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# A Hodge-theoretic proof of Hwang’s theorem on base manifolds of Lagrangian
fibrations

Benjamin Bakker  B. Bakker: Dept. of Mathematics, Statistics, and Computer
Science, University of Illinois at Chicago, Chicago, USA.
<EMAIL_ADDRESS>and  Christian Schnell  Ch. Schnell: Department of
Mathematics, Stony Brook University, Stony Brook, NY, USA.
<EMAIL_ADDRESS>
###### Abstract.

We give a Hodge-theoretic proof of Hwang’s theorem, which says that if the
base of a Lagrangian fibration of an irreducible holomorphic symplectic
manifold is smooth, it must be projective space.

##  1\. Introduction

Let $X$ be an irreducible holomorphic symplectic manifold of dimension $2n$,
and let $f:X\to B$ be a Lagrangian fibration over a base of dimension $n$. In
this note, we reprove the following well-known result of Hwang [Hwa08].

######  Theorem 1.1 (Hwang).

Let $f:X\to B$ be a Lagrangian fibration of an irreducible holomorphic
symplectic manifold of dimension $2n$. If $B$ is smooth, then
$B\cong\mathbb{P}^{n}$.

The proof that we present below is inspired by Hwang’s original argument, but
perhaps clarified by the consistent use of Hodge theory, especially the theory
of variations of Hodge structure. There are two other important inputs. The
first is the following enhancement of Mori’s characterization of projective
space, due to Cho, Miyaoka, and Shepherd-Barron [CMSB02] and Kebekus [Keb02].

######  Theorem 1.2 (Cho–Miyaoka–Shepherd-Barron, Kebekus).

Let $Y$ be a uniruled smooth projective variety and suppose there is a dense
Zariski open subset $Y^{\circ}\subseteq Y$ such that every nonconstant
rational curve $g\colon C\to Y$ meeting $Y^{\circ}$ satisfies $-\deg
g^{*}\omega_{Y}\geq\dim Y+1$. Then $Y$ is isomorphic to projective space.

The second input is a result of Voisin [Voi18] about the variation of Hodge
structure on the first cohomology of the fibers of a Lagrangian fibration. Let
$B^{\circ}\subseteq B$ be the open set over which the Lagrangian fibration $f$
is smooth, and $f^{\circ}:X^{\circ}\to B^{\circ}$ the base change of $f$ to
$B^{\circ}$. It is known that the discriminant locus $D=B\setminus B^{\circ}$
is a nonempty divisor. We denote by
$V=R^{1}f^{\circ}_{*}\mathbb{Q}_{X^{\circ}}$ the polarized variation of Hodge
structure of weight $1$ on $B^{\circ}$; by $\mathcal{V}$ the underlying flat
vector bundle; by $\nabla$ the Gauss-Manin connection on $\mathcal{V}$; and by
$F^{1}\mathcal{V}=f^{\circ}_{*}\Omega_{X^{\circ}/B^{\circ}}^{1}$ the Hodge
bundle of type $(1,0)$.

######  Theorem 1.3 (Voisin).

$V\otimes\mathbb{R}$ is irreducible as a real variation of Hodge structure.

The anticanonical degree of a curve (as well as its deformation theory) is
linked to the variation $V$ since contraction with the symplectic form yields
a natural isomorphism $\mathscr{T}_{B^{\circ}}\cong F^{1}\mathcal{V}$. Work of
Matsushita [Mat05] extends this to an isomorphism between $\mathscr{T}_{B}$
and the “canonical extension” of $F^{1}\mathcal{V}$ in codimension one, and
this will be an important ingredient in our approach.

It is known by [Mat99] that $B$ is a Fano manifold (and therefore uniruled).
The strategy is to assume the condition in 1.2 is not satisfied, so that there
is a low degree rational curve meeting $B^{\circ}$, and conclude from this
that: (1) the curve deforms to cover $B$; and (2) the variation $V$ must have
a nontrivial fixed part when restricted to $C$. We then obtain a nontrivial
splitting of $V$ on the universal family of curves and thus on a finite cover
of $B$. To finish, we show that this splitting descends to $B^{\circ}$, where
it of course contradicts 1.3. In our approach, the first step is proved using
a Hodge module version of Matsushita’s result combined with a straightforward
functoriality property of Hodge modules to show that $g^{*}\mathscr{T}_{B}$ is
semi-positive (see 2.2) and that its maximal trivial quotient comes from the
fixed part of the variation (see 2.4). As the low degree curve which is
assumed to exist may a priori not deform, it can intersect the discriminant
divisor in its singular locus, and the Hodge module perspective provides a
version of Matsushita’s theorem that works in higher codimension. The descent
step is straightforward in the case that $f$ is non-isotrivial (using the
result of [Bak22], see Remark 6.3), but we do not actually use this. Instead,
we prove the descent in general using a rather tricky argument which is a
reinterpretation of an argument from Hwang’s paper using Hodge theory to
simplify the steps. Note that the descent step must in some sense be subtle—in
fact, it is the most difficult part of our proof—as it is easy to construct
examples where $V$ splits off a factor on a finite cover that is constant on a
covering family of rational curves (see Remark 7.1).

######  Remark 1.4.

The recent preprint by Li and Tosatti [LT23] contains another proof of Hwang’s
theorem that follows the same overall strategy, but has a more analytic
flavor. Instead of Hodge theory, Li and Tosatti rely on the theory of special
Kähler manifolds [Fre99]. They use the curvature of the special Kähler metric
on $B^{\circ}$ to show that the restriction of $\mathscr{T}_{B}$ to a rational
curve of minimal degree is semipositive, and that any trivial quotient of
$g^{*}\mathscr{T}_{B}$ actually comes from a subbundle that is flat with
respect to the Chern connection of the metric. For the descent step at the end
of the proof, however, they depend on one of the results from our paper.

### Outline

In Section 2 we prove the results on the semipostivity of
$g^{*}\mathscr{T}_{B}$ and the relation between the maximal trivial quotient
and the fixed part of the variation. We give two proofs: the first in the
projective case is more geometric, using results of Kollár; the second as a
consequence of the functoriality of Hodge module extensions. In Section 3 we
make some general remarks relating subvariations to foliations on $B$. In
Section 4 we use the existence of a low degree curve to construct a nontrivial
splitting of the variation on a finite cover. In Section 5, we study the
geometry of the subvarieties swept out by connected components of the family
of rational curves through a given point. (For the sake of exposition, we
reprove all the necessary results, which are due to Araujo [Ara06] and Hwang
[Hwa07], even though this makes the paper longer.) In Section 6, we describe
the main ingredient for descending the splitting to $B$, using ideas from
[Hwa08]. Finally, we prove 1.1 in Section 7.

### Notation and terminology

By a _rational curve_ $g:C\to B$ we always mean a nonconstant morphism from
$C\cong\mathbb{P}^{1}$. If the image meets the open subset $B^{\circ}$, we
denote by $g^{\circ}:C^{\circ}:=g^{-1}(B^{\circ})\to B^{\circ}$ the
restriction of $g$, and in general the superscript “$\circ$” always stands for
objects that are related to $B^{\circ}$. We denote by
$V_{C}:=(g^{\circ})^{*}V$ the pullback of the variation, and likewise for any
pullback. We typically use Roman letters $V,W$ to denote local systems and
script letters $\mathcal{V},\mathcal{W}$ to denote the associated flat vector
bundles.

##  2\. Restriction of the tangent bundle to rational curves

In this section, we use results from Hodge theory to analyze the splitting
behavior of the tangent bundle $\mathscr{T}_{B}$ on rational curves in $B$
that meet the smooth locus $B^{\circ}$ of the Lagrangian fibration $f\colon
X\to B$. Here it will be more convenient to work with the variation of Hodge
structure on $R^{2n-1}f_{*}^{\circ}\mathbb{Q}_{X^{\circ}}$ instead of the one
on $R^{1}f_{*}^{\circ}\mathbb{Q}_{X^{\circ}}$. This makes no essential
difference, because the Hard Lefschetz theorem says that, after tensoring by
$\mathbb{R}$, the two variations are isomorphic up to a Tate twist by
$\mathbb{R}(-n+1)$; the isomorphism is induced by cup product with the
$(n-1)$-th power of a Kähler form on $X$. We will therefore change the
notation slightly, and—in this section only—use the symbol $V$ for the
variation of Hodge structure on $R^{2n-1}f_{*}^{\circ}\mathbb{Q}_{X^{\circ}}$.
The relevant Hodge bundle is then
$F^{n}\mathcal{V}=R^{n-1}f_{*}^{\circ}\omega_{X^{\circ}/B^{\circ}}$; it is of
course isomorphic to $f_{*}^{\circ}\Omega_{X^{\circ}/B^{\circ}}^{1}$.

Our starting point is the following result by Matsushita [Mat05], which
describes the tangent bundle of $B$ in terms of Hodge theory.

######  Theorem 2.1.

There is an isomorphism $R^{n-1}f_{*}\omega_{X/B}\cong\mathscr{T}_{B}$, whose
restriction to $B^{\circ}$ is the natural isomorphism induced by the
holomorphic symplectic form.

###### Proof.

Matsushita proves that the natural isomorphism on $B^{\circ}$ extends to
$R^{n-1}f_{*}\omega_{X}\cong\Omega_{B}^{n-1}\cong\omega_{B}\otimes\mathscr{T}_{B}$.
Tensoring both sides by $\omega_{B}^{-1}$ then gives the desired result.
Matsushita only states the theorem for projective $X$, but his proof carries
over to the case where $X$ is a compact Kähler manifold. For an alternative
proof, see [Sch23, §4]. ∎

Let $g\colon C\to B$ be a nonconstant morphism from $C\cong\mathbb{P}^{1}$
whose image intersects $B^{\circ}$. Like any vector bundle on
$\mathbb{P}^{1}$, the pullback $g^{*}\mathscr{T}_{B}$ decomposes as

$g^{*}\mathscr{T}_{B}\cong\mathscr{O}(a_{1})\oplus\dotsb\oplus\mathscr{O}(a_{n})$
(1)

for certain integers $a_{1},\dotsc,a_{n}$. Our first result is that
$g^{*}\mathscr{T}_{B}$ is semi-positive.

######  Proposition 2.2.

The pullback $g^{*}\mathscr{T}_{B}$ is semi-positive, in the sense that
$a_{1},\dotsc,a_{n}\geq 0$.

We will first present a geometric proof in the case when $X$ is projective;
afterwards, we will give a Hodge-theoretic proof that also works when $X$ is a
compact Kähler manifold. Let us denote by $h\colon Y\to C$ the base change of
the Lagrangian fibration, and let $r\colon\tilde{Y}\to Y$ be a resolution of
singularities of $Y$ that is an isomorphism over the preimage of
$C^{\circ}=g^{-1}(B^{\circ})$. For dimension reasons, the composition
$\tilde{h}=r\circ h\colon\tilde{Y}\to C$ is still flat of relative dimension
$n$. We get a commutative diagram

${\tilde{Y}}$${Y}$${X}$${C}$${B}$$\scriptstyle{r}$$\scriptstyle{\tilde{h}}$$\scriptstyle{h}$$\scriptstyle{f}$$\scriptstyle{g}$

in which the square is Cartesian. The key step in the proof is the following
lemma.

######  Lemma 2.3.

There is a morphism of sheaves

$R^{n-1}\tilde{h}_{*}\omega_{\tilde{Y}/C}\to g^{*}\mathscr{T}_{B}$

that restricts to the natural isomorphism over the open subset
$C^{\circ}=g^{-1}(B^{\circ})$.

###### Proof.

Before giving the proof, we need to review a few facts about relative
dualizing sheaves and base change. Here we probably have to assume that $X$ is
projective (although it is known by [Cam21] that Lagrangian fibrations are
projective locally in the analytic topology on the base). In this algebraic
setting, a good reference for duality theory is [Sta18, Tag ODWE]. To begin
with, $B$ is smooth and $f$ is flat, and so all fibers of $f$ are local
complete intersections in $X$ and therefore Gorenstein. Since $f$ is proper
and flat, it follows that the relative dualizing sheaf $\omega_{X/B}$ is
isomorphic to $\omega_{X}\otimes f^{*}\omega_{B}^{-1}$, and that the relative
dualizing complex is $\omega_{X/B}[n]$. Moreover, in this situation, the
relative dualizing sheaf commutes with arbitrary base change.

Now we begin the proof. Flatness of $f$ and the base change property for the
relative dualizing sheaf give us an isomorphism

$g^{*}\mathbf{R}f_{*}\omega_{X/B}\cong\mathbf{R}h_{*}\omega_{Y/C}.$

Since all the higher direct images $R^{i}f_{*}\omega_{X/B}$ are locally free
(by Matsushita’s theorem), we get in particular that

$g^{*}\mathscr{T}_{B}\cong g^{*}R^{n-1}f_{*}\omega_{X/B}\cong
R^{n-1}h_{*}\omega_{Y/C}.$ (2)

Consider now the resolution of singularities $r\colon\tilde{Y}\to Y$. The
trace map gives us a morphism $r_{*}\omega_{\tilde{Y}}\to\omega_{Y}$ that is
an isomorphism over the smooth locus of $Y$ and therefore injective. The
induced morphism

$R^{n-1}\tilde{h}_{*}\omega_{\tilde{Y}/C}\to R^{n-1}h_{*}\omega_{Y/C}$ (3)

is of course still an isomorphism over $C^{\circ}$. We then get the desired
morphism by composing (3) and (2); it is clearly the natural isomorphism over
$C^{\circ}$. ∎

We can now prove that $g^{*}\mathscr{T}_{B}$ is always semi-positive.

###### Proof of 2.2.

From 2.3, we get a short exact sequence

$0\to R^{n-1}\tilde{h}_{*}\omega_{\tilde{Y}/C}\to
g^{*}\mathscr{T}_{B}\to\mathscr{Q}\to 0,$

in which $\mathscr{Q}$ is supported on the finite set $C\setminus C^{\circ}$
and therefore a torsion sheaf. This reduces the problem to the semi-positivity
of $R^{n-1}\tilde{h}_{*}\omega_{\tilde{Y}/C}$. Since
$\tilde{h}\colon\tilde{Y}\to C$ is a Kähler fiber space over a curve, this is
the content of a classical result by Fujita [Fuj78]. Alternatively, one can
argue using Kollár’s results [Kol86a] about higher direct images of dualizing
sheaves. Since $C\cong\mathbb{P}^{1}$, we have

$R^{i}h_{*}\omega_{\tilde{Y}/C}\otimes\mathscr{O}(-1)\cong
R^{i}h_{*}\omega_{\tilde{Y}}\otimes\mathscr{O}(1),$

and this bundle has no higher cohomology by Kollár’s vanishing theorem (which
applies to any morphism from a compact Kähler manifold to a projective
variety). But on $\mathbb{P}^{1}$, this is only possible if
$R^{i}h_{*}\omega_{\tilde{Y}/C}$ is semi-positive. ∎

We also need to understand the trivial part in the decomposition (1), at least
in the case of generic rational curves. Let us therefore assume from now on
that $g\colon C\to B$ is an immersed rational curve that intersects the
discriminant divisor $D=B\setminus B^{\circ}$ transversely. The preimage
$Y=X\times_{C}B$ is then already smooth, and (2) gives us an isomorphism

$g^{\ast}\mathscr{T}_{B}\cong R^{n-1}h_{*}\omega_{Y/C}.$

Suppose that in the decomposition (1), we have $a_{1},\dotsc,a_{r}=0$ and
$a_{r+1},\dotsc,a_{n}>0$. In that case, the bundle $g^{\ast}\mathscr{T}_{B}$
has a canonical trivial quotient

$g^{\ast}\mathscr{T}_{B}\to\mathscr{O}_{C}^{\oplus r}.$

On the open subset $C^{\circ}=g^{-1}(B^{\circ})$, we have a polarized
variation of Hodge structure of weight $2n-1$, whose underlying local system
is $R^{2n-1}h^{\circ}_{*}\mathbb{Q}_{Y^{\circ}}$; here $h^{\circ}\colon
Y^{\circ}\to C^{\circ}$ is the restriction of $h$. Our next result identifies
the maximal trivial quotient of $g^{\ast}\mathscr{T}_{B}$ as coming from the
fixed part of the variation of Hodge structure on
$R^{2n-1}h^{\circ}_{*}\mathbb{Q}_{Y^{\circ}}$.

######  Proposition 2.4.

The fixed part of the variation of Hodge structure on
$R^{2n-1}h^{\circ}_{*}\mathbb{Q}_{Y^{\circ}}$ has rank exactly $2r$, and the
Hodge bundle of type $(n,n-1)$ of the fixed part projects isomorphically to
the maximal trivial quotient of $g^{\ast}\mathscr{T}_{B}$.

###### Proof.

To simplify the notation, let us set
$V_{C}=R^{2n-1}h^{\circ}_{*}\mathbb{Q}_{Y^{\circ}}$, and let us denote by
$\mathcal{V}_{C}$ the underlying flat vector bundle, and by
$F^{n}\mathcal{V}_{C}$ the Hodge bundle of type $(n,n-1)$. Since $h$ has
relative dimension $n$, the restriction of $R^{n-1}h_{*}\omega_{Y/C}$ to the
open subset $C^{\circ}$ is canonically isomorphic to $F^{n}\mathcal{V}_{C}$.
In fact, more is true. Let $\tilde{\mathcal{V}}_{C}^{>-1}$ denote Deligne’s
canonical extension of the flat vector bundle $\mathcal{V}_{C}$; it is
uniquely characterized by the condition that the connection on
$\mathcal{V}_{C}$ has logarithmic poles on $\tilde{\mathcal{V}}_{C}^{>-1}$ and
that the residues of the connection at each point of $C\setminus C^{\circ}$
have eigenvalues contained in the interval $(-1,0]$. With this notation,
Kollár [Kol86a] proves that

$R^{n-1}h_{*}\omega_{Y/C}\cong\tilde{\mathcal{V}}_{C}^{>-1}\cap
j_{*}(F^{n}\mathcal{V}_{C}),$ (4)

where $j\colon C^{\circ}\to C$ is the inclusion.

The fixed part of $V_{C}$ is a $\mathbb{Q}$-Hodge structure of type
$(n,n-1)+(n-1,n)$, and so its dimension is an even number, say $2s$. The fixed
part then contributes a trivial summand of rank $2s$ to the canonical
extension $\tilde{\mathcal{V}}_{C}^{>-1}$, and, due to (4), a trivial summand
of rank $s$ to the bundle $R^{n-1}h_{*}\omega_{Y/C}$. This gives $r\geq s$. To
prove that $r=s$, we need to argue that any nontrivial morphism of the form

$R^{n-1}h_{*}\omega_{Y/C}\to\mathscr{O}_{C}$ (5)

must come from the fixed part of $V_{C}$. This is a straightforward
computation with duality. First, we observe that since $C$ is a curve, the
derived pushforward of the line bundle $\omega_{Y/C}$ decomposes as

$\mathbf{R}h_{*}\omega_{Y/C}\cong\bigoplus_{i=0}^{n}R^{i}h_{*}\omega_{Y/C}[-i]$

in the derived category of coherent sheaves on $C$. (This kind of result is
due to Kollár [Kol86b] in general, but only needs elementary homological
algebra in the case of curves.) Consequently, $R^{n-1}h_{*}\omega_{Y/C}$ is a
direct summand of $\mathbf{R}h_{*}\omega_{Y/C}[n-1]$.

Grothendieck duality for the morphism $h\colon Y\to C$ yields

$\operatorname{Hom}_{C}\Bigl{(}\mathbf{R}h_{*}\omega_{Y/C}[n-1],\mathscr{O}_{C}\Bigr{)}\cong\operatorname{Hom}_{Y}\Bigl{(}\omega_{Y/C}[n-1],\omega_{Y/C}[n]\Bigr{)}\cong
H^{1}(Y,\mathscr{O}_{Y}).$

After substituting in the above formula for the direct image and remembering
that $H^{1}(C,\mathscr{O}_{C})=0$ (because $C\cong\mathbb{P}^{1}$), we get an
isomorphism

$\operatorname{Hom}_{C}\Bigl{(}R^{n-1}h_{*}\omega_{Y/C},\mathscr{O}_{C}\Bigr{)}\cong
H^{1}(Y,\mathscr{O}_{Y}).$

Now $Y$ is a compact Kähler manifold, and so by classical Hodge theory, the
morphism $H^{1}(Y,\mathbb{C})\to H^{1}(Y,\mathscr{O}_{Y})$ is surjective.
After some compatibility checking, it follows that any morphism as in (5)
comes about in the following way: at any point $c\in C^{\circ}$, the
restriction of (5) is the linear functional $H^{n,n-1}(Y_{c})\to\mathbb{C}$
obtained by integration against the restriction of a fixed class in
$H^{1}(Y,\mathbb{C})$ to the fiber $Y_{c}=h^{-1}(c)$. Since restriction to the
smooth fibers of $h\colon Y\to C$ determines a morphism of variations of Hodge
structure

$H^{1}(Y,\mathbb{Q})\otimes R^{2n-1}h^{\circ}_{*}\mathbb{Q}_{Y^{\circ}}\to
R^{2n}h^{\circ}_{*}\mathbb{Q}_{Y^{\circ}}\cong\mathbb{Q}_{C}(-n),$

we conclude (by the semi-simplicity of polarized variations of Hodge
structure) that the fixed part of
$R^{2n-1}h^{\circ}_{*}\mathbb{Q}_{Y^{\circ}}$ must have rank at least $2r$. ∎

######  Remark 2.5.

It is possible that the construction with relative dualizing sheaves and base
change still works in the analytic setting, but we have not been able to find
a reference for the necessary technical results.

In the remainder of this chapter, we explain how one can modify the argument
above to make it work when $X$ is just a compact Kähler manifold. The proof
relies on a few basic properties of Hodge modules [Sai88, Sai90]. Let $M$
denote the polarized Hodge module of weight $\dim B+(2n-1)$ with strict
support $B$, associated to the variation of Hodge structure
$V=R^{2n-1}f_{*}^{\circ}\mathbb{Q}_{X^{\circ}}$ on $B^{\circ}$ by [Sai90, Thm.
3.21]. Let $\mathcal{M}$ denote the (regular holonomic) left
$\mathscr{D}_{B}$-module underlying $M$, and let $F_{\bullet}\mathcal{M}$ be
its Hodge filtration. By Saito’s direct image theorem [Sai88, Thm. 5.3.1] (and
[Sai91] for the constant Hodge module on a compact Kähler manifold), we have
$F_{-n-1}\mathcal{M}=0$, and

$\omega_{B}\otimes F_{-n}\mathcal{M}\cong R^{n-1}f_{*}\omega_{X}.$

Together with 2.1, this implies that $F_{-n}\mathcal{M}\cong\mathscr{T}_{B}$,
and so we can also think of the tangent sheaf as being part of a Hodge module
on $B$. We may sometimes refer to the coherent $\mathscr{O}_{B}$-module
$F_{-n}\mathcal{M}$ as the “Hodge module extension” of $F^{n}\mathcal{V}$.

There is an alternative description of the coherent sheaf $F_{-n}\mathcal{M}$
using resolution of singularities. Let $\mu\colon\tilde{B}\to B$ be a log
resolution of the discriminant divisor $D=B\setminus B^{\circ}$ that is an
isomorphism over the open subset over which $D$ is a divisor with normal
crossings, hence in particular over $B^{\circ}$. Let $\mathcal{V}$ denote the
flat vector bundle underlying the variation of Hodge structure
$V=R^{2n-1}f_{*}^{\circ}\mathbb{Q}_{X^{\circ}}$, and let
$F^{n}\mathcal{V}=R^{n}f_{*}^{\circ}\omega_{X^{\circ}/B^{\circ}}$ be the Hodge
bundle of type $(n,n-1)$. As before, let $\tilde{\mathcal{V}}^{>-1}$ denote
Deligne’s canonical extension of the flat vector bundle $\mathcal{V}$; this is
the unique extension of $\mathcal{V}$ to a vector bundle on $\tilde{B}$ on
which the connection has logarithmic poles with residues having eigenvalues in
the interval $(-1,0]$. By Schmid’s nilpotent orbit theorem, the Hodge bundle
$F^{n}\mathcal{V}$ extends to a subbundle
$F^{n}\tilde{\mathcal{V}}^{>-1}=\tilde{\mathcal{V}}^{>-1}\cap
j_{*}(F^{n}\mathcal{V})$, where $j\colon B^{\circ}\to\tilde{B}$ is the
inclusion. According to [Sai91, Thm. 2.4 and (2.2.2)], we then have

$\omega_{B}\otimes
F_{-n}\mathcal{M}\cong\mathbf{R}\mu_{*}\Bigl{(}\omega_{\tilde{B}}\otimes
F^{n}\tilde{\mathcal{V}}^{>-1}\Bigr{)}.$ (6)

This morphism is an isomorphism over the open subset where $D=B\setminus
B^{\circ}$ is a divisor with normal crossings.

We can now present the alternative proof of 2.2. Let $g\colon C\to B$ be a
rational curve meeting the smooth locus $B^{\circ}$ of the Lagrangian
fibration. Let $V_{C}$ be the variation of Hodge structure obtained by pulling
back $V=R^{2n-1}f^{\circ}_{\ast}\mathbb{Q}_{X^{\circ}}$ to the open subset
$C^{\circ}=g^{-1}(B^{\circ})$, and let $M_{C}$ be its extension to a polarized
Hodge module of weight $\dim C+(2n-1)$ with strict support $C$. Let
$(\mathcal{M}_{C},F_{\bullet}\mathcal{M}_{C})$ be the underlying filtered left
$\mathscr{D}_{C}$-module. If we again write $\tilde{\mathcal{V}}_{C}^{>-1}$
for Deligne’s canonical extension, then for the same reason as above, we have

$F_{-n}\mathcal{M}_{C}\cong F^{n}\tilde{\mathcal{V}}_{C}^{>-1},$ (7)

Since $V_{C}$ is a polarized variation of Hodge structure with quasi-unipotent
local monodromy on a curve, this vector bundle is known to be semi-positive
[Pet84].111Peters claims to be working with the canonical extension for the
interval $[0,1)$, but his signs in the formula for the Hodge norm are wrong,
and so he should really be using the interval $(-1,0]$. Alternatively, using
the notation from the proof of 2.2, we can observe that
$F^{n}\tilde{\mathcal{V}}_{C}^{>-1}\cong R^{n-1}h_{*}\omega_{\tilde{Y}/C}$,
which is semi-positive for the reason explained above. Either way, to relate
this semi-positivity to that of $g^{*}\mathscr{T}_{B}$, we need the following
variant of our earlier 2.3.

######  Lemma 2.6.

There is a morphism of sheaves $F_{-n}\mathcal{M}_{C}\to
g^{*}\bigl{(}F_{-n}\mathcal{M}\bigr{)}$ that restricts to the natural
isomorphism over $C^{\circ}=g^{-1}(B^{\circ})$.

###### Proof.

This kind of morphism is constructed in [Sch12, §13-14] in much greater
generality. We present here a more elementary argument that reduces the input
from the theory of Hodge modules to a minimum. Recall from (6) that we have an
isomorphism

$\mathbf{R}\mu_{*}\Bigl{(}\omega_{\tilde{B}}\otimes
F^{n}\tilde{\mathcal{V}}^{>-1}\Bigr{)}\cong\omega_{B}\otimes
F_{-n}\mathcal{M}.$

The right-adjoint of the functor $\mathbf{R}\mu_{*}$ is the exceptional
inverse image functor
$\mu^{!}=\omega_{\tilde{B}}\otimes\mu^{*}\omega_{B}^{-1}\otimes\mathbf{L}\mu^{*}$.
After applying this to the above isomorphism, we get a morphism

$F^{n}\tilde{\mathcal{V}}^{>-1}\to\mathbf{L}\mu^{*}\bigl{(}F_{-n}\mathcal{M}\bigr{)}$

in the derived category; since $F_{-n}\mathcal{M}\cong\mathscr{T}_{B}$ is
locally free, this is a morphism of sheaves

$F^{n}\tilde{\mathcal{V}}^{>-1}\to\mu^{*}\bigl{(}F_{-n}\mathcal{M}\bigr{)}.$
(8)

By construction, this morphism is an isomorphism outside the exceptional
divisor of $\mu$, and therefore injective.

The rational curve $g\colon C\to B$ intersects the smooth locus $B^{\circ}$,
and therefore lifts uniquely to a rational curve $\tilde{g}\colon
C\to\tilde{B}$, making the following diagram commute:

${C}$${\tilde{B}}$${B}$$\scriptstyle{\tilde{g}}$$\scriptstyle{g}$$\scriptstyle{\mu}$

After pulling back (8) along $\tilde{g}$, we obtain a morphism

$\tilde{g}^{*}\bigl{(}F^{n}\tilde{\mathcal{V}}^{>-1}\bigr{)}\to
g^{*}\bigl{(}F_{-n}\mathcal{M}\bigr{)}.$

The properties of the canonical extension give us a natural morphism
$\tilde{\mathcal{V}}_{C}^{>-1}\to\tilde{g}^{*}\bigl{(}\tilde{\mathcal{V}}^{>-1}\bigr{)}$
that is an isomorphism over $C^{\circ}$. Consequently, we have a morphism

$F^{n}\tilde{\mathcal{V}}_{C}^{>-1}\to\tilde{g}^{*}\bigl{(}F^{n}\tilde{\mathcal{V}}^{>-1}\bigr{)},$

and after composing the two morphisms and remembering the identity in (7), we
get the desired morphism

$F_{-n}\mathcal{M}_{C}\to g^{*}(F_{-n}\mathcal{M}).$

By construction, it is an isomorphism over the open subset
$C^{\circ}=g^{-1}(B^{\circ})$. ∎

This gives us an injective morphism from the semi-positive vector bundle
$F_{-n}\mathcal{M}_{C}\cong F^{n}\tilde{\mathcal{V}}_{C}^{>-1}$ into
$g^{*}(F_{-n}\mathcal{M})\cong g^{*}\mathscr{T}_{B}$, and since this morphism
is an isomorphism over $C^{\circ}$, we can argue as before to finish the proof
of 2.2.

Now suppose that the rational curve $g\colon C\to B$ intersects the
discriminant divisor $D$ transversely. This means that it stays inside the
open subset where $F_{-n}\mathcal{M}$ is described by the canonical extension,
and because the intersection is transverse, the morphism

$F^{n}\tilde{\mathcal{V}}_{C}^{>-1}\to g^{*}\mathscr{T}_{B}$

is an isomorphism. Moreover, the fiber product $Y=C\times_{B}X$ is again
smooth, and so by (4), we have $F^{n}\tilde{\mathcal{V}}_{C}^{>-1}\cong
R^{n-1}h_{*}\omega_{Y/C}$, where $h\colon Y\to C$ is the base change of the
Lagrangian fibration. 2.4 therefore still applies in this setting.

##  3\. Foliations and variations of Hodge structure

The purpose of this section is to prove a couple of small technical results
that relate the variation of Hodge structure on
$V=R^{1}f_{*}^{\circ}\mathbb{Q}_{X^{\circ}}$ to the theory of foliations. Here
$f\colon X\to B$ is again a Lagrangian fibration from a compact Kähler
manifold $X$.

Recall that $\mathcal{V}$ denotes the underlying flat vector bundle,
$\nabla\colon\mathcal{V}\to\Omega_{B^{\circ}}^{1}\otimes\mathcal{V}$ the
Gauss-Manin connection, and
$F^{1}\mathcal{V}=f_{*}^{\circ}\Omega_{X^{\circ}/B^{\circ}}^{1}$ the Hodge
bundle. We have $F^{1}\mathcal{V}\cong\mathscr{T}_{B^{\circ}}$, with the
isomorphism induced by contraction with the holomorphic symplectic form
$\sigma\in H^{0}(X,\Omega_{X}^{2})$. We are going to need the following result
by Freed [Fre99] that describes the Lie bracket on the tangent bundle in terms
of the connection.

######  Lemma 3.1.

For any two holomorphic vector fields $\xi,\eta\in\mathscr{T}_{B^{\circ}}$,
one has

$\nabla_{\xi}(\eta)-\nabla_{\eta}(\xi)=[\xi,\eta].$

###### Proof.

Let $U\subseteq B^{\circ}$ be an open subset and $\xi,\eta\in
H^{0}(U,\mathscr{T}_{B})$ be two holomorphic vector fields. Since
$f^{\circ}\colon X^{\circ}\to B^{\circ}$ is a fiber bundle, we can lift them
to smooth vector fields $\tilde{\xi},\tilde{\eta}\in
A^{0}\bigl{(}f^{-1}(U),\mathscr{T}_{X}\bigr{)}$; then
$[\tilde{\xi},\tilde{\eta}]$ is a smooth lifting of the Lie bracket
$[\xi,\eta]$. The isomorphism $\mathscr{T}_{B^{\circ}}\cong F^{1}\mathcal{V}$
takes the holomorphic vector field $\xi$ to the fiberwise cohomology class of
the smooth $(1,0)$-form

$\Omega_{\tilde{\xi}}=\operatorname{int}(\tilde{\xi})(\sigma)\in
A^{1,0}\bigl{(}f^{-1}(U)\bigr{)},$

obtained by contracting the vector field $\tilde{\xi}$ against the holomorphic
symplectic form. In these terms, the Gauss-Manin connection
$\nabla_{\xi}(\eta)\in H^{0}(U,\mathcal{V})$ is given by taking the fiberwise
cohomology class of the smooth $1$-form

$\operatorname{int}(\tilde{\xi})(d\Omega_{\tilde{\eta}})\in
A^{1}\bigl{(}f^{-1}(U)\bigr{)}.$

Now we compute that, for any smooth vector field $\lambda$ on $f^{-1}(U)$, one
has

$\displaystyle\Bigl{(}$
$\displaystyle\operatorname{int}(\tilde{\xi})(d\Omega_{\tilde{\eta}})-\operatorname{int}(\tilde{\eta})(d\Omega_{\tilde{\xi}})\Bigr{)}(\lambda)=d\Omega_{\tilde{\eta}}(\tilde{\xi},\lambda)-d\Omega_{\tilde{\xi}}(\tilde{\eta},\lambda)$
$\displaystyle=\Bigl{(}\tilde{\xi}\cdot\sigma(\tilde{\eta},\lambda)-\lambda\cdot\sigma(\tilde{\eta},\tilde{\xi})-\sigma\bigl{(}\tilde{\eta},[\tilde{\xi},\lambda]\bigr{)}\Bigr{)}-\Bigl{(}\tilde{\eta}\cdot\sigma(\tilde{\xi},\lambda)-\lambda\cdot\sigma(\tilde{\xi},\tilde{\eta})-\sigma\bigl{(}\tilde{\xi},[\tilde{\eta},\lambda]\bigr{)}\Bigr{)}$
$\displaystyle=\sigma\bigl{(}[\tilde{\xi},\tilde{\eta}],\lambda\bigr{)}+\lambda\cdot\sigma(\tilde{\xi},\tilde{\eta}),$

using the coordinate-free formula for the exterior derivative, and the
identity

$\displaystyle 0=(d\sigma)(\lambda,\tilde{\xi},\tilde{\eta})$
$\displaystyle=\lambda\cdot\sigma(\tilde{\xi},\tilde{\eta})+\tilde{\xi}\cdot\sigma(\tilde{\eta},\lambda)+\tilde{\eta}\cdot\sigma(\lambda,\tilde{\xi})$
$\displaystyle+\sigma\bigl{(}\lambda,[\tilde{\xi},\tilde{\eta}]\bigr{)}+\sigma\bigl{(}\tilde{\xi},[\tilde{\eta},\lambda]\bigr{)}+\sigma\bigl{(}\tilde{\eta},[\lambda,\tilde{\xi}]\bigr{)}.$

This proves that
$\operatorname{int}(\tilde{\xi})(d\Omega_{\tilde{\eta}})-\operatorname{int}(\tilde{\eta})(d\Omega_{\tilde{\xi}})=\Omega_{[\tilde{\xi},\tilde{\eta}]}+d\bigl{(}\sigma(\tilde{\xi},\tilde{\eta})\bigr{)}$,
which then gives the desired identity in the first cohomology of the fibers of
the Lagrangian fibration. ∎

######  Remark 3.2.

Since $\mathcal{V}^{1,0}\cong\mathscr{T}_{B^{\circ}}$ and
$\mathcal{V}^{0,1}\cong\Omega_{B^{\circ}}^{1}$, this also implies that the
Higgs field

$\overline{\nabla}\colon\mathscr{T}_{B^{\circ}}\otimes\mathcal{V}^{1,0}\to\mathcal{V}^{0,1}$

is symmetric in its two arguments. This fact is a special case of the “cubic
condition” of Donagi and Markman [DM96]. The identity in Lemma 3.1 is actually
equivalent to the statement that the hermitian metric on $B^{\circ}$ induced
by the Hodge metric on the bundle $\mathcal{V}^{1,0}$ is a Kähler metric
[Fre99].

The formula for the Lie bracket allows us to produce foliations from
subvariations of $V$, using the fact that
$F^{1}\mathcal{V}\cong\mathscr{T}_{B^{\circ}}$ is the tangent bundle of
$B^{\circ}$.

######  Lemma 3.3.

Let $p\colon U\to B^{\circ}$ be a holomorphic mapping that is locally
biholomorphic, and let $W$ be a subvariation of Hodge structure of
$V_{U}=p^{*}V$. Then the distribution given by the Hodge bundle
$F^{1}\mathcal{W}\subseteq F^{1}\mathcal{V}_{U}$ is integrable, hence defines
a foliation on $U$.

###### Proof.

Since $p$ is locally biholomorphic, we have
$F^{1}\mathcal{V}_{U}=p^{*}(F^{1}\mathcal{V})\cong
p^{*}\mathscr{T}_{B^{\circ}}\cong\mathscr{T}_{U}$. Because $W$ is a
subvariation of Hodge structure, the vector bundle $\mathcal{W}$ is preserved
by the Gauss-Manin connection on $\mathcal{V}_{U}$. From $\mathcal{W}\cap
F^{1}\mathcal{V}_{U}=F^{1}\mathcal{W}$, we get

$[\xi,\eta]=\nabla_{\xi}(\eta)-\nabla_{\eta}(\xi)\in H^{0}(U,\mathcal{W})\cap
H^{0}(U,F^{1}\mathcal{V}_{U})=H^{0}(U,F^{1}\mathcal{W})$

for any two holomorphic vector fields $\xi,\eta$ in the subspace
$H^{0}(U,F^{1}\mathcal{W})\subseteq H^{0}(U,\mathscr{T}_{U})$. This proves
that the distribution is integrable. ∎

We are also going to need the following simple fact about the integrability of
the sum of two integrable distributions.

######  Lemma 3.4.

Let $F$ and $G$ be holomorphic foliations of rank $d$ on a complex manifold,
whose sum $F+G$ is a foliation of rank $2d$. Take a leaf $L$ of $F$, and form
the union of the leaves of $G$ through points of $L$. Then the resulting
submanifold is a leaf of $F+G$.

###### Proof.

Fix a point on $L$, and let $S$ be the unique leaf of $F+G$ through that
point; then $S$ is a (locally defined) complex submanifold of dimension $2d$,
and since $F\subseteq F+G$, it contains $L$. The restriction $G|_{S}$ is a
foliation on $S$, and so $S$ is locally the union of leaves of $G$. For
dimension reasons, those leaves of $G|_{S}$ that pass through points of $L$
fill up all of $S$, and this gives the desired result. ∎

######  Remark 3.5.

The proof of the lemma shows that the construction is actually symmetric in
$F$ and $G$: if we start from a leaf of $G$ that is contained in $S$, and form
the union of the leaves of $F$ through its points, we get the same submanifold
$S$ as before.

Lastly, we need to analyze the behavior of the tangent bundle at smooth points
of the discriminant locus $D$. It is known that $D$ is always a divisor in $B$
[HO09, Prop. 3.1], and that the local monodromy at any smooth point of $D$ is
always nontrivial [Hwa08, Prop. 3.2]. Our proof does not actually rely on
these two facts, but it is nevertheless useful to keep them in mind. The next
result says that if the local monodromy at a smooth point of $D$ has finite
order, then it looks like the local monodromy in an isotrivial elliptic
fibration on a K3 surface.

######  Lemma 3.6.

Suppose that, in a neighborhood of a given point $b\in B$, the discriminant
locus $D$ is smooth of dimension $n-1$, and that the local monodromy around
$D$ has finite order. Then the local monodromy transformation has exactly two
nontrivial eigenvalues, which are a $4$th or $6$th root of unity and its
complex conjugate.

###### Proof.

Let $U$ be a neighborhood of the point in question, and suppose that $D\cap U$
is smooth and defined by a nontrivial holomorphic function
$h\in\Gamma(U,\mathscr{O}_{B})$. Let $V_{U}$ denote the restriction of the
variation of Hodge structure $V=R^{1}f_{*}^{\circ}\mathbb{Q}_{X^{\circ}}$ to
this open subset. By Matsushita’s theorem, the isomorphism
$\mathscr{T}_{U^{\circ}}\cong F^{1}\mathcal{V}$ extends to an isomorphism

$\mathscr{T}_{U}\cong F^{1}\tilde{\mathcal{V}}_{U}^{>-1}$

between the tangent bundle of $U$ and the subbundle
$F^{1}\tilde{\mathcal{V}}_{U}^{>-1}=\tilde{\mathcal{V}}_{U}^{>-1}\cap
j_{*}(F^{1}\mathcal{V}_{U})$ of the canonical extension of
$(\mathcal{V}_{U},\nabla)$ along the open embedding $j\colon U^{\circ}\to U$.

Since the monodromy transformation $T$ has finite order, all of its
eigenvalues are roots of unity. Moreover, the weight filtration defined by the
logarithm of the unipotent part of $T$ is trivial, and so the limiting mixed
Hodge structure at $b$ is a polarized Hodge structure of type $(1,0)+(0,1)$,
and $T$ is an automorphism of this Hodge structure [Sch73, Thm. 6.16]. By
choosing a basis that is adapted to the Hodge decomposition and the eigenspace
decomposition of $T$, we obtain a local frame

$v_{1},\dotsc,v_{n},v_{n+1},\dotsc,v_{2n}\in\Gamma\bigl{(}U,\tilde{\mathcal{V}}_{U}^{>-1}\bigr{)}$

for the canonical extension, with the property that

$\nabla(v_{j})=\alpha_{j}\frac{dh}{h}\otimes v_{j},$

where $\alpha_{1},\dotsc,\alpha_{n}\in\mathbb{Q}\cap(-1,0]$. As explained in
[CK89, §2] (and in [SS22] for the case of complex variations of Hodge
structure), we can then find a collection of holomorphic functions $f_{j,k}$
near the point $b$, such that the $n$ sections

$s_{j}=v_{j}+\sum_{k=n+1}^{2n}f_{j,k}v_{k},\quad j=1,\dotsc,n,$

define a local frame for the bundle $F^{1}\tilde{\mathcal{V}}_{U}^{>-1}$,
after shrinking the open set $U$ if necessary. Because of the isomorphism
$\mathscr{T}_{U}\cong F^{1}\tilde{\mathcal{V}}_{U}^{>-1}$, we may also view
$s_{1},\dotsc,s_{n}$ as holomorphic vector fields on $U$ that generate the
tangent bundle $\mathscr{T}_{U}$.

We now apply the formula for the Lie bracket in Lemma 3.1. It gives

$[s_{i},s_{j}]=\nabla_{s_{i}}(s_{j})-\nabla_{s_{j}}(s_{i})\equiv\alpha_{j}\frac{dh(s_{i})}{h}\cdot
s_{j}-\alpha_{i}\frac{dh(s_{j})}{h}\cdot s_{i}\mod v_{n+1},\dotsc,v_{2n},$

and because of the special shape of the sections $s_{1},\dotsc,s_{n}$, the
congruence is strong enough to imply that

$[s_{i},s_{j}]=\alpha_{j}\frac{dh(s_{i})}{h}\cdot
s_{j}-\alpha_{i}\frac{dh(s_{j})}{h}\cdot s_{i}.$ (9)

Now the left-hand side is again a holomorphic vector field on $U$, and so we
deduce that, for every pair of indices $1\leq i<j\leq n$, either
$\alpha_{i}=0$, or $h$ divides $dh(s_{j})$. This last condition means that the
vector field $s_{j}$ is tangent to the hypersurface $D\cap U$. Since not all
the vector fields can be tangent to $D$, we conclude that at most one of
$\alpha_{1},\dotsc,\alpha_{n}$ can be nonzero; in fact, exactly one of them
has to be nonzero, because the monodromy transformation $T$ is nontrivial.

To prove the remaining assertion, we observe that $T$ is defined over
$\mathbb{Z}$ and that $\det T=1$. Therefore $T$ has exactly two nontrivial
eigenvalues, which are roots of unity and complex conjugates of each other. By
checking the possible values of Euler’s $\phi$-function, it is then easy to
see that the two eigenvalues must be either $4$th or $6$th roots of unity. ∎

######  Remark 3.7.

Suppose that $\alpha_{1}\neq 0$ and $\alpha_{2}=\dotsb=\alpha_{n}=0$. The
identity in (9) shows that the $n-1$ holomorphic vector fields
$s_{2},\dotsc,s_{n}$ are tangent to $D$ and commute with each other. This
observation will play an important role later on.

######  Remark 3.8.

By the same method, one can show that if $T=T_{s}\cdot e^{2\pi iN}$ is the
Jordan decomposition of the monodromy transformation (with $N^{2}=0$), then
$\operatorname{rk}N\leq 1$, and the limiting mixed Hodge structure contains a
pure Hodge structure of weight $1$ and dimension $\geq 2n-2$. This is related
to the classification of the general singular fiber of a Lagrangian fibration
in [HO09].

##  4\. Deformation theory and the universal family of curves

We first explain why 2.2 and 2.4, together with some basic deformation theory,
imply that $V$ splits off a factor universally computing the fixed part of the
variation of Hodge structure on any rational curve of minimal degree, at least
on a certain finite cover. We recommend Kollár’s book [Kol96] for the
necessary background on rational curves. Throughout, $B$ is the smooth base of
a Lagrangian fibration, $B^{\circ}$ is the complement of the discriminant
locus, and $n=\dim B$.

Recall that $\omega^{\vee}_{B}=\det\mathscr{T}_{B}$ is ample [Mat99]. Suppose
$g_{0}:C_{0}\to B$ is a rational curve meeting $B^{\circ}$ and of minimal
anticanonical degree among such curves. We know from 2.2 that
$g_{0}^{\ast}\mathscr{T}_{B}$ is semi-positive; consequently, we must have

$\deg g_{0}^{\ast}\mathscr{T}_{B}=d+1$ (10)

for some integer $d\in\\{1,\dotsc,n\\}$. (If the degree is $\geq n+2$, then
Mori’s Bend and Break technique produces a rational curve of smaller degree
through every point of $g_{0}(C)$, contradicting minimality.) In view of 1.2,
our goal will be to show that $d=n$. As $B$ is Fano, it is known that a very
general rational curve of minimal anticanonical degree deforms to cover $B$
and moreover the pullback of the tangent bundle decomposes as

$g_{0}^{\ast}\mathscr{T}_{B}\cong\mathscr{O}(2)\oplus\mathscr{O}(1)^{\oplus(d-1)}\oplus\mathscr{O}^{\oplus(n-d)}.$
(11)

By 2.4, the restriction of the variation of Hodge structure $V$ to such a
curve therefore has a fixed part of rank exactly $2(n-d)$.

We work with the (normalized) moduli space of rational curves
$\operatorname{RatCurves}^{n}(B)$, following Kollár’s notation [Kol96,
II.2.5]. Let $M$ be the open subset of $\operatorname{RatCurves}^{n}(B)$
consisting of all rational curves of anticanonical degree $d$ that intersect
$B^{\circ}$ and belong to the same connected component as $g_{0}$. The
universal family $\pi\colon\mathcal{C}\to M$ is a $\mathbb{P}^{1}$-bundle over
$M$, and we denote by $G\colon\mathcal{C}\to B$ the evaluation morphism. We
use the notation $g\colon C\to B$ for the individual curves in our family;
here $C$ is the fiber of $\pi\colon\mathcal{C}\to M$ over the point in
question, and $g=G|_{C}$.

For a point $g:C\to B$ in our moduli space $M$, we have an exact sequence

$0\to\mathscr{T}_{C}\xrightarrow{dg}g^{*}\mathscr{T}_{B}\to N_{g}\to 0$

and the obstructions for $M$ lie in $H^{1}(C,N_{g})$. This group vanishes by
2.2, and so $M$ is smooth at such a point, of dimension

$\dim H^{0}(C,N_{g})=\chi(C,N_{g})=n+d-2.$

Moreover, as $g^{*}\mathscr{T}_{B}$ is globally generated and by the minimal
degree condition, a generic deformation of our initial curve $g_{0}$
intersects the discriminant locus $D=B\setminus B^{\circ}$ transversely
[Kol96, II.3.7]. Finally, for any $g\colon C\to B$ in the space $M$, and for
any point $c\in C$, we have a short exact sequence

${0}$${H^{0}(C,\mathscr{T}_{C})}$${T_{c}C\oplus
H^{0}(C,g^{*}\mathscr{T}_{B})}$${T_{c}\,\mathcal{C}}$${0}$$\scriptstyle{r_{c}-dg}$$\scriptstyle{\tau}$

where $r_{c}$ is restriction to $c$. The map $\tau$ is naturally interpreted
as the derivative of the map $\nu\colon
C\times\operatorname{Hom}(C,B)\to\mathcal{C}$ from the universal framed curve,
and as the map $dg_{c}+r_{c}\colon T_{c}C\oplus
H^{0}(C,g^{*}\mathscr{T}_{B})\to T_{g(c)}B$ is the derivative of the universal
framed map $G\circ\nu$ [Kol96, II.3.4], it follows that the induced map
$T_{c}\,\mathcal{C}\to T_{g(c)}B$ is the derivative of $G$ at the point
$c\in\mathcal{C}$. Thus, again since $g^{*}\mathscr{T}_{B}$ is globally
generated, the evaluation map $G\colon\mathcal{C}\to B$ is smooth at every
point of $\mathcal{C}$.

######  Remark 4.1.

The rational curves in our family are free, and so a generic curve $g\colon
C\to B$ is an immersion [Kol96, II.3.14], but not necessarily an embedding. Of
course, we will eventually show that $B\cong\mathbb{P}^{n}$ and that $M$ is
the space of lines in $\mathbb{P}^{n}$, but one has to be careful not to make
any unjustified assumptions during the proof.

Let us summarize the conclusions that we have drawn from 2.2 and the
deformation theory of rational curves.

######  Lemma 4.2.

The moduli space $M$ is smooth of dimension $n+d-2$, the universal curve
$\pi\colon\mathcal{C}\to M$ is a $\mathbb{P}^{1}$-bundle over $M$, and the
evaluation morphism $G\colon\mathcal{C}\to B$ is smooth of relative dimension
$d-1$.

We denote by $G^{\circ}\colon\mathcal{C}^{\circ}\to B^{\circ}$ the base change
to $B^{\circ}$, and by $\pi^{\circ}\colon\mathcal{C}^{\circ}\to M$ the
projection to the moduli space. By our choice of $d$, any rational curve of
anticanonical degree $d$ that meets $B^{\circ}$ must be irreducible and
therefore cannot degenerate; this implies that $G^{\circ}$ is proper.

Now the general idea is the following. Suppose that $d<n$, which means that
the variation of Hodge structure $V$ on $B^{\circ}$ has a nontrivial fixed
part on a general curve in our family. We are going to argue that there is a
finite covering space of $B^{\circ}$, constructed from the Stein factorization
of $G^{\circ}\colon\mathcal{C}^{\circ}\to B^{\circ}$, on which the pullback of
$V$ decomposes in a nontrivial way. As a first step, we are going to construct
such a decomposition on the universal curve $\mathcal{C}^{\circ}$ itself.

Consider the pullback $V_{\mathcal{C}}=(G^{\circ})^{*}V$ of the variation of
Hodge structure to $\mathcal{C}^{\circ}$, and its direct image
$\pi_{*}^{\circ}V_{\mathcal{C}}$. By general theory, this is a constructible
sheaf on $M$. Let $T\subseteq M$ be the (dense) Zariski open subset where this
constructible sheaf is a local system (and therefore a variation of Hodge
structure) of rank $2(n-d)$; this includes all those curves $g\colon C\to B$
on which $g^{*}\mathscr{T}_{B}$ decomposes as in (11). (In fact, we will show
later that this is true for every curve, and so actually $T=M$; but this does
not matter for the time being.) By the semi-simplicity of polarized variations
of Hodge structure, we get a canonical splitting of rational variations of
Hodge structure

$V_{\mathcal{C}}\cong V^{\mathit{fix}}_{\mathcal{C}}\oplus W_{\mathcal{C}}$
(12)

where $V^{\mathit{fix}}_{\mathcal{C}}$ is the unique subvariation of Hodge
structure that agrees with the pullback
$(\pi^{\circ})^{*}(\pi^{\circ})_{*}V_{\mathcal{C}}$ on the open subset
$\mathcal{C}_{T}=(\pi^{\circ})^{-1}(T)$; this exists because the fundamental
group of this open subset surjects onto that of the complex manifold
$\mathcal{C}^{\circ}$. Note that $V^{\mathit{fix}}_{\mathcal{C}}$ is itself a
variation of Hodge structure of weight $1$ and rank $2(n-d)$ that represents
the “universal” fixed part of the restriction to the fibers. In fact, the
decomposition in (12) descends to a finite covering of $B^{\circ}$, as we now
explain.

The fiber $G^{-1}(b)$ over a point $b\in B$, or rather its image in the moduli
space $M$, parametrizes all those rational curves in our family that pass
through $b$. In the case of projective space, the space of lines through a
given point is irreducible, but this might (a priori) not be the case here. We
therefore consider the Stein factorization

${\mathcal{C}^{\circ}}$${Z^{\circ}}$${B}$$\scriptstyle{G^{\circ}}$$\scriptstyle{s^{\circ}}$$\scriptstyle{p^{\circ}}$

of the proper smooth morphism $G^{\circ}\colon\mathcal{C}^{\circ}\to
B^{\circ}$; here $s^{\circ}$ is smooth with connected fibers (of constant
dimension $d-1$), and $p^{\circ}$ is finite étale. Let $s\colon Z\to B$ be the
unique extension of this finite covering space to a finite morphism from a
normal projective variety $Z$. Because the universal curve $\mathcal{C}$ is
smooth by Lemma 4.2, we get the following commutative diagram:

${\mathcal{C}}$${Z}$${B}$${M}$$\scriptstyle{\pi}$$\scriptstyle{G}$$\scriptstyle{s}$$\scriptstyle{p}$
(13)

######  Remark 4.3.

The fact that every rational curve $g\colon C\to B$ in our family factors
through the finite morphism $p\colon Z\to B$ means that the fundamental group
of $C^{\circ}$ does not generate the fundamental group of $B^{\circ}$ if the
degree of $p$ is positive, unlike in the case of lines on projective space.

We again summarize all the relevant facts about the commutative diagram.

######  Lemma 4.4.

With the notation introduced above, the following is true:

  1. (1)

The variety $Z$ is smooth at every point of the open subset $s(\mathcal{C})$,
and the map $p\colon Z\to B$ is étale there.

  2. (2)

There is a splitting of variations of Hodge structure

$V_{Z}:=(p^{\circ})^{*}V=V^{\mathit{fix}}_{Z}\oplus W_{Z}$ (14)

of weight $1$ that pulls back to (12).

  3. (3)

The decomposition in (14) induces a decomposition of the tangent bundle

$\mathscr{T}_{s(\mathcal{C})}\cong\mathscr{T}_{s(\mathcal{C})}^{\mathit{fix}}\oplus\mathscr{T}_{s(\mathcal{C})}^{\mathit{pos}}$

on $s(\mathcal{C})$, and the two summands define foliations of rank $n-d$
respectively $d$.

  4. (4)

All fibers of $\pi\colon\mathcal{C}\to M$ are tangent to the foliation
$\mathscr{T}_{s(\mathcal{C})}^{\mathit{pos}}$.

###### Proof.

(1) follows from the fact that $G\colon\mathcal{C}\to B$ is smooth. For (2),
observe that because $\mathcal{C}^{\circ}$ and $Z^{\circ}$ are smooth, the
composition
$\pi_{1}(\mathcal{C}^{\circ}_{T})\to\pi_{1}(\mathcal{C}^{\circ})\to\pi_{1}(Z^{\circ})$
is surjective; this implies the claim. The restriction of $p\colon Z\to B$ to
the open subset $s(\mathcal{C})$ is étale, and so

$\mathscr{T}_{s(\mathcal{C})}\cong(p^{*}\mathscr{T}_{B})|_{s(\mathcal{C})}.$

Because $\mathscr{T}_{B}$ is the Hodge module extension of $F^{1}\mathcal{V}$,
it follows that $\mathscr{T}_{s(\mathcal{C})}$ is the Hodge module extension
of $F^{1}\mathcal{V}_{Z}$. The decomposition of the variation of Hodge
structure therefore induces a decomposition of $\mathscr{T}_{Z}$ into a vector
bundle of rank $n-d$ and a vector bundle of rank $d$. The two summands are
foliations on $s(\mathcal{C})$ by Lemma 3.3. It remains to prove (4). By
construction, the pullback of $\mathscr{T}_{s(\mathcal{C})}^{\mathit{fix}}$ to
a general curve $g\colon C\to B$ in our moduli space is a trivial bundle of
rank $n-d$. Since $g^{*}\mathscr{T}_{B}$ is always semipositive, it follows
(by the deformation invariance of the degree) that the pullback of
$\mathscr{T}_{s(\mathcal{C})}^{\mathit{fix}}$ is trivial for _every_ curve in
$M$. For degree reasons, $\mathscr{T}_{C}\cong\mathscr{O}(2)$ must therefore
be contained in the pullback of the other summand, and so all curves in the
universal family are indeed tangent to the foliation
$\mathscr{T}_{s(\mathcal{C})}^{\mathit{pos}}$. ∎

##  5\. Connected components of the family of rational curves

We are going to need two additional results about the connected components of
the family of rational curves through a given point. We keep the notation
introduced in (13). At least generically, the fibers of $p\colon Z\to B$
represent the different connected components of the family of rational curves
through a point. (This interpretation is slightly handicapped by the fact that
a generic rational curve in our family can a priori have nodes.) Indeed, a
point $z\in Z^{\circ}$ can be thought of as the point $p(z)\in B^{\circ}$,
together with the connected component $s^{-1}(z)$ of the fiber
$G^{-1}\bigl{(}p(z)\bigr{)}$; of course, the connected component of the family
of curves through $b$ is really the image $M_{z}=\pi(s^{-1}(z))$ in the moduli
space $M$. For any $z\in Z$, we denote by

$P_{z}:=s\bigl{(}\pi^{-1}(M_{z})\bigr{)}\subseteq Z$ (15)

the locus in $Z$ swept out by the curves in the given component.222This
notation is different from Hwang’s notation in [Hwa08]. By construction, $z\in
P_{z}$. The following result, which is mostly due to Araujo [Ara06], shows
that $P_{z}$ is generically a projective space.

######  Lemma 5.1.

For any $z\in Z^{\circ}$, the following is true:

  1. (1)

The subvariety $P_{z}\subseteq Z$ is isomorphic to $\mathbb{P}^{d}$, and its
normal bundle in $Z$ is a trivial bundle of rank $n-d$.

  2. (2)

At any point $z^{\prime}\in P_{z}\cap Z^{\circ}$, the isomorphism

$T_{z^{\prime}}Z\cong T_{p(z^{\prime})}B\cong
F^{1}\mathcal{V}|_{p(z^{\prime})}\cong F^{1}\mathcal{V}_{Z}|_{z^{\prime}}$

maps the tangent space $T_{z^{\prime}}P_{z}$ isomorphically to the fiber
$F^{1}\mathcal{W}_{Z}|_{z^{\prime}}$.

  3. (3)

Two subvarieties $P_{z}$ and $P_{z^{\prime}}$ are either equal or disjoint.

###### Proof.

Since $G^{\circ}\colon\mathcal{C}^{\circ}\to B^{\circ}$ is proper, it is clear
that $P_{z}$ is projective. Let us first show that $\dim P_{z}=d$. Since
$\pi\colon\mathcal{C}\to M$ is a $\mathbb{P}^{1}$-bundle, the subvariety
$s^{-1}(z)$ must be finite over its image $M_{z}$ in $M$ (because otherwise
$G$ would be constant along the curve in question). Therefore $\dim M_{z}=d-1$
and $\dim\pi^{-1}(M_{z})=d$. Now observe that the projection from
$\pi^{-1}(M_{z})$ to $P_{z}$ is finite away from $p^{-1}(z)$. Indeed, if the
fiber over a point $z^{\prime}\in P_{z}$ with $p(z^{\prime})\neq p(z)$ had
positive dimension, we would get a nontrivial family of rational curves
through the two fixed points $p(z)$ and $p(z^{\prime})$, and by by Mori’s Bend
and Break technique, this would produce a rational curve through $p(z)$ of
smaller anticanonical degree, contradicting our initial choice.

By construction, we have $P_{z}\subseteq s(\mathcal{C})$. From Lemma 4.4, we
get a decomposition

$\mathscr{T}_{s(\mathcal{C})}\cong\mathscr{T}_{s(\mathcal{C})}^{\mathit{fix}}\oplus\mathscr{T}_{s(\mathcal{C})}^{\mathit{pos}},$

and the summand $\mathscr{T}_{s(\mathcal{C})}^{\mathit{pos}}$ is a foliation
of rank $d$. Since $P_{z}$ is a union of curves that are tangent to this
foliation, it follows that the tangent space to $P_{z}$ at each point is
contained in the fiber of $\mathscr{T}_{s(\mathcal{C})}^{\mathit{pos}}$. For
dimension reasons, this means that $P_{z}$ is smooth and
$T_{z^{\prime}}P_{z}=\mathscr{T}_{s(\mathcal{C})}^{\mathit{pos}}\big{|}_{z^{\prime}}$
for every $z^{\prime}\in P_{z}$, which gives (2). We also get

$\mathscr{T}_{P_{z}}\cong\mathscr{T}_{s(\mathcal{C})}^{\mathit{pos}}\big{|}_{P_{z}}\quad\text{and}\quad
N_{P_{z}|Z}\cong\mathscr{T}_{s(\mathcal{C})}^{\mathit{fix}}\big{|}_{P_{z}}.$

Moreover, any two leaves $P_{z}$ and $P_{z^{\prime}}$ are either equal or
disjoint, proving (3).

Now let $g\colon C\to B$ be any curve in the subset $M_{z}\subseteq M$, and
denote by $i\colon C\to P_{z}$ the induced morphism to $P_{z}$. Because we
already know that the restriction of
$\mathscr{T}_{s(\mathcal{C})}^{\mathit{fix}}$ to every curve in $M$ is
trivial, we get

$\deg i^{*}\mathscr{T}_{P_{z}}=\deg i^{*}\mathscr{T}_{s(\mathcal{C})}=\deg
g^{*}\mathscr{T}_{B}=d+1.$

This means that there is a rational curve of anticanonical degree $d+1$
through any two points of $P_{z}$ (namely the rational curves that sweep out
the subvariety $P_{z^{\prime}}$ for any $z^{\prime}\in P_{z}$), and by our
choice of $d$, these curves can obviously not degenerate. We can therefore
apply [CMSB02, Thm. 0.1] and conclude that $P_{z}\cong\mathbb{P}^{d}$ and that
the curves in the subset $M_{z}\subseteq M$ are exactly the lines through the
point $z\in P_{z}$. Because $p\colon Z\to B$ is étale on the open subset
$s(\mathcal{C})$, it also follows that

$g^{\ast}\mathscr{T}_{B}\cong
i^{*}\mathscr{T}_{P_{z}}\oplus\mathscr{O}^{\oplus(n-d)}\cong\mathscr{O}(2)\oplus\mathscr{O}^{\oplus(d-1)}\oplus\mathscr{O}^{\oplus(n-d)},$

and so the pullback of the tangent bundle of $B$ actually has the same
splitting behavior as in (11) on _every_ curve in the universal family.

Finally, the normal bundle $N_{P_{z}|Z}$ is a vector bundle of rank $n-d$ on
$P_{z}\cong\mathbb{P}^{d}$ whose restriction to every line is trivial; by
general facts about vector bundles on projective space, it follows that
$N_{P_{z}|Z}$ is a trivial bundle of rank $n-d$, and (1) is proved. ∎

######  Remark 5.2.

One can use the results above to show that the open subset
$s(\mathcal{C})\subseteq Z$ is actually $\mathbb{P}^{d}$-bundle over a smooth
variety of dimension $n-d$, whose fibers are the projective spaces $P_{z}$.
The key point is that relevant Hilbert scheme of $Z$ is smooth of dimension
$n-d$ at the point corresponding to $P_{z}$, due to the fact that the normal
bundle of $P_{z}$ in $Z$ is trivial. For details, see [Ara06, Lem. 3.3].

The second result is a proposition proved by Hwang in his earlier paper
[Hwa07]. As it plays an important role in our argument, we include the proof.
First a lemma.

######  Lemma 5.3.

Let $z_{1},z_{2}\in Z^{\circ}$ be two points. If $p(P_{z_{1}})=p(P_{z_{2}})$,
then $P_{z_{1}}=P_{z_{2}}$.

###### Proof.

Suppose that $p(P_{z_{1}})=p(P_{z_{2}})$, and denote by $P$ the normalization
of this subvariety of $B$. Since $P_{z_{j}}\subseteq s(\mathcal{C})$, the
induced morphism $P_{z_{j}}\to P$ is a finite immersion, and therefore a
finite covering space. But $P_{z_{j}}\cong\mathbb{P}^{d}$, and therefore
$P\cong\mathbb{P}^{d}$; this is simply connected, and so $P_{z_{j}}\to P$ is
an isomorphism. By composing one isomorphism with the inverse of the other, we
get an isomorphism between $P_{z_{1}}$ and $P_{z_{2}}$. Under the isomorphism
$P_{z_{j}}\cong\mathbb{P}^{d}$, any line in $\mathbb{P}^{d}$ determines a
rational curve in $B$ of anticanonical degree $d+1$. Since every automorphism
of $\mathbb{P}^{d}$ takes lines to lines, it follows that
$\pi\bigl{(}s^{-1}(P_{z_{1}})\bigr{)}=\pi\bigl{(}s^{-1}(P_{z_{2}})\bigr{)}$,
and therefore $P_{z_{1}}=P_{z_{2}}$. ∎

The following result is a slight generalization of [Hwa07, Prop. 2.2].

######  Proposition 5.4.

Let $z_{1},z_{2}\in Z^{\circ}$ be two points with $p(z_{1})=p(z_{2})=b$. If
the subspaces

$F^{1}\mathcal{W}_{Z}|_{z_{1}}\subseteq F^{1}\mathcal{V}_{Z}|_{z_{1}}\cong
F^{1}\mathcal{V}|_{b}\quad\text{and}\quad
F^{1}\mathcal{W}_{Z}|_{z_{2}}\subseteq F^{1}\mathcal{V}_{Z}|_{z_{2}}\cong
F^{1}\mathcal{V}|_{b}$

have a nontrivial intersection in $F^{1}\mathcal{V}|_{b}$, then
$P_{z_{1}}=P_{z_{2}}$.

###### Proof.

Since $p\colon Z\to B$ is étale at the point $z_{1}$, we can find an open
neighborhood $U\subseteq B$ of the point $b$, and and open neighborhood
$U_{1}\subseteq Z$ of the point $z_{1}$, such that $p|_{U_{1}}\colon U_{1}\to
U$ is biholomorphic. The pullback of $W_{Z}|_{U_{1}}$ along the inverse of
this biholomorphic mapping therefore defines a subvariation of Hodge structure
$W_{1}$ of $V|_{U}$, whose rank is $2d$. After applying the same construction
to the point $z_{2}$, we obtain another subvariation of Hodge structure
$W_{2}$ of $V|_{U}$, also of rank $2d$. Since the two Hodge bundles
$F^{1}\mathcal{W}_{1}$ and $F^{1}\mathcal{W}_{2}$ have a nontrivial
intersection at the point $b$, the intersection $W=W_{1}\cap W_{2}$ must be a
nontrivial subvariation. According to Lemma 3.3, the intersection
$F^{1}\mathcal{W}_{1}\cap F^{1}\mathcal{W}_{2}=F^{1}\mathcal{W}$ is therefore
a foliation of positive rank on $U$. The leaf of the foliation through the
point $b$ is contained in both $p(P_{z_{1}})$ and $p(P_{z_{2}})$, and so the
intersection $p(P_{z_{1}})\cap p(P_{z_{2}})$ must have positive dimension.

As $P_{z}\to p(P_{z})$ is always birational, we can therefore find a smooth
projective curve $Y$ and two morphisms $f_{1}\colon Y\to P_{z_{1}}$ and
$f_{2}\colon Y\to P_{z_{2}}$ such that $p\circ f_{1}=p\circ f_{2}$. We can
clearly arrange that the image of $Y$ passes through the point $b$. The
following commutative diagram shows the relevant morphisms:

${Y}$${P_{z_{1}}}$${P_{z_{2}}}$${B}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{2}}$$\scriptstyle{p}$$\scriptstyle{p}$

From Lemma 5.1, we know that
$p^{*}\mathscr{T}_{B}|_{P_{z}}\cong\mathscr{T}_{P_{z}}\oplus\mathscr{O}^{\oplus(n-d)}$,
and that the tangent bundle to $P_{z}\cong\mathbb{P}^{d}$ is the restriction
of the bundle $\mathscr{T}_{s(\mathcal{C})}^{\mathit{pos}}$. Since the tangent
bundle of projective space is ample, the identity

$f_{1}^{*}\mathscr{T}_{P_{z_{1}}}\oplus\mathscr{O}^{\oplus(n-d)}=f_{2}^{*}\mathscr{T}_{P_{z_{2}}}\oplus\mathscr{O}^{\oplus(n-d)}$

inside the pullback of $\mathscr{T}_{B}$ clearly implies that
$f_{1}^{*}\mathscr{T}_{P_{z_{1}}}$ and $f_{2}^{*}\mathscr{T}_{P_{z_{2}}}$ are
the same subbundle. But this means that $F^{1}\mathcal{W}_{Z}|_{z_{1}}$ and
$F^{1}\mathcal{W}_{Z}|_{z_{2}}$ are equal, as subspaces of
$F^{1}\mathcal{V}|_{b}$, from which it follows as above that $p(P_{z_{1}})\cap
p(P_{z_{2}})$ has dimension $d$, and hence that $p(P_{z_{1}})=p(P_{z_{2}})$.
We now reach the desired conclusion by applying 5.3. ∎

##  6\. Controlling the ramification of the finite covering

After the results in the four preceding sections, the proof of Hwang’s theorem
is reduced to showing that the splitting in Lemma 4.4 descends to $B^{\circ}$
(where it contradicts the irreducibility of $V$). Our proof of this fact is a
version of [Hwa08, §5], but becomes simpler because everything is controlled
by variations of Hodge structure. The crucial step is to control the
ramification of the finite morphism $p\colon Z\to B$ via the following result.

######  Proposition 6.1.

Let $g\colon C\to B$ be the rational curve corresponding to a general point in
the moduli space $M$. Then the base change of the finite covering space
$p^{\circ}:Z^{\circ}\to B^{\circ}$ along $g^{\circ}\colon C^{\circ}\to
B^{\circ}$ is a disjoint union of copies of $C^{\circ}$.

Let $g\colon C\to B$ be a generic rational curve (in the universal family
$\mathcal{C}$). Such a curve intersects the discriminant locus $D$
transversely (because all the rational curves under consideration are free),
and as $Z$ is normal, the base change

${C\times_{B}Z}$${Z}$${C}$${B}$$\scriptstyle{p}$$\scriptstyle{g}$

is a finite union of smooth projective curves, each with a finite morphism to
$C$. Since $G\colon\mathcal{C}\to B$ factors through $p\colon Z\to B$ by
construction, exactly one of these curves is (the image of)
$C\subseteq\mathcal{C}$. Let $p\colon R\to C$ be one of the other components.
If $\deg(R/C)=1$, then $p$ is an isomorphism, and so $R^{\circ}$ is isomorphic
to $C^{\circ}$. In order to prove 6.1, we may therefore assume from now on
that $\deg(R/C)\geq 2$; after several steps, we will see that this assumption
leads to a contradiction. The idea is the same as in Hwang’s paper: to first
reduce to the $d=1$ case (so the curves in $M$ don’t deform with a fixed
point), and then to show there is a deformation of $C$ for which $R\to C$
deforms trivially but for which constant sections deform along curves in the
family $M$. The condition $\deg(R/C)\geq 2$ then contradicts 5.4.

######  Lemma 6.2.

If $\deg(R/C)\geq 2$, then the pullback of the variation of Hodge structure
$V$ to $R^{\circ}$ is constant.

###### Proof.

We use the superscript “$\circ$” to denote the base change to the open subset
$B^{\circ}$. We have the following commutative diagram:

${R^{\circ}}$${Z^{\circ}}$${C^{\circ}}$${B^{\circ}}$$\scriptstyle{p^{\circ}}$$\scriptstyle{p^{\circ}}$$\scriptstyle{g^{\circ}}$

We obtain two decompositions as variations of Hodge structure of
$V_{R}:=V_{Z}|_{R^{\circ}}$. One decomposition $V_{R}\cong
V^{\mathit{fix}}_{R}\oplus W_{R}$ comes from restricting (14) to $R^{\circ}$;
the other comes from pulling back $V_{C}=(g^{\circ})^{*}V\cong
V^{\mathit{fix}}_{C}\oplus W_{C}$ along the map $p^{\circ}\colon R^{\circ}\to
C^{\circ}$. Note that both $W_{C}$ and $W_{R}$ are variations of Hodge
structure of rank $2d$, where $d$ is the integer from (10). We now play the
two decompositions off against each other.

Consider the intersection $(g^{\circ})^{*}W_{C}\cap W_{R}$, which is again a
subvariation of Hodge structure of $V_{R}$. If it is nontrivial, then the
intersection of the two Hodge bundles $(g^{\circ})^{*}F^{1}\mathcal{W}_{C}$
and $F^{1}\mathcal{W}_{R}$ is a vector bundle of positive rank on all of
$R^{\circ}$. According to 5.4, this gives us $P_{\tilde{g}(y)}=P$ for every
$y\in R^{\circ}$, where $P$ is the projective space containing our original
curve. But this is clearly impossible. The conclusion is that
$(g^{\circ})^{*}W_{C}$ and $W_{R}$ intersect trivially. This makes the
composition

$W_{R}\to V_{R}\to(p^{\circ})^{*}V^{\mathit{fix}}_{C}$

of the inclusion and the projection injective. As $V^{\mathit{fix}}_{C}$ is a
constant variation of Hodge structure on $C^{\circ}$, it follows that $W_{R}$
is constant, too.

Now we claim that the intersection of the subvariations $W_{R}$ and
$(p^{\circ})^{*}V^{\mathit{fix}}_{C}$ is also trivial. Otherwise, the Hodge
bundle $F^{1}\mathcal{W}_{R}$ contains a subbundle of positive rank that is
the pullback of a trivial bundle from $C^{\circ}$, which means that the two
subspaces $F^{1}\mathcal{W}_{R}|_{y}$ and $F^{1}\mathcal{W}_{R}|_{y^{\prime}}$
intersect inside $F^{1}\mathcal{V}_{p(y)}$, for any two points
$y,y^{\prime}\in R^{\circ}$ such that $p(y)=p(y^{\prime})$. By 5.4, it follows
that $P_{\tilde{g}(y)}=P_{\tilde{g}(y^{\prime})}$, and as $\deg(R/C)\geq 2$,
this is clearly impossible. The conclusion is that, for dimension reasons,

$V_{R}\cong W_{R}\oplus(p^{\circ})^{*}V^{\mathit{fix}}_{C}$

is the sum of two constant variations of Hodge structure on $R$. ∎

######  Remark 6.3.

6.2 in particular implies there is a covering family of curves on which $V$ is
isotrivial, which contradicts the main result of [Bak22] (and therefore proves
6.1) unless the Lagrangian fibration is isotrivial. In the isotrivial case the
foliations all come from a flat structure on $\mathscr{T}_{B^{\circ}}$, and
are therefore easier to think about, but we do not in fact use this reduction.

The next step is to show that $d=1$; this means that, in our family, the are
only finitely many rational curves through any given point of $B^{\circ}$.

######  Lemma 6.4.

In the above situation, we must have $d=1$.

###### Proof.

Because $p\colon R\to C$ is étale over $C^{\circ}$, it must ramify over at
least one point $x\in C$ such that $g(x)\in D$. Being generic, $g\colon C\to
B$ intersects the discriminant locus $D$ transversely, and so $g(x)$ is a
smooth point of $D$. The pullback of $V$ to $R^{\circ}$ has trivial monodromy
(by 6.2), hence the Hodge structures do not vary from point to point; as
$C^{\circ}$ is quasi-projective, this implies that the Hodge bundle
$F^{1}\mathcal{V}_{C}$ is a flat subbundle of $\mathcal{V}_{C}$. It also
follows that the local monodromy of $V_{C}$ around the point $x$ has finite
order; consequently, the local monodromy of $V$ near $g(x)$ has finite order
as well. According to Lemma 3.6, the monodromy transformation therefore has
exactly two nontrivial eigenvalues, and because the Hodge bundle
$F^{1}\mathcal{V}_{C}$ is a flat subbundle of $\mathcal{V}_{C}$, it contains
(in a punctured neighborhood of the point $x$) a subbundle of rank $n-1$ that
is preserved by the monodromy transformation around $x$. Now if $d\geq 2$,
then it follows as above that at any two points $y,y^{\prime}\in R^{\circ}$
with $p(y)=p(y^{\prime})$ sufficiently close to $x$, the two subspaces
$F^{1}\mathcal{W}_{R}|_{y}$ and $F^{1}\mathcal{W}_{R}|_{y^{\prime}}$ intersect
nontrivially. As this is impossible (by 5.4), we conclude that $d=1$. ∎

This reduces the problem to the case $d=1$. The smooth morphism
$G\colon\mathcal{C}\to B$ now has finite fibers, and so $s\colon\mathcal{C}\to
Z$ is an open embedding. We may therefore identify the universal curve
$\mathcal{C}$ with an open subset of $Z$ for the remainder of the argument. We
have $\dim\mathcal{C}=n$ and $\dim M=n-1$; because there is nothing to prove
when $n=1$, we shall assume from now on that $n\geq 2$.

Recall from Lemma 4.2 that $p\colon Z\to B$ is étale at every point of
$\mathcal{C}$. According to Lemma 4.4, we have a decomposition of the tangent
bundle

$\mathscr{T}_{\mathcal{C}}\cong\mathscr{T}_{\mathcal{C}}^{\mathit{fix}}\oplus\mathscr{T}_{\mathcal{C}}^{\mathit{pos}}$
(16)

into two foliations of rank $n-1$ respectively $1$, induced by the
decomposition

$V_{Z}=V^{\mathit{fix}}_{Z}\oplus W_{Z}$

of the variation of Hodge structure $V_{Z}=(p^{\circ})^{*}V$ on $Z^{\circ}$.
From the construction in (12), it is clear that
$\mathscr{T}_{\mathcal{C}}^{\mathit{fix}}$ is the pullback of a vector bundle
of rank $n-1$ from the moduli space $M$, whereas
$\mathscr{T}_{\mathcal{C}}^{\mathit{pos}}$ is tangent to the fibers of
$\pi\colon\mathcal{C}\to M$. By comparing (16) with the short exact sequence
for the relative tangent bundle

$0\to\mathscr{T}_{\mathcal{C}/M}\to\mathscr{T}_{\mathcal{C}}\to\pi^{*}\mathscr{T}_{M}\to
0,$

we deduce that
$\mathscr{T}_{\mathcal{C}}^{\mathit{fix}}\cong\pi^{*}\mathscr{T}_{M}$ and
$\mathscr{T}_{\mathcal{C}}^{\mathit{pos}}\cong\mathscr{T}_{\mathcal{C}/M}$.

Denote by $\tilde{g}\colon R\to Z$ the morphism induced by $p\colon R\to B$,
as in the following commutative diagram:

${R}$${Z}$${C}$${B}$$\scriptstyle{\tilde{g}}$$\scriptstyle{p}$$\scriptstyle{p}$$\scriptstyle{g}$

The intersection $R\cap\tilde{g}^{-1}(\mathcal{C})$ is a dense Zariski open
subset of the curve $R$ that contains $R^{\circ}$. Under the projection
$\pi\colon\mathcal{C}\to M$, which is proper, the image
$\pi(\tilde{g}(R)\cap\mathcal{C})$ is a quasi-projective curve inside the
moduli space $M$.

######  Lemma 6.5.

The morphism $\pi\circ\tilde{g}\colon
R\cap\tilde{g}^{-1}(\mathcal{C})\to\pi(\tilde{g}(R)\cap\mathcal{C})$ is an
immersion.

###### Proof.

Suppose to the contrary that $y\in R\cap\tilde{g}^{-1}(\mathcal{C})$ is a
branch point. Set $z=\tilde{g}(y)\in\tilde{g}(R)\cap\mathcal{C}$. This means
that $\tilde{g}(R)$ is tangent to the fiber
$P_{z}=\pi^{-1}\bigl{(}\pi(z)\bigr{)}$ at the point $z$. Since $p\colon Z\to
B$ is étale there, we have $T_{z}Z\cong T_{p(z)}B$. Under this isomorphism,
the tangent space $T_{y}R$ maps isomorphically to $T_{p(z)}C$, and the tangent
space $T_{z}P_{z}$ is isomorphic to
$\mathscr{T}_{\mathcal{C}}^{\mathit{pos}}|_{z}$. By 6.2, the variation of
Hodge structure $(p^{\circ})^{*}V_{C}$ on $R^{\circ}$ is constant. It
therefore extends uniquely to a constant variation of Hodge structure $V_{R}$
on the entire curve $R$, and both $W_{Z}|_{R^{\circ}}$ and
$(p^{\circ})^{*}W_{C}$ extend to constant subvariations of rank $2$. The
condition that $T_{z}Z$ and $T_{y}R$ have the same image inside $T_{p(z)}B$
means that these two subvariations are equal at the point $y$. Being constant,
they must then be equal everywhere, and as usual, this is in contradiction
with 5.4. ∎

We now investigate what happens when we deform the rational curve $g\colon
C\to B$. Because $\pi\colon\mathcal{C}\to M$ is a $\mathbb{P}^{1}$-bundle, we
can choose a small open neighborhood $M_{0}$ of the point $g\colon C\to B$ in
the moduli space $M$, say with $M_{0}$ biholomorphic to an open ball, such
that $\pi^{-1}(M_{0})\cong M_{0}\times\mathbb{P}^{1}$. The original rational
curve is now a morphism $g\colon\mathbb{P}^{1}\to B$, and the nearby
subvarieties $P_{z}$ are the different copies of $\mathbb{P}^{1}$. The
following diagram shows the relevant morphisms:

${M_{0}\times\mathbb{P}^{1}}$${B}$${M_{0}}$$\scriptstyle{\pi}$$\scriptstyle{G}$

By restricting the decomposition in Lemma 4.4 to the open subset
$M_{0}\times\mathbb{P}^{1}$, we obtain a decomposition of the tangent bundle

$\mathscr{T}_{M_{0}\times\mathbb{P}^{1}}\cong\mathscr{T}_{M_{0}\times\mathbb{P}^{1}}^{\mathit{fix}}\oplus\mathscr{T}_{M_{0}\times\mathbb{P}^{1}}^{\mathit{pos}}$

in which the two summands are foliations of rank $n-1$ respectively $1$. By
construction, the fibers of $\pi$ are leaves of
$\mathscr{T}_{M_{0}\times\mathbb{P}^{1}}^{\mathit{pos}}$, and the restriction
of $\mathscr{T}_{M_{0}\times\mathbb{P}^{1}}^{\mathit{fix}}$ to the fibers is
trivial.

Since the initial curve $g\colon\mathbb{P}^{1}\to B$ intersects $D$
transversely, the inverse image $G^{-1}(D)$ is a union of finitely many
sections $(\operatorname{id}\times h_{j})(M_{0})$, where
$h_{1},\dotsc,h_{m}\colon M_{0}\to\mathbb{P}^{1}$ are holomorphic mappings.
The next result says that each of these sections is actually a leaf of the
foliation $\mathscr{T}_{M_{0}\times\mathbb{P}^{1}}^{\mathit{fix}}$.

######  Lemma 6.6.

The foliation $\mathscr{T}_{M_{0}\times\mathbb{P}^{1}}^{\mathit{fix}}$ is
tangent to the divisor $G^{-1}(D)$.

###### Proof.

The argument is very similar to the proof of Lemma 3.6. The holomorphic
mapping from $M_{0}\times\mathbb{P}^{1}\to Z$ is an open embedding, and by
Lemma 4.4, the tangent bundle $\mathscr{T}_{M_{0}\times\mathbb{P}^{1}}$ is
therefore the canonical extension of the pullback of the Hodge bundle
$F^{1}\mathcal{V}_{Z}$. Since $V_{Z}=V^{\mathit{fix}}_{Z}\oplus W_{Z}$, we can
apply the construction in Lemma 3.6 in a neighborhood of any component of
$G^{-1}(D)$, but choosing the sections $v_{1}$ and $v_{n+1}$ in the canonical
extension of $W_{Z}$, and the remaining sections $v_{2},\dotsc,v_{n}$ and
$v_{n+2},\dotsc,v_{2n}$ in the canonical extension of $V^{\mathit{fix}}_{Z}$.
Then $s_{1}$ is a local frame for the canonical extension of
$F^{1}\mathcal{W}_{Z}$, and $s_{2},\dotsc,s_{n}$ are a local frame for the
canonical extension of $F^{1}\mathcal{V}^{\mathit{fix}}_{Z}$, and therefore
for the foliation $\mathscr{T}_{M_{0}\times\mathbb{P}^{1}}^{\mathit{fix}}$.
The same argument as before shows that $s_{2},\dotsc,s_{n}$ are tangent to the
given component of $G^{-1}(D)$, and this proves the claim. ∎

The tangent bundle of $M_{0}\times\mathbb{P}^{1}$ is globally generated
(because $\mathscr{T}_{\mathbb{P}^{1}}$ is globally generated and $M_{0}$ is a
Stein manifold); the same thing is therefore true for the foliation
$\mathscr{T}_{M_{0}\times\mathbb{P}^{1}}^{\mathit{fix}}$. After composing $G$
with an automorphism of $M_{0}\times\mathbb{P}^{1}$, we can arrange that the
three standard sections $M_{0}\times\\{0\\}$, $M_{0}\times\\{1\\}$, and
$M_{0}\times\\{\infty\\}$ are leaves of the foliation
$\mathscr{T}_{M_{0}\times\mathbb{P}^{1}}^{\mathit{fix}}$. From the isomorphism

$H^{0}\bigl{(}M_{0}\times\mathbb{P}^{1},\mathscr{T}_{M_{0}\times\mathbb{P}^{1}}\bigr{)}\cong
H^{0}(M_{0},\mathscr{T}_{M_{0}})\oplus H^{0}(M_{0},\mathscr{O}_{M_{0}})\otimes
H^{0}\bigl{(}\mathbb{P}^{1},\mathscr{T}_{\mathbb{P}^{1}}\bigr{)}$

we then conclude that _every_ leaf of
$\mathscr{T}_{M_{0}\times\mathbb{P}^{1}}^{\mathit{fix}}$ must be of the form
$M_{0}\times\\{x\\}$ for a point $x\in\mathbb{P}^{1}$. In particular, every
component of the divisor $G^{-1}(D)$ must be of this form, which means that
the holomorphic mappings $h_{1},\dotsc,h_{m}\colon M_{0}\to\mathbb{P}^{1}$ are
constant.

From this fact, we can easily deduce that the finite covering $p\colon
R\to\mathbb{P}^{1}$ does not change as we deform $g\colon\mathbb{P}^{1}\to B$.
Recall that $R$ is a smooth projective curve, and that $\tilde{g}\colon R\to
Z$ is the induced morphism. After shrinking $M_{0}$, if necessary, the base
change

${(M_{0}\times\mathbb{P}^{1})\times_{B}Z}$${Z}$${M_{0}\times\mathbb{P}^{1}}$${B}$$\scriptstyle{p}$$\scriptstyle{G}$

is also smooth; let $\mathcal{R}$ be the unique component containing $R$. We
have just shown that $\mathcal{R}^{\circ}\to
M_{0}\times(\mathbb{P}^{1})^{\circ}$ is a finite covering space; because
finite covering spaces of a fixed quasi-projective curve do not deform in a
nontrivial way, we get $\mathcal{R}\cong M_{0}\times R$. The following
commutative diagram shows the relevant morphisms:

${\mathcal{C}}$${M}$${M_{0}\times
R}$${Z}$${M_{0}\times\mathbb{P}^{1}}$${B}$$\scriptstyle{\pi}$$\scriptstyle{\tilde{G}}$$\scriptstyle{\operatorname{id}\times
p}$$\scriptstyle{p}$$\scriptstyle{G}$

According to 6.5, the morphism

$\pi\circ\tilde{g}\colon
R\cap\tilde{g}^{-1}(\mathcal{C})\to\pi\bigl{(}\tilde{g}(R)\cap\mathcal{C}\bigr{)}$

is unramified. Because unramified morphisms between two fixed quasi-projective
curves also do not deform in a nontrivial way, the following lemma will very
quickly lead to the desired contradiction.

######  Lemma 6.7.

There is a $1$-dimensional family of deformations of the initial rational
curve $g\colon\mathbb{P}^{1}\to B$ for which the curves
$\pi(\tilde{g}(R)\cap\mathcal{C})\subseteq M$ stay the same.

###### Proof.

Consider the quasi-projective curve $\pi(\tilde{g}(R)\cap\mathcal{C})$ inside
the moduli space $M$. Its preimage under $\pi$ is a quasi-projective surface
in $\mathcal{C}$. Since $p\colon\mathcal{C}\to B$ is smooth of relative
dimension $0$, it follows that

$S=p\Bigl{(}\pi^{-1}\bigl{(}\pi(\tilde{g}(R)\cap\mathcal{C})\bigr{)}\Bigr{)}\subseteq
B$ (17)

is a quasi-projective surface in $B$ that contains the intersection
$g(\mathbb{P}^{1})\cap B^{\circ}$. A priori, $S$ could be singular along this
curve, but we will show in a moment that it is actually smooth in a
neighborhood of $g(\mathbb{P}^{1})\cap B^{\circ}$. We can think of $S$ as
being swept out by those rational curves in our family that intersect $R$.

Let us first prove that $S$ is smooth at every point $b_{0}\in
g(\mathbb{P}^{1})\cap B^{\circ}$. Let $z_{0}\in\mathcal{C}^{\circ}$ be a point
on the curve $g\colon\mathbb{P}^{1}\to B$ such that $p(z_{0})=b_{0}$, and
enumerate the points in the fiber $p^{-1}(b_{0})\cap R^{\circ}$ as
$y_{1},\dotsc,y_{r}$, where $r=\deg(R/\mathbb{P}^{1})$; set
$z_{j}=\tilde{g}(y_{j})$. Choose an open neighborhood $U\subseteq B$ of the
point $b_{0}$, and an open neighborhood $U_{j}\subseteq\mathcal{C}$ of each
point $z_{j}$, such that $p|_{U_{j}}\colon U_{j}\to U$ is a biholomorphism.
Pulling back the $r+1$ variations of Hodge structure $W_{Z}|_{U_{j}}$, we get
$r+1$ subvariations $W_{0},\dotsc,W_{r}\subseteq V|_{U}$. From the proof of
6.2, we know that the subvariation $W_{R}+(p^{\circ})^{*}W_{C}$ of $V_{R}$ has
rank $4d=4$, and that its fibers at any two points $y,y^{\prime}\in R^{\circ}$
with $p(y)=p(y^{\prime})$ map to the same subspace of $T_{p(y)}B$. This
implies that $W=W_{0}+W_{1}+\dotsb+W_{r}$ has rank $4$ at the point $b_{0}$,
and therefore on all of $U$. Consequently, the Hodge bundle $F^{1}\mathcal{W}$
defines a foliation of rank $2$ on the open set $U$, and this foliation is the
sum of the two foliations $F^{1}\mathcal{W}_{0}$ and $F^{1}\mathcal{W}_{j}$ of
rank $1$, for every $j=1,\dotsc,r$.

From (17), it is easy to see that all those curves involved in the
construction of $S$ that intersect the open subset $U$ must be tangent to the
foliation $F^{1}\mathcal{W}$. For dimension reasons, this implies that $S\cap
U$ is the leaf of the foliation $F^{1}\mathcal{W}$ through the point $b_{0}$,
and therefore smooth.

Now let $C_{0}$ denote the branch of the curve $g\colon\mathbb{P}^{1}\to B$
corresponding to the point $z_{0}$; to be precise, we have $C_{0}=p(U_{0}\cap
P_{z_{0}})$. Similarly, let $C_{j}=p(U_{j}\cap P_{z_{j}})$ be the branches of
the other $r$ rational curves. By construction, each $C_{j}$ is the leaf of
the foliation $F^{1}\mathcal{W}_{j}$ through $b_{0}$. According to Lemma 3.4,
the surface germ $S\cap U$ is therefore swept out by the deformations of the
curve germ $C_{0}$ along each of the curves $C_{j}$. This gives us the desired
$1$-dimensional family of deformations of the rational curve
$g\colon\mathbb{P}^{1}\to B$. ∎

We can now finish the proof of 6.1 as follows. Let $\Delta$ be the unit disk,
and represent the $1$-dimensional family of deformations of
$g\colon\mathbb{P}^{1}\to B$ by a holomorphic mapping $h\colon\Delta\to M_{0}$
with $h(0)=[g]$. Let us denote by $g_{t}=G(h(t),-)\colon\mathbb{P}^{1}\to B$
the rational curves in this family, and by
$\tilde{g}_{t}=\tilde{G}(h(t),-)\colon R\to Z$ the resulting family of
morphisms from $R$. By 6.5, the composition

$\pi\circ\tilde{g}_{t}\colon
R\cap\tilde{g}_{t}^{-1}(\mathcal{C})\to\pi\bigl{(}\tilde{g}_{t}(R)\cap\mathcal{C}\bigr{)}$

is unramified for every $t\in\Delta$. Since the source and the target stay
fixed, it follows that $\pi\circ\tilde{g}_{t}$ is independent of $t$. This
means that if we let $y_{1},\dotsc,y_{r}$ be the points in the fiber
$p^{-1}(x)$ over a fixed point $x\in(\mathbb{P}^{1})^{\circ}$, and set
$z_{j}=\tilde{g}(y_{j})$, then as $t\in\Delta$ varies, the image
$\tilde{g}_{t}(y_{j})$ sweeps out a small open subset of the curve
$P_{z_{j}}$. Because $p\circ\tilde{g}=g\circ p$, it follows that
$p(P_{z_{1}})=\dotsb=p(P_{z_{r}})$; but then 5.3 gives
$P_{z_{1}}=\dotsb=P_{z_{r}}$, and this clearly contradicts our initial
assumption that $r\geq 2$.

##  7\. Proof of the main theorem

At this point, we can prove the main theorem very quickly by putting together
the results from the previous five sections.

###### Proof of 1.1.

By way of contradiction, assume $B$ is not isomorphic to $\mathbb{P}^{n}$. By
1.2, there must be a minimal degree rational curve $g_{0}:C_{0}\to B$ meeting
$B^{\circ}$ with $d+1=-\deg g_{0}^{*}\mathscr{T}_{B}\leq n$. By 2.2, since
$g_{0}^{*}\mathscr{T}_{B}$ contains a copy of
$\mathscr{T}_{C_{0}}\cong\mathscr{O}(2)$, it follows that
$g_{0}^{*}\mathscr{T}_{B}$ has a trivial factor, so by 2.4 and Lemma 4.4, we
obtain a nontrivial splitting of the variation of Hodge structure $V_{Z}$. A
general curve $g\colon C\to B$ meets the boundary transversely, and since
$b_{2}(B)=1$, it must meet every irreducible component of the boundary that
has dimension $n-1$. By 6.1 it follows that $p\colon Z\to B$ is unramified in
codimension 1, and because $Z$ is normal by construction, $p$ must be an étale
cover by purity of the branch locus [Nag59]. Since $B$ is simply connected, it
follows that $p$ is an isomorphism; this means that there is a nontrivial
splitting of $V$ itself, which contradicts Voisin’s result in 1.3. ∎

######  Remark 7.1.

The fact that $M$ was a full deformation space of a family of rational curves
admitting a splitting was crucial to the argument. Indeed, consider
$S\to\mathbb{P}^{1}$ an elliptic K3 surface, and $X=S^{[n]}\to
B=\mathbb{P}^{n}$ the Hilbert scheme of $n$ points. Then taking the cover
$\mathbb{P}^{1}\times\mathbb{P}^{n-1}\to\mathbb{P}^{n}$ obtained by
quotienting $(\mathbb{P}^{1})^{n}$ by $S_{n-1}$, we have a covering family of
rational curves (namely the $\mathbb{P}^{1}$ fibers) along which the pullback
of $V$ has a fixed part (of rank $2n-2$), and the cover
$\mathbb{P}^{1}\times\mathbb{P}^{n-1}\to\mathbb{P}^{n}$ splits completely over
the open part of these curves. These curves are all tangent to the diagonal of
$\mathbb{P}^{n}$ however, so this does not imply
$\mathbb{P}^{1}\times\mathbb{P}^{n-1}\to\mathbb{P}^{n}$ is unramified (and
indeed it is not), and it also does not imply that the splitting descends (and
indeed it does not).

######  Remark 7.2.

The proof of 1.1 works equally well for a Lagrangian fibration $f:X\to B$ of a
primitive symplectic variety $X$, provided we additionally assume $X$ is
simply connected and that the discriminant is divisorial.

## References

  * [Ara06] Carolina Araujo, _Rational curves of minimal degree and characterizations of projective spaces_ , Math. Ann. 335 (2006), no. 4, 937–951.
  * [Bak22] Ben Bakker, _A short proof of a conjecture by Matsushita_ , arXiv:2209.00604, 2022.
  * [Cam21] Frédéric Campana, _Local projectivity of Lagrangian fibrations on hyperkähler manifolds_ , Manuscripta Math. 164 (2021), no. 3-4, 589–591.
  * [CK89] Eduardo Cattani and Aroldo Kaplan, _Degenerating variations of Hodge structure_ , Astérisque (1989), no. 179-180, 9, 67–96, Actes du Colloque de Théorie de Hodge (Luminy, 1987).
  * [CMSB02] Koji Cho, Yoichi Miyaoka, and N. I. Shepherd-Barron, _Characterizations of projective space and applications to complex symplectic manifolds_ , Higher dimensional birational geometry (Kyoto, 1997), Adv. Stud. Pure Math., vol. 35, Math. Soc. Japan, Tokyo, 2002, pp. 1–88.
  * [DM96] Ron Donagi and Eyal Markman, _Cubics, integrable systems, and Calabi-Yau threefolds_ , Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc., vol. 9, Bar-Ilan Univ., Ramat Gan, 1996, pp. 199–221.
  * [Fre99] Daniel S. Freed, _Special Kähler manifolds_ , Comm. Math. Phys. 203 (1999), no. 1, 31–52.
  * [Fuj78] Takao Fujita, _On Kähler fiber spaces over curves_ , J. Math. Soc. Japan 30 (1978), no. 4, 779–794.
  * [HO09] Jun-Muk Hwang and Keiji Oguiso, _Characteristic foliation on the discriminant hypersurface of a holomorphic Lagrangian fibration_ , Amer. J. Math. 131 (2009), no. 4, 981–1007.
  * [Hwa07] Jun-Muk Hwang, _Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1_ , Ann. Inst. Fourier (Grenoble) 57 (2007), no. 3, 815–823.
  * [Hwa08] by same author, _Base manifolds for fibrations of projective irreducible symplectic manifolds_ , Invent. Math. 174 (2008), no. 3, 625–644.
  * [Keb02] Stefan Kebekus, _Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron_ , Complex geometry (Göttingen, 2000), Springer, Berlin, 2002, pp. 147–155.
  * [Kol86a] János Kollár, _Higher direct images of dualizing sheaves. I_ , Ann. of Math. (2) 123 (1986), no. 1, 11–42.
  * [Kol86b] by same author, _Higher direct images of dualizing sheaves. II_ , Ann. of Math. (2) 124 (1986), no. 1, 171–202.
  * [Kol96] János Kollár, _Rational curves on algebraic varieties_ , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996\.
  * [LT23] Yang Li and Valentino Tosatti, _Special Kähler geometry and holomorphic Lagrangian fibrations_ , 2023.
  * [Mat99] Daisuke Matsushita, _On fibre space structures of a projective irreducible symplectic manifold_ , Topology 38 (1999), no. 1, 79–83.
  * [Mat05] by same author, _Higher direct images of dualizing sheaves of Lagrangian fibrations_ , Amer. J. Math. 127 (2005), no. 2, 243–259.
  * [Nag59] Masayoshi Nagata, _On the purity of branch loci in regular local rings_ , Illinois J. Math. 3 (1959), 328–333.
  * [Pet84] Chris Peters, _A criterion for flatness of Hodge bundles over curves and geometric applications_ , Math. Ann. 268 (1984), no. 1, 1–19.
  * [Sai88] Morihiko Saito, _Modules de Hodge polarisables_ , Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849–995.
  * [Sai90] by same author, _Mixed Hodge modules_ , Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333.
  * [Sai91] by same author, _On Kollár’s conjecture_ , Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., vol. 52, Part 2, Amer. Math. Soc., Providence, RI, 1991, pp. 509–517.
  * [Sch73] Wilfried Schmid, _Variation of Hodge structure: the singularities of the period mapping_ , Invent. Math. 22 (1973), 211–319.
  * [Sch12] Christian Schnell, _Complex analytic Néron models for arbitrary families of intermediate Jacobians_ , Invent. Math. 188 (2012), no. 1, 1–81.
  * [Sch23] by same author, _Hodge theory and Lagrangian fibrations on holomorphic symplectic manifolds_ , arXiv:2303.05364, 2023.
  * [SS22] Claude Sabbah and Christian Schnell, _Degenerating complex variations of Hodge structure in dimension one_ , arXiv:2206.08166, 2022.
  * [Sta18] The Stacks Project Authors, _Stacks Project_ , https://stacks.math.columbia.edu, 2018.
  * [Voi18] Claire Voisin, _Torsion points of sections of Lagrangian torus fibrations and the Chow ring of hyper-Kähler manifolds_ , Geometry of moduli, Abel Symp., vol. 14, Springer, Cham, 2018, pp. 295–326.