and expresses the ($\mathbb{Q}$-linear equivalence class of the) canonical
bundle of the total space of a relatively minimal $K$-trivial fibration
$\pi:X\longrightarrow B$ in terms of three $\mathbb{Q}$-divisors on the base
$B$ (see [Kol07, Thm. 8.5.1]): the canonical class of the base $K_{B}$, the
moduli divisor $M_{B}$ (which only depends on the restriction of the fibration
to the smooth locus), and a discriminant divisor $D_{\pi}$ which accounts for
the singular fibers in codimension one (compare (4.22) below).

In the isotrivial case, both the local structure of the singular fibers and
the canonical bundle formula take a particularly simple form. Namely, the
local monodromy groups are automatically finite, and in fact cyclic of order
$m\in\\{2,3,4,6\\}$ (Corollary 4.12, [BS23, Lemma 3.6]), and $m>2$ is only
possible if the elliptic curve $E$ occurring in Theorem 1.3 has CM by
$\mathbb{Q}(\sqrt{-1})$ or $\mathbb{Q}(\sqrt{-3})$. Moreover, under the mild
assumption that there are no multiple fibers in codimension $1$, we show that
the only types of Hwang–Oguiso fibers that occur for isotrivial fibrations are
II, III, IV, $\text{I}_{0}^{*}$, $\text{II}^{*}$, $\text{III}^{*}$, and
$\text{IV}^{*}$. Also, as in the case of elliptic fibrations, the type of
singular fiber that can occur is controlled by the local monodromy. In
particular, if the local mondromy is of order $2$, the only possible fiber
type is $\text{I}_{0}^{*}$. Returning to the canonical bundle formula, in the
isotrivial case, the moduli divisor contribution is trivial (compare [DB22,
Thm 3.3]) and thus the only contribution to be determined is the discriminant
divisor $D_{\pi}$, which is supported on the reduced discriminant $\Sigma$,
and whose coefficients for the various components $D_{i}$ of $\Sigma$ depend
on the Hwang–Oguiso singular fiber type at the generic point of $D_{i}$. These
coefficients are familiar from the elliptic fibration setting (compare with
Appendix A), but to our knowledge the computations in this section (e.g.
Proposition 4.16), refining Hwang–Oguiso’s work, are new in the hyper-Kähler
setting. We also refer the reader to Section 2.4.1 for a concrete application
of these ideas in higher dimensions.

Note that most of the results in this section rely on the assumption that
_every general singular fiber of $\pi:X\longrightarrow B$ is a non-multiple
fiber_, which explains this hypothesis in Theorem 1.9. This assumption is
satisfied when $\pi$ has a rational section.

###  4.1. Compactification of the intermediate base $U$

Let us start with the following general result about the existence of a
preferred $G$-equivariant compactification $Y\supset U$, which we call the
_normal $G$-compactification_.

######  Proposition 4.2.

Let $G$ be a finite group and $U$ a smooth quasi-projective variety with a
free $G$-action. Then

  1. (1)

For any normal projective compactification $B$ of $U/G$, there exists a unique
$G$-equivariant normal projective compactification $Y$ of $U$ such that
$Y/G\cong B$.

  2. (2)

If $B$ is smooth, then $Y$ is $\mathbb{Q}$-Gorenstein.

###### Proof.

This is standard (e.g. [GKP16, Thm. 3.8], [KM98, Prop. 5.20]). The
compactification $Y$ is obtained as the normalization of $B$ in the function
field $K(U)$. ∎

Recall that the intermediate base $U$ is a Galois covering $U\longrightarrow
B_{0}\subset B$. Let $Y\supset U$ be the normal $G$-compactification whose
quotient is $B$ and consider the quotient map

$q:Y\longrightarrow B.$

We write $\Sigma\subset B$ for the (reduced) discriminant locus of the
isotrivial fibration $\pi$, and $\\{D_{i}\\}$ for its irreducible components.
By [HO09, Prop 3.1], $\Sigma$ is a closed subset of pure codimension $1$ and
$D_{i}$ is a prime divisors on $B$. The Hurwitz formula for the quotient
morphism $q:Y\longrightarrow B$ reads

$K_{Y}\sim_{\mathbb{Q}}q^{*}\big{(}K_{B}+D_{q}\big{)},$ (4.3)

where $D_{q}$ is an an effective $\mathbb{Q}$-divisor on $B$ supported on the
branch locus $\Sigma$.

######  Proposition 4.4.

Let $Y$ be the normal $G$-compactification of the intermediate base $U$.

  1. (1)

If $\pi$ is of type A, then $Y$ is an abelian torsor. In particular, $U$
compactifies to an abelian torsor.

  2. (2)

If $\pi$ is of type B, then $K_{Y}$ is ample.

###### Proof.

First, because $\rho(B)=1$ a $\mathbb{Q}$-Cartier divisor on $B$ is either
ample, anti-ample, or trivial. From the Hurwitz formula (4.3) and the fact
that $q$ is finite we deduce the second item, namely that the variety $Y$ is
of general type if and only if $K_{Y}$ is ample.

Similarly, if $\pi$ is of type A then $K_{Y}\sim_{\mathbb{Q}}0$. We have
already shown that for type A fibrations, $U$ (or $Y$) admits a birational map

$f:Y\dashrightarrow A$

to an abelian torsor $A$. Since $Y$ is normal, the complement of its smooth
locus $Y_{\operatorname{reg}}\subset Y$ has a codimension at least $2$. The
restriction of $f$ to $Y_{\operatorname{reg}}$ is a morphism
$f:Y_{\operatorname{reg}}\longrightarrow A$. As the canonical divisor of
$Y_{\operatorname{reg}}$ is trivial, this morphism is unramified and hence is
an open immersion. It follows that the map $Y/G\dashrightarrow A/G$ induced by
$f$ on the quotient does not contract any divisors.

We claim that $f$ is an isomorphism in codimension $1$, namely that the
complement of $f(Y_{\operatorname{reg}})\subset A$ has codimension at least
$2$. Assume on the contrary that $A\setminus f(Y_{\operatorname{reg}})$
contains a prime divisor $E$. Then the inverse birational map
$f^{-1}:A\dashrightarrow Y$ contracts $E$, and the quotient birational map
$A/G\dashrightarrow Y/G$ contracts the effective divisor $(G\cdot E)/G$. On
the other hand, we have seen that the map $Y/G\dashrightarrow A/G$ induced by
$f$ on the quotient does not contract any divisor. The conclusion is that
$\rho(A/G)\geq\rho(Y/G)+1=\rho(B)+1=2$, which implies that
$H^{2}(A,\mathbb{Q})^{G}$ has dimension $\geq 2$. This violates Corollary
2.20, thereby proving that $f$ is an isomorphism in codimension $1$. The
conclusion follows from Lemma 4.5 (we omit the proof) which states that there
are no flops on an abelian variety. ∎

######  Lemma 4.5.

Let $Y$ be a normal $\mathbb{Q}$-Gorenstein projective variety with
$K_{Y}\sim_{\mathbb{Q}}0$ and $A$ an abelian torsor. A birational map
$f:Y\dashrightarrow A$ which is an isomorphism in codimension $1$ extends to
an isomorphism. ∎

######  Remark 4.6.

In the type B case, we only have partial control on the singularities of the
compactification $Y\supset U$. We plan to return to this question in the
future in connection to the study of type $B$ isotrivial fibrations.

###  4.2. Tubular neighborhood of general singular fibers

In this subsection, we will study the geometry of the general singular fibers
of the isotrivial fibration $\pi:X\longrightarrow B$ along with the local
monodromy. Our main tools will be the work of [HO09, HO11] and the fact that
the monodromy of an isotrivial fibration respects the Hodge structure
(Proposition 2.8).

We let $D_{i}\subset\Sigma$ be an irreducible component of the discriminant
locus of $\pi$, a prime divisor on $B$. Its preimage $q^{-1}(D_{i})\subset Y$
is a (possibly reducible) divisor on $Y$.

######  Definition 4.7.

Let $D_{i}\subset\Sigma$ be an irreducible component of the discriminant locus
and $\eta_{i}\in S$ the generic point of any irreducible component of
$q^{-1}(D_{i})\subset Y$. The _local monodromy group_ around $D_{i}$ is the
stabilizer subgroup

$G_{i}=\operatorname{Stab}(\eta_{i})\ \ \subset\ \ G,$

which is well-defined up to conjugation in $G$. The _local monodromy index_
around $D_{i}$ is its order

$m_{i}=|G_{i}|.$

The following is a well-known equivalent definition of the local monodromy
group and index. Fix a general point $b\in D_{i}$ and consider a small
analytic open neighborhood $b\in V\subset B$. We may assume that there are
biholomorphisms

$V\cong\Delta^{n},\qquad V^{*}:=V\setminus
D_{i}\cong\Delta^{*}\times\Delta^{n-1},$

where $\Delta$ is a $1$-dimensional open disc,
$\Delta^{*}=\Delta\setminus\\{0\\}$ is a punctured $1$-dimensional open disc,
and $b\in V$ corresponds to $(0,\cdots,0)\in\Delta^{n}$ under the
biholomorphism $V\cong\Delta^{n}$. The fundamental group of
$\Delta^{*}\times\Delta^{n-1}$ is isomorphic to $\mathbb{Z}$, so we have
homomorphisms of groups

$\mathbb{Z}\cong\pi_{1}(V^{*})\longrightarrow\pi_{1}(B_{0})\longrightarrow\operatorname{GL}(H^{1}(F,\mathbb{Z})),$
(4.8)

where the last is the global monodromy representation with image $G$. The
local monodromy group $G_{i}$, a finite cyclic subgroup of $G$, is the image
of this composition (4.8). We thus have an isomorphism
$G_{i}\cong\mu_{m_{i}}$, where $m_{i}=|G_{i}|$ is the local monodromy index.

Now suppose that the fiber $X_{b}$ is not multiple. By the classification of
[HO09, Thm 1.4] and [HO11, Thm 2.4], has at least one reduced irreducible
component. Therefore, shrinking $V$ if necessary, we may assume that
$\pi:X_{V}\longrightarrow V$ admits a section passing through the reduced
component of $X_{b}$. Using the same argument as in Lemma 2.4, we can
construct a cartesian diagram

${F\times\tilde{V}^{*}}$${X_{V^{*}}}$${\tilde{V}^{*}}$${V^{*},}$$\scriptstyle{\operatorname{pr}_{2}}$$\scriptstyle{\pi}$
(4.9)

where $\tilde{V}^{*}\longrightarrow V^{*}$ is the Galois covering
corresponding to the local monodromy group $G_{i}$, and $G_{i}$ acts on
$F\times\tilde{V}^{*}$ diagonally and by fixing a point in the abelian torsor
$F$. We will regard $F$ as a $G_{i}$-abelian variety by fixing an origin.

A surprising fact about isotrivial Lagrangian fibrations is that the local
monodromy group $G_{i}$ is the automorphism group of an elliptic curve, and in
particular has order $2$, $3$, $4$, or $6$. This was tacitly discussed in
[HO11], though in a slightly different context. See also [BS23, Lem 3.6] for
an alternative approach.

######  Proposition 4.10.

Let $G_{i}$ be the local monodromy group around an irreducible component
$D_{i}$ of the discriminant locus $\Sigma$. There exists a $1$-dimensional
$G_{i}$-abelian subvariety $E_{i}\subset F$ such that $G_{i}$ acts faithfully
on $E_{i}$ and trivially on $F/E_{i}$.

###### Proof.

We will construct a short exact sequence of $G_{i}$-abelian varieties

$0\longrightarrow E_{i}\longrightarrow F\longrightarrow F/E_{i}\longrightarrow
0,$ (4.11)

where $\dim E_{i}=1$ and the quotient $F/E_{i}$ has a trivial $G_{i}$-action.
Since $H^{1}(F,\mathbb{Z})^{G_{i}}\subset H^{1}(F,\mathbb{Z})$ is a
(saturated) Hodge substructure (see Proposition 2.8), it is enough to prove
that the Hodge substructure $H^{1}(F,\mathbb{Q})^{G_{i}}\subset
H^{1}(F,\mathbb{Q})$ has dimension $2n-2$. Since $F$ is an abelian variety,
this is equivalent to showing that $H^{0}(F,T_{F})^{G_{i}}$ has dimension
$n-1$.

Fixing a general point $b\in D_{i}$, an analytic neighborhood $b\in V\subset
B$, and a section of $X_{V}\longrightarrow V$, we construct diagram (4.9) as
above. By [HO09, §2], there exists a $1$-dimensional disc $b\in\Delta\subset
V$ transversal to the discriminant divisor $D_{i}$, such that
$X_{\Delta}=\pi^{-1}(\Delta)$ is smooth and we have $n-1$ _pointwise_ linearly
independent holomorphic vector fields

$\xi_{j}\in H^{0}(X_{\Delta},\,T_{X_{\Delta}}),\qquad j=1,\cdots,n-1.$

The vector fields $\xi_{j}$ are vertical in the sense that given a smooth
fiber $F$ we have $(\xi_{j})_{|F}\in H^{0}(F,T_{F})$. We can pull back these
vector fields to vector fields $\tilde{\xi}_{j}$ on
$F\times\tilde{\Delta}^{*}$ by the top horizontal arrow of (4.9). Since the
zero section $\\{0\\}\times\tilde{\Delta}^{*}$ is $G_{i}$-invariant and its
quotient recovers the section of $X_{\Delta^{*}}\longrightarrow\Delta^{*}$,
the vector fields $\tilde{\xi}_{j}$ are $G_{i}$-invariant. In particular,
restricting $\tilde{\xi}_{j}$ to a fiber $F$ of the projection
$F\times\tilde{\Delta}^{*}\longrightarrow\tilde{\Delta}^{*}$, we see that
$\tilde{\xi}_{j}|_{F}\in H^{0}(F,T_{F})\cong T_{0}F$ is $G_{i}$-invariant. ∎

######  Corollary 4.12.

Let $m_{i}$ be the local monodromy index around an irreducible component
$D_{i}$ of the discriminant locus. Then

  1. (1)

$m_{i}=2$, $3$, $4$, or $6$.

  2. (2)

If the endomorphism field of $E$ is not $\mathbb{Q}(\sqrt{-1})$ or
$\mathbb{Q}(\sqrt{-3})$, then $m_{i}=2$.

###### Proof.

The first item follows from Proposition 4.10, since $G_{i}$ acts faithfully on
the elliptic curve $E_{i}$. The elliptic curve $E_{i}$ is an isogeny factor of
$F$ and therefore is isogenous to $E$. Hence, if the endomorphism field of $E$
is not $\mathbb{Q}(\sqrt{-1})$ or $\mathbb{Q}(\sqrt{-3})$ then
$\operatorname{Aut}_{0}(E_{i})=\\{\pm 1\\}$. ∎

Using the local monodromy group $G_{i}$ and the associated elliptic curve
$E_{i}\subset F$, we can understand the geometry of a tubular neighborhood
around the general singular fiber. Extend the Galois covering
$\tilde{V}^{*}\longrightarrow V^{*}$ to a ramified covering

$\displaystyle q_{i}:\tilde{V}=\tilde{\Delta}\times\Delta^{n-1}$
$\displaystyle\longrightarrow V=\Delta\times\Delta^{n-1}$ $\displaystyle(t,\
s_{1},\cdots,s_{n-1})$ $\displaystyle\longmapsto(t^{m_{i}},\
s_{1},\cdots,s_{n-1}).$

Here $\tilde{\Delta}$ also denotes the $1$-dimensional disc but we use a
different notation to distinguish it from $\Delta$. Complete diagram (4.9) to
the following

${F\times\tilde{V}}$${\bar{X}_{V}}$${X_{V}}$${\tilde{V}}$${V,}$$\scriptstyle{\operatorname{pr}_{2}}$$\scriptstyle{\operatorname{bir}}$$\scriptstyle{\bar{\pi}}$$\scriptstyle{\pi}$

where

$\bar{X}_{V}=(F\times\tilde{V})/G_{i}$

denotes the quotient of $F\times\tilde{V}$ by the diagonal $G_{i}$-action.
Note that $G_{i}=\langle g\rangle$ was cyclic and it acts on
$\tilde{V}=\tilde{\Delta}\times\Delta^{n-1}$ by
$g\cdot(t,s_{1},\ldots,s_{n-1})=(\zeta t,s_{1},\ldots,s_{n-1})$ for a
primitive $m_{i}^{\text{th}}$ root of unity $\zeta$.

Consider the short exact sequence (4.11) realizing $F\longrightarrow F/E_{i}$
as an $E_{i}$-torsor. By the Poincaré reducibility theorem, there is an
isogeny $A_{i}\longrightarrow F/E_{i}$ of abelian varieties of dimension $n-1$
and a cartesian diagram

${A_{i}\times E_{i}}$${A_{i}}$${F}$${F/E_{i}}$ (4.13)

trivializing the $E_{i}$-torsor $F\longrightarrow F/E_{i}$. The abelian
variety $A_{i}$ is isogenous to $E^{n-1}$ (or $E_{i}^{n-1}$).

######  Proposition 4.14 (Local structure of $\bar{\pi}$ around the general
singular fibers).

Let $D_{i}$ be an irreducible component of the discriminant locus $\Sigma$ and
$b\in D_{i}$ a general point. Then

  1. (1)

There exists an analytic open neighborhood $b\in V\subset B$ biholomorphic to
$\Delta^{n}$ such that $\bar{\pi}:\bar{X}_{V}\longrightarrow V$ factors as

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  2. (2)

There exists a cartesian diagram

${A_{i}\times\bar{S}_{i}\times\Delta^{n-1}}$${A_{i}\times\Delta\times\Delta^{n-1}}$${\bar{X}_{V}}$${F/E_{i}\times
V\;\;,}$$\scriptstyle{\operatorname{id}\times\bar{f}_{i}\times\operatorname{id}}$
(4.15)

where $A_{i}\longrightarrow F/E_{i}$ is the étale covering from (4.13) and
$\bar{f}_{i}:\bar{S}_{i}\longrightarrow\Delta$ is an isotrivial elliptic
fibration over a one-dimensional disc.

In other words, $\bar{\pi}:\bar{X}_{V}\longrightarrow V$ is biholomorphic to a
finite étale quotient of the product of
$\bar{f}_{i}:\bar{S}_{i}\longrightarrow\Delta$ and
$\operatorname{pr}_{2}:A_{i}\times\Delta^{n-1}\longrightarrow\Delta^{n-1}$.

###### Proof.

Let $V$ be an open analytic neighborhood of $b$ as in the discussion above.
Consider the product of the diagram (4.13) with
$\tilde{V}=\tilde{\Delta}\times\Delta^{n-1}$ and equip it with a diagonal
$G_{i}$-action. Here $G_{i}$ acts by deck transformations on $\tilde{\Delta}$
and trivially on $\Delta^{n-1}$. The $G_{i}$-quotient of this diagram is
(4.15) and the isotrivial elliptic fibration
$\bar{f}_{i}:\bar{S}_{i}\longrightarrow\Delta$ is the $G_{i}$-quotient of the
trivial family
$\operatorname{pr}_{2}:E_{i}\times\tilde{\Delta}\longrightarrow\tilde{\Delta}$,
i.e., $\bar{S}_{i}=(E_{i}\times\tilde{\Delta})/G_{i}$. Note that $\bar{S}_{i}$
is a singular surface.

It remains to prove that the resulting diagram is cartesian. Since the
$G_{i}$-action on $A_{i}$ and $\Delta^{n-1}$ are trivial, it is enough to
prove that the diagram

${(E_{i}\times\tilde{\Delta})/G_{i}}$${\Delta}$${(F\times\tilde{\Delta})/G_{i}}$${F/E_{i}\times\Delta}$

is cartesian. This can be done by an explicit computation. The fiber product
of $(F\times\tilde{\Delta})/G_{i}$ and $\Delta$ over $F/E_{i}\times\Delta$ is

$\Big{\\{}\big{(}(f,\tilde{t}),\ t\big{)}\
\in(F\times\tilde{\Delta})/G_{i}\times\Delta\ :\ f=0\ \in F/E_{i},\ \
t=\tilde{t}^{m_{i}}\Big{\\}}.$

One readily shows that it is isomorphic to
$(E_{i}\times\tilde{\Delta})/G_{i}$, which completes the proof. ∎

The elliptic fibration $\bar{f}_{i}:\bar{S}_{i}\longrightarrow\Delta$ was
constructed as the $G_{i}$-quotient of the trivial family
$\operatorname{pr}_{2}:E_{i}\times\tilde{\Delta}\longrightarrow\tilde{\Delta}$.
By classical surface theory, $\bar{f}_{i}$ admits a unique relative minimal
model $f_{i}:S_{i}\longrightarrow\Delta$ from a smooth surface $S_{i}$. The
central fiber of $f_{i}$ is classified by the Kodaira classification of
singular fibers.

${E_{i}\times\tilde{\Delta}}$${\bar{S}_{i}}$${S_{i}}$${\tilde{\Delta}}$${\Delta.}$$\scriptstyle{\operatorname{pr}_{2}}$$\scriptstyle{\operatorname{bir}}$$\scriptstyle{\bar{f}_{i}}$$\scriptstyle{f_{i}}$$\scriptstyle{t\longmapsto
t^{m_{i}}}$

Notice that this description of a minimal isotrivial elliptic fibration is
identical to the one we provide in Appendix A. That is, we understand the
relation between $\bar{S}_{i}$ and $S_{i}$. We will see that $\bar{X}_{V}$ and
$X_{V}$ are related in very much the same manner.

${\bar{S}_{i}}$${S_{i}}$${\Delta}$$\scriptstyle{\bar{f}_{i}}$bir$\scriptstyle{f_{i}}$
${\bar{X}_{V}}$${X_{V}}$${V}$$\scriptstyle{\bar{\pi}}$bir$\scriptstyle{\pi}$

The following can be understood as a refinement of Hwang–Oguiso’s description
of general singular fibers in [HO09] for the case of isotrivial fibrations.

######  Proposition 4.16 (Local structure of $\pi$ around the general singular
fibers).

We use the notation from Proposition 4.14.

  1. (1)

The original isotrivial fibration $\pi:X_{V}\longrightarrow V$ factors as

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  2. (2)

There exists a cartesian diagram

${A_{i}\times
S_{i}\times\Delta^{n-1}}$${A_{i}\times\Delta\times\Delta^{n-1}}$${X_{V}}$${F/E_{i}\times
V}$$\scriptstyle{\operatorname{id}\times f_{i}\times\operatorname{id}}$

where $A_{i}\longrightarrow F/E_{i}$ is an étale covering in (4.13) and
$f_{i}:S_{i}\longrightarrow\Delta$ is the relative minimal model of
$\bar{f}_{i}:\bar{S}_{i}\longrightarrow\Delta$.

In other words, the isotrivial fibration $\pi:X_{V}\longrightarrow V$ is
biholomorphic to a finite étale quotient of the product of
$f_{i}:S_{i}\longrightarrow\Delta$ and
$\operatorname{pr}_{2}:A_{i}\times\Delta^{n-1}\longrightarrow\Delta^{n-1}$.

###### Proof.

Consider the (rational) factorization of $\pi:X_{V}\longrightarrow V$ as

$\pi:X_{V}\dashrightarrow\bar{X}_{V}\longrightarrow F/E_{i}\times
V\xlongrightarrow{\operatorname{pr}_{2}}V,$

where the first map is a birational map and the latter morphisms are as in
Proposition 4.14. Since $X_{V}$ is smooth and $F/E_{i}$ is an abelian torsor,
the rational map $X_{V}\dashrightarrow F/E_{i}$ is in fact a morphism. Hence,
we have a factorization

$\pi:X_{V}\longrightarrow F/E_{i}\times
V\xlongrightarrow{\operatorname{pr}_{2}}V,$

and moreover $X_{V}$ and $\bar{X}_{V}$ are birational over $F/E_{i}\times V$.
By Proposition 4.14, the étale covering of $\bar{X}_{V}$ obtained by the base
change $A_{i}\longrightarrow F/E_{i}$ is isomorphic to
$A_{i}\times\bar{S}_{i}\times\Delta^{n-1}$. Let us denote by $\tilde{X}_{V}$ a
similar étale covering of $X_{V}$. We have a cartesian diagram

${A_{i}\times\bar{S}_{i}\times\Delta^{n-1}}$${\tilde{X}_{V}}$${A_{i}\times\Delta\times\Delta^{n-1}}$${\bar{X}_{V}}$${X_{V}}$${F/E_{i}\times
V.}$$\scriptstyle{\operatorname{bir}}$$\scriptstyle{\operatorname{bir}}$
(4.17)

Note that both $\tilde{X}_{V}$ and $A_{i}\times S_{i}\times\Delta^{n-1}$ are
smooth, and are relatively minimal elliptic fibrations over
$A_{i}\times\Delta\times\Delta^{n-1}$. They are birational by diagram (4.17),
so they are isomorphic by the uniqueness of relative minimal models of curves.
This completes the proof. ∎

######  Corollary 4.18.

Let $b\in D_{i}$ be a general point of an irreducible component of the
discriminant locus, $X_{b}:=\pi^{-1}(b)$ the fiber over $b$, and $S_{0}$ the
central fiber of the minimal elliptic fibration $S_{i}\longrightarrow\Delta$
from Proposition 4.16. Then there is a cartesian diagram

${S_{0}\times A_{i}}$${A_{i}}$${X_{b}}$${F/E_{i}.}$

In other words, a general singular fiber $X_{b}$ is a finite étale quotient of
$S_{0}\times A_{i}$. ∎

######  Definition 4.19.

The _Kodaira type_ of a general singular $X_{b}$ of $\pi$ is the Kodaira type
of the central fiber $S_{0}$ of the minimal isotrivial elliptic fibration
$S_{i}\longrightarrow\Delta$ constructed in Proposition 4.16.

We emphasize again that Definition 4.19 is limited to _isotrivial_ Lagrangian
fibrations _whose general singular fibers are non-multiple_. Definition 4.19
coincides with the original definition of [HO09], which uses the notion of a
characteristic cycle. However, with this definition at hand, it is clear from
the Kodaira classification table 1 that general singular fibers can only have
the type II, III, IV, $\text{I}_{0}^{*}$, $\text{II}^{*}$, $\text{III}^{*}$,
or $\text{IV}^{*}$.

######  Proposition 4.20.

A general singular fiber of an isotrivial fibration $\pi:X\longrightarrow B$
can be of Kodaira type II, III, IV, $\textup{I}_{0}^{*}$, $\textup{II}^{*}$,
$\textup{III}^{*}$ or $\textup{IV}^{*}$.∎

###  4.3. Log canonical thresholds and the canonical bundle formula

In this subsection we will discuss the canonical bundle formula and explain
its relation with Theorem 3.2 (Proposition 4.24). In short, $\pi$ is of type
$A$ if and only if for every irreducible component $D_{i}\subset\Sigma$, the
fiber of $\pi$ over the generic point of $D_{i}$ is of $*$-type, namely of
type $\text{I}_{0}^{*},\text{II}^{*},\text{III}^{*},$ or $\text{IV}^{*}$. When
the endomorphism algebra of $E$ is different from $\mathbb{Q}(\sqrt{-1})$ or
$\mathbb{Q}(\sqrt{-3})$ the local monodromy is $\mu_{2}$ (Corollary 4.12)
forcing the general singular fibers to be of type $\text{I}_{0}^{*}$. Theorem
1.9 will follows easily.

The $\mathbb{Q}$-divisor

$D_{q}=\sum_{i}\big{(}1-\tfrac{1}{m_{i}}\big{)}D_{i}.$ (4.21)

on $B$ governs the behavior of the canonical divisor of $Y$ via the Hurwitz
formula $K_{Y}\sim_{\mathbb{Q}}q^{*}\big{(}K_{B}+D_{q}\big{)}$ (4.3). Let us
introduce another $\mathbb{Q}$-divisor on $B$ supported on the discriminant
locus. Following [DB22, §2.11], we define the _discriminant
$\mathbb{Q}$-divisor_ $D_{\pi}$ of the morphism $\pi:X\longrightarrow B$ by

$D_{\pi}=\sum_{i}\big{(}1-c_{i}\big{)}D_{i},\qquad
c_{i}=\operatorname{lct}_{D_{i}}(X,X_{D_{i}})\ \in\mathbb{Q}.$ (4.22)

Here $c_{i}$ is the log canonical threshold of an effective divisor
$X_{D_{i}}=\pi^{-1}(D_{i})$ in $X$ _over the generic point of $D_{i}$_.

The importance of these $\mathbb{Q}$-divisors stems from the canonical bundle
formula, a higher-dimensional generalization of Kodaira’s canonical bundle
formula for elliptic fibrations, which is due to Ambro [Amb04, Amb05],
Kawamata [Kaw98] and Fujino [Fuj03] (see also Kollár [Kol07, Thm. 8.3.7]). In
the case of isotrivial Lagrangian fibrations this formula takes the following
simple form:

######  Theorem 4.23 ([DB22, Thm 3.3]).

There exists a $\mathbb{Q}$-linear equivalence of $\mathbb{Q}$-divisors

$K_{X}\sim_{\mathbb{Q}}\pi^{*}\big{(}K_{B}+D_{\pi}\big{)}.$

In other words, $K_{B}+D_{\pi}\sim_{\mathbb{Q}}0$ on $B$.

Proposition 4.16 is describing the local structures around a general singular
fiber of $\pi$ and allow us to compute the log canonical thresholds $c_{i}$,
and thus $D_{\pi}$.

######  Proposition 4.24.

Assume that all general singular fibers of $\pi$ are non-multiple. Then we
have

$D_{q}\geq D_{\pi}.$

Moreover, the following are equivalent.

  1. (1)

$D_{q}=D_{\pi}$.

  2. (2)

$\pi$ is of type A.1 or A.2.

  3. (3)

The general singular fibers of $\pi$ are of Kodaira type $\textup{I}_{0}^{*}$,
$\textup{II}^{*}$, $\textup{III}^{*}$, or $\textup{IV}^{*}$.

###### Proof.

To show the inequality, we need to prove $\frac{1}{m_{i}}\leq c_{i}$ for all
irreducible components $D_{i}\subset\Sigma$. The log canonical threshold of a
pair is invariant under étale covering, so the étale covering $A_{i}\times
S_{i}\times\Delta^{n-1}\longrightarrow X_{V}$ in Proposition 4.16 gives us the
equality

$c_{i}=\operatorname{lct}(S_{i},S_{0}),$

where $f:S_{i}\longrightarrow\Delta$ is the minimal isotrivial elliptic
fibration constructed in the proposition and $S_{0}\subset S_{i}$ its
(singular) central fiber. The log canonical thresholds of the singular fibers
of a minimal isotrivial elliptic fibration are listed in Table 1. One checks
from the table that the inequality $\frac{1}{m_{i}}\leq c_{i}$ is satisfied
for every Kodaira fiber type.

In Table 1, the equality $c_{i}=\frac{1}{m_{i}}$ holds exactly when the
singular fiber $S_{0}$ in the elliptic fibration is of Kodaira $*$-type. From
the étale covering $A_{i}\times S_{i}\times\Delta^{n-1}\longrightarrow X_{V}$
in Proposition 4.16, we obtain the equivalence (1) $\Longleftrightarrow$ (3).
Recall that we had $K_{B}+D_{\pi}\sim_{\mathbb{Q}}0$ from the canonical bundle
formula in Theorem 4.23. The Hurwitz formula (4.3) then implies that
$K_{Y}\sim_{\mathbb{Q}}0$ if and only if $D_{q}\sim_{\mathbb{Q}}D_{\pi}$.
Since $D_{q}\geq D_{\pi}$ this is in fact equivalent to the equality
$D_{q}=D_{\pi}$. This proves (1) $\Longleftrightarrow$ (2). ∎

######  Remark 4.25.

One can add an additional equivalent condition in Proposition 4.24, namely
that $\pi$ is of type A if and only if $\bar{S}_{i}$ has canonical
singularities, or equivalently if and only if $\bar{X}_{V}$ has canonical
singularities (see Proposition 4.14). In this case $\bar{X}_{V}$ has
symplectic singularities and $X_{V}$ is a symplectic resolution of
$\bar{X}_{V}$.

Theorem 1.9 is now an easy consequence of the description of the local
monodromy group from Corollary 4.12.

###### Proof of Theorem 1.9.

If the endomorphism field of $E$ is not $\mathbb{Q}(\sqrt{-1})$ or
$\mathbb{Q}(\sqrt{-3})$, Corollary 4.12 states that the local monodromy index
$m_{i}$ is $2$ for all $i$. It follows from Table 1 that the general singular
fibers of $\pi$ are of type $\textup{I}_{0}^{*}$. Proposition 4.24 implies
that $\pi$ is of type $A$.

Using the same argument as in the proof of Theorem 1.9 above, we obtain that
for every irreducible component $D_{i}\subset\Sigma$ the log canonical
threshold $c_{i}=\operatorname{lct}_{D_{i}}(X,X_{D_{i}})$ is equal to $1/2$.
Theorem 4.23 implies that

$\deg\Sigma=2\deg D_{q}=-2\deg(K_{\mathbb{P}^{n}})=2(n+1).\qed$

##  5\. Classification of type A fibrations

In this section we prove Theorem 1.8, which describes type A.1 fibrations
admiting a rational section, conditional on Conjecture 1.6. Recall from
Theorem 3.2 that the intermediate trivializiation $p:Z\longrightarrow U$ is
equivariantly birational to a morphism between

  * •

two $G$-abelian torsors (type A.1), or

  * •

a $G$-variety of Kodaira dimension $n$ and a $G$-abelian torsor (type A.2), or

  * •

two $G$-varieties of Kodaira dimension $n$ (type B).

In particular, if $\pi$ is a type A.1 fibration then $X$ is birational to the
quotient of an abelian torsor of dimension $2n$ by the finite group $G$. As an
intermediate step we introduce Proposition 5.1, which does not require any
sections and does not rely on Conjecture 1.6.

######  Proposition 5.1.

Let $\pi:X\longrightarrow B$ be a type A.1 isotrivial fibration of a hyper-
Kähler $2n$-fold, so that $U$ and $Z$ are equivariantly birational to
$G$-abelian torsors $\bar{F}^{\prime}$ and $T$. Then

  1. (1)

$T/G$ is a primitive symplectic variety admitting a symplectic resolution to a
hyper-Kähler manifold $X^{\prime}$.

  2. (2)

There exists a commutative diagram

${X}$${X^{\prime}}$${\bar{F}/G=B^{\prime},}$${B}$${\bar{F}^{\prime}/G}$$\scriptstyle{\pi}$$\scriptstyle{\operatorname{bir}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\cong}$
(5.2)

where $\pi^{\prime}$ is an isotrivial fibration of a hyper-Kähler manifold
$X^{\prime}$.

  3. (3)

There is an elliptic curve $E^{\prime}$ such that
$\bar{F}^{\prime}\cong(E^{\prime})^{n}$. The group $G$ together with its
action on $\bar{F}^{\prime}$ is either:

    1. (a)

$G=\mathfrak{S}_{n+1}$ acting on $\ker\big{(}(E^{\prime})^{n+1}\to
E^{\prime}\big{)}\cong(E^{\prime})^{n}$ by permuting the factors.

    2. (b)

$G=\mu^{\times n}\rtimes\mathfrak{S}_{n}$ acting on $(E^{\prime})^{n}$ in the
obvious way, where $\mu=\mu_{2},\mu_{3},\mu_{4}$ or $\mu_{6}$ acts on the
elliptic curve $E^{\prime}$ faithfully and fixes the origin.

    3. (c)

(Only when $n=2$) The Pauli group acting on $(E^{\prime})^{2}$ for
$E^{\prime}=\mathbb{C}/\mathbb{Z}[\sqrt{-1}]$.

The _Pauli group_ is the following non-abelian group of order $16$:

$\left\\{\begin{pmatrix}\sqrt{-1}^{a}&0\\\
0&\sqrt{-1}^{b}\end{pmatrix}:a\equiv b\mod
2\right\\}\rtimes\left\langle\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\right\rangle\quad\subset\
\operatorname{GL}_{2}(\mathbb{Z}[\sqrt{-1}]).$

We refer the reader to [ALAQ22, §3.2] for the Pauli group action on
$(E^{\prime})^{2}$ appearing in (3) case (c) of Proposition 5.1. Note that
this proposition does not depend on Conjecture 1.6. Assuming the conjecture
allows us to classify of $\bar{F}$ as in (3) and will lead to a proof of
Theorem 1.8.

The rest of this section is devoted to the proof of the proposition. The
following three lemmas will play a central role. We will use the notion of a
primitive symplectic variety following [BL22, Def 3.1] (see also [Sch20]). A
_(projective) primitive symplectic variety_ is a normal projective symplectic
variety $(Y,\sigma)$ such that $H^{0}(Y,\Omega_{Y}^{[1]})=0$ and
$H^{0}(Y,\Omega_{Y}^{[2]})=\mathbb{C}\cdot\sigma$, where
$\Omega_{\bar{X}}^{[k]}$ is the sheaf of reflexive $k$-forms.

######  Lemma 5.3.

Let $G$ be a finite group and $T$ be a $G$-abelian variety of dimension $2n$.
Suppose that there is a simple $G$-module $V$ so that

$H^{1,0}(T)\cong V\oplus V^{\vee}.$

Then

  1. (1)

The quotient variety $T/G$ is a symplectic variety.

  2. (2)

If we further assume that $V$ is not a symplectic $G$-module, then $T/G$ is a
primitive symplectic variety.

###### Proof.

The projective variety $\bar{X}=T/G$ has quotient singularities so it is well
behaved from the point of view of Hodge theory. In particular, we have
$H^{1,0}(\bar{X})=H^{0}(\bar{X},\Omega_{\bar{X}}^{[1]})$ and
$H^{2,0}(\bar{X})=H^{0}(\bar{X},\Omega_{\bar{X}}^{[2]})$ as well as the
equalities

$H^{1,0}(\bar{X})=H^{1,0}(T)^{G}=\big{(}V\oplus V^{\vee}\big{)}^{G}=0$

and

$H^{2,0}(\bar{X})=H^{2,0}(T)^{G}=\Big{(}\wedge^{2}V\oplus\wedge^{2}V^{\vee}\oplus\big{(}V\otimes
V^{\vee}\big{)}\Big{)}^{G}.$

Note that the last term is $(V\otimes
V^{\vee})^{G}=\operatorname{End}_{G}(V)=\mathbb{C}\cdot\operatorname{id}$ by
Schur’s lemma. Hence there is a $G$-invariant holomorphic symplectic form
$\sigma$ on $T$, which descends to a symplectic form on
$\bar{X}_{\operatorname{sm}}$. The above identity also shows that if $V$ is
not a symplectic module then $T/G$ is a primitive symplectic variety since
$(\wedge^{2}V)^{G}=0\implies h^{2,0}(\bar{X})=1$.

To complete the proof that $\bar{X}$ is symplectic, it is enough to show that
it has rational singularities and is Gorenstein by Namikawa’s theorem (see
[Nam01, Thm 4] or [Fu06, Thm 1.2]). Since $\bar{X}$ is a global quotient
variety it has rational singularities. To prove that it is Gorenstein, fix a
closed point $x\in T$ and its image $\bar{x}\in\bar{X}$. We claim that
$\bar{X}$ is Gorenstein at $\bar{x}$. By [Wat74], it is enough to prove that
the representation of the stabilizer $G_{x}$ on the tangent space $T_{x}(T)$
factors through $\operatorname{SL}(T_{x}(T))$. Since the tangent bundle of the
abelian variety $T$ is trivial, this condition is equivalent to $G_{x}$ acting
trivially on $\wedge^{2n}H^{1,0}(T)=\wedge^{2n}\big{(}V\oplus
V^{\vee}\big{)}$. This is the case since

$\wedge^{2n}\big{(}V\oplus
V^{\vee}\big{)}=\wedge^{n}V\otimes\wedge^{n}V^{\vee}=\wedge^{n}V\otimes\left(\wedge^{n}V\right)^{\vee}$

is the trivial $G$-module. This proves that $\bar{X}$ is Gorenstein, and
therefore a symplectic variety. ∎

######  Lemma 5.4.

Any primitive symplectic variety that is birational to a hyper-Kähler manifold
admits a symplectic resolution to a hyper-Kähler manifold.

###### Proof.

Any primitive symplectic variety $\bar{X}$ admits a partial resolution
$X\longrightarrow\bar{X}$ from a $\mathbb{Q}$-factorial, terminal and
primitive symplectic variety $X$ by [Sch20, Prop 11]. If we assume $X$ is
birational to a hyper-Kähler manifold $X^{\prime}$, then $X$ itself is
necessarily a hyper-Kähler manifold by [GLR13, Prop 6.5]. ∎

The next lemma is a direct consequence of a series of recent results of
Auffarth, Lucchini Arteche, and Quezada which provide a classification of
smooth quotients of abelian varieties.

######  Lemma 5.5 ([ALA20, ALAQ22, ALA22]).

Let $F$ be a $G$-abelian torsor. Assume that (1) $G$ acts on $F$ faithfully
and without any translations, and (2) $F/G\cong\mathbb{P}^{n}$. Then there
exists a choice of an origin for $F$ making it into a $G$-abelian variety
satisfying:

  1. (1)

$F\cong E^{n}$ for an elliptic curve $E$.

  2. (2)

$G$ acts faithfully on $E^{n}$ by fixing the origin and this action belongs to
the following list:

    1. (a)

$\mathfrak{S}_{n+1}$ acting on $\ker(E^{n+1}\longrightarrow E)$ by permuting
the factors.

    2. (b)

$\mu^{\times n}\rtimes\mathfrak{S}_{n}$ acting on $E^{n}$ in the obvious way,
where $\mu=\mu_{2},\mu_{3},\mu_{4}$ or $\mu_{6}$ acts on $E$ faithfully by
fixing the origin.

    3. (c)

(Only when $n=2$) The Pauli group acting on $E^{2}$ for
$E=\mathbb{C}/\mathbb{Z}[\sqrt{-1}]$.

###### Proof.

Consider a normal subgroup of $N\subset G$ and its quotient

$N=\\{g\in G:F^{g}\neq\varnothing\\},\qquad H=G/N.$

By [ALA22, Thm 1.1], there exists at least one point of $F$ fixed by all of
$N$, i.e., $F^{N}\neq\varnothing$. We may thus choose an origin for $F$ so
that $N$ acts on $F$ by fixing the origin, i.e., so that $F$ is an $N$-abelian
variety. Moreover $F/N$ is a smooth projective fiber bundle

$F/N\longrightarrow Z,$

over an abelian variety $Z$ and the fiber is positive dimensional of the form
$\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{i}}$. Moreover, $H$ acts
freely on both $F/N$ and $Z$ so that the morphism is $H$-equivariant. Taking
the $H$-quotient, we obtain a fiber bundle $F/G\to Z/H$ whose fiber is of the
form $\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{i}}$.

The cohomology $H^{*}(F/G,\mathbb{Q})$ can be computed in terms of the
cohomology of the base $H^{*}(Z/H,\mathbb{Q})$ by the Leray–Hirsch theorem:

$H^{*}(F/G,\mathbb{Q})\cong H^{*}(Z/H,\mathbb{Q})\otimes
H^{*}(\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{i}},\mathbb{Q}).$

By assumption $F/G\cong\mathbb{P}^{n}$, so this forces the base $Z/H$ to be a
point. Since $Z$ is an abelian variety and $H$ was a finite group acting
freely on $Z$, this means that $Z$ is a point and $H$ is trivial. That is, $F$
is a $G$-abelian variety. The result now follows from the main results of
[ALA20, ALAQ22], where they classified all possible $G$-abelian varieties $F$
with smooth quotient $F/G$. ∎

###### Proof of Proposition 5.1.

Let $\pi:X\longrightarrow B$ be a type A isotrivial fibration. Then $X$ is
birational to a finite quotient of an abelian variety $T$ by the finite group
$G$. Combining Lemma 2.16, 5.3 and 5.4, $T/G$ is a primitive symplectic
variety admitting a symplectic resolution $X^{\prime}\longrightarrow T/G$ to a
hyper-Kähler manifold $X^{\prime}$. Note that we have two equivariant
morphisms $T\longrightarrow\bar{F}$ and $T\longrightarrow\bar{F}^{\prime}$;
the former extends the intermediate trivialization $p:Z\longrightarrow U$ and
the latter comes from the fibration $p^{\prime}:Z\longrightarrow\bar{F}$ from
Theorem 3.1. Taking their $G$-quotients, we obtain the commutative diagram
(5.2) but it remains to prove that the birational map
$B\dashrightarrow\bar{F}^{\prime}/G$ is an isomorphism. This follows from
[Mat14, Cor 2].

Since the $G$-action on $U$ is free the $G$-action on $\bar{F}^{\prime}$ is
faithful. Moreover, we have proved $H^{1,0}(\bar{F}^{\prime})=H^{1,0}(U)$ is
isomorphic to a simple faithful $G$-module $V^{\vee}$ (Lemma 2.23), so the
$G$-action has no translations. We can thus apply Lemma 5.5 to obtain the last
item. ∎

We are now ready to conclude the proof of Theorem 1.8.

###### Proof of Theorem 1.8.

In Proposition 5.1, we have constructed two isotrivial fibrations

${X^{\prime}}$${B^{\prime}=F/G}$${B=F^{\prime}/G,}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$

where $X^{\prime}$ is a hyper-Kähler resolution of $T/G$ for a $G$-abelian
torsor $T$. Moreover, since $F^{\prime}/G$ is isomorphic to $\mathbb{P}^{n}$,
we have deduced from Lemma 5.5 that $F^{\prime}\cong(E^{\prime})^{n}$ admits a
$G$-action corresponding to one of three possibilities outlined therein.

If we assume Conjecture 1.6 holds for $X^{\prime}$, we obtain that $F/G$ is
isomorphic to $\mathbb{P}^{n}$, and hence similarly that $F$ is isomorphic to
$E^{n}$. We also obtain a classification of the possible $G$-actions on $F$
and all such actions fix the origin, as does the $G$-action on $F^{\prime}$.
If we further assume $\pi$ has a rational section, then Lemma 2.4 implies that
$T$ is $G$-equivariantly isomorphic to $F\times F^{\prime}$.

The case $G=\mathfrak{S}_{n+1}$ corresponds to $\text{Kum}_{n}$-fibrations and
the case $G=\mu^{\times n}\rtimes\mathfrak{S}_{n}$ corresponds to
$\text{K3}^{[n]}$-fibrations, as we have explicitly described in §2.4. When
$n=2$, there is one additional possibility for which $G$ is the Pauli group.
We will show that the corresponding quotient does not admit a symplectic
resolution, thereby contradicting Proposition 5.1. Since $G$ acts on $T\cong
E^{2}\times(E^{\prime})^{2}$ by fixing the origin, around $0$ the variety
$T/G$ is analytically isomorphic to $(V\oplus V^{\vee})/G$. It is shown in
[Bel09] that such a singularity admits a symplectic resolution if and only if
(1) $G$ is of the form $\mu_{m}^{\times n}\rtimes\mathfrak{S}_{n}$ or (2) $G$
is the group $G_{4}$, a group of order $24$ appearing in the Shephard–Todd
classification of complex reflection groups. It follows that $T/G$ cannot
admit a symplectic resolution around $0$ if $G$ is the Pauli group. ∎

##  6\. Additional remarks

In this closing section we collect some remarks about the locus of hyper-
Kähler manifolds admitting an isotrivial Lagrangian fibration. Our main result
is Theorem 1.13, (restated here as Theorem 6.2), which states that an
isotrivial Lagrangian fibration of a hyper-Kähler variety $X$ can be deformed
into a Lagrangian fibration with maximal variation provided that the second
Betti number of $X$ is at least $7$. Let us first observe that isotriviality
of a Lagrangian fibration is a closed condition in moduli.

######  Proposition 6.1.

Let $\mathcal{X}\xlongrightarrow{\pi}\mathcal{B}\longrightarrow M$ be a family
of Lagrangian fibered hyper-Kähler $2n$-folds over a complex manifold $M$.
Then the following is a Zariski closed subset of $M$:

$\textup{Iso}(M)=\\{t\in M:\pi:X_{t}\longrightarrow B_{t}\mbox{ is
isotrivial}\\}.$

###### Proof.

For simplicity we assume that $\pi$ is projective; the general case when $\pi$
is proper can be dealt using the notion of the relative automorphism scheme
and torsor. Let $\mathcal{B}_{0}\subset\mathcal{B}$ be a Zariski open subset
over which $\pi$ is smooth and proper and pick a large positive integer $N$.
We can then find a finite étale cover $\tilde{\mathcal{B}}_{0}$ of
$\mathcal{B}_{0}$ such that the base change
$\tilde{\pi}:\mathcal{X}_{\tilde{\mathcal{B}}_{0}}\longrightarrow\tilde{B}_{0}$
has a section and such that there is a compatible choice of level $N$
structure on the fibers of $\tilde{\pi}$. We thus obtain a moduli map
$\mu:\tilde{\mathcal{B}}_{0}\longrightarrow\mathcal{A}$, where $\mathcal{A}$
is a fine moduli space of abelian $n$-folds with level $N$ structure and given
polarization type.

Write $f:\widetilde{\mathcal{B}}_{0}\longrightarrow M$ for the composition of
the étale cover with the map $\mathcal{B}_{0}\longrightarrow M$. Since fibers
of the map $(\mu,f):\tilde{\mathcal{B}}_{0}\longrightarrow\mathcal{A}\times M$
have dimension at most $n$, upper semicontinuity of fiber dimension implies
that the union of $n$-dimensional fibers is a Zariski closed subset
$W_{0}\subset\tilde{\mathcal{B}}_{0}$. Moreover, a point
$t\in\tilde{\mathcal{B}}_{0}$ is in $W_{0}$ if and only if $\mu$ is constant
on $f^{-1}(f(t))\subset\tilde{\mathcal{B}}_{0}$, so that $W_{0}$ is the
$f$-preimage of $\textup{Iso}(M)\subset M$. Since $f$ is flat and $W_{0}$ is
closed in $\tilde{\mathcal{B}}_{0}$, $\textup{Iso}(M)$ is closed in $M$. ∎

######  Theorem 6.2 (= Theorem 1.13).

Let $\pi:X\longrightarrow B$ be a Lagrangian fibration of a hyper-Kähler
manifold with $b_{2}(X)\geq 7$. There exists a deformation of the Lagrangian
fibration $\pi$ which is not isotrivial.

###### Proof.

There exists a local universal deformation family $\operatorname{Def}(\pi)$ of
the Lagrangian fibration $\pi$, an analytic open ball of dimension
$b_{2}(X)-3$ [Mat16]. Assume that every Lagrangian fibration over
$\operatorname{Def}(\pi)$ is isotrivial. Taking an appropriate
Noether–Lefschetz locus of $\operatorname{Def}(\pi)$ parametrizing Lagrangian
fibrations $\pi^{\prime}:X^{\prime}\longrightarrow B^{\prime}$ with
$X^{\prime}$ projective, we can obtain a $3$-dimensional family of
$\pi^{\prime}$ whose generic Mumford–Tate group is isomorphic to
$\operatorname{SO}(T,q)$, where $T$ is the generic transcendental lattice and
$\dim_{\mathbb{Q}}T=5$. Note that by [Zar83], this means the endomorphism
field $\operatorname{End}_{\mathsf{HS}}(T)$ is $\mathbb{Q}$. This violates the
following lemma. ∎

######  Lemma 6.3.

Let $\pi:X\longrightarrow B$ be an isotrivial Lagrangian fibration of a hyper-
Kähler manifold and let $T\subset H^{2}(X,\mathbb{Q})$ be the transcendental
Hodge structure. If the endomorphism field of $T$ is $\mathbb{Q}$, then
$\dim_{\mathbb{Q}}T=3$ or $4$.

###### Proof.

Recall from the proof of Proposition 2.19 that there is a generically finite
$G$-equivariant morphism $U\longrightarrow A$ to a $G$-abelian torsor $A$ such
that the holomorphic symplectic form $\sigma\in H^{2,0}(Z)$ is contained in
the image of $H^{1}(F,\mathbb{Q})\otimes H^{1}(A,\mathbb{Q})\longrightarrow
H^{2}(Z,\mathbb{Q})$. This yields an inclusion
$T\subset\big{(}H^{1}(F,\mathbb{Q})\otimes H^{1}(A,\mathbb{Q})\big{)}^{G}$.
Since the smooth fiber $F$ of $\pi$ is isogenous to $E^{n}$ and $A$ is
isogenous to the power of a simple abelian torsor, there is an inclusion

$T\subset H^{1}(E,\mathbb{Q})\otimes W,$

where $W$ is a simple polarizable Hodge structure of weight $1$. The following
lemma appeared in a special case in [vGV16, Prop 12] and more generally in
[Moo18, Thm 7.2, Rmk 7.3]. It implies that $H^{1}(E,\mathbb{Q})$ must be
isomorphic to the _simple Kuga–Satake partner_ $\operatorname{KS}(T)$ of $T$
defined in [Moo18, §7.1]:

######  Lemma 6.4 ([vGV16, Prop 12] and [Moo18, Thm 7.2]).

Consider two weight $1$ polarizable simple Hodge structures $W_{1}$ and
$W_{2}$, and a weight $2$ simple polarizable Hodge structure $T$ of K3 type.
Assume

  1. (1)

there exists an inclusion of Hodge structures $T\subset W_{1}\otimes W_{2}$;
and

  2. (2)

$\operatorname{End}_{\mathsf{HS}}(T)=\mathbb{Q}$.

Then $W_{1}\cong W_{2}\cong\operatorname{KS}(T)$.

The simple Kuga–Satake partner $\operatorname{KS}(T)$ is a polarizable simple
Hodge structure of weight $1$. Its behavior depends on the parity of
$\dim_{\mathbb{Q}}T$ because the (half) spin representations of type B and D
simple algebraic groups behave differently. If $\dim_{\mathbb{Q}}T=2r+1$ is
odd then $\dim_{\mathbb{Q}}\operatorname{KS}(T)=2^{r}$ or $2^{r+1}$. By the
lemma, we have an isomorphism $H^{1}(E,\mathbb{Q})\cong\operatorname{KS}(T)$
so $\dim\operatorname{KS}(T)=2$. This implies that $r=1$ and $\dim T=3$. On
the other hand, if $\dim T=2r$ is even then $\dim\operatorname{KS}(T)=2^{r-1}$
or $2^{r}$, and we deduce that $r=1$ or $2$. Note that $\dim T$ cannot be $2$,
because any $2$-dimensional weight $2$ Hodge structure of K3 type is
necessarily CM. Hence $\dim T=4$. ∎

######  Lemma 6.5.

Let $E$ be an elliptic isogeny factor of $F$ and suppose that
$K=\operatorname{End}_{\mathbb{Q}}(E)$ is a CM field. Then the transcendental
Hodge structure $H^{2}_{\operatorname{tr}}(X)$ has weak CM by $K$.

###### Proof.

Since $K\cong\operatorname{End}_{\mathsf{HS}}(H^{1}(E,\mathbb{Q}))$ and $F$ is
isogenous to $E^{n}$, we have an isomorphism
$\operatorname{End}_{\mathsf{HS}}(H^{1}(F,\mathbb{Q}))\cong\operatorname{Mat}_{n}(K)$.
Moreover, Proposition 2.8 gives inclusions

$G\hookrightarrow\operatorname{Aut}_{0}(F)\cong\operatorname{Aut}_{\mathsf{HS}}(H^{1}(F,\mathbb{Z}))\
\subset\
\operatorname{End}_{\mathsf{HS}}(H^{1}(F,\mathbb{Q}))\cong\operatorname{Mat}_{n}(K).$

Since $K$ is the center of $\operatorname{Mat}_{n}(K)$, the $G$-module and
$K$-vector space structures on $H^{1}(F,\mathbb{Q})$ are compatible, namely
$H^{1}(F,\mathbb{Q})$ is a $G$-representation over the field $K$.

Recall the inclusion
$H^{2}_{\operatorname{tr}}(X)\subset\big{(}H^{1}(F,\mathbb{Q})\otimes
H^{1}(A,\mathbb{Q})\big{)}^{G}$ from Lemma 6.3. We divide the proof into two
cases according to whether $\dim A=n$ or $\dim A>n$. If $\dim A=n$, by
Corollary 2.20 and the fact that $A/G$ is $\mathbb{Q}$-Fano, $A$ is isogenous
to $(E^{\prime})^{n}$ for an elliptic curve $E^{\prime}$. Hence we get

$H^{2}_{\operatorname{tr}}(X)\subset H^{1}(E,\mathbb{Q})\otimes
H^{1}(E^{\prime},\mathbb{Q}),$

and in fact

$H^{2}_{\operatorname{tr}}(X)\cong\left(H^{1}(E,\mathbb{Q})\otimes
H^{1}(E^{\prime},\mathbb{Q})\right)_{\text{tr}}.$

Since the right hand side is a $K$-vector space, so is the left hand side.

The second case is when $\dim A>n$. In this case, we claim that

$H^{2}_{\operatorname{tr}}(X)\cong\big{(}H^{1}(F,\mathbb{Q})\otimes
H^{1}(A,\mathbb{Q})\big{)}^{G}.$ (6.6)

This equality endows $H^{2}_{\operatorname{tr}}(X)$ with a $K$-vector space
structure. To show the equality, notice first that $\big{(}H^{1,0}(F)\otimes
H^{1,0}(A)\big{)}^{G}$ is $1$-dimensional (Lemma 2.22), so the right hand side
is a weight $2$ Hodge structure of K3 type. It cannot have any Hodge classes
since

$\Big{(}\big{(}H^{1}(F,\mathbb{Q})\otimes
H^{1}(A,\mathbb{Q})\big{)}^{G}\Big{)}^{\operatorname{MT}}=\big{(}H^{1}(F,\mathbb{Q})\otimes
H^{1}(A,\mathbb{Q})\big{)}^{G\times\operatorname{MT}}=0,$

where the last equality follows from the fact that $H^{1}(F,\mathbb{Q})$ and
$H^{1}(A,\mathbb{Q})$ are simple $(G,\operatorname{MT})$-bimodule (Proposition
2.11) which are not dual. Indeed, they are of different dimensions since $\dim
A>n$. Therefore, the right hand side of (6.6) is a simple Hodge structure and
the equality follows. ∎

######  Proposition 6.7.

Let $\mathcal{X}\xlongrightarrow{\pi}\mathcal{B}\longrightarrow M$ be a family
of isotrivial Lagrangian fibered hyper-Kähler $2n$-folds over a Zariski closed
subset $M\subset\operatorname{Def}(\pi)$. Assume $\pi$ has a rational section
and the elliptic curve factor $E$ of its general smooth fibers has no CM by
$\mathbb{Q}(\sqrt{-1})$ or $\mathbb{Q}(\sqrt{-3})$. Then $M$ has dimension at
most $2$.

###### Proof.

Under the hypotheses, isotrivial fibrations $\pi_{t}:X_{t}\longrightarrow
B_{t}$ over a general point $t\in M$ are of type A.1 by Theorem 1.9 (note: the
assumption in the theorem is satisfied because $\pi_{t}$ has a rational
section) and hence $X_{t}$ is birational to the quotient of a $2n$-dimensional
abelian variety $T$ by a finite group $G$. Moreover, $T$ is isogenous to
$E^{n}\times(E^{\prime})^{n}$ for elliptic curves $E$ and $E^{\prime}$. The
result follows from the fact that such a $T$ can only vary in a two-parameter
family. ∎

##  Appendix A Isotrivial K3 surfaces

In this appendix we classify isotrivial elliptic K3 surfaces with a section.
We make no claim to originality – in fact, much of what we have to say already
appears in [FM94, §1.4.2] and [Moo18]. However, since we could not find
Theorem A.2 in the literature and our main goal is to generalize this result
to hyper-Kähler manifolds we feel this appendix is useful and necessary.

Let $\pi:S\longrightarrow B\cong\mathbb{P}^{1}$ be an isotrivial elliptic
fibration of a projective K3 surface, $B_{0}\subset B$ the complement of the
discriminant locus, and $S_{0}:=S\times_{B}B_{0}$ the complement of the
singular fibers of $\pi$. Write $\pi_{0}:S_{0}\longrightarrow B_{0}$ for the
restriction of $\pi$ to $S_{0}$. The fibers of $\pi_{0}$ are isomorphic to a
fixed curve $F$ of genus one. Let $G$ be the image of the monodromy
representation
$\pi_{1}(B_{0})\longrightarrow\operatorname{GL}(H^{1}(F,\mathbb{Z}))$
corresponding to the local system $R^{1}(\pi_{0})_{*}\underline{\mathbb{Z}}$,
and consider the minimal Galois cover $U\longrightarrow B_{0}$ trivializing
this local system. We consider the base change diagram

${Z}$${S_{0}}$${U}$${B_{0}.}$$\scriptstyle{p}$$\scriptstyle{\pi}$

As in Proposition 2.8, the monodromy representation
$\pi_{1}(B_{0})\longrightarrow\operatorname{GL}(H^{1}(F,\mathbb{Z}))$ factors
through the subgroup $\operatorname{Aut}_{0}F$, so $G$ is forced to be
isomorphic to one of four cyclic groups as follows:

$G=\begin{cases}\mu_{2},\mu_{3},\text{ or }\mu_{6}&\text{if }j(F)=0,\\\
\mu_{2}\text{ or }\mu_{4}&\text{if }j(F)=1728,\\\
\mu_{2}&\text{otherwise.}\end{cases}$ (A.1)

######  Theorem A.2 (= Theorem 1.1).

Let $\pi:S\longrightarrow\mathbb{P}^{1}$ be an isotrivial elliptic fibration
of a projective K3 surface. Then

  1. (1)

If $G=\mu_{2}$, there is an abelian surface $A$ and a fibration
$p:A\longrightarrow E$ to an elliptic curve $E$ with fiber $F$, such that
$\pi$ is birational to the quotient of $p$ by the involution $[-1]$. In
particular, $S$ is a Kummer surface.

  2. (2)

If $\pi$ admits a section, there is a curve $C$ with a $G$-action such that
$\pi$ is birational to the quotient of the projection $F\times
C\longrightarrow C$ by a diagonal $G$-action. The possible curves $C$ for
$G=\mu_{3},\mu_{4}$, and $\mu_{6}$ are classified in tables 2, 3, and 4
respectively.

The remainder of this appendix will be devoted to a few preliminaries and the
proof of Theorem A.2.

Note that the smooth open curve $U$ can be uniquely compactified to a smooth
projective curve $C$, and the $G$-action on $U$ uniquely extends to $C$. The
quotient $C/G$ is normal and birational to the base $B\cong\mathbb{P}^{1}$ of
the elliptic fibration, so it is isomorphic to $B$. Let $\tilde{Z}\supset Z$
be a projective compactification to which the $G$-action extends. Then
$\pi:S\longrightarrow\mathbb{P}^{1}$ is birational to the map

$\tilde{Z}/G\longrightarrow C/G=\mathbb{P}^{1}$

and can be obtained as its (unique) _relative minimal model_.

We will make repeated use of the following table which describes Euler
characteristic contributions, monodromy, and local models, for the singular
fibers arising in the Kodaira classification. Let us explain the meaning of
the last row. We can realize each Kodaira singular fiber as the central fiber
of the relative minimal model of a family

$(F\times\tilde{\Delta})/G\longrightarrow\Delta,$

where $\Delta$ is an open disc,
$\tilde{\Delta}=\\{(z,t)\in\mathbb{C}\times\Delta:z^{m}=t^{d}\\}$ is a cyclic
branched covering of $\Delta$ of degree $m$, and $G$ acts diagonally. When
$d>1$, $\tilde{\Delta}$ is singular and it may be replaced by its
normalization.

Kodaira type | $\text{I}_{0}$ | $\text{I}_{n\geq 1}$ | II | III | IV | $\text{I}_{0}^{*}$ | $\text{I}_{n\geq 1}^{*}$ | $\text{II}^{*}$ | $\text{III}^{*}$ | $\text{IV}^{*}$
---|---|---|---|---|---|---|---|---|---|---
Euler characteristic | $0$ | $n$ | $2$ | $3$ | $4$ | $6$ | $n+6$ | $10$ | $9$ | $8$
Monodromy order $m$ | $1$ | $\infty$ | $6$ | $4$ | $3$ | $2$ | $\infty$ | $6$ | $4$ | $3$
$d$ for $z^{m}=t^{d}$ | $1$ | - | $1$ | $1$ | $1$ | $1$ | - | $5$ | $3$ | $2$
Log canonical threshold $c$ | $1$ | $1$ | $\frac{5}{6}$ | $\frac{3}{4}$ | $\frac{2}{3}$ | $\frac{1}{2}$ | - | $\frac{1}{6}$ | $\frac{1}{4}$ | $\frac{1}{3}$
Table 1. Euler characteristic contributions, monodromy, local models, and log
canonical thresholds for singular fibers in the Kodaira classification.

We note that the quotient $(F\times\tilde{\Delta})/G$ has canonical
singularities in the cases $\text{I}_{0}^{*}$, $\text{II}^{*}$,
$\text{III}^{*}$, and $\text{IV}^{*}$, and its crepant resolution is then the
relative minimal model for
$(F\times\tilde{\Delta})/G\longrightarrow\tilde{\Delta}/G$. The quotient for
the other cases is only klt, and its relative minimal model should be
constructed by a sequence of blowups and blowdowns.

###### Proof of Theorem A.2 (1).

Consider an isotrivial elliptic fibration $\pi:S\longrightarrow B$ for which
$G=\mu_{2}$. We will show that there is an elliptic curve $E$, an abelian
surface $A$ with a fibration $f:A\longrightarrow E$ and fiber $F$, and a
commutative diagram as follows, where the top horizontal arrow is a minimal
resolution of singularities:

${S}$${A/[-1]}$${\mathbb{P}^{1}}$${E/[-1]\;.}$$\scriptstyle{\pi}$$\scriptstyle{f/{[-1]}}$$\scriptstyle{\cong}$

In particular, $S$ is the Kummer surface associated to the abelian surface
$A$. Note that $\pi$ admits a section if and only if $f$ does.

######  Lemma A.3.

$C$ is a genus $1$ curve $E$ and the quotient $E\longrightarrow\mathbb{P}^{1}$
is the quotient by a hyperelliptic involution.

###### Proof.

Since the quotient of $C$ by $G=\mu_{2}$ is rational, $G$ acts by
multiplication by $(-1)$ on $H^{1,0}(C)$. Similarly, since $q(S)=0$ the local
system $R^{1}(\pi_{0})_{*}\underline{\mathbb{Z}}$ does not contain a non-zero
$G$-invariant local system. It follows that $G$ acts by multiplication by
$(-1)$ on $H^{1,0}(F)$. Accordingly, one computes that

$1=p_{g}(S)=h^{1,0}(C)\cdot h^{1,0}(E)=h^{1,0}(C)=g(C).$

Any involution on a genus $1$ curve is hyperelliptic. ∎

According to Table 1, the Kodaira fiber $\text{I}_{0}^{*}$ is the only
possible singular fiber with local monodromy of order $2$. This means that all
the singular fibers of $\pi$ are of type $\text{I}_{0}^{*}$. Since each
$\text{I}_{0}^{*}$ fiber contributes $6$ to the Euler characteristic of $S$,
there must be precisely four singular fibers. (This method was already used in
[Saw14].)

The four copies of $\mathbb{P}^{1}$ with multiplicity one in a fiber of type
$\text{I}_{0}^{*}$ are $(-2)$-curves. Contracting these sixteen $(-2)$-curves
gives a singular symplectic surface $\bar{S}$ and an isotrivial elliptic
fibration $\bar{\pi}:\bar{S}\longrightarrow\mathbb{P}^{1}$. Each singular
fiber is $\mathbb{P}^{1}$ with multiplicity $2$, and $\bar{S}$ is locally the
quotient of the trivial family $F\times\Delta\longrightarrow\Delta$ by the
involution $(x,t)\mapsto(\iota(x),-t)$, where $\iota$ is a hyperelliptic
involution. We can thus locally and equivariantly compactify $Z\longrightarrow
U$ into a trivial family. Gluing these partial compactifications gives a
smooth surface $\tilde{Z}$ and an isotrivial elliptic fibration
$p:\tilde{Z}\longrightarrow E$ . By the canonical bundle formula,
$\kappa(\tilde{Z})=0$. Finally, $q(\tilde{Z})\neq 0$ implies that $\tilde{Z}$
is an abelian surface. ∎

In the rest of this appendix we assume that the elliptic fibration
$\pi:S\longrightarrow B\cong\mathbb{P}^{1}$ admits a section and describe the
curve $C$ obtained as a smooth compactification of $U$ for all possibilities
of the group $G$.

We will make repeated use of the fact that such a section gives a
trivialization of the elliptic torsor $Z\longrightarrow U$ for which the
$G$-action is diagonal (Lemma 2.4). Indeed, if
$\pi:S\longrightarrow\mathbb{P}^{1}$ admits a section, then
$p:Z\longrightarrow U$ is an isotrivial elliptic fibration with an equivariant
section $\sigma$ and trivial monodromy. Since the moduli space of elliptic
curves with an appropriate level structure is a fine moduli space, there is a
commutative diagram

${Z}$${F\times
U}$${U}$,$\scriptstyle{\cong}$$\scriptstyle{p}$$\scriptstyle{\operatorname{pr}_{2}}$

such that the $G$-action transported to $F\times U$ is diagonal.

A.1. Classification for $G=\mu_{3}$. As shown above, $p:Z\longrightarrow U$ is
isomorphic to $\operatorname{pr}_{2}:E\times U\to U$ with a diagonal
$G$-action on $E\times U$, and the $G$-action on $E$ fixes the origin of $E$.
Recall that $C$ is the unique smooth projective compactification of $U$. The
$G$-action uniquely extends to $C$ and hence to $F\times C$. As a result,
$\pi:S\longrightarrow\mathbb{P}^{1}$ is the relative minimal model of

$(F\times C)/G\longrightarrow C/G=\mathbb{P}^{1}.$

It remains to classify all possible curves $C$ which can arise. Each singular
fibers of $\pi$ has local monodromy of order $3$. By Table 1, all the singular
fibers are of type IV or $\text{IV}^{*}$. Type IV and $\text{IV}^{*}$ fibers
contribute $4$ and $8$ respectively to the Euler characteristic of $S$. Thus
if there are $f_{\text{IV}}$ type IV fibers and $f_{\text{IV}^{*}}$ type
$\text{IV}^{*}$ fibers we must have

$4f_{\text{IV}}+8f_{\text{IV}^{*}}=24.$

Hence there are four possibilities for $(f_{\text{IV}},f_{\text{IV}^{*}})$ as
in Table 2. We note that these examples already appeared in [DM86] and
[Moo18]. The last column refers to the index in Table A1 of [Moo18].

No. | IV | $\text{IV}^{*}$ | $g(C)$ | Ramification degree | No. in [Moo18]
---|---|---|---|---|---
1 | 6 | 0 | 4 | $1^{6}$ | 115
2 | 4 | 1 | 3 | $1^{4},2$ | 84
3 | 2 | 2 | 2 | $1^{2},2^{2}$ | 1
4 | 0 | 3 | 1 | $2^{3}$ | -
Table 2. Classification of possible singular fibers configurations and the
genus of $C$ when $G=\mu_{3}$.

Type IV and $\text{IV}^{*}$ fibers can be described as the central fibers of
the relative minimal models of $(F\times\tilde{\Delta})/\mu_{3}$, where
$\tilde{\Delta}$ is a ramified degree $3$ covering
$\tilde{\Delta}\longrightarrow\Delta$ given by $z^{3}=t$ and $z^{3}=t^{2}$
respectively. From this we obtain a global description of $C$. For example,
let us describe the second item of Table 2. The curve $C$ is the normalization
of the degree $3$ cyclic covering of $\mathbb{P}^{1}$ given by

$z^{3}=(x-a_{1})(x-a_{2})(x-a_{3})(x-a_{4})(x-b_{1})^{2}$

for five distinct points $a_{1},\cdots,a_{4},b_{1}\in\mathbb{P}^{1}$. For such
a curve $C$, we construct $(F\times C)/\mu_{3}\longrightarrow\mathbb{P}^{1}$
as above. Taking the relative minimal model modifies the four singular fibers
over $a_{1},\cdots,a_{4}\in\mathbb{P}^{1}$ through a sequence of blowups and
blowdowns, resulting in fibers of Kodaira type IV. The singular fiber over
$b_{1}\in\mathbb{P}^{1}$ is different; we only need blowups to obtain a fiber
of type $\text{IV}^{*}$. The resulting surface is the minimal elliptic surface
over $\mathbb{P}^{1}$ with four type IV singular fibers and one type
$\text{IV}^{*}$ singular fiber, which is an elliptic K3 surface.

A.2. Classification for $G=\mu_{4}$. In the case $G=\mu_{4}$, the local
monodromy around singular fibers is of order $2$ or $4$. By Table 1, singular
fibers with local monodromy of order $2$ are of type $\text{I}_{0}^{*}$ and
singular fibers with local monodromy of order $4$ are of type III or
$\text{III}^{*}$. If the number of singular fibers of each type is
$f_{\text{III}}$, $f_{\text{I}_{0}^{*}}$, and $f_{\text{III}^{*}}$, then

$3f_{\text{III}}+6f_{\text{I}_{0}^{*}}+9f_{\text{III}^{*}}=24.$

There are $10$ possible triples
$(f_{\text{III}},f_{\text{I}_{0}^{*}},f_{\text{III}^{*}})$ satisfying this
condition, which are listed in Table 3. Note that item 9 gives a curve $C$
which is a disjoint union of two elliptic curves obtained in the $G=\mu_{2}$
case, so it can safely be ignored.

No. | III | $\text{I}_{0}^{*}$ | $\text{III}^{*}$ | $g(C)$ | Ramification degree | No. in [Moo18]
---|---|---|---|---|---|---
1 | 8 | 0 | 0 | 9 | $1^{8}$ | 138
2 | 6 | 1 | 0 | 7 | $1^{6},2$ | 130
3 | 5 | 0 | 1 | 5 | $1^{5},3$ | 116
4 | 4 | 2 | 0 | 5 | $1^{4},2^{2}$ | 117
5 | 3 | 1 | 1 | 5 | $1^{3},2,3$ | 85
6 | 2 | 3 | 0 | 3 | $1^{2},2^{3}$ | 86
7 | 2 | 0 | 2 | 3 | $1^{2},3^{2}$ | 2
8 | 1 | 2 | 1 | 5 | $1,2^{2},3$ | 3
9 | 0 | 4 | 0 | 1 ($\times 2$) | $2^{4}$ | -
10 | 0 | 1 | 2 | 1 | $2,3^{2}$ | -
Table 3. Classification of possible singular fiber configurations and the
genus of $C$ when $G=\mu_{4}$.

A.3. Classification for $G=\mu_{6}$. The order of the local monodromy around
singular fibers can be $2$, $3$, or $6$. By Table 1, type $\text{I}_{0}^{*}$
fibers have local monodromy of order $2$, while type IV and $\text{IV}^{*}$
fibers have local monodromy of order $3$, and type II and $\text{II}^{*}$
fibers have local monodromy of order $6$. Again the Euler characteristic
computation gives the following constraint on the numbers
$f_{\text{II}},f_{\text{IV}},f_{\text{I}_{0}^{*}},f_{\text{IV}^{*}},f_{\text{II}^{*}}$
of fibers of each type:

$2f_{\text{II}}+4f_{\text{IV}}+6f_{\text{I}_{0}^{*}}+8f_{\text{IV}^{*}}+10f_{\text{II}^{*}}=24.$

The classification for all such tuples
$(f_{\text{II}},\cdots,f_{\text{II}^{*}})$ is done in Table 4. There are 47
cases, with the items 38, 39, 42, 45, and 47 corresponding to disjoint unions
of examples which arose for the Galois groups $\mu_{2}$ and $\mu_{3}$. In this
sense, item 43 is the only connected example with $g(C)>1$ that is not covered
in Moonen’s list [Moo18, Table A1].

No. | II | IV | $\text{I}_{0}^{*}$ | $\text{IV}^{*}$ | $\text{II}^{*}$ | $g(C)$ | Ramification degree | No. in [Moo18]
---|---|---|---|---|---|---|---|---
1 | 12 | 0 | 0 | 0 | 0 | 25 | $1^{12}$ | 150
2 | 10 | 1 | 0 | 0 | 0 | 22 | $1^{10},2$ | 149
3 | 9 | 0 | 1 | 0 | 0 | 19 | $1^{9},3$ | 147
4 | 8 | 2 | 0 | 0 | 0 | 19 | $1^{8},2^{2}$ | 148
5 | 8 | 0 | 0 | 1 | 0 | 17 | $1^{8},4$ | 144
6 | 7 | 1 | 1 | 0 | 0 | 16 | $1^{7},2,3$ | 145
7 | 7 | 0 | 0 | 0 | 1 | 15 | $1^{7},5$ | 139
8 | 6 | 3 | 0 | 0 | 0 | 16 | $1^{6},2^{3}$ | 146
9 | 6 | 1 | 0 | 1 | 0 | 14 | $1^{6},2,4$ | 140
10 | 6 | 0 | 2 | 0 | 0 | 13 | $1^{6},3^{2}$ | 142
11 | 5 | 2 | 1 | 0 | 0 | 13 | $1^{5},2^{2},3$ | 141
12 | 5 | 1 | 0 | 0 | 1 | 12 | $1^{5},2,5$ | 131
13 | 5 | 0 | 1 | 1 | 0 | 11 | $1^{5},3,4$ | 132
14 | 4 | 4 | 0 | 0 | 0 | 13 | $1^{4},2^{4}$ | 143
15 | 4 | 2 | 0 | 1 | 0 | 11 | $1^{4},2^{2},4$ | 133
16 | 4 | 1 | 2 | 0 | 0 | 10 | $1^{4},2,3^{2}$ | 134
17 | 4 | 0 | 1 | 0 | 1 | 9 | $1^{4},3,5$ | 118
18 | 4 | 0 | 0 | 2 | 0 | 9 | $1^{4},4^{2}$ | 119
19 | 3 | 3 | 1 | 0 | 0 | 10 | $1^{3},2^{3},3$ | 135
20 | 3 | 2 | 0 | 0 | 1 | 9 | $1^{3},2^{2},5$ | 120
21 | 3 | 1 | 1 | 1 | 0 | 8 | $1^{3},2,3,4$ | 121
22 | 3 | 0 | 3 | 0 | 0 | 7 | $1^{3},3^{3}$ | 123
23 | 3 | 0 | 0 | 1 | 1 | 7 | $1^{3},4,5$ | 88
24 | 2 | 5 | 0 | 0 | 0 | 10 | $1^{2},2^{5}$ | 136
25 | 2 | 3 | 0 | 1 | 0 | 8 | $1^{2},2^{3},4$ | 124
26 | 2 | 2 | 2 | 0 | 0 | 7 | $1^{2},2^{2},3^{2}$ | 125
27 | 2 | 1 | 1 | 0 | 1 | 6 | $1^{2},2,3,5$ | 89
28 | 2 | 1 | 0 | 2 | 0 | 6 | $1^{2},2,4^{2}$ | 90
29 | 2 | 0 | 2 | 1 | 0 | 5 | $1^{2},3^{2},4$ | 92
30 | 2 | 0 | 0 | 0 | 2 | 5 | $1^{2},5^{2}$ | 6
31 | 1 | 4 | 1 | 0 | 0 | 7 | $1,2^{4},3$ | 126
32 | 1 | 3 | 0 | 0 | 1 | 6 | $1,2^{3},5$ | 93
33 | 1 | 2 | 1 | 1 | 0 | 5 | $1,2^{2},3,4$ | 94
34 | 1 | 1 | 3 | 0 | 0 | 4 | $1,2,3^{3}$ | 96
35 | 1 | 1 | 0 | 1 | 1 | 4 | $1,2,4,5$ | 7
36 | 1 | 0 | 1 | 2 | 0 | 3 | $1,3,4^{2}$ | 11
37 | 1 | 0 | 2 | 0 | 1 | 3 | $1,3^{2},5$ | 9
38 | 0 | 6 | 0 | 0 | 0 | 4 ($\times 2$) | $2^{6}$ | 115 ($\times 2$)
39 | 0 | 4 | 0 | 1 | 0 | 3 ($\times 2$) | $2^{4},4$ | 84 ($\times 2$)
40 | 0 | 3 | 2 | 0 | 0 | 4 | $2^{3},3^{2}$ | 97
41 | 0 | 2 | 1 | 0 | 1 | 3 | $2^{2},3,5$ | 12
42 | 0 | 2 | 0 | 2 | 0 | 2 ($\times 2$) | $2^{2},4^{2}$ | 1 ($\times 2$)
43 | 0 | 1 | 0 | 0 | 2 | 2 | $2,5^{2}$ | -
44 | 0 | 1 | 2 | 1 | 0 | 2 | $2,3^{2},4$ | 14
45 | 0 | 0 | 4 | 0 | 0 | 1 ($\times 3$) | $3^{4}$ | -
46 | 0 | 0 | 1 | 1 | 1 | 1 | $3,4,5$ | -
47 | 0 | 0 | 0 | 3 | 0 | 1 ($\times 2$) | $4^{3}$ | -
Table 4. Classification of the possible singular fibers configurations and the
genus of $C$ when $G=\mu_{6}$.

##  Appendix B Background

B.1. Albanese torsors and morphisms. The aim of this section is to clarify the
notions of Albanese varieties and morphisms used in this article.

######  Theorem-Definition B.1.

We can associate to a smooth quasi-projective variety $Y$ a triple
$(\operatorname{Alb}_{Y},\operatorname{Alb}_{Y}^{\operatorname{tor}},\textup{alb}_{Y})$
consisting of:

  1. (1)

an abelian variety $\operatorname{Alb}_{Y}$ called the Albanese variety of
$Y$,

  2. (2)

an $\operatorname{Alb}_{Y}$-torsor
$\operatorname{Alb}_{Y}^{\operatorname{tor}}$ called the Albanese torsor of
$Y$, and

  3. (3)

a morphism
$\textup{alb}_{Y}:Y\longrightarrow\operatorname{Alb}_{Y}^{\operatorname{tor}}$
called the Albanese morphism of $Y$.

These objects satisfy the following universal property: for any abelian
variety $A$, an $A$-torsor $A^{\operatorname{tor}}$, and a morphism
$Y\longrightarrow A^{\operatorname{tor}}$, there exists a unique homomorphism
of abelian varieties $\operatorname{Alb}_{Y}\longrightarrow A$ and an
equivariant morphism
$\operatorname{Alb}_{Y}^{\operatorname{tor}}\longrightarrow
A^{\operatorname{tor}}$ sitting in the commutative diagram

${\operatorname{Alb}_{Y}^{\operatorname{tor}}}$${\operatorname{Alb}_{Y}}$${Y}$${A^{\operatorname{tor}}}$${A.}$

######  Remark B.2.

Typically, in the literature, the Albanese variety of a smooth quasi-
projective variety $Y$ is a semi-abelian variety (e.g., [Wit08]). Strictly
speaking, we work with the Albanese variety of a smooth compactification of
$Y$, which is a birational invariant of smooth quasi-projective varieties

Given a smooth quasi-projective variety $Y$ there is a mixed Hodge structure
on $H^{1}(Y,\mathbb{Q})$ with weights $\geq 1$. We can obtain a free
$\mathbb{Z}$-module by letting

$W_{1}H^{1}(Y,\mathbb{Z}):=H^{1}(Y,\mathbb{Z})\cap W_{1}H^{1}(Y,\mathbb{Q}).$

For any smooth projective compactification $j:Y\hookrightarrow\tilde{Y}$, the
restriction map $j^{*}:H^{1}(\tilde{Y},\mathbb{Z})\longrightarrow
H^{1}(Y,\mathbb{Z})$ is injective with image $W_{1}H^{1}(Y,\mathbb{Z})$.

######  Proposition B.3.

Let $Y$ be a smooth quasi-projective variety. Then there exist isomorphisms of
Hodge structures (induced by the Albanese morphism)

$H^{1}(\operatorname{Alb}_{Y},\mathbb{Z})\cong
H^{1}(\operatorname{Alb}_{Y}^{\operatorname{tor}},\mathbb{Z})\cong
W_{1}H^{1}(Y,\mathbb{Z}).$

Moreover, if $Y$ and $Y^{\prime}$ are birational then their Albanese
varieties, torsors, and morphisms are isomorphic. ∎

######  Example B.4.

Let $C$ be a smooth projective curve. Then its Albanese variety is
$\operatorname{Alb}_{C}=\operatorname{Pic}^{0}_{C}$, its Albanese torsor is
$\operatorname{Alb}_{C}^{\operatorname{tor}}=\operatorname{Pic}^{1}_{C}$, and
its Albanese morphism is

$\displaystyle C$ $\displaystyle\longrightarrow\;\;\operatorname{Pic}^{1}_{C}$
$\displaystyle x$ $\displaystyle\longmapsto\big{[}\mathcal{O}_{C}(x)\big{]}.$

######  Proposition B.5.

Let $X$ and $Y$ be smooth quasi-projective varieties, $A$ an abelian variety
and $f:X\times Y\longrightarrow A$ a morphism. Then there are morphisms
$g:X\longrightarrow A$ and $h:Y\longrightarrow A$ such that
$f(x,y)=g(x)+h(y)$.

###### Proof.

The map $f$ factorizes as $X\times
Y\longrightarrow\operatorname{Alb}_{X}^{\operatorname{tor}}\times\operatorname{Alb}_{Y}^{\operatorname{tor}}\longrightarrow
A$, where the latter map is a morphism between abelian torsors, and therefore
is the sum of the two factors
$\operatorname{Alb}_{X}^{\operatorname{tor}}\longrightarrow A$ and
$\operatorname{Alb}_{Y}^{\operatorname{tor}}\longrightarrow A$. ∎

######  Proposition B.6.

Let $Y$ be a smooth quasi-projective variety with a $G$-action. Then

  1. (1)

There are $G$-actions on both $\operatorname{Alb}_{Y}$ and
$\operatorname{Alb}_{Y}^{\operatorname{tor}}$.

  2. (2)

The Albanese morphism $Y\to\operatorname{Alb}_{Y}^{\operatorname{tor}}$ is
$G$-equivariant.

###### Proof.

The existence of the $G$-action on both $\operatorname{Alb}_{Y}$ and
$\operatorname{Alb}_{Y}^{\operatorname{tor}}$ comes from the universal
property of the Albanese morphism. For any $g\in G$, there exists a unique
commutative diagram

${Y}$${\operatorname{Alb}_{Y}^{\operatorname{tor}}}$${\operatorname{Alb}_{Y}}$${Y}$${\operatorname{Alb}_{Y}^{\operatorname{tor}}}$${\operatorname{Alb}_{Y}.}$$\scriptstyle{g}$$\scriptstyle{g}$$\scriptstyle{g}$

The $G$-actions on
$A^{\operatorname{tor}}=\operatorname{Alb}_{Y}^{\operatorname{tor}}$ and
$A=\operatorname{Alb}_{Y}$ are related by the following morphism

$G\longrightarrow\operatorname{Aut}(A^{\operatorname{tor}})\cong
A\rtimes\operatorname{Aut}_{0}(A)\twoheadrightarrow\operatorname{Aut}_{0}(A).\qed$

######  Remark B.7.

Though $\operatorname{Alb}_{Y}$ and
$\operatorname{Alb}_{Y}^{\operatorname{tor}}$ are isomorphic as varieties,
they may not be isomorphic at $G$-varieties because some elements of $G$ may
act on $\operatorname{Alb}_{Y}^{\operatorname{tor}}$ by translation. For
example, this is the case if $Y$ is an abelian variety and $G$ acts on $Y$ by
translations. The Albanese morphism is then the ($G$-equivariant) identity
map. However, the $G$-action on the Albanese variety $\operatorname{Alb}_{Y}$
is trivial.

B.2. Rational fibrations. A proper morphism $f:Y\longrightarrow Z$ between
irreducible normal varieties is called a fibration if it is surjective and has
connected fibers. This is equivalent to the equality of coherent sheaves
$\mathcal{O}_{Z}=f_{*}\mathcal{O}_{Y}$. One can generalize this notion to
rational maps as follows.

######  Definition B.8 ([Laz04, Ex 2.1.12]).

A rational map $f:Y\dashrightarrow Z$ between irreducible normal varieties is
called a _fibration_ (or _rational fibration_ to emphasize the fact that $f$
is a rational map) if it is dominant and the subfield $f^{*}K(Z)\subset K(Y)$
is algebraically closed.

######  Example B.9.

If $f:Y\longrightarrow Z$ is a proper morphism, then this definition coincides
with the usual one by [Laz04, Ex 2.1.12].

######  Example B.10.

If a fibration $f:Y\longrightarrow Z$ is not proper, then not all fibers of
$f$ need to be connected. For example, given a proper fibration
$\bar{f}:\bar{Y}\longrightarrow Z$ with a reducible fiber $F=F_{1}\cup F_{2}$
over $z\in Z$, consider the composition of the inclusion $Y=\bar{Y}\setminus
F_{1}\cap F_{2}\subset\bar{F}$ with $\bar{f}$.

##  Appendix C Equivariant versions of theorems of Ueno and Kawamata

This section is devoted to generalizing results of Ueno [Uen75, Thm 10.9]
[Mor87, Thm 3.7] and Kawamata [Kaw81, Thm 13] to the equivariant setting.

######  Theorem C.1 (Equivariant version of Ueno’s theorem).

Let $G$ be a finite group and $X\subset A^{\operatorname{tor}}$ a closed
irreducible $G$-subvariety of a $G$-abelian torsor $A^{\operatorname{tor}}$.
Then there exists a $G$-abelian subvariety $B\subset A$, a closed
$G$-subvariety $Y\subset A^{\operatorname{tor}}/B$, and a $G$-equivariant
commutative diagram

${X}$${A^{\operatorname{tor}}}$${Y}$${A^{\operatorname{tor}}/B}$

with the following properties:

  1. (1)

The diagram is cartesian, i.e., $X\longrightarrow Y$ is a $B$-torsor.

  2. (2)

$X\longrightarrow Y$ is the Iitaka fibration of (the normalization of) $X$.

###### Proof.

Ueno’s theorem is a characterization of the Iitaka fibration of subvarieties
of abelian varieties. We show that the proof works $G$-equivariantly. Ueno
defines the abelian subvariety $B\subset A$ as the identity component of

$B^{\prime}=\\{a\in A:a+X\subset X\\},$

and sets $Y:=X/B$. It thus suffices to show that $B$ is a $G$-abelian
subvariety of $A$. This follows from the fact that $B^{\prime}\subset A$ is an
algebraic subgroup closed under the $G$-action, and that the $G$-action fixes
$0\in A$, thus taking the identity component of $B^{\prime}$ to itself. ∎

######  Theorem C.2 (Equivariant version of Kawamata’s theorem).

Consider a finite group $G$ and a finite $G$-equivariant morphism
$f:X\longrightarrow A^{\operatorname{tor}}$ from a normal projective
irreducible $G$-variety $X$ to a $G$-abelian torsor $A^{\operatorname{tor}}$.
Then there exists a $G$-abelian subvariety $B\subset A$, a finite
$G$-equivariant morphism $h:Y\longrightarrow A^{\operatorname{tor}}/B$ from a
normal $G$-variety $Y$, and a $\,G$-equivariant commutative diagram

${X}$${A^{\operatorname{tor}}}$${Y}$${A^{\operatorname{tor}}/B}$$\scriptstyle{f}$$\scriptstyle{h}$

with the following properties.

  1. (1)

$X\longrightarrow Y$ is generically $\tilde{B}$-isotrivial for an abelian
variety $\tilde{B}$ isogenous to $B$.

  2. (2)

$X\longrightarrow Y$ is the Iitaka fibration of $X$.

###### Proof.

The proof is essentially the same as Kawamata’s original proof from [Kaw81,
Thm 13]. We provide details for the sake of completeness. Start with an Iitaka
fibration $\phi:X\dashrightarrow Y^{\prime}$ of $X$. Since the Iitaka
fibration is defined up to birational equivalence, we may assume that
$Y^{\prime}$ is smooth projective. By realizing the Iitaka fibration as a
stable image of the pluricanonical maps, $Y^{\prime}$ admits a rational
$G$-action and $\phi$ is $G$-equivariant.

Let $F_{y}=\overline{\phi^{-1}(y)}$ be the Zariski closure of a very general
fiber of $\phi$. It is an irreducible subvariery of $X$ with
$\kappa(F_{y})=0$. The image $f(F_{y})\subset A^{\operatorname{tor}}$ has
Kodaira dimension $0$ since it is a subvariety of $A^{\operatorname{tor}}$ and
is dominated by $F_{y}$. By Ueno’s theorem, it is isomorphic to an abelian
torsor $B_{y}^{\operatorname{tor}}\subset A^{\operatorname{tor}}$ for some
abelian subvariety $B_{y}\subset A$. Since the set of abelian subvarieties of
$A$ is discrete, $B:=B_{y}$ does not depend on the choice of a very general
point $y\in Y^{\prime}$.

We claim that the abelian subvariety $B\subset A$ is a $G$-abelian subvariety.
For $y\in Y^{\prime}$ very general, the automorphism $g:X\longrightarrow X$
induces an isomorphism $g:B_{y}^{\operatorname{tor}}\to B_{g\cdot
y}^{\operatorname{tor}}$. Since $B_{y}^{\operatorname{tor}},B_{g\cdot
y}^{\operatorname{tor}}\subset A^{\operatorname{tor}}$ are translations of one
another, they are canonically isomorphic up to translation. Ignoring the
translation part, we obtain an isomorphism $g:B\longrightarrow B$. This
realizes $B$ as a $G$-abelian subvariety of $A$.

Consider the quotient $p:A^{\operatorname{tor}}\longrightarrow
A^{\operatorname{tor}}/B$. A very general fiber of $\phi$ is sent to a point
via

$p\circ f:X\longrightarrow A^{\operatorname{tor}}\longrightarrow
A^{\operatorname{tor}}/B.$

By the rigidity lemma [Kaw81, Lem 14], this induces an equivariant rational
map $h^{\prime}:Y^{\prime}\dashrightarrow A^{\operatorname{tor}}/B$ making the
following diagram commutative:

${X}$${A^{\operatorname{tor}}}$${Y^{\prime}}$${A^{\operatorname{tor}}/B.}$$\scriptstyle{f}$$\scriptstyle{\phi}$$\scriptstyle{h^{\prime}}$

As $Y^{\prime}$ is smooth and $A^{\operatorname{tor}}/B$ is an abelian torsor,
the rational map $h^{\prime}$ is defined everywhere. Denote by $\bar{X}$ and
$\bar{Y}$ the image of $f$ and $h^{\prime}$:

${X}$${\bar{X}}$${A^{\operatorname{tor}}}$${Y^{\prime}}$${\bar{Y}}$${A^{\operatorname{tor}}/B.}$$\scriptstyle{f}$$\scriptstyle{\phi}$$\scriptstyle{p}$$\scriptstyle{h^{\prime}}$

By construction $\bar{X}\longrightarrow\bar{Y}$ is an equivariant and
surjective morphism. We claim that it is in fact a $B$-torsor, i.e., that
$\bar{X}=p^{-1}(\bar{Y})$. Since both $\bar{X}$ and $p^{-1}(\bar{Y})$ are
projective and irreducible, the claim is equivalent to the dimension count
$\dim\bar{X}=\dim\bar{Y}+\dim B$. This follows from a sequence of inequalities

$\dim\bar{Y}+\dim B\leq\dim Y^{\prime}+\dim B=\dim
X=\dim\bar{X}\leq\dim\bar{Y}+\dim B.$

This shows that $\bar{X}\to\bar{Y}$ is a $B$-torsor and that $\dim\bar{Y}=\dim
Y^{\prime}=\kappa(X)$.

Finally, consider the Stein factorization $X\longrightarrow Y$ of the
composition $X\longrightarrow A^{\operatorname{tor}}\longrightarrow
A^{\operatorname{tor}}/B$:

${X}$${\bar{X}}$${A^{\operatorname{tor}}}$${Y}$${\bar{Y}}$${A^{\operatorname{tor}}/B.}$$\scriptstyle{f}$$\scriptstyle{p}$$\scriptstyle{h}$

Since $\bar{X}\longrightarrow\bar{Y}$ is a $B$-torsor, the Stein factorization
$X\longrightarrow Y$ is generically isotrivial with general fiber $\tilde{B}$,
a finite étale cover of $B$, hence an abelian variety. It follows that a very
general fiber of $X\longrightarrow Y$ has Kodaira dimension $0$ and $\dim
Y=\kappa(X)$. These two properties characterize the Iitaka fibration (see
[Mor87, Def-Thm 1.11]), so $X\longrightarrow Y$ is again an Iitaka fibration
of $X$. The diagram is $G$-equivariant. ∎

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