin{pmatrix}f_{1}&f_{2}\\\
f_{3}&f_{4}\end{pmatrix}\colon
A_{1}\times\widehat{A}_{1}\xrightarrow{\sim}A_{2}\times\widehat{A}_{2}\Big{|}f^{-1}=\widetilde{f}\coloneqq\begin{pmatrix}\hat{f_{4}}&-\hat{f_{2}}\\\
-\hat{f_{3}}&\hat{f_{1}}\end{pmatrix}\right\\}.$

Let $\text{Eq}(\operatorname{D}^{b}(A_{1}),\operatorname{D}^{b}(A_{2}))$ be
the set of exact equivalences from $\operatorname{D}^{b}(A_{1})$ to
$\operatorname{D}^{b}(A_{2})$. Then the derived Torelli theorem for abelian
varieties asserts

######  Theorem 3.10 (Orlov–Polishchuk).

There is a morphism

$\gamma_{A_{1},A_{2}}\colon\text{Eq}\left(\operatorname{D}^{b}(A_{1}),\operatorname{D}^{b}(A_{2})\right)\to
U(A_{1},A_{2}).$ (3.4.1)

If $U(A_{1},A_{2})\neq\emptyset$, then $A_{1}$ and $A_{2}$ are Fourier-Mukai
partners (cf. [19] or [21]).

The following lemma shows that derived equivalence is preserved under smooth
specialization.

######  Lemma 3.11.

For two abelian schemes $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ over $V$, any
derived equivalence
$\operatorname{D}^{b}(\mathcal{A}_{1,F})\xrightarrow{\sim}\operatorname{D}^{b}(\mathcal{A}_{2,F})$
has a specialization
$\operatorname{D}^{b}(\mathcal{A}_{1,k})\to\operatorname{D}^{b}(\mathcal{A}_{2,k})$,
which is also a derived equivalence.

###### Proof.

It is known that any derived equivalence
$\operatorname{D}^{b}(\mathcal{A}_{1,F})\xrightarrow{\sim}\operatorname{D}^{b}(\mathcal{A}_{2,F})$
induces a symplectic isomorphism
$f_{F}\colon\mathcal{A}_{1,F}\times\widehat{\mathcal{A}}_{1,F}\xrightarrow{\sim}\mathcal{A}_{2,F}\times\widehat{\mathcal{A}}_{2,F}$.
By using the Matsusaka-Mumford theorem (cf. [13, Chapter I. Corollary 1]),
$f_{F}$ can be extended to an isomorphism
$f\colon\mathcal{A}_{1}\times\widehat{\mathcal{A}}_{1}\xrightarrow{\sim}\mathcal{A}_{2}\times\widehat{\mathcal{A}_{2}}$
such that the restriction $f_{k}$ is a specialization of $f_{F}$. On the other
hand, we can also take the specialization of $\widetilde{f}_{F}$. As
$(\widetilde{f}\circ f)_{F}=\widetilde{f}_{F}\circ
f_{F}=\operatorname{id}_{\mathcal{A}_{1,F}\times\widehat{\mathcal{A}}_{1,F}}$.
The graph of $\widetilde{f}\circ f$ is the diagonal of
$\mathcal{A}\times\widehat{\mathcal{A}}$. Thus the specialization $f_{k}$ is
also a symplectic isomorphism. Hence the derived equivalence
$\operatorname{D}^{b}(\mathcal{A}_{1,k})\xrightarrow{\sim}\operatorname{D}^{b}(\mathcal{A}_{2,k})$
is from Theorem 3.10. ∎

###  3.5. Proof of Theorem 1.2 (iii): finite height case

Given two abelian surfaces $A_{1}$ and $A_{2}$, if there is an exact derived
equivalence $\operatorname{D}^{b}(A_{1})\simeq\operatorname{D}^{b}(A_{2})$,
then we have $\operatorname{D}^{b}(\operatorname{\rm
Km}(A_{1}))\cong\operatorname{D}^{b}(\operatorname{\rm Km}(A_{2}))$ by
Proposition 3.6. Then we have

$\operatorname{\rm Km}(A_{1})\cong\operatorname{\rm Km}(A_{2})$

because the Kummer surface does not have non-trivial Fourier-Mukai partners
(cf. [12, Theorem 1.1]).

Recall that an abelian surface is called of finite height if it is not
supersingular. Assume that $A_{1}$ and $A_{2}$ are of finite heights. In this
case, if $X$ is a Kummer surface isomorphic to $\operatorname{\rm
Km}(A_{1})\cong\operatorname{\rm Km}(A_{2})$, then $X$ is a K3 surface with
finite height. There is a Néron-Severi lattice preserving lifting
$\mathcal{X}$ of $X$ over some $V$ (cf. [11, Corollary 4.2]). By Lemma 3.9,
there exist relative abelian surfaces $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$
over $V$ such that

$\operatorname{\rm Km}(\mathcal{A}_{1})\cong\mathcal{X}\cong\operatorname{\rm
Km}(\mathcal{A}_{2}).$

Due to [7, Theorem 0.1 (1)], the generic fibers of $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$ are geometrically derived equivalent. By using the standard
spreading out argument and [19, Lemma 2.12], we can find a finite extension
$V^{\prime}$ of $V$ such that
$\operatorname{D}^{b}(A_{1,F^{\prime}})\xrightarrow{\sim}\operatorname{D}^{b}(A_{2,F^{\prime}})$,
where $F^{\prime}$ is the fraction field of $V^{\prime}$. The assertion
follows from Lemma 3.11.

##  4\. Supersingular derived Torelli theorem

In this section, we will focus on derived categories of supersingular abelian
varieties. There is a cohomological realization of the derived Torelli theorem
for supersingular abelian varieties using supersingular abelian crystals and
supersingular K3 crystals.

Throughout the rest part, we fix an algebraically closed field $k$ in
characteristic $p$. We also denote $W=W(k)$ for the ring of Witt vectors of
$k$ and $\sigma\colon W\to W$ for the Frobenius morphism.

###  4.1. Abelian crystals and K3 crystals

For supersingular abelian varieties, we can give a cohomological realization
of Theorem 3.10 via Ogus’ supersingular Torelli theorem ([14]).

Let $A$ be an abelian variety of dimension $g$ over $k$. Recall that $A$ is
supersingular if the crystalline cohomology
$\mathrm{H}_{\operatorname{crys}}^{1}(A/W)$ is purely of slope $\frac{1}{2}$
as a weight one $F$-crystal with natural Frobenius. Over an algebraically
closed field, it is also equivalent to say that $A$ is isogenous to a $g$-fold
product of supersingular elliptic curves (cf. [16, Theorem (4.2)]).

######  Definition 4.1.

An abelian crystal of genus $g$ is a $F$-crystal $(H,\varphi)$ of rank $2g$
endowed with an isomorphism between $F$-crystals
$\operatorname{tr}\colon\Lambda^{2g}H\to W(-n)$, whose Hodge numbers of Hodge
polygon are both equal to $g$. The isomorphism $\operatorname{tr}$ is called
the trace of abelian crystal $(H,\varphi)$.

As its name indicates, for a $g$-dimensional abelian variety $A$, the
$F$-crystal $\mathrm{H}^{1}_{\operatorname{crys}}(A/W)$ is an example of
abelian crystal. The trace map of $\mathrm{H}^{1}_{\operatorname{crys}}(A/W)$
is given by

$\Lambda^{2g}\mathrm{H}^{1}_{\operatorname{crys}}(A/W)\cong\mathrm{H}^{2g}_{\operatorname{crys}}(A/W)\cong
W(-n).$

Here the first isomorphism is coming from the multiplication of $A$ and the
cup-product of its crystalline cohomology.

The crystalline Torelli theorem of supersingular abelian varieties states
that, for any integer $g\geq 2$, there is a bijection

$\left\\{\begin{matrix}\text{isomorphism classes of }\\\ \text{ supersingular
abelian varieties}\\\ \text{of genus
$g$}\end{matrix}\right\\}\xrightarrow[H^{1}_{\operatorname{crys}}(-/W)]{\sim}\left\\{\begin{matrix}\text{isomorphism
classes of}\\\ \text{supersingular abelian crystals}\\\ \text{ of genus
$g$}\end{matrix}\right\\}_{.}$ (4.1.1)

See [14, Theorem 6.2].

The second crystalline cohomology of an abelian surface is also equipped with
a structure called _K3-crystal_ which is defined as follows.

######  Definition 4.2.

We say a $F$-crystal $(H,\varphi)$ is K3-crystal with rank $n$, if there is a
symmetric bilinear form $\langle,\rangle_{H}$ on $H$, satisfying:

  1. (i)

$H$ is of weight $2$, that means $p^{2}H\subset\operatorname{Im}(\varphi)$;

  2. (ii)

the Hodge number $h^{0}(H)=1$, that means
$\varphi\otimes\operatorname{id}_{k}$ is of rank 1;

  3. (iii)

$\langle,\rangle_{H}$ is perfect;

  4. (iv)

$\langle\varphi x,\varphi y\rangle_{H}=p^{2}\sigma\langle x,y\rangle_{H}$ for
any $x,y\in H$.

Inside a K3-crystal $(H,\varphi)$, there is a natural $\mathbb{Z}_{p}$-lattice
defined as

$T_{H}\coloneqq H^{\varphi=p}=\left\\{x\in H\big{|}\varphi(x)=px\right\\}$

equipped with bilinear form $\langle,\rangle_{T_{H}}$ induced from $H$. A
K3-crystal is called _supersinguar_ if it is purely of slope 1. If
$(H,\varphi)$ is supersingular of rank $n$, then its Tate module $T_{H}$ is a
free $\operatorname{\mathbb{Z}}_{p}$-module of rank $n$, which comes from the
definition of pure of slope $1$. Its discriminant is equal to
$p^{-2\sigma_{0}}$ for some integer $\sigma_{0}\geq 1$, which is called the
_Artin invariant_ of $H$.

Let $A$ be an abelian surface over $k$. Then
$\mathrm{H}^{2}_{\operatorname{crys}}(A/W)$ is naturally endowed with a
K3-crystal structure. Consider the canonical isomorphism

$\mathrm{H}^{2}_{\operatorname{crys}}(A/W)\cong\Lambda^{2}\mathrm{H}^{1}_{\operatorname{crys}}(A/W),$

one can see that $A$ is supersingular if and only if
$\mathrm{H}^{2}_{\operatorname{crys}}(A/W)$ is supersingular as a K3-crystal.
An important observation due to Ogus is that the functor $\Lambda^{2}$ forms
an equivalence from the category of supersingular abelian crystals of genus 2
to the category of supersingular K3 crystals of rank 6 (cf. [14, Proposition
6.9]). Thus the crystalline Torelli theorem for supersingular abelian surfaces
can be rephrased in terms of K3-crystals, that means two supersingular abelian
surfaces $A_{1}$ and $A_{2}$ are isomorphic if and only if

$\mathrm{H}^{2}_{\operatorname{crys}}(A_{1}/W)\cong\mathrm{H}^{2}_{\operatorname{crys}}(A_{2}/W).$

######  Remark 4.3.

This also implies that any supersingular abelian surface $A$ is principal
polarized, since there is a natural isomorphism of K3-crystals
$\mathrm{H}^{2}_{\operatorname{crys}}(A/W)\xrightarrow{\sim}\mathrm{H}^{2}_{\operatorname{crys}}(\widehat{A}/W)$.
This can also deduced from the classification of Néron-Severi lattices of
supersingular abelian surfaces (cf. [6, Lemma 6.2]).

One can characterize $K3$-crystals via the characteristic space. Let $H$ be a
supersingular K3 crystal with Artin invariant $\sigma_{0}$. We have the an
orthogonal decomposition (not unique!) for its Tate module:

$(T_{H},\langle,\rangle_{T_{H}})\cong(T_{0},p\langle,\rangle_{T_{0}})\oplus(T_{1},\langle,\rangle_{T_{1}}),$
(4.1.2)

such that $T_{0}$ is of rank $2\sigma_{0}$, $\langle,\rangle_{T_{0}}$ and
$\langle,\rangle_{T_{1}}$ are both perfect, since the cokernel of
$T_{H}\hookrightarrow T_{H}^{\vee}$ is killed by $p$. The kernel

$\overline{H}\coloneqq\ker(T_{H}\otimes_{\mathbb{Z}_{p}}k\to H\otimes_{W}k)$

forms a $\sigma_{0}$-dimensional $\mathbb{F}_{p}$-vector space which is
totally isotropic with respect ot $\langle,\rangle_{T_{H}}$ and it is
isomorphic to the image of

$H\cong H^{\vee}\to T_{H}^{\vee}\otimes_{\mathbb{Z}_{p}}W$

It also forms a _strictly characteristic subspace_ of
$T_{0}\otimes_{\mathbb{Z}_{p}}k$ (cf. [14, Definition 3.19] for the definition
and loc.cit.  Remark 3.16 for the explanation). Let
$\mathbb{K}_{H}=\varphi^{-1}(\overline{H})\subset
T_{0}\otimes_{\mathbb{Z}_{p}}k$, which is another strictly characteristic
subspace of $T_{0}\otimes_{\mathbb{Z}_{p}}k$. We also call it the
characteristic space of the $K3$-crystal $H$.

The classification of $K3$-crystals (see [14, Theorem 3.20]) asserts that

######  Theorem 4.4 (Ogus).

For two supersingular $K3$-crystals $H$ and $H^{\prime}$ of the same rank,
$H\cong H^{\prime}$ if and only if there is an isomorphism of pairs
$(T_{0}\otimes k,\mathbb{K}_{H})\cong(T_{0}^{\prime}\otimes
k,\mathbb{K}_{H^{\prime}})$.

###  4.2. Supersingular derived Torelli theorem

Let $\mathcal{P}$ be the Poincaré line bundle on $A\times\widehat{A}$
associated to the polarization $\varphi_{\mathcal{L}}$. There is a canonical
isomorphism between Dieudonné modules:

$\Phi_{A}\colon\mathrm{H}^{1}_{\operatorname{crys}}(A/W)^{*}\xrightarrow{}\mathrm{H}^{1}_{\operatorname{crys}}(\widehat{A}/W),$
(4.2.1)

which corresponds to the first crystalline Chern class

$c_{1}(\mathcal{P})\in\mathrm{H}^{1}_{\operatorname{crys}}(A/W)\otimes_{W}\mathrm{H}^{1}_{\operatorname{crys}}(\widehat{A}/W)\subseteq\mathrm{H}^{2}_{\operatorname{crys}}(A\times\widehat{A}/W);$

see [1, Théorèm 5.1.2]. We have a natural quadratic form on the $F$-crystal
$\mathrm{H}^{1}_{\operatorname{crys}}(A\times\widehat{A})$:

$q_{A}(a,\alpha)=2\Phi_{A}^{-1}(\alpha)(a)\quad\text{for
}(a,\alpha)\in\mathrm{H}^{1}_{\operatorname{crys}}(A/W)\oplus\mathrm{H}^{1}_{\operatorname{crys}}(\widehat{A}/W)$

With abelian crystals, we are able to provide a cohomological description of
Theorem 3.10 for supersingular abelian varieties.

It is clear that any symplectic isomorphism $f\colon
A_{1}\times\widehat{A}_{1}\xrightarrow{\sim}A_{2}\times\widehat{A}_{2}$ will
induce an isomorphism of abelian crystals

$\varphi\colon\mathrm{H}^{1}_{\operatorname{crys}}(A_{1}/W)\oplus\mathrm{H}^{1}_{\operatorname{crys}}(\widehat{A}_{1}/W)\xrightarrow{\sim}\mathrm{H}^{1}_{\operatorname{crys}}(A_{2}/W)\oplus\mathrm{H}^{1}_{\operatorname{crys}}(\widehat{A}_{2}/W),$

by taking Künneth decomposition.

Conversely, we can write the isomorphism $\varphi$ into a $2\times 2$-matrix

$\mathrm{H}^{1}_{\operatorname{crys}}(A_{1}/W)\oplus\mathrm{H}^{1}_{\operatorname{crys}}(\widehat{A}_{1}/W)\xrightarrow{\begin{pmatrix}\varphi_{1}&\varphi_{2}\\\
\varphi_{3}&\varphi_{4}\end{pmatrix}}\mathrm{H}^{1}_{\operatorname{crys}}(A_{2}/W)\oplus\mathrm{H}^{1}_{\operatorname{crys}}(\widehat{A}_{2}/W)$

in which $\varphi_{i}$ are all morphisms between $F$-crystals. Then $\varphi$
being isometry in terms of the quadratic form $q_{A}$ is equivalent to
satisfying matrix equation

$\begin{pmatrix}\varphi_{1}^{t}&\varphi_{3}^{t}\\\
\varphi_{2}^{t}&\varphi_{4}^{t}\end{pmatrix}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\begin{pmatrix}\varphi_{1}&\varphi_{2}\\\
\varphi_{3}&\varphi_{4}\end{pmatrix}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}.$

It implies that

$\varphi^{-1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\begin{pmatrix}\varphi_{1}^{t}&\varphi_{3}^{t}\\\
\varphi_{2}^{t}&\varphi_{4}^{t}\end{pmatrix}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}=\begin{pmatrix}\varphi_{4}^{t}&\varphi_{2}^{t}\\\
\varphi_{3}^{t}&\varphi_{1}^{t}\end{pmatrix}$

Now suppose $A_{1}$ or $A_{2}$ is supersingular. By Ogus’s crystalline Torelli
theorem for supersinsgular abelian varieties (cf. [14, Theorem 6.2]), we can
find an isomorphism
$f\in\operatorname{Hom}(A_{1}\times\widehat{A}_{1},A_{2}\times\widehat{A}_{2})$
such that $\mathrm{H}^{1}_{\operatorname{crys}}(f)=\varphi$ . It also
satisfies $H^{1}_{\operatorname{crys}}(f_{i})=\varphi_{i}$ if we rewrite $f$
into the following matrix form uniquely:

$f=\begin{pmatrix}f_{1}&f_{2}\\\ f_{3}&f_{4}\end{pmatrix}.$

Note that

$c_{1}(\widehat{\mathcal{P}})=-c_{1}(\mathcal{P})\in\mathrm{H}^{1}_{\operatorname{crys}}(\widehat{A}/W)\otimes\mathrm{H}^{1}_{\operatorname{crys}}(A/W)$

by the identification
$\mathrm{H}^{1}_{\operatorname{crys}}(\widehat{\widehat{A}}/W)\cong\mathrm{H}^{1}_{\operatorname{crys}}(A/W)$.
Thus we have

$\varphi_{1}^{t}=\mathrm{H}^{1}_{\operatorname{crys}}(\widehat{f}_{1})\quad\varphi_{2}^{t}=-\mathrm{H}^{1}_{\operatorname{crys}}(\widehat{f}_{2})\quad\varphi_{3}^{t}=-\mathrm{H}^{1}_{\operatorname{crys}}(\widehat{f}_{3})\quad\varphi_{4}^{t}=\mathrm{H}^{1}_{\operatorname{crys}}(\widehat{f}_{4}).$

Thus $f\in U(A_{1},A_{2})$. Therefore, we have the following

######  Proposition 4.5.

For arbitrary supersingular abelian varieties $A_{1}$ and $A_{2}$ over $k$,
the following statements are equivalent.

  1. (i)

$\operatorname{D}^{b}(A_{1})\simeq\operatorname{D}^{b}(A_{2})$.

  2. (ii)

There is a isometry of abelian crystals

$\mathrm{H}^{1}_{\operatorname{crys}}(A_{1}\times\widehat{A}_{1}/W)\xrightarrow{\sim}\mathrm{H}^{1}_{\operatorname{crys}}(A_{2}\times\widehat{A}_{2}/W).$

with respect to $q_{A_{i}}$.

For supersingular abelian surfaces, we have the following consequence via
K3-crystals.

######  Theorem 4.6.

Let $A_{1}$ and $A_{2}$ be two supersingular abelian surfaces over $k$. Then
$\operatorname{D}^{b}(A_{1})\cong\operatorname{D}^{b}(A_{2})$ if and only if
$A_{1}\cong A_{2}$.

###### Proof.

It is clear that when $\dim A_{1}=2$ then $\dim A_{2}=2$ if they are derived
equivalent. The given derived equivalence $\Phi^{P}$ induces an isometry
between K3 crystals
$\widetilde{\mathrm{H}}(A_{1}/W)\cong\widetilde{\mathrm{H}}(A_{2}/W)$, where

$\widetilde{\mathrm{H}}(A_{i}/W)\coloneqq
W(-1)\oplus\mathrm{H}^{2}_{\operatorname{crys}}(A_{i}/W)\oplus W(-1)$

is the Mukai K3-crystal of $A_{i}$ equipped with the Mukai pairing. Since
$\mathrm{H}^{2}_{\operatorname{crys}}(A_{i}/W)^{\varphi_{i}=p}\cong\operatorname{NS}(A_{i})\otimes\mathbb{Z}_{p}$
(cf. [14, (1.6)]), we have

$\widetilde{\mathrm{H}}(A_{i}/W)^{\varphi_{i}=p}\cong\mathbb{Z}_{p}^{\oplus
2}\oplus\operatorname{NS}(A_{i})\otimes\mathbb{Z}_{p}.$

Therefore, the characteristic subspace of $\widetilde{\mathrm{H}}(A_{i}/W)$ is
equal to

$\ker\left(\operatorname{NS}(A_{i})\otimes
k\to\mathrm{H}^{2}_{\operatorname{dR}}(A_{i}/k)\right),$

which is isomorphic to the characteristic subspace of
$\mathrm{H}^{2}_{\operatorname{crys}}(A_{i}/W)$. Let

$\operatorname{NS}(A_{i})\otimes\mathbb{Z}_{p}\cong\left(T_{0}^{(i)},p\langle,\rangle\right)\oplus\left(T_{1}^{(i)},\langle,\rangle\right)$

be decomposition of $\mathbb{Z}_{p}$-lattice as in (4.1.2). Then there is a
decomposition of the Tate module of $\widetilde{\mathrm{H}}(A_{i}/W)$:

$\left(T_{0}^{(i)},p\langle,\rangle\right)\oplus\left(T_{1}^{(i)}\oplus\mathbb{Z}_{p}^{\oplus
2},\langle,\rangle\right),$

where the second inner product is from the restriction of Mukai pairing.
Therefore

$(T_{0}^{(1)},\mathbb{K}_{A_{1}})\cong(T_{0}^{(2)},\mathbb{K}_{A_{2}}),$

which implies an isomorphism between K3-crystals:

$\mathrm{H}^{2}_{\operatorname{crys}}(A_{1}/W)\cong\mathrm{H}^{2}_{\operatorname{crys}}(A_{2}/W).$

Then the isomorphism $A_{1}\cong A_{2}$ follows from the crystalline Torelli
theorem. ∎

Now we can give a summary of the known equivalence relations between
supersingular abelian surfaces.

######  Corollary 4.7.

Let $A_{1},A_{2}$ be two abelian surfaces. If $A_{1}$ is supersingular, then
the following statements are equivalent:

  1. (a)

There is an isomorphism $A_{1}\cong A_{2}$;

  2. (b)

There is an isomorphism between K3-crystals
$\mathrm{H}^{2}_{\operatorname{crys}}(A_{1}/W)\cong\mathrm{H}^{2}_{\operatorname{crys}}(A_{2}/W)$;

  3. (c)

There is an isomorphism between Kummer surfaces $\operatorname{\rm
Km}(A_{1})\cong\operatorname{\rm Km}(A_{2})$;

  4. (d)

There is a derived equivalence $\operatorname{D}^{b}(\operatorname{\rm
Km}(A_{1}))\xrightarrow{\sim}\operatorname{D}^{b}(\operatorname{\rm
Km}(A_{2}))$;

  5. (e)

There is a derived equivalence
$\operatorname{D}^{b}(A_{1})\xrightarrow{\sim}\operatorname{D}^{b}(A_{2})$.

###### Proof.

We firstly note that the supersingularity is invariant under the relations
listed in the statements. Hence $A_{2}$ is supersingular in each statement.

Then the equivalence between (a), (b) is just the Ogus’s crysalline Torelli
theorem for supersingular abelian surfaces. The statements (c), (d) and (e)
are equivalent by the Theorem 1.2. The (a) and (e) are equivalent by Theorem
4.6. ∎

Therefore, the statement (iii) in Theorem 1.2 is true for supersingular
abelian surfaces.

######  Remark 4.8.

We suspect whether every supersingular abelian variety $A$ has only trivial
(in the strong sense) Fourier-Mukai partners, i.e. $\mathrm{FM}(A)=\\{A\\}$.
This requires that $A$ admits a principal polarization, which holds if $\dim
A=2$ (cf. Remark 4.3). But it may fail in higher dimensional case. See [10,
§10] for the discussion of non-principal polarizations on supersingular
abelian varieties of any genus. It will be very interesting to know if
$\mathrm{FM}(A)=\\{A,\widehat{A}\\}$.

###  4.3. Proof of Theorem 1.3

We can see there is a quasi-liftably birational map

$K_{v}(A)\dashrightarrow K_{n}(A^{\prime}),$ (4.3.1)

with $n=\frac{v^{2}}{2}-1$, $p\nmid n$ and some (supersingular) abelian
surface $A^{\prime}$ derived equivalent to $A$ (cf. [6, Theorem 6.12]). Then
(i) follows from the fact $A$ does not have non-trivial Fourier-Mukai partners
(cf. 4.7).

To prove (ii), our strategy is to endow
$\mathrm{H}^{2}_{\operatorname{crys}}(K_{n}(A)/W)$ with a natural222Here the
“natural” means the structure should be at least functorial with respect to
algebraic correpsondences. K3-crystal structure, which is closely related to
that of $A$. Combining the algebraic correspondence given by the birational
map (4.3.1), one can show that there is an isomorphism between K3-crystals of
$A$ and $A^{\prime}$. In fact,

######  Lemma 4.9.

Assume that $p\nmid n+1$.

  1. (i)

For any abelian surface $A$ and positive integer $n$, we can endow the
$F$-crystal $\mathrm{H}^{2}_{\operatorname{crys}}(K_{n}(A)/W)$ with the
Beauville-Bogomolov form $q$, which makes
$(\mathrm{H}^{2}_{\operatorname{crys}}(K_{n}(A)/W),q)$ a K3 crystal of rank 7.

  2. (ii)

There is an orthogonal decomposition with respect to $q$:

$\mathrm{H}^{2}_{\operatorname{crys}}(K_{n}(A)/W)\cong\mathrm{H}^{2}_{\operatorname{crys}}(A/W)\oplus
W(-1).$ (4.3.2)

###### Proof.

Let $\mathcal{A}$ be a lifting of abelian surface $A$ to the Witt vector ring
$W$, which is an abelian scheme over $W$ (cf. [17, (11.1)]). Consider the
summation of points $s\colon\mathcal{A}^{[n]}\to\mathcal{A}$. The fiber of $s$
at the origin point $\operatorname{Spec}(W)\to\mathcal{A}$ is a lifting of
generalized Kummer variety $K_{n}(A)$, which will be denoted by
$\mathcal{K}_{n}(\mathcal{A})$ as a $W$-scheme. For the geometric generic
fiber $\mathcal{K}_{n}(\mathcal{A})_{\bar{K}}$, we have the Beauville-
Bogomolov form $q_{\bar{K}}$ on $p$-adic étale cohomology
$\mathrm{H}^{2}_{\operatorname{\text{\'{e}t}}}(\mathcal{K}_{n}(\mathcal{A})_{\bar{K}},\mathbb{Z}_{p})$.
There is an orthogonal decomposition

$\mathrm{H}^{2}_{\operatorname{\text{\'{e}t}}}(\mathcal{K}_{n}(\mathcal{A})_{\bar{K}},\mathbb{Z}_{p})\cong\mathrm{H}^{2}_{\operatorname{\text{\'{e}t}}}(\mathcal{A}_{\bar{K}},\mathbb{Z}_{p})\oplus\mathbb{Z}_{p}\delta,$
(4.3.3)

such that $q_{\bar{K}}(\delta)=-2(n+1)$. The Beauville-Bogomolov form on
$\mathrm{H}^{2}_{\operatorname{crys}}(K_{n}(A)/W)$ can be defined by applying
integral crystalline-étale comparison (cf. [2, Theorem 1]) and denoted by $q$.
It is followed by a decomposition of $F$-crystals

$\mathrm{H}^{2}_{\operatorname{crys}}(K_{n}(A)/W)\cong\mathrm{H}^{2}_{\operatorname{crys}}(A/W)\oplus
W(-1),$

with respect to $q$.

The bilinear form induced by $q$ is perfect since $p\nmid n+1$. A direct
computation shows that

$q(\varphi^{*}{\alpha})=p^{2}\sigma q(\alpha).$

The canonical decomposition (4.3.2) implies the Hodge number
$h^{0}(\mathrm{H}^{2}_{\operatorname{crys}}(K_{n}(A)/W))=1$. Therefore,
$(\mathrm{H}^{2}_{\operatorname{crys}}(K_{n}(A)/W),q)$ is a K3-crystal. ∎

By the decomposition (4.3.2), we have

$\mathrm{H}^{2}_{\operatorname{crys}}(K_{n}(A)/W)^{\varphi=p}\cong\mathrm{H}^{2}_{\operatorname{crys}}(A/W)^{\varphi=p}\oplus\mathbb{Z}_{p}\delta.$

Thus there is an isomorphism between Tate modules (as
$\mathbb{Z}_{p}$-lattices)

$\operatorname{NS}(K_{n}(A))\otimes\mathbb{Z}_{p}\cong\left(\operatorname{NS}(A)\otimes\mathbb{Z}_{p}\right)\oplus\langle-2(n+1)\rangle.$
(4.3.4)

######  Lemma 4.10.

Under the same assumption in 4.9, if $A$ is supersingular, then we have an
isomorphism of characteristic subspaces

$\mathbb{K}_{\mathrm{H}^{2}_{\operatorname{crys}}(A/W)}\xrightarrow{\sim}\mathbb{K}_{\mathrm{H}^{2}_{\operatorname{crys}}(K_{n}(A)/W)}.$

###### Proof.

Consider the following commutative diagram

${\operatorname{NS}(A)\otimes\mathbb{Z}_{p}}$${\mathrm{H}^{2}_{\operatorname{crys}}(A/W)}$${\operatorname{NS}(K_{n}(A))\otimes\mathbb{Z}_{p}}$${\mathrm{H}^{2}_{\operatorname{crys}}(K_{n}(A)/W)}$$\scriptstyle{c_{1}}$$\scriptstyle{c_{1}}$

The horizontal injective morphisms are isometric embeddings from (4.3.4). Thus
it is not hard to see that they induces isomorphic characteristic subspaces. ∎

######  Remark 4.11.

The Lemma 4.10 can be viewed as a higher dimensional analogue to the
relationship between supersingular abelian surfaces and their associated
supersingular Kummer surfaces, which is a miracle used in Ogus’s proof of
crystalline Torelli theorem for supersingular Kummer surfaces.

######  Lemma 4.12.

If $K_{n}(A)$ and $K_{n}(A^{\prime})$ are quasi-liftably birational
equivalent, then there is an isomorphism of $F$-crystals
$\mathrm{H}^{2}_{\operatorname{crys}}(K_{n}(A)/W)\xrightarrow{\sim}\mathrm{H}^{2}_{\operatorname{crys}}(K_{n}(A^{\prime})/W)$
preserving Beauville-Bogomolov forms.

###### Proof.

We may assume that $K_{n}(A)$ and $K_{n}(A^{\prime})$ are liftable as
$\mathcal{K}$ and $\mathcal{K}^{\prime}$ over some finite extension $V$ of
$W$, whose geometric generic fibers are birational equivalent irreducible
symplectic varieties. Then there is a correspondence
$Z_{F}\subset(\mathcal{K}_{F}\times\mathcal{K}_{F}^{\prime})$ for some totally
ramified finite field extension $F$ of $K=\operatorname{Frac}(V)$, such that
$[Z_{F}]_{*}\colon\mathrm{H}^{2}_{\operatorname{\text{\'{e}t}}}(\mathcal{K}_{\bar{K}},\mathbb{Z}_{p})\xrightarrow{\sim}\mathrm{H}^{2}_{\operatorname{\text{\'{e}t}}}(\mathcal{K}^{\prime}_{\bar{K}},\mathbb{Z}_{p})$
is a $G_{F}$-equivariant isomorphism, compatible with Beauville-Bogomolov
forms (cf. [8, Lemma 2.6]). The construction in Lemma 4.9 implies that there
is an isomorphism of $F$-crystals as we required. ∎

If $K_{v}(A)$ and $K_{v^{\prime}}(A^{\prime})$ are quasi-liftably birational
equivalent, then $K_{n}(A)$ and $K_{n}(A^{\prime})$ are also quasi-liftably
birational equivalent by combining (4.3.1). This implies that there exists
isomorphisms between characteristic subspaces

$\mathbb{K}_{A}\xrightarrow[\text{Lemma \ref{lemma:abelian-
Kummer}}]{\sim}\mathbb{K}_{K_{n}(A)}\xrightarrow[\text{Lemma
\ref{lemma:isocrystalH2}}]{\sim}\mathbb{K}_{K_{n}(A^{\prime})}\xrightarrow[\text{Lemma
\ref{lemma:abelian-Kummer}}]{\sim}\mathbb{K}_{A^{\prime}}.$

Therefore, by Theorem 4.4 there is an isomorphism of K3-crystals

$\mathrm{H}^{2}_{\operatorname{crys}}(A/W)\cong\mathrm{H}^{2}_{\operatorname{crys}}(A^{\prime}/W).$

Now we can conclude that $A\cong A^{\prime}$ by Ogus’s crystalline Torelli
theorem for supersingular abelian surfaces.

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f^{*}(\omega_{B}\otimes\mathscr{N})$ for some invertible sheaf $\mathscr{N}$
on $B$. Since $\mathscr{O}_{E}(E)=\mathscr{L}^{\otimes-1}$, the Adjunction
Formula for the effective Cartier divisor $E\subset P$ yields

$\omega_{B}=(\omega_{P}\otimes\mathscr{O}_{P}(E))|E=\mathscr{L}^{\otimes
2}\otimes\omega_{B}\otimes\mathscr{N}\otimes\mathscr{L}^{\otimes-1}.$

The assertion follows. ∎

Recall that a scheme $V$ is called a _Fano variety_ if it is proper, with
$h^{0}(\mathscr{O}_{V})=1$, all local rings $\mathscr{O}_{V,a}$ are
Gorenstein, and the dualizing sheaf $\omega_{V}$ is anti-ample. Note that we
make no other assumptions on the singularities. Fano varieties of dimension
$n\geq 1$ come with two important numerical invariants: the _degree_ and the
_index_

$\deg(V)=(-K_{V})^{n}=c_{1}^{n}(\omega_{V}^{\otimes-1})>0\quad\text{and}\quad\operatorname{ind}(V)>0.$

The latter is defined as the divisibility of $\omega_{V}$ in the numerical
group $\operatorname{Num}(V)$.

######  Theorem 6.5.

Suppose the following:

  1. (i)

The projective scheme $B$ is a Fano variety of dimension $n\geq 1$.

  2. (ii)

The ample sheaf $\mathscr{L}$ has the property $\mathscr{L}^{\otimes
m}=\omega_{B}^{\otimes-1}$ for some $m\geq 1$.

  3. (iii)

The group $H^{i}(B,\mathscr{L}^{\otimes t})$ vanishes for all integers
$i,t\geq 1$.

Then the projective scheme $X$ is Gorenstein, and we have the the Canonical
Bundle Formula $K_{X}=-(m+1)A$. In particular, $X$ is a Fano variety of
dimension $n+1$, with numerical invariants

$\deg(X)=\frac{(m+1)^{n+1}}{m^{n}}\deg(B)\quad\text{and}\quad\operatorname{ind}(X)=m+1.$

###### Proof.

Assumption (iii) guarantees that the cone $X$ is Cohen–Macaulay, according to
[Kol13, Proposition 3.13]. In light of Condition (ii) and Proposition 6.3, the
dualizing sheaf on the $\mathbb{P}^{1}$-bundle is

$\omega_{P}=\mathscr{O}_{P}(-2E)\otimes f^{*}(\mathscr{L}^{\otimes-m-1}).$

Furthermore, we have $f^{*}(\mathscr{L})=\mathscr{O}_{P}(A-E)$, which follows
from (6) and the exact sequence (5). This gives $K_{P}=-(m+3)E-(m+1)A$. Since
$A$ is disjoint from the exceptional locus $E$, we see that $X$ is Gorenstein,
having $K_{X}=-(m+1)A$. With Proposition 6.3 we conclude that $X$ is a Fano
variety with index $m+1$. The degree is $(-K_{X})^{n+1}=(m+1)^{n+1}A^{n+1}$.
In light of (6), we have
$A^{n+1}=c_{1}^{n}(\mathscr{L})=\frac{1}{m^{n}}(-K_{B})^{n}$. This concludes
the proof. ∎

## 7\. Fano threefolds with unusual torsion

Let $k$ be an algebraically closed ground field of characteristic $p=2$, and
$S$ be a simply-connected Enriques surface endowed with an ADE-curve $E\subset
X$ as in Theorem 4.2. This actually exists, as we saw in Section 5. The
resulting contraction $f:S\rightarrow Z$ yields a normal Enriques surface,
coming with a $G$-torsor ${\tilde{Z}}\rightarrow Z$, with respect to the
Cartier dual $G$ for the unipotent group scheme
$P=\operatorname{Pic}^{\tau}_{Z/k}$ of order two. As explained in Section 5,
the normalization $Z^{\prime}\rightarrow{\tilde{Z}}$ gives a del Pezzo surface
with $\operatorname{Pic}(Z^{\prime})=\mathbb{Z}K_{Z^{\prime}}$ and
$h^{1}(\mathscr{O}_{Z^{\prime}})=0$, of degree
$K_{Z^{\prime}}^{2}\in\\{2,4,6,8\\}$. It comes with an inseparable double
covering $\nu:Z^{\prime}\rightarrow Z$.

We now consider the $\mathbb{P}^{1}$-bundle $P=\mathbb{P}(\mathscr{E})$ with
$\mathscr{E}=\mathscr{O}_{Z^{\prime}}\oplus\omega_{Z^{\prime}}^{\otimes-1}$,
and the ensuing contraction

$r:P\longrightarrow X=\operatorname{Proj}\bigoplusop\displaylimits_{i\geq
0}H^{0}(P,\mathscr{O}_{P}(iA))$

of the negative section $E=\mathbb{P}(\omega_{Z^{\prime}}^{\otimes-1})$,
defined via the positive section $A=\mathbb{P}(\mathscr{O}_{Z^{\prime}})$, as
explained in the previous section. The singular loci can be written as

$\operatorname{Sing}(P)=\mathbb{P}^{1}_{\operatorname{Sing}(Z^{\prime})}\quad\text{and}\quad\operatorname{Sing}(X)=r(\mathbb{P}^{1}_{\operatorname{Sing}(Z^{\prime})}).$

######  Proposition 7.1.

The scheme $X$ is a normal Fano threefold of degree $(-K_{X})^{3}=8\cdot
K_{Z^{\prime}}^{2}$, index $\operatorname{ind}(X)=2$, and
$h^{i}(\mathscr{O}_{X})=0$ for all integers $i\geq 1$. The Picard scheme is
$\operatorname{Pic}_{X/k}=\mathbb{Z}$, which is generated by
$\mathscr{O}_{X}(A)$, such that $\omega_{X}=\mathscr{O}_{X}(-2A)$.

###### Proof.

The statement about the Picard scheme follows from Proposition 6.3. According
to Corollary 4.4, we have $h^{1}(\omega_{Z^{\prime}}^{\otimes t})=0$. We now
can apply Theorem 6.5 with $B=Z^{\prime}$, $\mathscr{L}=\omega_{Z^{\prime}}$
and $m=1$ and conclude that $X$ is a Fano threefold of index two and degree
$(-K_{X})^{3}=8\cdot K_{Z^{\prime}}^{2}$.

It remains to compute the cohomological invariants. The group
$H^{1}(X,\mathscr{O}_{X})$ is the Lie algebra for the Picard scheme, hence
vanishes. Since $X$ is integral and $\omega_{X}^{\otimes-1}$ is ample, the
group $H^{0}(X,\omega_{X})$ vanishes. Serre duality gives
$h^{3}(\mathscr{O}_{X})=0$. To see that $h^{2}(\mathscr{O}_{X})$ vanishes, it
suffices to check $\chi(\mathscr{O}_{X})=1$. The Leray–Serre spectral
sequences for the fibration $f:P\rightarrow Z^{\prime}$ and the contraction
$r:P\rightarrow X$ give

$1=\chi(\mathscr{O}_{P})=\chi(\mathscr{O}_{X})-\chi(R^{1}r_{*}(\mathscr{O}_{P}))+\chi(R^{2}r_{*}(\mathscr{O}_{P})).$

Recall that $E\subset P$ is a negative section that is contracted. The short
exact sequence

$0\rightarrow\mathscr{O}_{E}(-nE)\rightarrow\mathscr{O}_{(n+1)E}\rightarrow\mathscr{O}_{nE}\rightarrow
0$

yields

$H^{i}(\mathscr{O}_{E}(-nE))\longrightarrow
H^{i}(\mathscr{O}_{(n+1)E})\longrightarrow
H^{i}(\mathscr{O}_{nE})\longrightarrow H^{i+1}(\mathscr{O}_{E}(-nE)).$

Using the identification $E=Z^{\prime}$ and
$\mathscr{O}_{E}(-E)=\omega_{Z^{\prime}}^{\otimes-1}$, we see that
$H^{2}(\mathscr{O}_{E}(-nE))$ vanishes for all $n\geq 1$. According to
Corollary 4.4, the groups $H^{1}(\mathscr{O}_{E}(-nE))$ vanish as well. With
the Formal Function Theorem we conclude that
$R^{1}r_{*}(\mathscr{O}_{P})=R^{2}r_{*}(\mathscr{O}_{P})=0$, and hence
$\chi(\mathscr{O}_{X})=1$. ∎

Using the inclusion $Z^{\prime}\subset X$ coming from $Z^{\prime}\subset
P\rightarrow X$ and the inseparable double covering $\nu:Z^{\prime}\rightarrow
Z$, we now form the cocartesian and cartesian square

(7) $\begin{CD}Z^{\prime}@>{{\rm\text{can}}}>{}>X\\\ @V{\nu}V{}V@V{}V{}V\\\
<EMAIL_ADDRESS>
Then $Y=X\amalg_{Z}Z^{\prime}$ is an integral proper threefold, with
normalization $X$. A priori, the amalgamated sum exists as an algebraic space
([Art70, Theorem 6.1]). Since the normalization map $\nu:X\rightarrow Y$ is a
universal homeomorphism, the algebraic space $Y$ must be a scheme ([Ols16,
Theorem 6.2.2]). It contains the normal Enriques surface $Z$ as a closed
subscheme. By construction, the singular locus is given by
$\operatorname{Sing}(Y)=Z\cup
r(\mathbb{P}^{1}_{\operatorname{Sing}(Z^{\prime})})$.

######  Proposition 7.2.

The integral proper threefold $Y$ has the following properties:

  1. (i)

The dualizing sheaf $\omega_{Y}$ is invertible and anti-ample, with
$(-K_{Y})^{3}=K_{Z^{\prime}}^{2}$.

  2. (ii)

We have $\operatorname{Num}(Y)=\mathbb{Z}$.

  3. (iii)

The Euler characteristic is $\chi(\mathscr{O}_{Y})=1$.

  4. (iv)

For each closed point $y\in Y$, the local ring $\mathscr{O}_{Y,y}$ satisfies
Serre’s Condition $(S_{2})$. It is actually Cohen–Macaulay provided that
$\mathscr{O}_{Z,y}\subset\mathscr{O}_{Z^{\prime},y}$ is flat.

###### Proof.

The conductor square (7) yields the short exact sequence

(8)
$0\longrightarrow\mathscr{O}_{Y}\longrightarrow\mathscr{O}_{X}\oplus\mathscr{O}_{Z}\longrightarrow\mathscr{O}_{Z^{\prime}}\longrightarrow
0.$

The normal Enriques surface $Z$ and the normal del Pezzo surface $Z^{\prime}$
both have Euler characteristic $\chi=1$, and the same holds for the normal
Fano threefold $X$, by Proposition 7.1. This gives $\chi(\mathscr{O}_{Y})=1$.

Since the map $\nu:X\rightarrow Y$ is surjective, each integral curve on $Y$
is the image of an integral curve on $X$, hence the induced map
$\nu^{*}\colon\operatorname{Num}(Y)\to\operatorname{Num}(X)$ is injective, and
it follows that the group $\operatorname{Num}(Y)$ is free of rank one.

The singular locus $\operatorname{Sing}(Y)=Z\cup
r(\mathbb{P}^{1}_{\operatorname{Sing}(Z^{\prime})})$ consists of an
irreducible surface and a curve. Let $\zeta\in Y$ be the generic point of the
conductor locus $Z$, and write $\zeta^{\prime}\in X$ for the corresponding
point on $X$. Then $\kappa(\zeta)\subset\kappa(\zeta^{\prime})$ is an
inseparable field extension of degree two, and the subring
$\mathscr{O}_{Y,\zeta}\subset\mathscr{O}_{X,\zeta}$ comprises all ring
elements whose class in the residue field $\kappa(\zeta^{\prime})$ lies in the
subfield $\kappa(\zeta)$. It follows that the local ring
$\mathscr{O}_{Y,\zeta}$ is Gorenstein, compare the discussion in [FS20,
Appendix A]. Thus the dualizing sheaf $\omega_{Y}$ is invertible on some open
subset containing all codimension-one points. Now let $y\in Y$ be an arbitrary
point, and write $x\in X$ for the corresponding point. Since both $\omega_{X}$
and $\mathscr{O}_{X}(Z^{\prime})$ are invertible, [FS20, Proposition A.4]
applies, and we conclude that the dualizing sheaf $\omega_{Y}$ is invertible.

According to Theorem 6.5, the dualizing sheaf on the Fano threefold $X$ is
given by $\omega_{X}=\mathscr{O}_{X}(-2Z^{\prime})$. The relative dualizing
sheaf for the finite birational morphism $\nu:X\rightarrow Y$ is defined by
the equality
$\nu_{*}(\omega_{X/Y})=\underline{\operatorname{Hom}}(\nu_{*}(\mathscr{O}_{X}),\mathscr{O}_{Y})=\nu_{*}\mathscr{O}_{X}(-Z^{\prime})$.
From $\omega_{X}=\nu^{*}(\omega_{Y})\otimes\omega_{X/Y}$ we conclude
$\nu^{*}(\omega_{Y})=\mathscr{O}_{X}(-Z^{\prime})$. In particular,
$K_{Y}^{3}=(-Z^{\prime})^{3}=-K_{Z^{\prime}}^{2}$ holds by Lemma 6.1.

Now fix a closed point $y\in Z$, and consider the three-dimensional local ring
$\mathscr{O}_{Y,y}$. It is Cohen–Macaulay by Theorem 6.5, provided that
$y\notin Z^{\prime}$. Now suppose that $y\in Z$. The short exact sequence (8)
induces a long exact sequence

(9) $H^{i-1}_{y}(\mathscr{O}_{Z^{\prime}})\longrightarrow
H^{i}_{y}(\mathscr{O}_{Y})\longrightarrow H^{i}_{y}(\mathscr{O}_{X})\oplus
H^{i}_{y}(\mathscr{O}_{Z})\longrightarrow H^{i}_{y}(\mathscr{O}_{Z^{\prime}})$

of local cohomology groups. The two-dimensional schemes $Z$ and $Z^{\prime}$
are Cohen–Macaulay, so their local cohomology groups vanish in degree $i<2$.
Likewise, the three-dimensional scheme $X$ is Cohen–Macaulay, so we have
vanishing in degree $i<3$. It follows that $H_{y}^{i}(\mathscr{O}_{Y})=0$ for
$i\leq 1$, hence the local ring $\mathscr{O}_{Y,y}$ satisfies Serre’s
Condition $(S_{2})$.

Finally suppose that $\mathscr{O}_{Z,y}\subset\mathscr{O}_{Z^{\prime},y}$ is
flat. Then the inclusion is a direct summand, in particular the induced map
$H^{2}_{y}(\mathscr{O}_{Z})\rightarrow H^{2}_{y}(\mathscr{O}_{Z^{\prime}})$ is
injective. This ensures $H^{2}_{y}(\mathscr{O}_{Y})=0$, so the local ring
$\mathscr{O}_{Y,y}$ is Cohen–Macaulay. ∎

We see that the scheme $Y$ qualifies as a Fano variety, except that it is not
necessarily Cohen–Macaulay. By definition, a local noetherian ring is called
_Gorenstein_ if it is Cohen–Macaulay, and the dualizing module is invertible.
Without the former condition, one should use the term _quasi-Gorenstein_
instead. Thus it is natural to call our scheme $Y$ a _quasi-Fano variety_ , or
a _Fano variety that is not necessarily Cohen–Macaulay_. Our construction
yields unusual torsion in the Picard scheme:

######  Theorem 7.3.

The integral Fano threefold $Y$ that is not necessarily Cohen–Macaulay has the
following property:

${}_{Y/k}=\operatorname{Pic}^{\tau}_{Y/k}=\begin{cases}\mathbb{Z}/2\mathbb{Z}&\text{if
the Enriques surface $S$ is classical;}\\\ \alpha_{2}&\text{if $S$ is
supersingular.}\end{cases}$

Moreover, $h^{1}(\mathscr{O}_{Y})=h^{2}(\mathscr{O}_{Y})=0$ in the former
case, and $h^{1}(\mathscr{O}_{Y})=h^{2}(\mathscr{O}_{Y})=1$ in the latter.

###### Proof.

The conductor square (7) yields a short exact sequence of multiplicative
abelian sheaves

$1\rightarrow\mathscr{O}_{Y}^{\times}\rightarrow\mathscr{O}_{X}^{\times}\oplus\mathscr{O}_{Z}^{\times}\rightarrow\mathscr{O}_{Z^{\prime}}^{\times}\rightarrow
1$

as explained in [SS02, Proposition 4.1]. This holds not only in the Zariski
topology, but also in the finite flat topology. In turn, we get an exact
sequence of group schemes

$0\longrightarrow\operatorname{Pic}_{Y/k}\longrightarrow\operatorname{Pic}_{X/k}\oplus\operatorname{Pic}_{Z/k}\longrightarrow\operatorname{Pic}_{Z^{\prime}/k}.$

The map on the left is indeed injective, because
$H^{0}(Z,\mathscr{O}_{Z})\rightarrow
H^{0}(Z^{\prime},\mathscr{O}_{Z^{\prime}})=k$ is surjective. Since
$\rho(Y)=1$, an invertible sheaf $\mathscr{L}$ on $Y$ is numerically trivial
if and only if its restriction to $X$, or equivalently to $Z$, is numerically
trivial. Since $\rho(X)=1$, the same holds for invertible sheaves on $X$ and
their restriction to $Z^{\prime}$. In turn, we get an induced exact sequence

$0\longrightarrow\operatorname{Pic}^{\tau}_{Y/k}\longrightarrow\operatorname{Pic}^{\tau}_{X/k}\oplus\operatorname{Pic}^{\tau}_{Z/k}\longrightarrow\operatorname{Pic}^{\tau}_{Z^{\prime}/k}.$

The term for the del Pezzo surface $Z^{\prime}$ vanishes, by Theorem 4.2. Also
the term for the normal Fano threefold $X$ is zero, according to Proposition
7.1. We thus get an identification
$\operatorname{Pic}^{\tau}_{Y/k}=\operatorname{Pic}^{\tau}_{Z/k}$. We saw in
Proposition 4.1 that the morphism $S\rightarrow Z$ from the simply-connected
Enriques surface $S$ to the normal Enriques surface $Z$ induces an
identification
$\operatorname{Pic}^{\tau}_{Z/k}=\operatorname{Pic}^{\tau}_{S/k}$ and the
assertion on Picard schemes and its maximal unipotent quotient follows.

It remains to check the assertions on $h^{i}(\mathscr{O}_{Y})$. This follows
from the long exact sequence for (8), together with
$h^{i}(\mathscr{O}_{X})=h^{i}(\mathscr{O}_{Z^{\prime}})=0$ for $i\geq 1$. ∎

Let us now examine a concrete example for the construction of $Y$: Suppose
that our simply-connected Enriques surface $S$ is an exceptional Enriques
surface of type $T_{2,3,7}$, and that $f:S\rightarrow Z$ is the contraction of
the ADE-curves $C_{1}+\ldots+C_{8}$ and $C_{9}$, as analyzed in Section 5.
Recall that $\operatorname{Sing}(Z)=\\{a,b\\}$ and
$\operatorname{Sing}(Z^{\prime})=\\{b^{\prime}\\}$, where the corresponding
local rings are rational double points of type $E_{8},A_{1}$ and $D_{5}$,
respectively. Moreover, the resulting threefold $Y$ has
$\operatorname{Sing}(Y)=Z\cup r(\mathbb{P}^{1}_{b^{\prime}})$. It turns out
that the closed point $a\in Y$, which comes from the $E_{8}$-singularity
$\mathscr{O}_{Z,a}$, plays a special role:

######  Proposition 7.4.

In the above situation, our quasi-Fano threefold $Y$ has degree
$(-K_{Y})^{3}=4$ and $\operatorname{Num}(Y)=\mathbb{Z}K_{Y}$. Moreover, $Y$ is
Cohen–Macaulay outside $a\in Y$, whereas the local ring $\mathscr{O}_{Y,a}$
satisfies $(S_{2})$ but not $(S_{3})$.

###### Proof.

By Corollary 5.4, the double covering $Z^{\prime}\rightarrow Z$ is flat
precisely outside $a\in Z$. So Proposition 7.2 tells us that the local rings
$\mathscr{O}_{Y,y}$ are Cohen–Macaulay for $y\neq a$. It remains to understand
the case $y=a$. Now the cokernel $M$ for the inclusion
$\mathscr{O}_{Z,a}\subset\mathscr{O}_{Z^{\prime},a}$ still is torsion-free of
rank one, but fails to be invertible. Since the local ring $\mathscr{O}_{Z,a}$
is a factorial, the bidual $M^{\vee\vee}$ is invertible, and the cokernel
$F=M^{\vee\vee}/M$ is finite and non-zero. Using the long exact sequence for
the short exact sequence $0\rightarrow M\rightarrow M^{\vee\vee}\rightarrow
F\rightarrow 0$, one easily infers $H^{1}_{a}(M)\neq 0$. With the exact
sequence (9) we infer that the map $H^{2}_{a}(\mathscr{O}_{Z})\rightarrow
H^{2}_{a}(\mathscr{O}_{Z^{\prime}})$ is not injective, and hence
$H^{2}_{a}(\mathscr{O}_{Y})\neq 0$. In turn, the local ring
$\mathscr{O}_{Y,a}$ is not Cohen–Macaulay.

Combining Proposition 5.1 and 7.2 we get the degree
$(-K_{Y})^{3}=K_{Z^{\prime}}^{2}=4$. This is not a multiple of eight, and it
follows that $\operatorname{Num}(Y)$ is generated by the canonical class. ∎

By construction, our integral Fano threefold $Y$ is not Cohen–Macaulay in
codimension three, and not regular in codimension one. In light of
Salomonsson’s equations [Sal03], the construction works over any ground field
of characteristic $p=2$. It would be interesting to know if there are
imperfect fields $F$ over which $Y$ admits a twisted form $Y^{\prime}$ whose
local rings are normal, $\mathbb{Q}$-factorial klt singularities. Such twisted
forms could appear as generic fibers in Mori fiber spaces. Recall that there
are indeed examples of three-dimensional normal, $\mathbb{Q}$-factorial
terminal singularities that are not Cohen–Macaulay [Tot19]. In [Sch07],
related problems where considered for non-normal del Pezzo surface. Del Pezzo
surfaces over $F$ with $\operatorname{pdeg}(F)=1$ were studied in [FS20]. Here
the _p-degree_ is defined as
$\operatorname{pdeg}(F)=\dim_{F}({}^{1}_{F/F^{p}})$.

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