# Reduced Tate–Shafarevich group

Hayato Morimura  SISSA, via Bonomea 265, 34136 Trieste, Italy
<EMAIL_ADDRESS>
###### Abstract.

Given two generic elliptic Calabi–Yau $3$-folds over a common base in the
sense of Căldăraru, we prove that if their derived categories are equivalent
as a triangulated category linear over the base, then they share the common
relative Jacobian. It follows that one of them is isomorphic to a relative
moduli space of semistable sheaves of rank $1$ with an appropriate degree on
the fibers of the other. As a consequence, up to linear derived equivalence,
generic elliptic Calabi–Yau $3$-folds are classified by a certain quotient set
of the Brauer group of their relative Jacobian.

##  1\. Introduction

Let $\pi\colon X\to S$ be a _generic elliptic $3$-fold_ in the sense of
[Căl00, Definition 6.1.6], which is an elliptic fibration with $X,\dim X=3$
and $S,\dim S=2$ smooth over $\mathbb{C}$ satisfying:

  * •

$\pi$ is flat;

  * •

$\pi$ does not have multiple fibers;

  * •

$\pi$ admits a multisection;

  * •

the discriminant locus $\Delta\subset S$ is an integral curve in $S$ with only
nodes and cusps as singularities;

  * •

the fiber over a general point of $\Delta$ is a rational curve with one node.

It is known that these properties are shared by general members of many
families of elliptic $3$-folds. When $X$ is Calabi–Yau in the strict sense, we
call $\pi$ a _generic elliptic Calabi–Yau $3$-fold_. There exists an étale
cover $\\{U_{i}\\}_{i\in I}$ of $S$ such that to each $U_{i}$ the base change
$\pi_{i}\colon X_{i}=X\times_{S}U_{i}\to U_{i}$ admits a section
$\sigma_{i}\colon U_{i}\to X_{i}$.

Fix a line bundle $\mathscr{O}_{X/S}(1)$ on $X$ relatively ample to $S$. Let
$\varpi\colon M\to S$ be the relative coarse moduli space of semistable
sheaves of rank $1$, degree $0$ on the fibers $X_{s},s\in S$ of $\pi$ with
respect to the restrictions $\mathscr{O}_{X/S}(1)|_{X_{s}}$. By universality
of $M$, the morphism $\varpi$ admits a section $\sigma\colon S\to M$ whose
image is contained in a unique component, called the _relative Jacobian_

$\displaystyle f\colon J\to S.$

In general, $J$ has singularities all of which are ordinary double points.
They arise from the base changes $\pi_{i}$ by contracting all components of
the fibers which do not intersect $\sigma_{i}(U_{i}),i\in I$. One can not
always expect to resolve the singularities in the algebraic category. Let

$\displaystyle\rho\colon\bar{J}\to J$

be any analytic small resolution.

The _smooth part_ of $\pi$ is the base change

$\displaystyle\pi_{U}\colon X_{U}=X\times_{S}U\to U=S\setminus\Delta.$

Due to Ogg–Shafarevich theory developped by Dolgachev–Gross [DG94], smooth
parts of generic elliptic $3$-folds with relative Jacobian $f_{U}\colon
J_{U}\to U$ are classified up to $U$-isomorphism by the Tate–Shafarevich
group, which in our setting is isomorphic to the quotient

(1.1) $\displaystyle\operatorname{Br}(J_{U})/\operatorname{Br}(U)$

of the Brauer group $\operatorname{Br}(J_{U})$, where we regard
$\operatorname{Br}(U)$ as a subgroup via the pullback along $\pi_{U}$. The
goal of this paper is to give a categorical interpretation of the
Tate–Shafarevich group.

Let $\alpha_{U}\in\operatorname{Br}(J_{U})$ be a Brauer class representing
$\pi_{U}$. It uniquely lifts to $\alpha\in\operatorname{Br}(J)$ along the
pullback by the open immersion $J_{U}\hookrightarrow J$. Note that the Brauer
group is a birational invariant. We denote by
$\bar{\alpha}\in\operatorname{Br}(\bar{J})$ the image of $\alpha$ under
$\rho^{*}$, which is injective. In [Căl02, Theorem 5.1] Căldăraru constructed
an $S$-linear twisted Fourier–Mukai functor

(1.2) $\displaystyle\Phi_{\iota_{*}\mathscr{U}_{\bar{\alpha}}}\colon
D^{b}(X)\to D^{b}(\bar{J},\bar{\alpha}),$

where $\mathscr{U}_{\bar{\alpha}}$ is a
$\operatorname{pr}^{*}_{2}\bar{\alpha}^{-1}$-twisted pseudo-universal sheaf on
$X\times_{S}\bar{J}$ and $\iota\colon X\times_{S}\bar{J}\hookrightarrow
X\times\bar{J}$ is the closed immersion. The equivalence (1.2) tells us that
any $S$-linear triangulated category equivalent to $D^{b}(X)$ is represented
by $\alpha$.

He also constructed an $S$-linear Fourier–Mukai functor

(1.3) $\displaystyle\Phi_{\tau_{*}\mathscr{U}_{d}}\colon D^{b}(X)\to
D^{b}(X^{d}),$

for any $d$ coprime to the order $\operatorname{ord}(\alpha_{U})$ of
$\alpha_{U}$ in $\operatorname{Br}(J_{U})$, where $\pi_{d}\colon X^{d}\to S$
is another generic elliptic $3$-fold called the _$d$ -th twisted power_ whose
smooth part is represented by $\alpha^{d}_{U}$, $\mathscr{U}_{d}$ is a
universal sheaf on $X\times_{S}X^{d}$, and $\tau\colon
X\times_{S}X^{d}\hookrightarrow X\times X^{d}$ is the closed immersion. Recall
that $\pi_{d}$ is the relative moduli space of semistable sheaves of rank $1$,
degree $d$ on the fibers $X_{s},s\in S$ of $\pi$ with respect to
$\mathscr{O}_{X/S}(1)|_{X_{s}}$. The moduli problem is fine if and only if $d$
is coprime to $\operatorname{ord}(\alpha_{U})$ [Căl00, Theorem 4.5.2]. In the
sequel, we will call such $\pi_{d}$ a _coprime $d$-th twisted power_ of $\pi$.
The equivalence (1.3) tells us that the group (1.1) is too big to classify, up
to $S$-linear triangle equivalence, derived categories of generic elliptic
$3$-folds with relative Jacobian $f$.

We prove a reconstruction theorem of generic elliptic $3$-folds with a common
relative Jacobian modulo coprime twisted powers. Namely, all generic elliptic
$3$-folds over a fixed base $S$ with relative Jacobian $f$ are determined,
modulo coprime twisted powers, by their $S$-linear derived categories.

######  Theorem 1.1 (Theorem3.4).

Let $\pi\colon X\to S,\pi^{\prime}\colon X^{\prime}\to S$ be generic elliptic
$3$-folds with relative Jacobian $f\colon J\to S$. Assume that there exists an
$S$-linear triangle equivalence $\Phi\colon D^{b}(X)\to D^{b}(X^{\prime})$.
Then $\pi^{\prime}$ is isomorphic to a coprime $d$-th twisted power of $\pi$
for some $d\in\mathbb{Z}$.

In particular, two generic elliptic $3$-folds are $S$-linear derived-
equivalent if and only if one of them is isomorphic to a coprime $d$-th
twisted power of the other. Define an equivalence relation $\sim_{U}$ in
$\operatorname{Br}(J_{U})$ as

(1.4)
$\displaystyle\alpha_{U}\sim\beta_{U}\Leftrightarrow\beta_{U}=\alpha^{d}_{U}\
\rm{for}\ \rm{some}\ d\ \rm{coprime}\ \rm{to}\
\operatorname{ord}(\alpha_{U}),$

which descends to that $\sim$ in the quotient group (1.1). Then the quotient
set

$\displaystyle(\operatorname{Br}(J_{U})/\operatorname{Br}(U))/\sim$

classifies up to $S$-linear triangle equivalence derived categories of generic
elliptic $3$-folds over $S$ with relative Jacobian $f$.

Our proof works without assuming $\dim X=3$. However, we are not interested in
other cases mainly because this condition was responsible in [Căl02, Theorem
5.1] for construction of $\mathscr{U}_{\bar{\alpha}}$. Antieau–Krashen–Ward
proved a similar statement exploiting the twisted Brauer space [AKW17, Theorem
1.5, Remark 6.5]. It relies on the existence of a twisted relative
Fourier–Mukai functor

$\displaystyle\Phi_{\mathcal{P}}\colon D^{b}(\bar{J},\bar{\alpha})\simeq
D^{b}(\bar{J},\bar{\alpha}^{\prime})$

with kernel $\mathcal{P}\in D^{b}(\bar{J}\times_{S}\bar{J}^{\prime})$, which
practically would be obtained by concatenating (1.2) with (1.3). Here, we only
assume the existence of an $S$-linear triangle equivalence $\Phi$.
Essentially, our contribution is to deduce the derived equivalence of the
generic fibers from $\Phi$. After that the argument in the proof of [AKW17,
Theorem 1.5] would complete the proof. However, we exploit Ogg–Shafarevich
theory to yield a geometric proof, following the original idea by Căldăraru.

The reconstruction problem will be more interesting when $X$ is Calabi–Yau, as
the derived category of a Calabi–Yau manifold might have nontrivial
autoequivalences. Moreover, most of Calabi–Yau $3$-folds admit elliptic
fibrations. Some of them are connected with their mirrors admitting dual
fibrations via relative Fourier–Mukai functors [HLS09, HM02], which can be
seen as $S$-duality in string theory [Don98, DP08, DP12]. In this case, we
strengthen Theorem 1.1 as follows.

######  Corollary 1.2 (Corollary 4.3).

Let $\pi\colon X\to S,\pi^{\prime}\colon X^{\prime}\to S$ be generic elliptic
Calabi–Yau $3$-folds. Assume that there exists an $S$-linear triangle
equivalence $\Phi\colon D^{b}(X)\to D^{b}(X^{\prime})$. Then $\pi^{\prime}$ is
isomorphic to a coprime $d$-th twisted power of $\pi$ for some
$d\in\mathbb{Z}$.

Here, we could remove the assumption from Theorem 1.1 on $\pi,\pi^{\prime}$ to
share the common relative Jacobian due to our main result below.

######  Theorem 1.3 (Theorem 4.2).

Let $\pi\colon X\to S,\pi^{\prime}\colon X^{\prime}\to S$ be generic elliptic
Calabi–Yau $3$-folds. Assume that there exists an $S$-linear triangle
equivalence $\Phi\colon D^{b}(X)\to D^{b}(X^{\prime})$. Then
$\pi,\pi^{\prime}$ share the relative Jacobian $f\colon J\to S$.

In particular, two generic elliptic Calabi–Yau $3$-folds are $S$-linear
derived-equivalent if and only if one of them is isomorphic to a coprime
$d$-th twisted power of the other. Note that if $X$ is Calabi–Yau in the
strict sence, so is $X^{\prime}$. The key step is reconstruction of fibers
based on [LT17]. Also [AKW17, Lemma 2.4] is important to find the generic
fibers of $X,X^{\prime}$ sharing the common Jacobian. We use isomorphisms
$\omega_{X}\cong\mathscr{O}_{X},\omega_{X^{\prime}}\cong\mathscr{O}_{X^{\prime}}$
to guarantee the coincidence of base changes of relative Jacobians of
$X,X^{\prime}$ to any sufficiently small open subset.

Now, since $\operatorname{Br}(S)$ vanishes [Gro94, Proposition 2.3] and the
Brauer group is a birational invariant [Gro68], we obtain an isomorphism

$\displaystyle\operatorname{Br}(J)=\operatorname{Br}(J)/\operatorname{Br}(S)\to\operatorname{Br}(J_{U})/\operatorname{Br}(U)=\operatorname{Br}(J_{U})$

induced by the pullback along $J_{U}\hookrightarrow J$. Thus the
Tate–Shafarevich group (1.1) gets reduced to the quotient set of
$\operatorname{Br}(J)$ by the equivalence relation induced by $\sim_{U}$ to
classify $S$-linear derived categories of generic elliptic Calabi–Yau
$3$-folds.

We emphasize that a generic elliptic Calabi–Yau $3$-fold $\pi$ is $S$-linaer
derived-equivalent to another one $\pi^{\prime}$ if and only if $\pi^{\prime}$
is the relative moduli space of semistable sheaves of rank $1$ with an
appropriate degree. It would be interesting to find such a moduli structure
via our result in, for example, the several pairs of Fourier–Mukai partners
constructed by Inoue [Ino22], whose derived equivalences follow from
homological projective duality for categorical joins developed by Kuznetsov
and Perry [KP21].

######  Notations and conventions.

Throughout the paper, all $\mathbb{C}$-varieties are integral separated scheme
of finite type over $\mathbb{C}$. Via Serre’s GAGA theorem we will go back and
forth from the algebraic to analytic categories for smooth proper
$\mathbb{C}$-varieties, abusing notations without much comments. For any
morphism $\pi\colon X\to S$ of nonsingular $\mathbb{C}$-varieties, we consider
the canonical $S$-linear structure on $D^{b}(X)\simeq\operatorname{perf}(X)$
given by the action

$\displaystyle\operatorname{perf}(X)\times\operatorname{perf}(S)\to\operatorname{perf}(X),\
(F,G)\mapsto F\otimes_{\mathscr{O}_{X}}\pi^{*}G.$

Then for morphisms $\pi\colon X\to S,\pi^{\prime}\colon X^{\prime}\to S$ of
nonsingular $\mathbb{C}$-varieties, a triangle functor $\Phi\colon D^{b}(X)\to
D^{b}(X^{\prime})$ is $S$-linear if $\Phi$ respects the $S$-linear structure,
i.e., we have functorial isomorphisms

$\displaystyle\Phi(F\otimes_{\mathscr{O}_{X}}\pi^{*}G)\cong\Phi(F)\otimes_{\mathscr{O}_{X^{\prime}}}\pi^{\prime*}G,\
F\in D^{b}(X),G\in D^{b}(S).$

Since up to isomorphism all Brauer groups we consider are only of smooth
quasi-projective $\mathbb{C}$-varieties, we use a single symbol
$\operatorname{Br}$ to denote the Brauer group and the cohomological Brauer
group with respect to both étale and analytic topology. See [Jon, Theorem 1.1]
and [Căl00, Theorem 1.1.4].

######  Acknowledgements.

The author is supported by SISSA PhD scholarships in Mathematics.

##  2\. Reconstruction of fibers

######  Lemma 2.1.

Let $\pi\colon X\to S,\pi^{\prime}\colon X^{\prime}\to S$ be morphisms of
$\mathbb{C}$-varieties with $X^{\prime},S$ smooth over $\mathbb{C}$ and
$\pi_{Z}\colon X_{Z}\to Z,\pi^{\prime}_{Z}\colon X^{\prime}_{Z}\to Z$ their
base changes to $Z=S\setminus U$ for any open subset $U\subset S$. Then every
$S$-linear triangle functor $\Phi\colon D^{b}(X)\to D^{b}(X^{\prime})$
restricts to

$\displaystyle\Phi_{Z}\colon D^{b}_{\operatorname{coh}_{X_{Z}}(X)}(X)\to
D^{b}_{\operatorname{coh}_{X^{\prime}_{Z}}(X^{\prime})}(X^{\prime}),$

where $D^{b}_{\operatorname{coh}_{X_{Z}}(X)}(X)\subset
D^{b}(X),D^{b}_{\operatorname{coh}_{X^{\prime}_{Z}}(X^{\prime})}(X^{\prime})\subset
D^{b}(X^{\prime})$ are the full $S$-linear triangulated subcategories with
cohomology supported on $X_{Z},X^{\prime}_{Z}$ respectively.

###### Proof.

For each $E\in D^{b}(X)$ we have

$\displaystyle\Phi(E)\cong\Phi(\bigoplus_{i}\mathcal{H}^{i}(E)[-i])\cong\bigoplus_{i}\Phi(\mathcal{H}^{i}(E))[-i]$

as $\Phi$ is an $S$-linear triangle functor. Consider the pullback diagrams

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If $E$ is supported on $X_{Z}$ then we have

$\displaystyle\Phi(\mathcal{H}^{i}(E))$
$\displaystyle\cong\Phi(\mathcal{H}^{i}(E)\otimes_{\mathscr{O}_{X}}\bar{\iota}_{*}\mathscr{O}_{X_{Z}})$
$\displaystyle\cong\Phi(\mathcal{H}^{i}(E)\otimes_{\mathscr{O}_{X}}\pi^{*}\iota_{*}\mathscr{O}_{Z})$
$\displaystyle\cong\Phi(\mathcal{H}^{i}(E))\otimes_{\mathscr{O}_{X^{\prime}}}\pi^{\prime*}\iota_{*}\mathscr{O}_{Z}$
$\displaystyle\cong\Phi(\mathcal{H}^{i}(E))\otimes_{\mathscr{O}_{X^{\prime}}}\bar{\iota}^{\prime}_{*}\mathscr{O}_{X^{\prime}_{Z}},$

where the first, the second, the third, and the forth isomorphisms follow from
$E$ being supported on $X_{Z}$, the isomorphism
$\bar{\iota}_{*}\bar{\pi}^{*}\mathscr{O}_{Z}\cong\pi^{*}\iota_{*}\mathscr{O}_{Z}$,
$S$-linearity of $\Phi$, and the isomorphism
$\bar{\iota}^{\prime}_{*}\bar{\pi}^{\prime*}\mathscr{O}_{Z}\cong\pi^{\prime*}\iota_{*}\mathscr{O}_{Z}$
respectively. Since $X^{\prime}$ is smooth over $\mathbb{C}$, replacing it
with a quasi-isomorphic object if necessary, we may assume that
$\Phi(\mathcal{H}^{i}(E))$ are perfect. Then the last term becomes
$\bar{\iota}^{\prime}_{*}(\Phi(\mathcal{H}^{i}(E))|_{X^{\prime}_{Z}})$ by the
projection formula. Note that $\bar{\iota}^{\prime}_{*}$ is exact as
$\bar{\iota}^{\prime}$ is affine. Hence $\Phi(\mathcal{H}^{i}(E))$ is
supported on $X^{\prime}_{Z}$, which completes the proof. ∎

######  Lemma 2.2.

Let $\pi\colon X\to S,\pi^{\prime}\colon X^{\prime}\to S$ be flat morphisms of
nonsingular $\mathbb{C}$-varieties. Assume that there exists an $S$-linear
triangle equivalence $\Phi\colon D^{b}(X)\to D^{b}(X^{\prime})$. Then every
closed fibers $X_{s},X^{\prime}_{s}$ over $s\in S$ are derived-equivalent.

###### Proof.

Take an affine open subset $U=\operatorname{Spec}R\subset S$. First, we show
that $\Phi$ induces an $R$-linear triangle equivalence

$\displaystyle\Phi_{U}\colon D^{b}(X_{U})\to D^{b}(X^{\prime}_{U}),$

where $\pi_{U}\colon X_{U}\to U,\pi^{\prime}_{U}\colon X^{\prime}_{U}\to U$
are the base changes to $U$. For their complements $X_{Z}=X\setminus
X_{U},X^{\prime}_{Z}=X^{\prime}\setminus X^{\prime}_{U}$ we have

$\displaystyle\operatorname{coh}(X_{U})\simeq\operatorname{coh}(X)/\operatorname{coh}_{X_{Z}}(X),\
\operatorname{coh}(X^{\prime}_{U})\simeq\operatorname{coh}(X^{\prime})/\operatorname{coh}_{X^{\prime}_{Z}}(X^{\prime}),$

where the right hand sides are the quotients by the Serre subcategories

$\displaystyle\operatorname{coh}_{X_{Z}}(X)\subset\operatorname{coh}(X),\
\operatorname{coh}_{X^{\prime}_{Z}}(X)\subset\operatorname{coh}(X^{\prime})$

of coherent sheaves supported on $X_{Z},X^{\prime}_{Z}$ respectively. Passing
to their derived categories, via [Miy91, Theorem 3.2] we obtain

$\displaystyle D^{b}(X_{U})\simeq
D^{b}(X)/D^{b}_{\operatorname{coh}_{X_{Z}}(X)}(X),\
D^{b}(X^{\prime}_{U})\simeq
D^{b}(X^{\prime})/D^{b}_{\operatorname{coh}_{X^{\prime}_{Z}}(X^{\prime})}(X^{\prime}).$

Since $X^{\prime},S$ are smooth over $\mathbb{C}$, one can apply Lemma 2.1 to
obtain the induced equivalence

$\displaystyle\Phi_{U}\colon D^{b}(X_{U})\simeq
D^{b}(X)/D^{b}_{\operatorname{coh}_{X_{Z}}(X)}(X)\to
D^{b}(X^{\prime}_{U})\simeq
D^{b}(X^{\prime})/D^{b}_{\operatorname{coh}_{X^{\prime}_{Z}}(X^{\prime})}(X^{\prime})$

by universality of Verdier quotients.

Next, we show that $\Phi_{U}$ induces an equivalence
$\Phi_{s}\colon\operatorname{perf}(X_{s})\to\operatorname{perf}(X^{\prime}_{s})$
of categories of perfect complexes on every closed fibers
$X_{s},X^{\prime}_{s}$ over $s\in S$. Consider dg enhancements
$\operatorname{Perf}(X),\operatorname{Perf}(X^{\prime})$ of
$\operatorname{perf}(X)\simeq D^{b}(X),\operatorname{perf}(X^{\prime})\simeq
D^{b}(X^{\prime})$, which are unique up to quasi-equivalence [CS18,
Proposition 6.10]. Here, we use the assumption on $X$ to be smooth over
$\mathbb{C}$. Since $\pi,\pi^{\prime}$ are flat, one can apply [BFN10, Theorem
1.2] and [Töe, Exercise 32] to the pullback diagrams

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to obtain a quasi-equivalence
$\operatorname{Perf}(X_{s})\to\operatorname{Perf}(X^{\prime}_{s})$. Passing to
their homotopy, we obtain the induced equivalence

$\displaystyle\Phi_{s}\colon\operatorname{perf}(X_{s})\to\operatorname{perf}(X^{\prime}_{s}).$

Now, the claim follows from [CS18, Proposition 7.4]. ∎

######  Lemma 2.3 ([Căl00, Theorem 6.1.9]).

Let $\pi\colon X\to S$ be a generic elliptic $3$-fold. Then the closed fiber
$X_{s}$ over $s\in S$ is:

  * •

a smooth elliptic curve when $s\in S\setminus\Delta$;

  * •

a rational curve with one node when $s\in\Delta$ is a smooth point;

  * •

two copies of $\mathbb{P}^{1}$ intersecting transversely at two points when
$s\in\Delta$ is a node;

  * •

a rational curve with one cusp when $s\in\Delta$ is a cusp.

######  Proposition 2.4.

Let $\pi\colon X\to S,\pi^{\prime}\colon X^{\prime}\to S$ be generic elliptic
$3$-folds. Assume that there exists an $S$-linear triangle equivalence
$\Phi\colon D^{b}(X)\to D^{b}(X^{\prime})$. Then every closed fibers
$X_{s},X^{\prime}_{s}$ over $s\in S$ are isomorphic.

###### Proof.

By Lemma 2.2 every closed fibers $X_{s},X^{\prime}_{s}$ over $s\in S$ are
derived-equivalent. Since by Lemma 2.3 $X_{s},X^{\prime}_{s}$ are either a
smooth elliptic curve, a rational curve, two copies of $\mathbb{P}^{1}$
intersecting transversely at two points, or a rational curve with one cusp,
one can apply [LT17, Theorem 7.4(2)] to obtain $X_{s}\cong X^{\prime}_{s}$. ∎

##  3\. Reconstruction of fibrations

######  Lemma 3.1.

Let $\pi\colon X\to S,\pi^{\prime}\colon X^{\prime}\to S$ be separated
morphisms of nonsingular $\mathbb{C}$-varieties. Then every $S$-linear
triangle equivalence $\Phi\colon D^{b}(X)\to D^{b}(X^{\prime})$ induces an
$K(S)$-linear triangle equivalence $\Phi_{K(S)}\colon D^{b}(X_{\eta})\to
D^{b}(X^{\prime}_{\eta})$ on the generic fibers $X_{\eta},X^{\prime}_{\eta}$,
where $K(S)$ is the function filed of $S$.

###### Proof.

By the same argument as in the proof of Lemma 2.2 and [Har77, Cororallry
III10.7] we may assume that $\pi,\pi^{\prime}$ are smooth over
$S=\operatorname{Spec}R$. One can apply [Mor, Theorem 1.1] to obtain
$K(S)$-linear triangle equivalences

$\displaystyle D^{b}(X_{\eta})\to D^{b}(X)/D^{b}_{0}(X),\
D^{b}(X^{\prime}_{\eta})\to D^{b}(X^{\prime})/D^{b}_{0}(X^{\prime}).$

to the Verdier quotients by the full $S$-linear triangulated subcategories
spanned by complexes with coherent $R$-torsion cohomology. Since $\Phi$ is
$S$-linear, universality of Verdier quotients induces the desired equivalence.
∎

######  Corollary 3.2.

Let $\pi\colon X\to S,\pi^{\prime}\colon X^{\prime}\to S$ be generic elliptic
$3$-folds. Assume that there exists an $S$-linear triangle equivalence
$\Phi\colon D^{b}(X)\to D^{b}(X^{\prime})$. Then their generic fibers
$X_{\eta},X^{\prime}_{\eta}$ share the common Jacobian.

###### Proof.

The generic fibers $X_{\eta},X^{\prime}_{\eta}$ are derived-equivalent by
Lemma 3.1. Now, the claim follows from [AKW17, Lemma 2.4]. ∎

######  Lemma 3.3.

Let $\pi\colon X\to S,\pi^{\prime}\colon X^{\prime}\to S$ be smooth elliptic
$3$-folds with relative Jacobian $f\colon J\to S$. Assume that there exists an
$S$-linear triangle equivalence $\Phi\colon D^{b}(X)\to D^{b}(X^{\prime})$.
Then $\pi^{\prime}$ is isomorphic to a coprime $d$-th twisted power of $\pi$
for some $d\in\mathbb{Z}$.

###### Proof.

Let $\alpha,\alpha^{\prime}\in\operatorname{Br}(J)$ be representatives of the
Tate–Shafarevich group for $\pi,\pi^{\prime}$. Take an étale cover
$\\{U_{i}\\}_{i\in I}$ of $S$ representing both $\alpha$ and
$\alpha^{\prime}$. Namely, to each $U_{i}$ the base changes
$\pi_{i},\pi^{\prime}_{i}$ admit sections

$\displaystyle\sigma_{i}\colon U_{i}\to X_{i}=X\times_{S}U_{i},\
\sigma^{\prime}_{i}\colon U_{i}\to X^{\prime}_{i}=X^{\prime}\times_{S}U_{i}$

and $\alpha,\alpha^{\prime}$ give the gluing data for $\\{X_{i}\\}_{i\in
I},\\{X^{\prime}_{i}\\}_{i\in I}$ to yield $X,X^{\prime}$ from
$\\{J_{i}\\}_{i\in I}$ respectively [Căl00, Chapter 4.2, 4.4] via the
description of Brauer classes as gerbes [Căl00, Chapter 1.1]. Let
$\alpha_{\eta},\alpha^{\prime}_{\eta}\in\operatorname{Br}(J_{\eta})$ be the
images of $\alpha,\alpha^{\prime}$ under the injective map induced by the
pullback along the base change $J_{\eta}\to J$ to the function field $K(S)$ of
$S$. Refining it if necessary, we may assume that the induced étale cover
$\\{U_{i}\times_{S}K(S)\\}$ of $\operatorname{Spec}K(S)$ represents both
$\alpha_{\eta}$ and $\alpha^{\prime}_{\eta}$.

By assumption and Lemma 3.1 the generic fibers $X_{\eta},X^{\prime}_{\eta}$
are derived-equivalent. From [AKW17, Lemma 2.3] it follows that
$X^{\prime}_{\eta}$ is isomorphic to the moduli space of degree $d$ line
bundles on $X_{\eta}$ for some $d$ coprime to the order
$\operatorname{ord}(\alpha_{\eta})$ in $\operatorname{Br}(J_{\eta})$. Hence
both $\alpha^{d}_{\eta}$ and $\alpha^{\prime}_{\eta}$ represent
$\pi^{\prime}_{\eta}$. Since the map
$\operatorname{Br}(J)\to\operatorname{Br}(J_{\eta})$ is injective, both
$\alpha^{d}$ and $\alpha^{\prime}$ represent $\pi^{\prime}$. As explained in
[Căl00, Chapter 4.5], the Brauer class $\alpha^{d}$ represents the $d$-th
twisted power of $X$. Thus $\pi^{\prime}$ is isomorphic to a coprime $d$-th
twisted power of $\pi$. ∎

######  Remark 3.1.

If the equivalence $\Phi$ was given by a relative Fourier–Mukai functor, one
could have used [AKW17, Theorem 1.5]. However, still one should deduce the
derived equivalence $D^{b}(J_{\eta},\alpha_{\eta})\simeq
D^{b}(J_{\eta},\alpha^{\prime}_{\eta})$ on the twisted generic fibers via a
different way from [AKW17, Proposition 6.2]. We will discuss and remedy this
issue in the forthcoming work [MU].

######  Theorem 3.4.

Let $\pi\colon X\to S,\pi^{\prime}\colon X^{\prime}\to S$ be generic elliptic
$3$-folds with relative Jacobian $f\colon J\to S$. Assume that there exists an
$S$-linear triangle equivalence $\Phi\colon D^{b}(X)\to D^{b}(X^{\prime})$.
Then $\pi^{\prime}$ is isomorphic to a coprime $d$-th twisted power of $\pi$
for some $d\in\mathbb{Z}$.

###### Proof.

Let $\alpha_{U},\alpha^{\prime}_{U}\in\operatorname{Br}(J_{U})$ be the
representatives of the Tate–Shafarevich group for the smooth parts
$\pi_{U},\pi^{\prime}_{U}$. By the same argument as in the proof of Lemma 2.2
there is an $U$-linear triangle equivalence

$\displaystyle D^{b}(X_{U})\to D^{b}(X^{\prime}_{U}).$

One can apply Lemma 3.3 to find some $d$ coprime to
$\operatorname{ord}(\alpha_{U})$ in $\operatorname{Br}(J_{U})$ such that both
$\alpha^{d}_{U}$ and $\alpha^{\prime}_{U}$ represent $\pi^{\prime}_{U}$. By
[Căl00, Theorem 6.5.1] or birational invariance of the Brauer group,
$\alpha_{U},\alpha^{\prime}_{U}$ uniquely lift to
$\alpha,\alpha^{\prime}\in\operatorname{Br}(J)$ along the pullback by the open
immersion $J_{U}\hookrightarrow J$.

Let $\bar{f}\colon\bar{J}\to S$ be the composition of the resolution
$\bar{J}\to J$ with $f$. Take an analytic open cover $\\{U_{i}\\}_{i\in I}$ of
$S$ representing both $\alpha$ and $\alpha^{\prime}$ such that each $U_{i}$
contains at most one node. By [Căl00, Theorem 6.4.6] there are
$U_{i}$-isomorphisms

$\displaystyle X_{i}\cong\bar{J}_{i}=\bar{J}\times_{S}U_{i}\cong
X^{\prime}_{i}$

of analytic spaces. As explained in the proof of [Căl00, Theorem 6.5.1], both
$\alpha$ and $\alpha^{\prime}$ give the gluing data for $\\{X_{i}\\}_{i\in
I},\\{X^{\prime}_{i}\\}_{i\in I}$ to yield $X,X^{\prime}$ from
$\\{\bar{J}_{i}\\}_{i\in I}$. See, also [Căl00, Theorem 4.4.1].

Let $\pi^{\prime\prime}\colon X^{\prime\prime}\to S$ be the $d$-th twisted
power of $\pi$. Refining the analytic open cover $\\{U_{i}\\}_{i\in I}$ if
necessary, we may assume that to each $U_{i}$ the base change
$\pi^{\prime\prime}_{i}$ admits a section

$\displaystyle\sigma^{\prime\prime}_{i}\colon U_{i}\to
X^{\prime\prime}_{i}=X^{\prime\prime}\times_{S}U_{i}.$

Again, by [Căl00, Theorem 6.4.6] there are $U_{i}$-isomorphisms

$\displaystyle X^{\prime\prime}_{i}\cong\bar{J}_{i}$

of analytic spaces. Then $\alpha^{d}$ gives the gluing data for
$\\{X^{\prime\prime}_{i}\\}_{i\in I}$ to yield $X^{\prime\prime}$ from
$\\{\bar{J}\\}_{i\in I}$. Hence $\pi^{\prime}$ and $\pi^{\prime\prime}$ are
isomorphic. ∎

######  Remark 3.2.

In the above proof we have not used the assumption on $X$ to be
$3$-dimensional. However, this is crucial for Lemma 2.3 on the classification
of the fibers, which in turn is responsible for the construction of local
universal sheaves $\mathscr{U}_{i}$ on $W_{i}=X_{i}\times_{U_{i}}\bar{J}_{i}$
[Căl00, Theorem 6.4.2]. The collection defines the
$\operatorname{pr}^{*}_{2}\bar{\alpha}^{-1}$-twisted pseudo-universal sheaf
$\mathscr{U}_{\bar{\alpha}}=(\\{\mathscr{U}_{i}\\},\\{\varphi_{ij}\\})$ on
$W=X\times_{S}\bar{J}$, where
$\varphi_{ij}\colon\mathscr{U}_{j}|_{W_{i}\times_{W}W_{j}}\to\mathscr{U}_{i}|_{W_{i}\times_{W}W_{j}}$
is an isomorphism on each double intersection $W_{ij}=W_{i}\times_{W}W_{j}$.
Hence without the assumption one would not have the equivalence (1.2).

######  Remark 3.3.

For the convenience of readers we add a little more explaination to the above
remark. The Brauer class $\bar{\alpha}$ appeared in (1.2) gives the
obstruction against the collection $\mathscr{U}_{\bar{\alpha}}$ to glue to
yield a universal sheaf. The discrepancy between the composition
$\varphi_{ij}\circ\varphi_{jk}\circ\varphi_{ki}$ and the identity on each
triple intersection $W_{ijk}=W_{i}\times_{W}W_{j}\times_{W}W_{k}$ defines
precisely the $\operatorname{pr}^{*}_{2}\bar{\alpha}^{-1}$-twisting of
$\mathscr{U}_{\bar{\alpha}}$. Recall that $\bar{\alpha}$ is the image of
$\alpha$ under the injection $\rho^{*}$. Consider the image $\alpha_{X}$ of
$\alpha$ under the composition of the isomorphism
$\operatorname{Br}(J)\cong\operatorname{Br}(J_{U})$ with the projection
$\operatorname{Br}(J_{U})\to\operatorname{Br}(J_{U})/\operatorname{Br}(U)$.
Then [Căl00, Theorem 4.4.1] claims that $\alpha_{X}$ corresponds to $\pi_{U}$
via Ogg–Shafarevich theory. Namely, $\alpha_{X}$ gives the gluing data for
$J_{U}$ cut into pieces to yield $X_{U}$ via the description of Brauer classes
as gerbes. Hence up to elements of $\operatorname{Br}(U)$ the obstruction
$\bar{\alpha}$ can be interpreted geometrically, which was tacitly used to
obtain the equivalence (1.2)

##  4\. Calabi–Yau case

######  Lemma 4.1.

Let $\pi\colon X\to S$ be a projective morphism of nonsingular
$\mathbb{C}$-varieties with $\omega_{X}\cong\mathscr{O}_{X}$ such that every
closed fiber $X_{s},s\in S$ is connected. Then the relative canonical sheaf
$\omega_{X/S}$ is invertible and isomorphic to $\pi^{*}\omega^{-1}_{S}$. In
particular, we have $\pi_{*}\omega_{X/S}\cong\omega^{-1}_{S}$.

###### Proof.

Consider the first fundamental exact sequence

(4.1)
$\displaystyle\pi^{*}\Omega_{S}\xrightarrow{u}\Omega_{X}\to\Omega_{X/S}\to 0.$

If $u$ is injective, then we obtain $\omega_{X/S}\cong\pi^{*}\omega^{-1}_{S}$,
which is invertible. Since $X,S$ are smooth over $\mathbb{C}$, for any $x\in
X$ there is an open neighborhood $V\subset X$ on which $u$ can be expressed as
a homomorphism $\mathscr{O}^{\oplus\dim S}_{V}\to\mathscr{O}^{\oplus\dim
X}_{V}$ of free $\mathscr{O}_{V}$-modules. As $X$ is integral, it suffices to
check the injectivity on the generic point. However, (4.1) restricts to yield
the short exact sequence

$\displaystyle
0\to\pi^{*}_{U}\Omega_{U}\xrightarrow{u|_{X_{U}}}\Omega_{X_{U}}\to\Omega_{X_{U}/U}\to
0.$

on the smooth part $\pi_{U}\colon X_{U}\to U$ of $\pi$ for some open subset
$U\subset S$ [Har77, Corollary III10.7].

The second claim follows from the projection formula, since every closed fiber
$X_{s},s\in S$ is connected, by [Har77, Corollary III 12.9] the coherent sheaf
$\pi_{*}\mathscr{O}_{X}$ is invertible and hence is isomorphic to
$\mathscr{O}_{S}$. ∎

######  Theorem 4.2.

Let $\pi\colon X\to S,\pi^{\prime}\colon X^{\prime}\to S$ be generic elliptic
Calabi–Yau $3$-folds. Assume that there exists an $S$-linear triangle
equivalence $\Phi\colon D^{b}(X)\to D^{b}(X^{\prime})$. Then
$\pi,\pi^{\prime}$ share the common relative Jacobian $f\colon J\to S$.

###### Proof.

By Proposition 2.4 every closed fibers $X_{s},X^{\prime}_{s}$ over $s\in S$
are isomorphic. It follows that $\pi,\pi^{\prime}$ share the common
discriminant locus $\Delta\subset S$. Let $f\colon J\to S,f^{\prime}\colon
J^{\prime}\to S$ be the relative Jacobians of $\pi,\pi^{\prime}$. For any
analytic small resolutions $\rho\colon\bar{J}\to
J,\rho^{\prime}\colon\bar{J}^{\prime}\to J^{\prime}$ of singularities, we
denote by $\bar{f},\bar{f}^{\prime}$ the compositions
$f\circ\rho,f^{\prime}\circ\rho^{\prime}$ respectively. Take an analytic open
cover $\\{U_{i}\\}_{i\in I}$ of $S$ such that each $U_{i}$ contains at most
one node and to each $U_{i}$ the base changes
$\pi_{i},\pi^{\prime}_{i},\bar{f}_{i},\bar{f}^{\prime}_{i}$ admit sections

$\displaystyle\sigma_{i}\colon U_{i}\to X_{i},\ \sigma^{\prime}_{i}\colon
U_{i}\to X^{\prime}_{i},\ \bar{\sigma}_{i}\colon
U_{i}\to\bar{J}_{i}=\bar{J}\times_{S}U_{i},\ \bar{\sigma}^{\prime}_{i}\colon
U_{i}\to\bar{J}^{\prime}_{i}=\bar{J}^{\prime}\times_{S}U_{i}.$

Applying [Nak87, Theorem 2.1] or [DG94, Theorem 2.3], we obtain birational
morphisms

$\displaystyle g_{i}\colon\bar{J}_{i}\to W(\mathscr{L}_{i},a_{i},b_{i}),\
g^{\prime}_{i}\colon\bar{J}^{\prime}_{i}\to
W(\mathscr{L}^{\prime}_{i},a^{\prime}_{i},b^{\prime}_{i}),$

where
$W(\mathscr{L}_{i},a_{i},b_{i}),W(\mathscr{L}^{\prime}_{i},a^{\prime}_{i},b^{\prime}_{i})$
are the Weierstrass fibrations associated with line bundles
$\mathscr{L}_{i},\mathscr{L}^{\prime}_{i}$ on $U_{i}$, global sections
$a_{i},a^{\prime}_{i}$ of $\mathscr{L}^{\otimes
4}_{i},\mathscr{L}^{\prime\otimes 4}_{i}$, and $b_{i},b^{\prime}_{i}$ of
$\mathscr{L}^{\otimes 6}_{i},\mathscr{L}^{\prime\otimes 6}_{i}$ such that
$4a^{3}_{i}+27b^{2}_{i},4a^{\prime 3}_{i}+27b^{\prime 2}_{i}$ are nonzero
global sections of $\mathscr{L}^{\otimes 12}_{i},\mathscr{L}^{\prime\otimes
12}_{i}$. Since $g_{i},g^{\prime}_{i}$ are the morphism which blow down all
components of fibers not intersecting
$\bar{\sigma}_{i}(U_{i}),\bar{\sigma}^{\prime}_{i}(U_{i})$ respectively, the
fibrations $W(\mathscr{L}_{i},a_{i},b_{i})\to
S,W(\mathscr{L}^{\prime}_{i},a^{\prime}_{i},b^{\prime}_{i})\to S$ coincide
with the base changes of the relative Jacobians $f_{i},f^{\prime}_{i}$ to
$U_{i}$. See also [Căl02, Theorem 6.4.3].

From Lemma 4.1 it follows

$\displaystyle\mathscr{L}_{i}\cong\bar{f}_{i*}\omega_{J_{i}/U_{i}}\cong\omega^{-1}_{U_{i}}\cong\bar{f}^{\prime}_{i*}\omega_{J^{\prime}_{i}/U_{i}}\cong\mathscr{L}^{\prime}_{i}$

as we have $U_{i}$-isomorphisms
$X_{i}\cong\bar{J}_{i},X^{\prime}_{i}\cong\bar{J}^{\prime}_{i}$ of analytic
spaces. Note that $\bar{f}_{i},\bar{f}^{\prime}_{i}$ are locally projective,
as they are proper and $\bar{J}_{i},\bar{J}^{\prime}_{i}$ have relatively
ample line bundles [SP, Tag 01WC]. Refining $\\{U_{i}\\}_{i\in I}$ if
necessary, we may assume that $\bar{f}_{i},\bar{f}^{\prime}_{i}$ are
projective. Since by Corollary 3.2 the generic fibers
$J_{\eta},J^{\prime}_{\eta}$ are isomorphic, we obtain $U_{i}$-isomorphisms

(4.2) $\displaystyle W(\mathscr{L}_{i},a_{i},b_{i})\cong
W(\mathscr{L}^{\prime}_{i},a^{\prime}_{i},b^{\prime}_{i}),$

as two Weierstrass fibrrations
$W(\mathscr{L}_{i},a_{i},b_{i}),W(\mathscr{L}_{i},a^{\prime}_{i},b^{\prime}_{i})$
must coincide whenever they share the common generic fiber.

Now, $f,f^{\prime}$ are locally isomorphic elliptic fibrations over the common
base $S$. Since the construction of a Weierstrass fibration is functorial, we
have the $U_{ij}$-isomorphism

$\displaystyle
W(\mathscr{L}_{i}|_{U_{ij}},a_{i}|_{U_{ij}},b_{i}|_{U_{ij}})\cong
W(\mathscr{L}_{i},a_{i},b_{i})|_{U_{ij}}$

for each double intersection $U_{ij}=U_{i}\times_{S}U_{j}$. Then the
$U_{i}$-isomorphisms (4.2) canonically glue to yield an $S$-isomorphism $J\to
J^{\prime}$, which completes the proof. ∎

######  Corollary 4.3.

Let $\pi\colon X\to S,\pi^{\prime}\colon X^{\prime}\to S$ be generic elliptic
Calabi–Yau $3$-folds. Assume that there exists an $S$-linear triangle
equivalence $\Phi\colon D^{b}(X)\to D^{b}(X^{\prime})$. Then $\pi^{\prime}$ is
isomorphic to a coprime $d$-th twisted power of $\pi$ for some
$d\in\mathbb{Z}$.

###### Proof.

The claim follows immediately from Theorem 4.2 and Theorem 3.4. ∎

######  Remark 4.1.

It sufficed to assume either $X$ or $X^{\prime}$ is Calabi–Yau. Without loss
of generrality we may assume that $X$ is Calabi–Yau. The triviality of the
canonical bundle of $X^{\prime}$ follows from the uniqueness of Serre
functors. The vanishing of $H^{1}(X^{\prime},\mathscr{O}_{X^{\prime}})$
follows from
$H^{2}(X,\mathscr{O}_{X})=H^{2}(X^{\prime},\mathscr{O}_{X^{\prime}})$ and
Serre duality. Note that the $S$-linear triangle equivalence is in particular
$\mathbb{C}$-linear triangle equivalence and hence naturally isomorphic to a
unique Fourier Mukai functor with kernel belonging to $D^{b}(X\times
X^{\prime})$ unique up to isomorphism [Orl97, Section 2]. One obtains the
induced isometry

$\displaystyle H^{2}(X,\mathscr{O}_{X})\oplus H^{4}(X,\mathscr{O}_{X})\cong
H^{2}(X^{\prime},\mathscr{O}_{X^{\prime}})\oplus
H^{4}(X^{\prime},\mathscr{O}_{X^{\prime}})$

from [Căl00, Corollary 3.1.13, 3.1.14].

######  Remark 4.2.

If $X$ is Calabi–Yau, then $\bar{J}$ is also Calabi–Yau. Again, the triviality
of the canonical bundles of $\bar{J}$ follows from the uniqueness of Serre
functors. Refining it if necessary, we may assume that to each $U_{i}$ the
base change $\bar{f}_{i}=f_{i}\circ\rho_{i}$ is projective. Then by [CR11,
Corollary 3.2.10] the higher direct image $R^{j}\rho_{*}\mathscr{O}_{\bar{J}}$
vanishes for each $j>0$. Hence we obtain
$R\bar{f}_{*}\mathscr{O}_{\bar{J}}=\mathscr{O}_{S}$ and Leray spectral
sequence implies

$\displaystyle H^{1}(\bar{J},\mathscr{O}_{\bar{J}})=H^{1}(S,\mathscr{O}_{S}).$

Since $X$ is Calabi–Yau the base $S$ must be either a rational or Enriques
surface by [Gro94, Proposition 2.3]. In both cases $H^{1}(S,\mathscr{O}_{S})$
vanishes.

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