# Compact fibrations with hyperkähler fibers

Rodion N. Déev111The article was prepared within the framework of the Academic
Fund Program at the National Research University Higher School of Economics
(HSE) in 2015 (grant 15-05-0033) and supported within the framework of a
subsidy granted to the HSE by the Government of the Russian Federation for the
implementation of the Global Competitiveness Program.

###### Abstract

Essential dimension of a family of complex manifolds is the dimension of the
image of its base in the Kuranishi space of the fiber. We prove that any
family of hyperkähler manifolds over a compact simply connected base has
essential dimension not greater than $1$. A similar result about families of
complex tori is also obtained.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Bogomolov–Beauville–Fujiki form
    2. 2.2 Global Torelli theorem
  3. 3 Geometry of the period space
    1. 3.1 Positive Grassmannians
    2. 3.2 Period space
    3. 3.3 Fibrations by complex tori
    4. 3.4 Borcherds—Katzarkov—Pantev—Shepherd-Barron and Oguiso theorems
  4. 4 Appendix: Lorentzian Kähler metric on the period space

##  1 Introduction

Let $\mathbb{H}$ be the skew-field of quaternions.

######  Definition 1.1.

A manifold $M$ with a left $\mathbb{H}$-action in the tangent bundle is called
almost hypercomplex. Any quaternion $q$ with $q^{2}=-1$ defines an almost
complex structure on $M$, and if all these complex structures are integrable,
$M$ is called hypercomplex.

One does not need to check the integrability condition for all points $q$ in
the 2-sphere $q^{2}=-1$, but only for three quaternions spanning the space of
imaginary ones. Any hypercomplex structure admits a unique torsion-free
connection preserving the hypercomplex structure, called Obata connection [Ka,
Ob].

######  Definition 1.2.

A Riemannian hypercomplex manifold $(M,g)$ is called hyperkähler, if its Obata
and Levi-Civita connections are equal.

One can think of hyperkähler manifolds as of Riemannian manifolds with three
integrable almost complex structures $I$, $J$, $K$ such that $IJ=-JI=K$ which
are Kähler with respect to the Riemannian metric, or as of Riemannian
manifolds with holonomy in the group $\operatorname{Sp}(n)$. When $n=1$,
$\operatorname{Sp}(1)=\operatorname{SU}(2)$ and all the hyperkähler manifolds
are either K3 surfaces or tori.

One can wonder what are the families of hyperkähler manifolds, i. e.
submersions of complex manifolds with hyperkähler fibers. This question is
ultimately closely related to the variations of Hodge structures, because any
family of complex manifolds gives rise to a variation of Hodge structures.
Some complex manifolds are completely determined by their Hodge structures;
statements of such type are known as “Torelli theorems”. Although the global
Torelli theorem for hyperkähler manifolds was proved in 2009 [V1], Torelli
theorem for K3 surfaces has much longer history: its local version was
established in 1964 by G. N. Tjurina [T], whereas its global version was
obtained in 1977 by Vik. S. Kulikov [Ku] (earlier advances in this topic
include a 1971 paper by I. I. Pyatetski-Shapiro and I. R. Shafarevich [PShSh];
a 1978 paper [Bo1] of Bogomolov is also worthy to mention). Because of that
study of the families of complex manifolds was reduced to study of variations
of Hodge structures since olden times. The first advance in this direction is
due to Deligne and Griffiths.

######  Fact 1.1 (P. R. Deligne, Ph. A. Griffiths, 1970).

Any variation of Hodge structures over a compact simply connected base is
trivial.

Of course, Griffiths has proved much stronger statement, called “Theorem of
the Fixed Part” [G, Ch. II, Application 7], but we shall not need it in its
full generality. It follows from this fact that any polarized family of
hyperkähler manifolds over a compact simply connected base is trivial. The
following definition asserts that one cannot drop out the polarization
condition in the Deligne’s and Griffiths’ statement.

######  Definition 1.3.

If $X$ is a hypercomplex (for example, hyperkähler) manifold, then any
imaginary unit quaternion $q$ defines a complex structure on $X$. This gives
rise to a nontrivial almost complex structure on the space
$X\times\mathbb{C}P^{1}$, where we identify $\mathbb{C}P^{1}$ with the unit
sphere in the space of imaginary quaternions: namely, tangent space at point
$(x,q)$ splits as $T_{x}X\oplus T_{q}\mathbb{C}P^{1}$, and one can put the
complex structure $q$ on $T_{x}X$ and the standard one on
$T_{q}\mathbb{C}P^{1}$.

######  Theorem (M. Obata, 1953 [Ob], S. Salamon, 1982 [S], D. Kaledin, 1996
[Ka]).

This almost complex structure is integrable.

The product $X\times\mathbb{C}P^{1}$ with this complex structure is denoted by
$\operatorname{Tw}(X)$ and called the twistor space.

The projection $\operatorname{Tw}(X)\to\mathbb{C}P^{1}$ is holomorphic and
defines a nontrivial family of hyperkähler manifolds. It is not hard to prove
that the twistor space cannot bear any Kähler form [Hi, V3, Proposition 2.13].
The concept of twistors has appeared in works of R. Penrose in the late
1960ies [P], and has been developed by M. F. Atiyah, N. J. Hitchin and I. M.
Singer [AHS] (in context of 4-dimensional Riemannian geometry) and S. Salamon
[S] (in context of quaternionic Kähler geometry).

Seeking for a way to generalize a well-known fact about isotriviality of the
complete families of elliptic curves, R. E. Borcherds, L. Katzarkov, T. Pantev
and N. I. Shepherd-Barron proved in 1997 the following theorems, which allow
to drop out the condition of simply connectedness in Griffiths’ statement at
least for fibrations with fibers K3 surfaces (but need some extra restrictions
on the fibration).

######  Fact 1.2 (R. E. Borcherds, L. Katzarkov, T. Pantev and N. I. Shepherd-
Barron, 1997, [BKPShB]).

Any complete family of minimal Kähler surfaces of Kodaira dimension $0$ and
constant Picard number is isotrivial.

######  Fact 1.3 (R. E. Borcherds, L. Katzarkov, T. Pantev and N. I. Shepherd-
Barron, 1997, [BKPShB]).

Any family of smooth polarized K3 surfaces with polarization of degree 2 over
a projective variety is isotrivial.

They dealt separately with the cases of hyperelliptic, Enriques and K3 fiber,
and the latter was the essential one. Their technique heavily uses the theory
of automorphic forms, namely the properties of the Borcherds’ automorphic form
$\Phi_{12}$ on the Cartan symmetric space for the group
$\mathrm{O}(\mathrm{II}_{2,26})$, and one cannot prove similar results for
fibrations with hyperkähler fibers of dimension greater than 28 in the same
way. It seems that the only generalization of their result to arbitrary
hyperkähler manifolds is the following theorem of K. Oguiso.

######  Fact 1.4 (K. Oguiso, 2000, [Og]).

Let $\mathfrak{X}\to\Delta$ be a nontrivial family of hyperkähler manifolds
over a disc. Then the set of points where the Picard number of the fiber jumps
is a dense countable subset of a disc.

In this paper we give alternative proofs of the Facts 1.2 and 1.4 via newly
developed techniques (whereas the Fact 1.3 cannot be proved with them).

The global Torelli theorem is known for curves, tori, hyperkähler manifolds
and some exotic cases such as cubic threefolds [G, Ch. XII and XIII]. Curves
and threefolds are automatically polarized, so we are not interested in them.
Recall that the essential dimension of a smooth complex fibration is the
dimension of the base in the Kuranishi space of the fiber. The main theorem we
are going to prove is the following.

######  Theorem.

Let $\mathfrak{X}\to B$ be a smooth fibration of compact complex manifolds
with smooth hyperkähler fibers and simply connected base. Then its essential
dimension is no greater than one.

For the fibrations by complex tori a similar result holds.

######  Theorem.

Let $\mathfrak{X}\to B$ be a smooth fibration of compact complex manifolds
with fibers $n$-dimensional complex tori and simply connected base. Then its
essential dimension is no greater than $\frac{n(n-1)}{2}$.

Here is a brief outline of the paper. In Section 2 we give some well-known
facts about hyperkähler manifolds, such as the global Torelli theorem. In
Section 3 we describe some geometry of the moduli space of hyperkähler
manifolds, which is also widely known. Following those facts, we prove the
main theorems and derive the Borcherds–Katzarkov–Pantev–Shepherd-Barron and
Oguiso theorems. Subsequently, in Section 4 we give some other relevant
observations on the fibrations with hyperkähler fibers.

##  2 Preliminaries

It follows from a straightforward calculation that a form

$\Omega_{I}(u,v)=g(Ju,v)+\sqrt{-1}g(Ku,v)$

on a compact hyperkähler manifold $(M,g,I,J,K)$ is holomorphic with respect to
the complex structure $I$.

Hyperkähler manifolds with $h^{2,0}(M,I)=1$ are known as simple. For simple
hyperkähler manifolds the cohomology class $[\Omega_{I}]$ spans the line
$H^{2,0}(M,I)\subset H^{2}(M,\mathbb{C})$. The famous Bogomolov decomposition
theorem [Bo2] states that any hyperkähler manifold admits a finite cover which
is a product of torus and simple hyperkähler manifolds, so we are interested
in the simple ones. From now onwards, until otherwise stated, while speaking
of a hyperkähler manifold $(M,g,I,J,K)$ as of complex manifold, we would
consider the complex structure $I$. Consider a complex family
$(\mathfrak{X},\mathcal{I})$ of simple hyperkähler manifolds over a simply
connected base $B$ (so that $R^{2}\pi_{*}\mathbb{C}$ is a trivial bundle), and
let $X_{b}$ denote the fiber over a point $b\in B$. Then we can define the
period map $B\xrightarrow{per}\mathbb{P}(H^{2}(X,\mathbb{C}))$ which sends the
point $b$ to the line spanned by the class $[\Omega_{\mathcal{I}|_{X_{b}}}]$.

If the base $B$ is not simply connected, the period map is defined as a map
from the universal cover $\widetilde{B}$. The fundamental group of the base
acts on the universal cover and on the period space (as the monodromy group of
the local system $R^{2}\pi_{*}\mathbb{C}$), and the period map is equivariant
with respect to these two actions. One can try to obtain a period map from the
base $B$ into the quotient of the period space by this action, but this
quotient can have very poor topology. It is known to be an orbifold in the
case of the polarized fibration with fiber K3 surface. However, in other cases
the topology on the factor can be very non-Hausdorff, e. g. every two non-
empty subsets intersect.

###  2.1 Bogomolov–Beauville–Fujiki form

For a K3 surface $X$ the intersection form is a bilinear form on the space
$H^{2}(X,\mathbb{R})$; by the Hodge index theorem, its signature is $(3,19)$.
We shall need a similar form on the second cohomology space of a hyperkähler
manifold, which would have the signature $(3,b_{2}-3)$.

######  Fact 2.1 (F. A. Bogomolov, 1978 [Bo1], A. Beauville, 1983 [Be], A.
Fujiki, 1985, 1987 [F]).

Let $X$ be a hyperkähler manifold of real dimension $4n$. There exists a
unique primitive quadratic form $q\colon H^{2}(X,\mathbb{Z})\to\mathbb{Z}$ and
a constant $c$ such that for any $\alpha\in H^{2}(X)$ one has

$\int_{X}\lambda^{2n}=cq(\alpha)^{n}$

and for non-zero $\sigma\in H^{2,0}(X)$ one has
$q(\sigma+\overline{\sigma})>0$.

For more details, see [OG].

This form is uniquely determined by this condition. One can write down an
explicit formula for $q$ (here we use same letter $q$ for the polarization of
the Bogomolov–Beauville–Fujiki quadratic form):

$cq(\alpha,\beta)=2\int_{X}\alpha\wedge\beta\wedge\Omega_{I}^{n-1}\wedge\overline{\Omega_{I}}^{n-1}-\\\
-\frac{n-1}{n}\frac{(\int_{X}\alpha\wedge\Omega_{I}^{n-1}\wedge\overline{\Omega_{I}}^{n})(\int_{X}\beta\wedge\Omega_{I}^{n}\wedge\overline{\Omega_{I}}^{n-1})}{\int_{X}\Omega_{I}^{n}\wedge\overline{\Omega_{I}}^{n}},$

where the positive constant on the left-hand side is needed for the form $q$
to be integer.

It is positive definite on the real part of the space spanned by $\Omega_{I}$,
$\overline{\Omega_{I}}$ and the Kähler form $\omega$, and negative definite on
the primitive forms (i. e. it has signature $(3,b_{2}-3)$). The image of the
period map lies in the set of lines spanned by the classes $\alpha$ such that
$q(\alpha)=0$ and $q(\alpha+\overline{\alpha})>0$ (or, equivalently,
$q(\alpha,\overline{\alpha})>0$).

###  2.2 Global Torelli theorem

The main reference for this section is [V1].

######  Definition 2.1.

Let $(X,g)$ be a Riemannian manifold, and $\mathfrak{I}$ be the set of complex
structures on $X$ which can be extended to hyperkähler structures. The group
$\operatorname{Diff}_{0}(X)$ of oriented diffeomorphisms of $X$ act on
$\mathfrak{I}$. The Teichmüller space $\operatorname{Teich}(X)$ of $X$ is the
factor $\mathfrak{I}/\operatorname{Diff}_{0}(X)$.

The space $\operatorname{Teich}(X)$ admits a period map $(X,I)\mapsto
H^{2,0}(X,I)\in\mathbb{P}(H^{2}(X,\mathbb{C}))$ into
$\mathbb{P}(H^{2}(X,\mathbb{C}))$ in the same way as the base of any fibration
with hyperkähler fibers (provided that the base is simply connected). The
image of the period map lies in the period space
$\operatorname{Per}(X)=\\{[\alpha]\in\mathbb{P}(H^{2}(X,\mathbb{C}))\mid
q(\alpha)=0,q(\alpha,\overline{\alpha})>0\\}$. Actually,
$\operatorname{Teich}$ is not Hausdorff, but its non-Hausdorff points
correspond to the bimeromorphically equivalent hyperkähler manifolds.
Moreover, there exists a Hausdorff space $\operatorname{Teich}_{b}$ with map
$\operatorname{Teich}\to\operatorname{Teich}_{b}$ such that any map from
$\operatorname{Teich}$ to a Hausdorff space factorizes through this map (so
the period map $\operatorname{Teich}_{b}(X)\to\operatorname{Per}(X)$ is well-
defined).

######  Fact 2.2 (M. Verbitsky, 2009 [V1]).

The map $per\colon\operatorname{Teich}_{b}\to\operatorname{Per}$ is a
diffeomorphism on each connected component of $\operatorname{Teich}_{b}$.

##  3 Geometry of the period space

###  3.1 Positive Grassmannians

######  Definition 3.1.

Let $V$ be a vector space over $\mathbb{R}$ with a non-degenerate bilinear
form $\langle\cdot\mid\cdot\rangle$. We shall consider a subset in the
oriented Grassmannian $\operatorname{Gr}_{2}(V)$ consisting of oriented
2-planes with positive definite restriction of the bilinear form (we shall
call such planes positive). This subset is called the positive Grassmannian
and denoted $\operatorname{Gr}_{++}(V)$.

######  Lemma (see e. g. C. LeBrun, 1993 [LB]).

The set $\operatorname{Gr}_{++}(V)$ is in one-to-one correspondence with the
projectivization of the set of vectors $v\in V\otimes\mathbb{C}$ such that
$\langle v\mid v\rangle=0$ and $\langle v\mid\overline{v}\rangle>0$ (we shall
call them positive null-vectors).

###### Proof.

If $W\subseteq V$ is a point in $\operatorname{Gr}_{++}(V)$, then the cone of
null-vectors in $W\otimes\mathbb{C}$ consists of two lines, which are
interchanged by the complex conjugation. If $w$ is a vector spanning one of
the lines, then one needs to be $\langle w\mid\overline{w}\rangle>0$ for the
inner product on $W$ to be positive definite, and vectors $w$ and
$\overline{w}$ correspond to two copies of the plane $W$ coming with different
orientations: if the basis $\\{w+\overline{w},i(w-\overline{w})\\}$ is
positively oriented, then the null-vector corresponding to the plane $W$ is
$w$, and $\overline{w}$ otherwise.

Conversely, if $v\in V\otimes\mathbb{C}$ is such that $\langle v\mid
v\rangle=0$ and $\langle v\mid\overline{v}\rangle>0$, then vectors
$v+\overline{v}$ and $i(v-\overline{v})$ are both real and linearly
independent, so they span a plane in $V$ with an orientation. It is easy to
see that the metric on this plane is positive definite. ∎

Let $v\in V$ be a non-zero vector. Then the subset of positive 2-planes
orthogonal to $v$ is exactly
$\operatorname{Gr}_{++}(v^{\perp})\subset\operatorname{Gr}_{++}(V)$. Indeed,
the positive 2-plane $W$ is orthogonal to $v$ if and only if both $w$ and
$\overline{w}$ are.

The tangent space to $\operatorname{Gr}_{++}(V)$ at the point $W$ is the same
as to the Grassmannian, $\operatorname{Hom}(W,V/W)$. However, $W$ is an
oriented plane with positive definite metric, so it can be regarded as a one-
dimensional complex vector space, turning the tangent space into a complex
one. It gives an almost complex structure on $\operatorname{Gr}_{++}(V)$,
which is the same as the restriction of the complex structure from the complex
quadric $\\{\langle v\mid v\rangle=0\\}\subset\mathbb{P}(V\otimes\mathbb{C})$.

######  Proposition 3.1.

$\operatorname{Gr}_{++}(\mathbb{R}^{2,n})$ is a Stein manifold.

###### Proof.

$\operatorname{Gr}_{++}(\mathbb{R}^{2,n})=\operatorname{SO}(2,n)/\operatorname{SO}(2)\times\operatorname{SO}(n)$,
so it is a symmetric domain of non-compact type. Due to a theorem of É. Cartan
[C] it can be holomorphically embedded into a complex space as a bounded
domain.

See [He, Theorem 7.1] for a complete proof. ∎

######  Proposition 3.2.

$\operatorname{Gr}_{++}(\mathbb{R}^{2,1})$ is a topological disc.

###### Proof.

Let $u,v,w$ be the orthogonal basis of $\mathbb{R}^{2,1}$ such that
$\|v\|^{2}=\|w\|^{2}=1$ and $\|u\|^{2}=-1$. Any 2-plane $W$ with positive
definite restriction of the metric projects along $u$ onto the plane
$W_{0}=\langle v,w\rangle$ isomorphically (just because $W$ cannot contain the
kernel of this projection, the line spanned by $u$). So $W=\langle
v+au,w+bu\rangle$ for some real numbers $a$ and $b$, and different pairs of
numbers define different planes. The restriction of metric on $W$ is positive
definite iff for any real numbers $x,y$ (at least one of which is not equal to
zero) one has $0<\|x(v+au)+y(w+bu)\|^{2}=x^{2}+y^{2}-(ax+by)^{2}$, which is
equivalent to the condition $a^{2}+b^{2}<1$.

One can also send a positive plane in $\mathbb{R}^{2,1}$ into its orthogonal,
which is a line spanned by a negative vector, and obtain a representation of
$\operatorname{Gr}_{++}(\mathbb{R}^{2,1})$ as the projectivization of the
negative cone in $\mathbb{R}^{2,1}$, which is precisely the Cayley–Klein model
for the Bolyai–Lobachevskian plane. ∎

######  Proposition 3.3.

$\operatorname{Gr}_{++}(\mathbb{R}^{n,m})$ can be retracted onto
$\operatorname{Gr}_{++}(\mathbb{R}^{n,0})$. In particular,
$\operatorname{Gr}_{++}(\mathbb{R}^{2,n})$ is contractible.

###### Proof.

Let $l\subset V$ be a line spanned by a negative vector. Then if $W$ is a
positive 2-plane, then its projection along $l$ in $l^{\perp}$ is also a
positive 2-plane. This defines a fibration
$\operatorname{Gr}_{++}(V)\to\operatorname{Gr}_{++}(l^{\perp})$, and its fiber
over a plane $W^{\prime}\subseteq l^{\perp}$ is a set of positive 2-planes in
the linear hull of $W^{\prime}$ and $l$, i. e.
$\operatorname{Gr}_{++}(\mathbb{R}^{2,1})$, which is contractible due to the
previous Proposition. ∎

###  3.2 Period space

Now we shall study the geometry of the period space of a hyperkähler manifold
$X$ itself, which is identified with the space
$\operatorname{Gr}_{++}(H^{2}(X,\mathbb{R}))$ because of LeBrun’s lemma. From
now onwards the letter $V$ stands for a real vector space with a metric $q$ of
signature $(3,n)$.

Let $U\subset V$ be a 3-dimensional space with $q|_{U}$ positive definite.
Then any 2-plane in it is positive, and they constitute a rational curve
$\mathbb{C}P^{1}=\operatorname{Gr}_{++}(U)\subset\operatorname{Gr}_{++}(V)$.
In the case $V=H^{2}(X,\mathbb{R})$ there exist a natural positive 3-subspace
in $V$ spanned by the Kähler forms $\omega_{I}$, $\omega_{J}$ and
$\omega_{K}$, and the corresponding family is the twistorial family. Because
of that we shall call such curves twistorial, and denote the rational curve
consisting of 2-planes in a positive 3-subspace $U$ by
$\operatorname{Gr}_{++}(U)$. Twistorial curves are parametrized by the
manifold $\operatorname{Gr}_{+++}(V)$ of positive 3-subspaces in the space
$V$. Unlike the positive Grassmannian of 2-planes $\operatorname{Gr}_{++}(V)$,
the Grassmannian of positive 3-subspaces $\operatorname{Gr}_{+++}(V)$ do not
carry a natural complex structure (for example, in the case of period space of
$K3$ surfaces its real dimension equals $57$). On the other hand, the rational
curves in $\operatorname{Gr}_{++}(V)$ are parametrized by the Hilbert scheme,
which is a scheme over $\mathbb{C}$, so twistorial curves can be deformed to
non-twistorial. It is also clear that the normal bundles of the twistorial
curves are ample. One can map a curve of any genus into twistorial curve via
ramified covering and then deform it into an embedded curve (that is possible
because its normal bundle is generated by global sections due to a theorem of
J. Kollár [Ko]). That gives examples of many families of hyperkähler manifolds
over curves.

Let $v\in V$ be a positive vector. Then the space $v^{\perp}$ has signature
$(2,n)$, and $\operatorname{Gr}_{++}(v^{\perp})$ can be identified with a
contractible bounded domain in a complex vector space. On the other hand, it
is a divisor in the space $\operatorname{Gr}_{++}(V)$. We shall call it a
Cauchy divisor (because of reasons explained in Section 4) and denote as
$\operatorname{Cau}_{v}$. For any 2-plane $W\in\operatorname{Gr}_{++}(V)$ the
set of Cauchy divisors passing through $W$ is the projectivization of the
positive cone in $W^{\perp}$. It is easy to see that twistorial curves
intersect the Cauchy divisors at one point.

######  Proposition 3.4.

The period space contains no compact submanifolds of dimension greater than
$1$.

###### Proof.

If $X\subset\operatorname{Per}$ is a compact submanifold, then
$\operatorname{Cau}_{v}\cap X$ is a divisor on $X$. If it is not empty, it is
a compact submanifold of the Stein manifold $\operatorname{Cau}_{v}$, i. e. a
set of points. Therefore the dimension of $X$ needs to be equal to $1$.

One can prove this theorem the other way around: the period space
$\operatorname{Per}$ is a subset of a quadric in a complex projective space,
so it carries a positive $(1,1)$-form $\omega$. Due to the positivity one has
$\int_{X}(\omega|_{X})^{\dim X}>0$. However, the period space
$\operatorname{Per}$ retracts onto the twistorial curve, so $\omega^{\dim
X}=d\eta$ for some $(2\dim X-1)$-form $\eta$ and the integral needs to vanish
unless $\dim X>1$. ∎

Now we can prove the main statement.

######  Proposition 3.5.

Any family of hyperkähler manifolds over a compact simply connected base has
essential dimension not greater than 1.

###### Proof.

The image of the period map $per(B)$ is a compact submanifold in the period
space $\operatorname{Per}$, thus a curve or a point. ∎

###  3.3 Fibrations by complex tori

In a way analogous to the stated above one can try to make a similar statement
for the fibrations by $n$-dimensional complex tori. However, families of tori
over a base of dimension greater that $1$ do exist, some of them were
described in [V2]. The Teichmüller space of the $n$-dimensional complex torus
is the space of linear complex structures on the real vector space $V$ of
dimension $2n$, hence isomorphic to
$\operatorname{GL}^{+}(2n,\mathbb{R})/\operatorname{GL}(n,\mathbb{C})$ (in
particular, its real dimension is $4n^{2}-2n^{2}=2n^{2}$). It contains plenty
of complex submanifolds – namely, for every Eucledian metric $g$ on $V$ one
can define the submanifold consisting of complex structures orthogonal with
respect to $g$. These submanifolds are the symmetric spaces
$\operatorname{SO}(2n)/\operatorname{U}(n)$, and the complex structure on
these symmetric spaces coincides with the restriction of the complex structure
on $\operatorname{GL}^{+}(2n,\mathbb{R})/\operatorname{GL}(n,\mathbb{C})$. The
real dimension of these submanifolds is equal to
$\frac{2n(2n-1)}{2}-n^{2}=n^{2}-n$.

On the other hand, any symplectic form $\psi$ on the vector space $V$
determines the submanifold $\operatorname{Cau}_{\psi}$ in the space of complex
structures on it as follows: the subvariety
$\operatorname{Cau}_{\psi}\subset\operatorname{GL}^{+}(2n,\mathbb{R})/\operatorname{GL}(n,\mathbb{C})$
is the set of complex structures $I$ for which the form
$\psi_{I}(u,v)=\psi(u,Iv)$ is symmetric and positive definite. A well-known
2-out-of-3 property of the unitary group asserts that a symplectic form and a
Eucledian metric on a vector space uniquely determine the complex structure on
it, in other words, the submanifold of complex structures orthogonal with
respect to fixed metric intersects the submanifold $\operatorname{Cau}_{\psi}$
by $0$-cycle, thence
$\dim_{\mathbb{C}}\operatorname{Cau}_{\omega}=n^{2}-\frac{n^{2}-n}{2}=\frac{n^{2}+n}{2}$.
Obviously, such submanifolds pass through each point in the Teichmüller space.
But these submanifolds are biholomorphic to the Siegel upper-half space, in
particular, they are Stein [M, Ch. 2]. Therefore the Teichmüller space of an
$n$-dimensional complex torus cannot contain compact submanifolds of dimension
greater than $\frac{n^{2}-n}{2}$.

So, we have proved the following

######  Proposition 3.6.

Any family of $n$-dimensional complex tori over a compact simply connected
base has essential dimension no greater than $\frac{n^{2}-n}{2}$.

###  3.4 Borcherds—Katzarkov—Pantev—Shepherd-Barron and Oguiso theorems

For an integral vector $v\in H^{2}(X,\mathbb{Z})$ the divisor
$\operatorname{Cau}_{v}$ consists of complex structures $I$ for which one has
$v\in H^{1,1}(X,I)$ (i. e., the corresponding 2-plane in $H^{2}(X,\mathbb{R})$
lies in the orthogonal to $v$). The subset $S_{Z}$ in the projectivization
$\mathbb{P}T^{*}{\operatorname{Per}}$ consisting of hyperplanes tangent to the
integral Cauchy divisors is dense in an open subset of
$\mathbb{P}T^{*}{\operatorname{Per}}$ consisting of tangent hyperplanes to all
Cauchy divisors $S$: hyperplanes tangent to Cauchy divisors are orthogonal to
positive vectors, and lines spanned by positive integral vectors are dense in
the positive cone.

Now, for any holomorphic disc $\Delta$ we can consider a point $p\in\Delta$
together with a Cauchy divisor $D$ passing through $p$. The tangent hyperplane
$t=T_{p}D\subset T_{p}{\operatorname{Per}}$ is a point in $S$, and we can pick
up a point $t^{\prime}\in S_{Z}$ arbitrarily close to $t$. The divisor
corresponding to $t^{\prime}$ would be an integral Cauchy divisor intersecting
$\Delta$. By shrinking $\Delta$, we can show that union of all integral Cauchy
divisors interset any holomorphic disc in a dense subset. On the other hand,
this union is the set where the Picard rank jumps. This proves Fact 1.4 (and
thence the Fact 1.2).

##  4 Appendix: Lorentzian Kähler metric on the period space

In the present section, we shall also deal with the fibrations by hyperkähler
manifolds over noncompact or non-simply connected base. All fibrations are
assumed to be such that the corresponding period map is an immersion (so that
the universal covers of their bases are subvarieties in the period space
$\operatorname{Per}$).

######  Definition 4.1.

Let $(X,g,I)$ be a complex manifold equipped with a metric of signature
$(+,+,-,-,\dots)$. If the 2-form $\omega(u,v)=g(Iu,v)$ is closed, then we
shall call such manifold a Lorentzian Kähler manifold.

Due to the LeBrun’s lemma stated in the Section 3, the period space
$\operatorname{Per}$ is in fact the positive Grassmannian
$\operatorname{Gr}_{++}(\mathbb{R}^{3,b_{2}-3})$. Its tangent space at the
point $W\subset V$ is equal to $\operatorname{Hom}(W,W^{\perp})$ and carries a
natural (and hence $\operatorname{SO}(3,b_{2}-3)$-invariant) metric of
indefinite signature $(2,2b_{2}-6)$. We shall denote in by
$g_{\operatorname{Per}}$ and put
$\omega_{\operatorname{Per}}(u,v)=g_{\operatorname{Per}}(Iu,v)$. This is a
non-degenerate 2-form.

######  Proposition 4.1.

$d\omega_{\operatorname{Per}}=0.$

###### Proof.

Stabilizer of a point $W$ is a subgroup
$\operatorname{SO}(W)\times\operatorname{SO}(W^{\perp})\subset\operatorname{SO}(3,b_{2}-3)$.
The group $\operatorname{SO}(W)=\operatorname{SO}(2)$ contains an operator
$-\operatorname{Id}$, so the $\operatorname{SO}(3,b_{2}-3)$-invariant form
$d\omega_{\operatorname{Per}}$ would be invariant under the fiberwise operator
$-\operatorname{Id}$. However, it is a 3-form and thence vanishes. ∎

Nevertheless, this form is not exact because of the following reason:

######  Proposition 4.2.

Let $U\subset V=\mathbb{R}^{3,b_{2}-3}$ be a 3-subspace with positive definite
metric, and $\operatorname{Gr}_{++}(U)\subset\operatorname{Gr}_{++}(V)$ be the
corresponding twistorial curve. Then the restriction
$\omega_{\operatorname{Per}}|_{\operatorname{Gr}_{++}(U)}$ is the Fubini–Study
form.

###### Proof.

Actually, one can naturally associate such a form to any positive Grassmannian
$\operatorname{Gr}_{++}$, and it would be compatible to its inclusions
obtained from ones of vector spaces. The fact that this form on the
$\mathbb{C}P^{1}=\operatorname{Gr}_{++}(\mathbb{R}^{3,0})$ is its Fubini–Study
form may be regarded as its definition. ∎

As $\operatorname{Gr}_{++}(\mathbb{R}^{3,n})$ retracts onto any twistorial
curve, its second cohomology is one-dimensional, and it is spanned by the
cohomology class $[\omega_{\operatorname{Per}}]$. Moreover, one holds the
following

######  Proposition 4.3.

Any $\operatorname{SO}(3,b_{2}-3)$-invariant 2-form on the period space
$\operatorname{Per}$ is a multiple of the form $\omega_{\operatorname{Per}}$.

###### Proof.

Invariant tensors on homogenuous spaces are determined by its value at one
point, which needs to be an invariant tensor in the tangent space under the
action of the stabilizer of this point. So, one needs to check that the form
$g_{\operatorname{Per}}|_{W}$ is a unique
$\operatorname{SO}(2)\times\operatorname{SO}(1,b_{2}-3)$-invariant form on the
space $\operatorname{Hom}(W,W^{\perp})$. This 2-form defines a representation
homomorphism
$\operatorname{Hom}(W,W^{\perp})\to\operatorname{Hom}(W^{\perp},W)$. But
$\operatorname{Hom}(W,W^{\perp})=W\otimes(W^{\perp})^{*}$. The space $W$
carries naturally the irreducible representation of $\operatorname{SO}(2)$,
and $(W^{\perp})^{*}$ is an irreducible representation of
$\operatorname{SO}(1,b_{2}-3)$, so $\operatorname{Hom}(W,W^{\perp})$ is an
irreducible representation of
$\operatorname{SO}(2)\times\operatorname{SO}(1,b_{2}-3)$. Schur’s lemma
implies that the space of homomorphisms between
$\operatorname{Hom}(W,W^{\perp})$ and its dual is one-dimensional. This proves
the Proposition. ∎

Because of invariance of the form $\omega_{\operatorname{Per}}$ on the period
space, its pullback
$per^{*}(\omega_{\operatorname{Per}})\in\Omega^{2}(\widetilde{B})$ is
invariant under the $\pi(B)$-action, so it descends to the base and thus
defines an invariant of a family $\mathfrak{X}\to B$ of hyperkähler manifolds
in the space of 2-forms on $B$. We shall denote the 2-form on $B$ obtained via
this construction as $\omega_{\mathfrak{X}}\in\Omega^{2}(B)$.

One can find some similarities between the Lorentzian Kähler geometry of the
positive Grassmannian $\operatorname{Gr}_{++}(\mathbb{R}^{3,n})$ and geometry
of usual Lorentzian manifolds. For example, twistorial curves resemble
timelike geodesics, whilst the divisors $\operatorname{Cau}_{v}$ are similar
to Cauchy hypersurfaces. Nevertheless, this similarity is not complete: for
example, while on a Lorentzian manifold $X$ containing a Cauchy hypersurface
$M$ a bunch of timelike geodesics startled orthogonally from $M$ defines a
decomposition $X=M\times\mathbb{R}$ [ChN, Section 3], twistorial curves
orthogonal to a Cauchy divisor $\operatorname{Cau}_{v}$ massively intersect,
forming somewhat like a Lefschetz pencil.

#### Acknowledgements.

I am grateful to M. Verbitsky for turning my mind to the problem, fruitful
discussions and help with preparation of the manuscript, and to D. Kaledin and
the anonymous referee of this paper on the Möbius Contest for pointing to
serious flaws.

This work was supported in part by the Möbius Contest Foundation for Young
Scientists.

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