# Extension properties of orbit spaces for proper actions revisited

Sergey A. Antonyan
Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional
Autónoma de México, 04510 México City, México<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a locally compact Hausdorff group. We study orbit spaces of
equivariant absolute neighborhood extensors ($G$-${\rm ANE}$’s) for the class
of all proper $G$-spaces that are metrizable by a $G$-invariant metric. We
prove that if a $G$-space $X$ is a $G$-${\rm ANE}$ and all $G$-orbits in $X$
are metrizable, then the $G$-orbit space $X/G$ is an ANE. If $G$ is either a
Lie group or an almost connected group, then for any closed normal subgroup
$H$ of $G$, the $H$-orbit space $X/H$ is a $G/H$-ANE provided that all
$H$-orbits in $X$ are metrizable.

2020 Mathematics Subject Classification. 54C55; 54C20; 54H15; 57S20

Key words and phrases. Proper $G$-space; $G$-ANE; Orbit space; Slice

The author was supported by grants IN-100123 from PAPIIT (UNAM) and A1-S-7897
from CONACYT

##  1\. Introduction

In [3, Question 4], the following problem about extension properties of orbit
spaces was posed.

######  Problem 1.1 (Orbit Space Problem).

Let $G$ be a compact group and $X$ a $G$-ANR. Is then the orbit space $X/G$ an
ANR?

This problem has been solved affirmatively first in [5, Theorem 8] for $G$ a
compact metrizable group. The result was extended in [9, Theorem 1.1] to the
case of any compact Hausdorff group $G$ and any $X\in G$-A(N)E with all orbits
$G(x)\subset X$ metrizable. More precisely, the following more general result
was proved in [9, Theorem 1.1].

######  Theorem 1.2 ([9]).

Let $G$ be a compact group and $H$ a closed normal subgroup of $G$. Suppose
$X$ is a $G$-space such that all $H$-orbits in $X$ are metrizable. If $X$ is a
$G$-${\rm ANE}$ (respectively, a $G$-${\rm AE}$), then the $H$-orbit space
$X/H$ is a $G/H$-${\rm ANE}$ (respectively, a $G/H$-${\rm AE}$).

This result has been widely applied to the study of the topology of Banach-
Mazur compacta (see [8], [10], [11]). Other applications can be found in [5],
[17] and [24].

The possibility of a further extension of Theorem 1.2 to the case of proper
actions (in the sense of R. Palais [23]) of locally compact groups has been
studied in several papers by the author [9], [12], [13], [14].

The main purpose of this article is to carefully review the proofs of various
subsequent generalizations of Theorem 1.2 and make corrections and
improvements where necessary. Along the way we will provide very short proofs
for the following three theorems about preservation of the extension
properties by the orbit space functor. More general versions of these theorems
are also presented in Section 6.

######  Theorem 1.3 (Proper actions of locally compact groups and $H=G$).

Let $G$ be a locally compact group and $X$ a proper $G$-space such that all
$G$-orbits in $X$ are metrizable. If $X$ is a $G$-${\rm ANE}$ then the
$G$-orbit space $X/G$ is an ANE.

A proof of this theorem was provided in [12, Theorem 6.4]. In that proof the
following affirmation, that we state here in the form of a proposition, was
used.

######  Proposition 1.4.

Let $G$ be a topological group and $K$ any closed subgroup of $G$. If $S$ is a
$K$-space, then the $G$-orbit space $({G\times_{K}S})/G$ is homeomorphic to a
retract of the $K$-orbit space $({G\times_{K}S})/K$.

Here ${G\times_{K}S}$ denotes the twisted product of $G$ and $S$ with respect
to the subgroup $K$ defined in Section 2.

The argument for the proof of this statement given in the proof of [12,
Theorem 6.4], unfortunately, works correctly only for an Abelian acting group
$G$. Namely, in that proof the formula $(G\times_{K}S)/G\cong G/K\times S/K$
was used which, however, is valid only for an Abelian group $G$ (see [9,
Proposition 2]).

Although we will need Proposition 1.4 here only in the case of a compact
subgroup $K\subset G$, we provide a detailed proof even for any closed
subgroup $K\subset G$ in the Appendix. Thus, the gap in the proof of [12,
Theorem 6.4] is easily filled.

The second theorem is the following.

######  Theorem 1.5 (Proper actions of almost connected groups).

Let $G$ be an almost connected locally compact group, $H$ a closed normal
subgroup of $G$, and $X$ a proper $G$-space that admits a $G$-invariant
metric. If $X$ is a $G$-${\rm ANE}$, then the $H$-orbit space $X/H$ is a
$G/H$-${\rm ANE}$.

The proof of this theorem given in [9, Theorem 3(1)] is correct only for
compact subgroups $H\subset G$. For an arbitrary closed subgroup $H\subset G$
our argument in [9, Theorem 3(3)] is based on [9, Proposition 8(1)] the proof
of which, unfortunately, contains a gap. The proof of Theorem 1.5 given in
Section 4 is based on the following well-known proposition.

######  Proposition 1.6.

Let $G$ be an almost connected locally compact group, $K$ a compact maximal
subgroup of $G$, and $S$ a $K$-space. Then $S$ is a $K$-equivariant retract of
the twisted product $G\times_{K}S$.

###### Proof.

This result follows from a result of Abels [1, Theorem 2.1] according to which
$G\times_{K}S$ is $K$-homeomorphic to a product $T\times S$ endowed with the
diagonal action of $K$, where $T$ is a finite-dimensional linear $K$-space. In
this case the map $(t,s)\mapsto(0,s)$ is a $K$-equivariant retraction of
$T\times S$ onto $\\{0\\}\times S$ which, in turn, is $K$-homeomorphic to $S$.
∎

The third theorem is the following.

######  Theorem 1.7 (Proper actions of any Lie groups).

Let $G$ be a Lie group, $H$ a closed normal subgroup of $G$, and $X$ a proper
$G$-space. If $X$ is a $G$-${\rm ANE}$, then the $H$-orbit space $X/H$ is a
$G/H$-${\rm ANE}$.

In [13, Theorem 1.1] a proof of this theorem was given even for any locally
compact acting group $G$. Again, this proof used a formula (see [13, formula
(3.3)]), which is only valid for Abelian groups. Below we give a very brief
proof of this theorem in the case of proper actions of arbitrary Lie groups,
which is practically the most important case. This proof is based on the
following result we proved in [13, Proposition 4.1].

######  Proposition 1.8 ([13]).

Let $G$ be a Lie group, $K$ a compact subgroup of $G$, and $S$ a $K$-space.
Then $S$ is a neighborhood $K$-equivariant retract of the twisted product
$G\times_{K}S$.

Regarding the “$G$-AE version” of the above results, we have the following
theorem proven in [14, Theorem 7.1].

######  Theorem 1.9.

Let $G$ be a locally compact group and $X$ any $G$-${\rm{\rm AE}}$. Assume
that $H$ is an almost connected normal subgroup of $G$ such that all
$H$-orbits in $X$ are metrizable. Then the $H$-orbit space $X/H$ is a
$G/H$-${\rm AE}$.

We note that almost connectedness of $H$ is essential in this theorem. Indeed,
let $G=\mathbb{R}$, the reals, $X=\mathbb{R}$ and $H=\mathbb{Z}$, the
integers. Then the translation action is a proper action of $G$ on $X$, and by
[2, Theorem 4.4], $X$ is a $G$-${\rm{\rm AE}}$. However $X/H$, being a circle,
is not an ${\rm{\rm AE}}$.

In Section 6 we strengthen Theorems 1.3, 1.5 and 1.7, discarding in their
statements the hypothesis about the properness of the $G$-space $X$. This is
achieved by using the lifting properties of equivariant embeddings.

Before passing to the details of the proofs, it is convenient to recall some
auxiliary notions and results.

##  2\. Some basic definitions and auxiliary results

Throughout the paper the letter $G$ will denote a locally compact Hausdorff
group unless otherwise is stated; by $e$ we denote the unity of $G$.

All topological spaces are assumed to be Tychonoff (= completely regular and
Hausdorff). The basic ideas and facts of the theory of $G$-spaces or
topological transformation groups can be found in Bredon [18] and in Palais
[22]. Our basic references on proper group actions are Palais [23] and Abels
[2]. For the equivariant theory of retracts the reader can see, for instance,
[4], [5] [9], [12] and [13].

For the convenience of the reader we recall, however, some more special
definitions and facts.

Here we deal with $G$-spaces. If $X$ and $Y$ are two $G$-spaces then a
continuous map $f:X\to Y$ is called a $G$-map, if $f(gx)=gf(x)$ for all $x\in
X$ and $g\in G$. If a $G$-map is a homemorphism then it is called a
$G$-homeomorphism.

If $X$ is a $G$-space and $H$ a subgroup of $G$ then, for a subset $S\subset
X$, $H(S)$ denotes the $H$-saturation of $S$, i.e., $H(S)$= $\\{hs|\ h\in H,\
s\in S\\}$. In particular, $H(x)$ denotes the $H$-orbit $\\{hx\in X|\ h\in
H\\}$ of $x$. The quotient space of all $H$-orbits is called the $H$-orbit
space and denoted by $X/H$.

If $H(S)$=$S$, then $S$ is said to be an $H$-invariant set. A $G$-invariant
set will simply be called an invariant set.

For a closed subgroup $H\subset G$, by $G/H$ we will denote the $G$-space of
cosets $\\{gH|\ g\in G\\}$ under the action induced by left translations.

If $X$ is a $G$-space and $H$ a closed normal subgroup of $G$, then the
$H$-orbit space $X/H$ will always be regarded as a $G/H$-space endowed with
the following action of the group $G/H$: $(gH)*H(x)=H(gx),\ \ \text{where}\ \
gH\in G/H,\ H(x)\in X/H$.

For any $x\in X$, the subgroup $G_{x}=\\{g\in G\mid gx=x\\}$ is called the
stabilizer (or stationary subgroup) at $x$.

Let $X$ be a $G$-space. Two subsets $U$ and $V$ in $X$ are called thin
relative to each other [23, Definition 1.1.1], if the set $\langle
U,V\rangle=\\{g\in G|\ gU\cap V\neq\emptyset\\}$ has a compact closure in $G$.
A subset $U$ of a $G$-space $X$ is called small, if every point in $X$ has a
neighborhood thin relative to $U$. A $G$-space $X$ is called proper (in the
sense of R. Palais), if every point in $X$ has a small neighborhood. We refer
to the seminal paper of R. Palais [23] for further information about proper
$G$-spaces.

In the present paper we are especially interested in the class
$G$-$\mathcal{M}$ of all metrizable proper $G$-spaces that admit a compatible
$G$-invariant metric. It is well-known that, for $G$ a compact group, the
class $G$-$\mathcal{M}$ coincides with the class of all metrizable $G$-spaces
(see [22, Proposition 1.1.12]). A fundamental result of R. Palais [23, Theorem
4.3.4] states that if $G$ is a Lie group, then $G$-$\mathcal{M}$ includes all
separable, metrizable proper $G$-spaces.

Let us recall the definition of a twisted product $G/H\times_{K}S$, where $H$
is a closed normal subgroup of $G$, $K$ any closed subgroup of $G$, and $S$ a
$K$-space.

$G/H\times_{K}S$ is the orbit space of the $K$-space $G/H\times S$, where $K$
acts on the Cartesian product $G/H\times S$ by $k(gH,s)=(gk^{-1}H,ks)$.
Furthermore, there is a natural action of $G$ on $G/H\times_{K}S$ given by
$g^{\prime}[gH,s]=[g^{\prime}gH,s]$, where $g^{\prime}\in G$ and $[gH,s]$
denotes the $K$-orbit of the point $(gH,s)$ in $G/H\times S$. The twisted
products of the form $G\times_{K}S$ (i.e., when $H$ is the trivial subgroup of
$G$) are of a particular interest in the theory of transformation groups (see
[18, Ch. II, § 2]).

A $G$-space $Y$ is called an equivariant absolute neighborhood extensor for
the class $G$-$\mathcal{M}$ (notation: $Y\in G$-ANE) if, for any $X\in
G$-$\mathcal{M}$ and any closed invariant subset $A\subset X$, every $G$-map
$f:A\to Y$ admits a $G$-map $\psi\colon U\to Y$ defined on an invariant
neighborhood $U$ of $A$ in $X$ such that $\psi|_{A}=f$. If, in addition, one
can always take $U=X$, then we say that $Y$ is an equivariant absolute
extensor for $G$-$\mathcal{M}$ (notation: $Y\in G$-AE). The map $\psi$ is
called a $G$-extension of $f$.

The following proposition was proved in [13, Proposition 3.3] and will be used
in the proofs of Theorems 1.5 and 1.7.

######  Proposition 2.1 ([13]).

Let $H$ be a closed normal subgroup of $G$, $K$ a compact large subgroup of
$G$, and $S$ a $K$-space. If $S$ is a $K$-$ANE$, and all $K\cap H$-orbits in
$S$ are metrizable, then the twisted product $G/H\times_{K}S$ is a
$G/H$-$ANE$.

Let us recall the well known definition of a slice [23, p. 305]:

######  Definition 2.2.

Let $X$ be a $G$-space and $H$ a closed subgroup of $G$. An $H$-invariant
subset $S\subset X$ is called an $H$-slice in $X$, if $G(S)$ is open in $X$
and there exists a $G$-map $f:G(S)\to G/H$ such that $S$=$f^{-1}(eH)$. The
saturation $G(S)$ is called a tubular set and $H$ is called a slicing group.

If $G(S)=X$, then we say that $S$ is a global $H$-slice for $X$.

The following result of R. Palais [23, Proposition 2.3.1] plays a central role
in the theory of topological transformation groups.

######  Theorem 2.3 (Slice Theorem).

Let $G$ be a Lie group, $X$ be a proper $G$-space and $x\in X$. Then there
exists a $G_{x}$-slice $S\subset X$ such that $x\in S$.

In our proofs we will also need the following approximate version of the Slice
Theorem proved in [12, Theorem 3.6] (see also [15, Theorem 6.1]) which is
valid for any locally compact group.

######  Theorem 2.4 (Approximate Slice Theorem).

Let $G$ be any group, $X$ a proper $G$-space and $x\in X$. Then for any
neighborhood $O$ of $x$ in $X$, there exist a compact large subgroup $K$ of
$G$ with $G_{x}\subset K$, and a $K$-slice $S$ such that $x\in S\subset O$.

Recall that here a subgroup $K\subset G$ is called large, if the quotient
space $G/K$ is locally connected and finite-dimensional (see [15]).

In the context of equivariant extension properties the notion of a large
subgroup was first singled out in [7] (for compact groups) and in [9] (for
locally compact groups). Although some geometric characterizations of this
notion were available much earlier (see [15, Section 3] and the literature
cited there), new characterizations through equivariant extension properties
of the coset space $G/K$ were given in [9, Proposition 6], [12, Proposition
3.2] and [15, Theorem 5.3].

The following result will be applied in the proofs of all three theorems
below.

######  Proposition 2.5 ([12, Proposition 3.4]).

Let $K$ be a compact large subgroup of $G$, and $X$ a $G$-${\rm ANE}$
(respectively, a $G$-${\rm AE}$). Then $X$ is a $K$-${\rm ANE}$ (respectively,
a $K$-${\rm AE}$).

######  Remark 2.6.

A careful analysis of the proof of [12, Proposition 3.4] shows that this
result is true also for any compact subgroup $K$ of $G$ such that the coset
space $G/K$ is just metrizable. Indeed, in the proof of [12, Proposition 3.4]
it is just needed that the twisted product $G\times_{K}S$ admits a
$G$-invariant metric provided that $G/K$ and $S$ are metrizable. But this is
true without assuming that $K$ is a large subgroup and this is proved
explicitly in [14, Lemma 6.5] and [14, Theorem 6.1].

The following proposition is well known (see, e.g. [2, Lemma 3.5]).

######  Proposition 2.7.

Let $H$ be a compact subgroup of $G$, $X$ a proper $G$-space and $S$ a global
$H$-slice of $X$. Then the map $\xi:G\times_{H}S\to X$ defined by
$\xi([g,s])=gs$ is a $G$-homeomorphism.

The following equivariant version of Hanner’s open union theorem [20, Theorem
19.2] is proved in [12, Corollary 5.7]. A short and beautiful proof of
Hanner’s theorem was given by J. Dydak [19, Corollary 1.5].

######  Theorem 2.8 ([12]).

Let $Z\in G$-$\mathcal{M}$. If a $G$-space $Y$ is the union of a family of
invariant open $G$-${\rm ANE}(Z)$ subsets $Y_{\mu}\subset Y$,
$\mu\in\mathcal{M}$, then $Y$ is a $G$-${\rm ANE}(Z)$.

##  3\.  Proof of Theorem 1.3

By Theorem 2.4, $X$ has an open invariant cover by tubular sets of the form
$G(S)$, where each $S$ is a $K$-slice with the slicing group $K$ a compact
large subgroup of $G$. Then the orbit space $X/G$ is the union of its open
subsets of the form $G(S)/G$. According to Hanner’s open union theorem in [20,
Theorem 19.2] or [19, Corollary 1.5] (see also Theorem 2.8), it suffices to
show that each $G(S)/G$ is an ANE.

To this end, we first observe that each $G(S)$ is $G$-homeomorphic to the
twisted product $G\times_{K}S$ (see Proposition 2.7). This implies that
$G(S)/G$ is homeomorphic to $(G\times_{K}S)/G$. Since $X\in G$-ANE, the
tubular set $G(S)$, being an open invariant subset of $X$, is itself a
$G$-ANE. Thus, $G\times_{K}S$ is a $G$-ANE. Since the slicing group $K$ is a
compact large subgroup of $G$, one can apply Proposition 2.5, according to
which $G\times_{K}S$ is a $K$-ANE. Each $K$-orbit in $X$ is contained in a
$G$-orbit, and hence, is metrizable. Since $K$ is compact, Theorem 1.2 implies
that $(G\times_{K}S)/K$ is an ANE. By Proposition 1.4, $(G\times_{K}S)/G$ is
homeomorphic to a retract of $(G\times_{K}S)/K$, and hence, is itself an ANE.
Consequently, $G(S)/G$ is an ANE, as required. ∎

##  4\. Proof of Theorem 1.5

Since $X\in G$-$\mathcal{M}$ the orbit space $X/G$ is metrizible, and hence,
by Abels [1, Main Theorem], $X$ admits a global $K$-slice $S$ where $K$ is a
maximal compact subgroup of $G$. Then, by Proposition 2.7, $X$ is
$G$-homeomorphic to the twisted product $G\times_{K}S$.

Observe that for every maximal compact subgroup $K\subset G$, the coset space
$G/K$ is metrizable. Moreover, $G/K$ is homeomorphic to a Euclidean space (see
[1, Corollary A6]).

Therefore, one can apply Proposition 2.5 and Remark 2.6, according to which
$G\times_{K}S$ is a $K$-ANE.

Since $G$ is almost connected, one can apply Proposition 1.6, according to
which $S$ is a $K$-equivariant retract of $G\times_{K}S$, and hence, $S$ is a
$K$-ANE.

Further, one has the following $G$-homeomorphism:

$(G\times_{K}S)/H\cong G/H\times_{K}S.$

Indeed, the map that sends the point $[g,s]_{H}$ of $(G\times_{K}S)/H$ to the
point $[gH,s]$ of $G/H\times_{K}S$ is a $G/H$-homeomorphism, where $[g,s]_{H}$
denotes the $H$-orbit of $[g,s]$ in $G\times_{K}S$ (the easy verification is
left to the reader).

Next we observe that every $K\cap H$-orbit in $S$ is metrizable since it is
contained in the corresponding $H$-orbit in $X$, which is metrizable by the
hypothesis. Further, since $S\in K$-ANE, it then follows from Proposition 2.1
that the twisted product $G/H\times_{K}S$ is a $G/H$-ANE. This yields that
$(G\times_{K}S)/H\in G/H$-ANE, and since $X/H$ is $G/H$-homeomorphic to
$(G\times_{K}S)/H$, we conclude that $X/H\in G/H$-ANE, as required. ∎

##  5\.  Proof of Theorem 1.7

By Theorem 2.3, $X$ has an open invariant cover by tubular sets of the form
$G(S)$, where each $S$ is a $K$-slice with the slicing group $K$ a compact
subgroup of $G$. Then the $G/H$-space $X/H$ is the union of its open
$G/H$-invariant subsets of the form $G(S)/H$. According to Theorem 2.8, it
suffices to show that each $G(S)/H$ is a $G/H$-ANE.

To this end, we first observe that each $G(S)$ is $G$-homeomorphic to the
twisted product $G\times_{K}S$ (see Proposition 2.7).

This yields that $G(S)/H$ is $G/H$-homeomorphic to $(G\times_{K}S)/H$. Since
$X\in G$-ANE, the tubular set $G(S)$, being an open invariant subset of $X$,
is itself a $G$-ANE. Thus, $G\times_{K}S$ is a $G$-ANE. Since $G$ is a Lie
group, we infer that $G/K$ is metrizable (moreover, evidently, $K$ is a
compact large subgroup in this case). Then one can apply Proposition 2.5,
according to which $G\times_{K}S$ is a $K$-ANE. By Proposition 1.8, $S$ is a
$K$-equivariant neighborhood retract of $G\times_{K}S$, and hence, $S$ is a
$K$-ANE.

Further, as we mentioned in the proof of Theorem 1.5, one has the following
$G$-homeomorphism:

$(G\times_{K}S)/H\cong G/H\times_{K}S.$

Since $S\in K$-ANE, it then follows from Proposition 2.1 that the twisted
product $G/H\times_{K}S$ is a $G/H$-ANE. This yields that $(G\times_{K}S)/H\in
G/H$-ANE, and since, $G(S)/H$ is $G/H$-homeomorphic to $(G\times_{K}S)/H$, we
conclude that $G(S)/H\in G/H$-ANE, as required. ∎

##  6\. Lifting of equivariant embeddings and extension properties of orbit
spaces

The lifting properties of $G$-equivariant closed embeddings for a compact
acting group $G$ were first established in [6]. Below, in Theorems 6.1 and 6.2
we generalize these results to the case of proper actions of non-compact
groups. In turn, this allows us to strengthen Theorems 1.3, 1.5 and 1.7,
discarding in their statements the hypothesis about the properness of the
$G$-space $X$.

######  Theorem 6.1.

Let $G$ be either a Lie group or an almost connected group, and let $H$ be a
closed normal subgroup of $G$. Suppose that $A\in G$-$\mathcal{M}$ and
$f:A/H\hookrightarrow B$ is a $G/H$-equivariant closed embedding into a
$G/H$-space $B\in G/H$-$\mathcal{M}$. Then there exist a $G$-space $Z\in
G$-$\mathcal{M}$ and a $G$-equivariant closed embedding $\phi:A\hookrightarrow
Z$ such that $Z/H$ is a $G/H$-invariant neighborhood of $A/H$ in $B$ and
$q\circ\phi=f\circ p$, where $p:A\to A/H$ and $q:Z\to Z/H$ are the $H$-orbit
maps.

###### Proof.

According to [16, Theorem 6.1], it can be assumed that A is a closed
$G$-invariant subset of a $G$-AE space $L\in G$-$\mathcal{M}$. Then $A/H$ is a
closed invariant subset of the $G/H$-space $L/H$.

$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{p}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{A/H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{L/H}$

Now, by Theorem 1.5 (for almost connected groups) and Theorem 1.7 (for Lie
groups), $L/H\in G/H$-ANE. Therefore, there exist a $G/H$-equivariant
extension. $F:U\to L/H$ of the $G/H$-map $f^{-1}:f(A/H)\to A/H\hookrightarrow
L/H$ defined on some $G/H$-neighborhood $U$ of the set $A/H$ in $B$.

Let $r:L\to L/H$ be the $H$-orbit projection. Denote by $Z$ the pull-back (or
fiber product) of $L$ with respect to the maps $F$ and $r$, i.e.,

$Z=\\{(u,x)\in U\times L\ |\ F(u)=r(x)\\}.$

We will consider the coordinate-wise defined action of the group $G$ on $Z$,
i.e., $g(u,x)=(gu,gx)$ for $g\in G$ and $(u,x)\in Z$. Let $h:Z/H\to U$ be the
map defined by the formula $h(q(u,x))=u$, where $q:Z\to Z/H$ is the $H$-orbit
projection and $(u,x)\in Z$. It is clear that $h$ is a well-defined
$G/H$-equivariant map. It can easily be shown (and this is well known, see
[21, Ch.4, Proposition 4.1]) that $h$ is a homeomorphism.

On the other hand the product $U\times L$ is a proper $G$-space because $L$ is
so. Besides, since $U$ and $L$ admit $G$-invariant metrics, we infer that
$U\times L$ also has a $G$-invariant metric. Thus $U\times L\in
G$-$\mathcal{M}$, which implies that $Z\in G$-$\mathcal{M}$.

It remains to define the $G$-equivariant embedding $\phi:A\hookrightarrow Z$
by the formula $\phi(a)=(f(p(a)),a)$ for $a\in$ A. This completes the proof. ∎

######  Theorem 6.2.

Let $G$ be any locally compact group. Assume that $A\in G$-$\mathcal{M}$ and
$f:A/G\hookrightarrow B$ is a closed embedding into a metrizable space $B$.
Then there exist a $G$-space $Z\in G$-$\mathcal{M}$ and a $G$-equivariant
closed embedding $\phi:A\hookrightarrow Z$ such that $Z/G$ is a neighborhood
of $A/G$ in $B$ and $q\circ\phi=f\circ p$, where $p:A\to A/G$ and $q:Z\to Z/G$
are the $G$-orbit maps.

###### Proof.

Repeat the proof of Theorem 6.1 with $H=G$, where you simply need to replace
the reference to Theorem 1.7 with a reference to Theorem 1.3. ∎

By virtue of Theorems 6.1 and 6.2, one can dropp the hypothesis about the
properness of the $G$-space $X$ in Theorems 1.3, 1.5 and 1.7.

######  Theorem 6.3 (Non-proper actions of Lie groups and almost connected
groups).

Let $G$ be either a Lie group or an almost connected group, and let $X$ be any
$G$-${\rm{\rm ANE}}$. Assume that $H$ is a closed normal subgroup of $G$ such
that all $H$-orbits in $X$ are metrizable. Then the $H$-orbit space $X/H$ is a
$G/H$-${\rm ANE}$.

###### Proof.

Let $B\in G/H$-$\mathcal{M}$. Let $L$ be a closed $G/H$-invariant subset of
$B$ and let $s:L\to X/H$ be a $G/H$-map. Define $A\subset L\times X$ to be the
pull-back of the $G$-space $X$ with respect to $s$ and $t$, where $t:X\to X/H$
is the $H$-orbit map. Then $A$ is a $G$-invariant subspace of $L\times X$
endowed with the diagonal action of $G$, and we have $A/H=L$ (see [21, Ch. 4,
Proposition 4.1]). Since the $H$-orbit of each point $a=(l,x)\in A$ lies in
the metrizable space $L\times H(x)$, we conclude that $H(a)$ is metrizable
too. So, all $H$-orbits of the $G$-space $A$, as well as its $H$-orbit space
$A/H=L$, are metrizable. By [14, Theorem 6.1], $A$ is metrizable. Now applying
Theorem 6.1, we get a $G$-space $Z\in G$-$\mathcal{M}$ with $Z/H$ a
$G/H$-invariant neighborhood of $L$ in $B$ such that $A$ is a closed
$G$-invariant subspace of $Z$.

Let $\psi:A\to X$ be the restriction of the projection $L\times X\to X$. Since
$X\in G$-ANE, there exist a $G$-invariant neighborhood $U$ of $A$ in $Z$ and a
$G$-extension $\alpha:U\to X$ of the $G$-map $\psi$. It is easy to see that
the induced $G/H$-map $\beta:U/H\to X/H$ is the desired $G/H$-extension of
$s$. This completes the proof. ∎

######  Theorem 6.4 (Non-proper actions of locally compact groups with $H=G$).

Let $G$ be a locally compact group and $X$ any $G$-space such that all
$G$-orbits in $X$ are metrizable. If $X$ is a $G$-${\rm ANE}$ then the
$G$-orbit space $X/G$ is an ANE.

###### Proof.

Repeat the proof of Theorem 6.3, where you simply need to replace the
reference to Theorem 6.1 with a reference to Theorem 6.2. ∎

##  7\. Appendix

In this section we will present in detail some simple and very useful
propositions that culminate in a proof of Proposition 1.4.

######  Proposition 7.1.

Let $G$ be a topological group, $H$ a closed subgroup of $G$, and $X$ a
$G$-space. Then for any closed subset $B\subset X$, the set
$A=\\{(h^{-1},hb)\mid h\in H,\ b\in B\\}$ is closed in the product $G\times
X$.

###### Proof.

Assume that $(g,x)$ is a closure point of $A$ and prove that $(g,x)\in A$.
There exist nets $(h_{i})\subset H$ and $(b_{i})\subset B$ such that the net
$(h_{i}^{-1},h_{i}b_{i})$ converges to $(g,x)$. This yields that
$(h_{i}^{-1})$ converges to $g$ and $(h_{i}b_{i})$ converges to $x$. Since $H$
is closed we infer that $g\in H$. Clearly, $b_{i}=h_{i}^{-1}(h_{i}b_{i})$
converges to $gx$. Since $b_{i}\in B$ and $B$ is closed, we infer that $gx\in
B$. Thus, $(g,x)=(g,g^{-1}gx)$ where $g\in H$ and $gx\in B$. This shows that
$(g,x)\in A$, as required.

∎

The following proposition is well-known in the literature only for the compact
subgroup $H$.

######  Proposition 7.2.

Let $G$ be a topological group and $H$ a closed subgroup of $G$. If $X$ is an
$H$-space, then the map $\iota:X\hookrightarrow G\times_{H}X,\;\iota(x)=[e,x]$
is a closed $H$-embedding, where $e\in G$ is the unit element.

###### Proof.

Since $\iota$ is the composition $X\stackrel{{\scriptstyle
j}}{{\rightarrow}}G\times X\stackrel{{\scriptstyle
p}}{{\rightarrow}}G\times_{H}X$, where $j$ is the closed embedding
$j(x)=(e,x)$ and $p$ is the $H$-orbit map, we infer that $\iota$ is
continuous.

Let $A$ be a closed subset of $X$. To prove that $\iota(A)$ is closed in
$G\times_{H}X$ it suffices to prove that the inverse image
$p^{-1}\big{(}\iota(A)\big{)}$ is closed in $G\times X$, where $p:G\times X\to
G\times_{H}X$ is the $H$-orbit map. We have that

$p^{-1}\big{(}\iota(A)\big{)}=\\{(g,x)\in G\times X\mid[g,x]=[e,a]\ \text{for
some}\ a\in A\\}.$

But the equality $[g,x]=[e,a]$ means that $(g,x)=(h^{-1},ha)$ for some $h\in
H$. Consequently,

$p^{-1}\big{(}\iota(A)\big{)}=\\{(h^{-1},ha)\in G\times X\mid h\in H,\ a\in
A\\},$

which, by Proposition 7.2, is closed in $G\times X$. Thus, $\iota$ is a closed
map.

Further, the map $\iota$ is injective since

$[e,x]=[e,y]\Longleftrightarrow(e,y)=(eh^{-1},hx)\quad\text{for some}\;h\in H$

and, in this case, $h=e$, so $y=x$. Hence, $\iota$ is a closed embedding.

If $x\in X$ and $h\in H$, then

$\iota(hx)=[e,hx]=[h,x]=h[e,x]=h\iota(x)$

showing that $\iota$ is $H$-equivariant, as required. ∎

######  Proposition 7.3.

Let $G$ be a topological group and $H$ any subgroup of $G$. If $X$ is an
$H$-space, then the $G$-orbit space $({G\times_{H}X})/G$ is homeomorphic to
the $H$-orbit space $X/H$.

###### Proof.

The projection $\pi:G\times X\to X$ is a continuous open map which induces a
continuous open map $\alpha:G\times_{H}X\to X/H,\;[g,x]\mapsto H(x)$ between
$H$-orbit spaces.

Observe that $\alpha$ is constant on the $G$-orbits of the $G$-space
$G\times_{H}X$. Indeed for every $g^{\prime}\in G$ one has
$\alpha(g^{\prime}[g,x])=\alpha([g^{\prime}\cdot g,x])=H(x)=\alpha([g,x])$, as
required. Then $\alpha$ induces a continuous bijective map
$r:({G\times_{H}X})/G\to X/H$ which makes to commute the following diagram:

$\textstyle{G\times_{H}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{q}$$\textstyle{X/H,}$$\textstyle{\frac{G\times_{H}X}{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$

where $q$ is the $G$-orbit map that sends $[g,x]$ to its orbit $G([g,x])$.
Since $\alpha$ is open, it follows from the commutativity of the diagram that
the induced map $r$ also is open, and hence, it is the desired homeomorphism.

∎

######  Proposition 7.4.

Let $G$ be a topological group and $H$ a closed subgroup of $G$. If $X$ is an
$H$-space, then the $H$-orbit space $X/H$ is homeomorphic to a retract of the
$H$-orbit space $({G\times_{H}X})/H$.

###### Proof.

By Proposition 7.2, the map $\iota:X\hookrightarrow G\times_{H}X$,
$x\mapsto[e,x]$ is a closed $H$-embbeding. This induces a closed embedding
$\tilde{\iota}:X/H\hookrightarrow(G\times_{H}X)/H$. Consequently, it suffices
to prove that the image $\mathcal{I}m\,\tilde{\iota}$ is a retract of
$(G\times_{H}X)/H$.

For every $[g,x]\in G\times_{H}X$ we will denote by $[g,x]_{H}$ the $H$-orbit
in the $G$-space $G\times_{H}X$. Clearly,
$\mathcal{I}m\,\tilde{\iota}=\\{[e,x]_{H}\mid x\in X\\}.$

Define a map $r:(G\times_{H}X)/H\to\mathcal{I}m\,\tilde{\iota}$ by the rule:
$r:[g,x]_{H}\mapsto[e,x]_{H}$. This map is well defined since for any $h\in H$
one has

$r:[gh^{-1},hx]_{H}\mapsto[e,hx]_{H}=[h,x]_{H}=(h[e,x])_{H}=[e,x]_{H},$

as required.

The projection $G\times X\to X$ is $H$-equivariant and thus induces a
continuous map $(G\times_{H}X)/H\to X/H$. This is given by $[g,x]\mapsto
H(x)$, and hence, factors as a continuous map
$f:\frac{G\times_{H}X}{H}\longrightarrow X/H$, $[g,x]_{H}\mapsto H(x)$.

The continuity of $r$ follows from the fact that it is the composition of the
following two continuous maps:

$\frac{G\times_{H}X}{H}\stackrel{{\scriptstyle
f}}{{\longrightarrow}}X/H\stackrel{{\scriptstyle\tilde{\iota}}}{{\longrightarrow}}\mathcal{I}m\,\tilde{\iota},$

$[g,x]_{H}\mapsto H(x)\mapsto[e,x]_{H}.$

Besides, if $[e,x]_{H}\in\mathcal{I}m\,\tilde{\iota}$, then
$r\big{(}[e,x]_{H}\big{)}=[e,x]_{H}$, so $r$ is the desired retraction. Thus,
$X/H$ is homeomorphic to $\mathcal{I}m\,\tilde{\iota}$ which is a retract of
the $H$-orbit space $({G\times_{H}X})/H$, as required. ∎

Now, as a simple combination of Propositions 7.3 and 7.4, we get Proposition
1.4 already stated in Section 1.

We conclude the paper with the following conjecture.

######  Conjecture 7.5.

Let $G$ be a locally compact group, $K$ a compact large subgroup of $G$, and
$S$ a $K$-space. Then $S$ is a neighborhood $K$-equivariant retract of the
twisted product $G\times_{K}S$.

This conjecture first appeared in [13, Question 4.4] in the form of a
question. Note that this is true for any Lie group $G$ (see Proposition 1.8)
and for any almost connected group $G$ (see Proposition 1.6). The validity of
this conjecture will allow us to extend the proof of Theorem 1.7 to the case
of proper actions of arbitrary locally compact groups.

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Ergodic complex structures on hyperkähler manifolds: an erratum

Misha Verbitsky111Misha Verbitsky is partially supported by the Russian
Academic Excellence Project ’5-100’.

Abstract

Let $M$ be a hyperkähler manifold, $\Gamma$ its mapping class group, and
$\operatorname{\sf Teich}$ the Teichmüller space of complex structures of
hyperkähler type. After we glue together birationally equivalent points, we
obtain the so-called birational Teichmüller space $\operatorname{\sf
Teich}_{b}$. Every connected component of $\operatorname{\sf Teich}_{b}$ is
identified with $\operatorname{{\mathbb{P}}\sf
er}=\frac{SO(3,b_{2}-3)}{SO(2)\times SO(1,b_{2}-3)}$ by global Torelli
theorem. The mapping class group of $M$ acts on $\operatorname{{\mathbb{P}}\sf
er}$ as a finite index subgroup of the group of isometries of the integer
cohomology lattice, that is, satisfies assumptions of Ratner theorem. We prove
that there are three classes of orbits, closed, dense and the intermediate
class which corresponds to varieties with
$\dim\operatorname{Re}(H^{2,0}(M))\cap H^{2}(M,{\mathbb{Q}})=1$. The closure
of the later orbits is a fixed point set of an anticomplex involution of
$\operatorname{{\mathbb{P}}\sf er}$. This fixes an error in the paper [V2],
where this third class of orbits was overlooked. We explain how this affects
the works based on [V2].

###### Contents

  1. 1 Introduction
    1. 1.1 Orbits of monodromy action: an extra orbit
    2. 1.2 Preliminaries
  2. 2 Subgroups of $SO(a,b)$ and Ratner theory
    1. 2.1 Lie subalgebras $\mathfrak{so}(a-2,b)\subsetneq\mathfrak{g}\subsetneq\mathfrak{so}(a,b)$
    2. 2.2 Ratner theory for $SO^{+}(a-2,b)\subset SO^{+}(a,b)$
    3. 2.3 Ratner theory and complex geometry of the period space
  3. 3 Mapping class group action on the Teichmüller space
    1. 3.1 Mapping class group action and non-Hausdorff points
    2. 3.2 Round segments on the boundary of the Kähler cone
    3. 3.3 Horocycles in the Kähler cone
  4. 4 Applications to hyperkähler geometry: Hyperbolicity and Kobayashi pseudometric

##  1 Introduction

###  1.1 Orbits of monodromy action: an extra orbit

This paper is essentially an erratum to [V2]. For an introduction to
Teichmüller theory, global Torelli theorem and Ratner theory as applied to the
Teichmüller space, please see [V2].

Let $M$ be a hyperkähler manifold, and $\operatorname{\sf Teich}$ the
Teichmüller space of complex structures of hyperkähler type. Fix a connected
component $\operatorname{\sf Teich}^{I}$ of $\operatorname{\sf Teich}$ (there
are finitely many of them, per [Hu]), and let $\Gamma$ be the subgroup of
mapping class group of $M$ mapping $\operatorname{\sf Teich}^{I}$ to itself.
If we glue together the points of $\operatorname{\sf Teich}^{I}$ which are
non-separable, we obtain a Hausdorff manifold $\operatorname{\sf Teich}_{b}$,
which is diffeomorphic to its period space $\operatorname{{\mathbb{P}}\sf
er}=\frac{SO(3,b_{2}-3)}{SO(1,b_{2}-3)\times SO(2)}$, as follows from the
global Torelli theorem ([V1]). The mapping class group of $M$ acts on
$\operatorname{{\mathbb{P}}\sf er}$ as an arithmetic lattice, that is, a
finite index subgroup in $SO(H^{2}(M,{\mathbb{Z}}))$.

In [V2], we classified the orbits of $\Gamma$ on $\operatorname{\sf Teich}$
and proved that a general orbit of $\Gamma$ is dense in $\operatorname{\sf
Teich}$, and an orbit which corresponds to a manifold of maximal Picard rank
is closed. However, one class of orbits was overlooked: the orbits of
$I\subset\operatorname{\sf Teich}$ such that
$\operatorname{Re}(H^{2,0}(M,I))\cap H^{2}(M,{\mathbb{Q}})$ has rank 1.

In the present erratum we give a complete classification of orbits of
$\Gamma$-action on $\operatorname{{\mathbb{P}}\sf er}$, and prove that the
closure of the intermediate orbit is totally real: each of these orbits is a
fixed point set of an anti-complex involution. In particular, it does not
contain positive-dimensional complex subvarieties, and is not contained in any
proper complex subvariety.

In the last section we check that the geometric applications obtained using
the older version of [V2] remain valid with exception of one statement from
[KLV] (not stated as a theorem, but implied in the abstract).

The third orbit was omitted in [V2] because here we actually classified the
orbits of the group $H:=SO(1,b_{2}-3)\times SO(2)$ on the space
$SO(3,b_{2}-3)/\Gamma$. However, one cannot apply Ratner’s theorem to the
group $H$, because it is not generated by unipotents. To use Ratner’s theorem,
we need to consider orbits of the smaller group
$H^{\prime}:=SO(1,b_{2}-3)\subsetneq H$.

Ratner’s theorem implies that a closure of an orbit of $H^{\prime}$ is an
orbit of an intermediate subgroup $S$, with $H^{\prime}\subset S\subset
SO(3,b_{2}-3)$. There are no proper intermediate subgroups between
$SO(1,b_{2}-3)\times SO(2)$ and $SO(3,b_{2}-3)$; however, there is an
intermediate subgroup $SO(2,b_{2}-3)$ between $SO(1,b_{2}-3)$ and
$SO(3,b_{2}-3)$, and it corresponds to the extra orbit.

###  1.2 Preliminaries

Throughout this paper, hyperkähler manifold means a compact holomorphically
symplectic manifold $M$ of Kähler type, satisfying
$\pi_{1}(M)=0,H^{2,0}(M)={\mathbb{C}}$. Its second cohomology are equipped
with a primitive, bilinear symmetric, non-degenerate integral form $q$ of
signature $(3,b_{2}-3)$, called Bogomolov-Beauville-Fujiki form, or “BBF
form”. The corresponding Teichmüller space $\operatorname{\sf Teich}$ is the
space of all complex structures of hyperkähler type up to isotopy. We shall
also speak of “Teichmüller spaces” for other geometric structures, defined in
the same way. The monodromy group associated with a component of
$\operatorname{\sf Teich}$ is the subgroup of $O(H^{2}(M,{\mathbb{Z}})$. The
period space $\operatorname{{\mathbb{P}}\sf er}$ is the Grassmanian of
positive, oriented 2-planes in $H^{2}(M,{\mathbb{R}})$. The period map
$\operatorname{\sf Per}:\;\operatorname{\sf
Teich}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf er}$ takes a manifold
$(M,I)$ to the 2-plane $\operatorname{Re}H^{0,2}(M,{\mathbb{R}})\subset
H^{2}(M,{\mathbb{R}})$. “Picard rank” of a hyperkähler manifold is
$\dim_{\mathbb{Q}}(H^{1,1}(M)\cap H^{2}(M,{\mathbb{Q}}))$. Global Torelli
theorem ([V1]) states that the map $\operatorname{\sf Per}$ is “the Hausdorff
quotient”: it is surjective, and satisfies $\operatorname{\sf
Per}(I)=\operatorname{\sf Per}(I^{\prime})$ if and only if $I$ and
$I^{\prime}$ are non-separable in $\operatorname{\sf Teich}$.

The monodromy group $\Gamma$ is the subgroup of $O(H^{2}(M,{\mathbb{Z}}))$
generated by monodromy of all Gauss-Manin local systems for all complex
deformations of $(M,I)$ over a compact base. This group acts on the
corresponding connected component of the Teichmüller space, denoted as
$\operatorname{\sf Teich}^{I}$, and the quotient set $\operatorname{\sf
Teich}^{I}/\Gamma$ is precisely the set of all equivalence classes of complex
structures on $M$ in the same deformation class as $I$. In [V2] it was shown
that $\Gamma$ is a finite index subgroup of $O(H^{2}(M,{\mathbb{Z}}))$, and
its action on $\operatorname{{\mathbb{P}}\sf er}$ is ergodic.

We use the notation $(M,I)$ to put emphasis on the complex structure $I$
defined on the underlying smooth manifold $M$.

The connected component subgroup of the group $SO(p,q)$ is denoted as
$SO^{+}(p,q)$.

##  2 Subgroups of $SO(a,b)$ and Ratner theory

###  2.1 Lie subalgebras
$\mathfrak{so}(a-2,b)\subsetneq\mathfrak{g}\subsetneq\mathfrak{so}(a,b)$

In this subsection, we classify connected Lie subgroups $G\subset SO(a,b)$
containing $SO^{+}(a-2,b)$. To classify connected subgroups it suffices to
classify Lie subalgebras
$\mathfrak{so}(a-2,b)\subsetneq\mathfrak{g}\subsetneq\mathfrak{so}(a,b)$. This
is done as follows.

Theorem 2.1:  Let $V={\mathbb{R}}^{p+q}$ be a vector space equipped with a
quadratic form of signature $(p,q)$, $V_{1}\subset V$ a positive 2-dimensional
subspace, and $V_{0}=V_{1}^{\bot}$. The Lie algebra $\mathfrak{so}(V_{0})$ is
identified with a subalgebra of $\mathfrak{so}(V)$ acting trivially on
$V_{1}$. Consider a proper Lie subalgebra
${\mathfrak{g}}\subsetneq\mathfrak{so}(V)$ such that
$\mathfrak{so}(V_{0})\subsetneq\mathfrak{g}\subsetneq\mathfrak{so}(V)$. Then
either $\mathfrak{g}=\mathfrak{so}(V_{2})$, where $V_{2}\subset V$ is a
$p+q-1$-dimensional subspace, $V_{0}\subsetneq V_{2}\subsetneq V$, or
$\mathfrak{g}=\mathfrak{so}(V_{0})\oplus\mathfrak{so}(V_{1})$.

Proof. Step 1: We prove that
${\mathfrak{g}}=\mathfrak{so}(V_{0})\oplus\mathfrak{so}(V_{1})$ or
${\mathfrak{g}}$ is isomorphic to $\mathfrak{so}(V_{0})\oplus V_{0}$ as an
$\mathfrak{so}(V_{0})$-representation. Consider the decomposition of
$\mathfrak{so}(V)$ as an $\mathfrak{so}(V_{0})$-representation:

$\mathfrak{so}(V)=\mathfrak{so}(V_{0})\oplus(V_{0}\otimes
V_{1})\oplus\mathfrak{so}(V_{1}).$

Since $\mathfrak{g}\subset\mathfrak{so}(V)$ is
$\mathfrak{so}(V_{0})$-invariant, it is an $\mathfrak{so}(V_{0})$-submodule in
$\mathfrak{so}(V)$. Since $\mathfrak{so}(V_{0})\oplus V_{0}\otimes
V_{1}\subset\mathfrak{so}(V)$ generates the Lie algebra $\mathfrak{so}(V)$,
one has $\mathfrak{g}\not\supset V_{0}\otimes V_{1}$. Then either
$\mathfrak{g}\cap V_{0}\otimes V_{1}=0$, in which case
$\mathfrak{g}=\mathfrak{so}(V_{0})\oplus\mathfrak{so}(V_{1})$ (the space
$\mathfrak{so}(V_{1})$ is 1-dimensional), or $\mathfrak{g}\cap V_{0}\otimes
V_{1}=0$ is a proper $\mathfrak{so}(V_{0})$-invariant subspace of
$V_{0}\otimes V_{1}$. Since $V_{1}$ is a trivial 2-dimensional representation
of the Lie algebra $\mathfrak{so}(V_{0})$, the space $V_{0}\otimes V_{1}$ is
isomorphic (as an $\mathfrak{so}(V_{0})$-module) to a direct sum of two copies
of $V_{0}$, which is an irreducible representation of $\mathfrak{so}(V_{0})$.
Then any proper subrepresentation of $V_{0}\otimes V_{1}$ is isomorphic to
$V_{0}$.

Step 2: Let ${\mathfrak{g}}=\mathfrak{so}(V_{0})\oplus R$, where $R\subset
V_{0}\otimes V_{1}$ is a $\mathfrak{so}(V_{0})$-invariant subspace. Then
$R=V_{0}\otimes R_{1}$, where $R_{1}\subset V_{1}$ is a 1-dimensional subspace
in a 2-dimensional subspace $V_{1}$. Denote by $S\subset V_{1}$ the orthogonal
complement to $S$ in $V_{1}$, and let $V_{2}=V_{0}\oplus S$. Then
$\mathfrak{g}$ fixes $S$, which gives a Lie algebra homomorphism
${\mathfrak{g}}{\>\longrightarrow\>}\mathfrak{so}(V_{2})$. Comparing
dimensions, we find that it is an isomorphism.

###  2.2 Ratner theory for $SO^{+}(a-2,b)\subset SO^{+}(a,b)$

To fix the notation, we state the Ratner orbit closure theorem in a particular
situation where we needed it. For a general version, see [V2] or [Mor]. As
usual, $SO^{+}(a,b)$ denotes the connected component of the special orthogonal
group $SO(a,b)$.

Theorem 2.2:  Consider an integer lattice $V_{\mathbb{Z}}={\mathbb{Z}}^{a+b}$
equipped with an integer scalar product $q$ of signature $(a,b)$, $a>2$,
$b>0$, $a+b>4$. Let $\Gamma\subset SO(V_{\mathbb{Z}})$ be a finite index
sublattice, $V_{\mathbb{R}}:=V_{\mathbb{Z}}\otimes_{\mathbb{Z}}{\mathbb{R}}$,
and $SO^{+}(V_{\mathbb{R}},q)$ the connected component of its orthogonal
group. Replacing $\Gamma$ by a finite index subgroup if necessarily, we may
assume that $\Gamma$ belongs to the connected component
$SO^{+}(V_{\mathbb{R}})$ of the group $SO(V_{\mathbb{R}})$. Let $H\subset
SO^{+}(V_{\mathbb{R}},q)$ be the group of all elements of
$SO^{+}(V_{\mathbb{R}})$ acting trivially on a 2-dimensional positive subspace
$V_{1}\subset V_{\mathbb{R}}$. Consider the left action of $H$ on $G/\Gamma$;
let $Hx$ be an orbit of $x\in G/\Gamma$, and $\overline{Hx}$ its closure in
standard topology. Then there exists a rational point $y\in\overline{Hx}$ and
a rational Lie subgroup $H_{1}$ generated by unipotents, $H\subset
H_{1}\subset G$, such that $\overline{Hx}=H_{1}y$.

Proof: This is the statement of Ratner theorem from [Mor]; to apply it, we
have to check that the group $H$ is generated by unipotents. The universal
cover of $SO^{+}(2,2)$ is $SL(2,{\mathbb{R}})\times SL(2,{\mathbb{R}})$, hence
it is generated by unipotents. For other values of $(a,b)$, $a+b>4,a\geqslant
3,b>1$, the group $H=SO^{+}(a-2,b)$ is simple, and its subgroup generated by
unipotents is normal, hence it is equal to $H$ or empty. There are unipotent
elements in $SO^{+}(a,b)$ for $a\geqslant 2,b\geqslant 1$, which gives the
statement of 2.2.

Remark 2.3: In [V2, Example 4.4] the Ratner Theorem was stated in the same way
and then applied to $H=SO(a-2,b)\times SO(2)$. However the group
$SO(a-2,b)\times SO(2)$ is not generated by unipotents. Its smallest subgroup
generated by unipotents is $SO(a-2,b)$. This is why in place of [V2, Example
4.4] we should use 2.2.

###  2.3 Ratner theory and complex geometry of the period space

Claim 2.4:  Let $V={\mathbb{R}}^{a,b}$ be a vector space equipped with a non-
degenerate scalar product $q$ of signature $(a,b)$, and
$\operatorname{Gr}_{++}=\frac{SO^{+}(a,b)}{SO^{+}(a-2,b)\times SO(2)}$ the
Grassmannian of positive, oriented 2-planes. Then $Gr_{++}$ is in bijective
correspondence with

$\operatorname{{\mathbb{P}}\sf er}:=\\{l\in{\mathbb{P}}V_{\mathbb{C}}\ \ |\ \
q(l,l)=0,q(l,\overline{l})>0\\}.$

where $V_{\mathbb{C}}$ is the complexification of $V$.

Proof: See, for example, [V2, Proposition 2.12].

Theorem 2.5:  Consider an integral lattice $V_{\mathbb{Z}}={\mathbb{Z}}^{a+b}$
equipped with an integer-valued scalar product $q$ of signature $(a,b)$,
$a>2$, $b>0$, $a+b>4$, $V:=V_{\mathbb{Z}}\otimes_{\mathbb{Z}}{\mathbb{R}}$,
$\Gamma\subset SO(V_{\mathbb{Z}})$ a finite index sublattice,
$l\in\operatorname{Gr}_{++}(V)$ a point, and $\Gamma
l\subset\operatorname{Gr}_{++}(V)$ its orbit. Then one of the following three
possibilities is true.

(i)

$\Gamma l$ is closed. This happens when the 2-plane $l\subset V$ is rational.

(ii)

$\Gamma l$ is dense in $\operatorname{Gr}_{++}(V)$. This happens when $l$
contains no rational vectors.

(iii)

The closure $\overline{\Gamma l}$ is the set of all 2-planes
$V_{1}\in\operatorname{Gr}_{++}(V)$ containing a 1-dimensional rational
subspace $v\subset V_{\mathbb{Q}}$.

Proof: Let $P:=SO(V)/SO(V_{0})$, where $V_{0}\subset V$ is a codimension 2
subspace of signature $(a-2,b)$. Clearly, $P$ is fibered over
$\operatorname{Gr}_{++}(V)$ with the fiber $SO(2)=S^{1}$:

$\pi:\;P\ext@arrow 0099\arrowfill@\relbar\relbar\longrightarrow{}{\ /SO(2)\
}{}\frac{SO^{+}(a,b)}{SO^{+}(a-2,b)\times SO(2)}=\frac{P}{SO(2)}.$

Since $SO(2)$ is compact, the closure of a $\Gamma$-orbit of $x\in P$
satisfies

$\pi(\overline{\Gamma\cdot l})=\overline{\Gamma\cdot\pi(l)}.$ (2.1)

Notice that an $H$-invariant subset $Z$ on $G/\Gamma$ is closed if and only if
its image is closed in the double quotient $H\backslash G/\Gamma$. Therefore,
the closure of an orbit $H\cdot x$ in $G/\Gamma$ can be obtained by taking the
closure of $x\cdot\Gamma$ in the left quotient $H\backslash G$. Using Ratner’s
theorem (2.2), we find that such closures correspond to intermediate subgroups
$H\subset S\subset G$. For $\Gamma$, $G$ and $H$ as in 2.2, the list of
possible $S$ is provided by 2.1.

To obtain a classification of $\Gamma$-orbits in $\operatorname{Gr}_{++}$, we
consider the $\Gamma$-action on $\frac{SO^{+}(a,b)}{SO^{+}(a-2,b)}$, and use
2.1, (2.1) and 2.2. From (2.1) it is clear that it suffices to classify the
orbits of $\Gamma$-action on $\frac{SO^{+}(a,b)}{SO^{+}(a-2,b)}$. From 2.2, it
follows that the classes of orbits in 2.2 correspond to classes of
intermediate rational subgroups $S\subset SO^{+}(a,b)$ generated by
unipotents, with $SO^{+}(a-2,b)\subset H\subset SO^{+}(a,b)$. From 2.1, we
obtain that $S$ is either $SO^{+}(a-2,b)$, $SO^{+}(a-1,b)$, $SO^{+}(a,b)$ or
$SO^{+}(a-2,b)\times SO(2)$; however, the later case is impossible, because
$S$ is generated by unipotents. The remaining three classes give the three
cases (i)-(iii) of 2.3.

The last class of orbits (2.3, (iii)) is what is missing from [V2]. However,
most applications of [V2], obtained since, remain valid, because the extra
orbit has sufficiently high codimension. This is explained in more detail in
Section 4. Also, the extra orbits are negligible from the complex-geometric
point of view, which is implied by the following result. Recall that a
subvariety $N\subset M$ of a complex manifold is called totally real if there
exists an anti-holomorphic involution $\tau:\;M{\>\longrightarrow\>}M$ such
that $N$ is its fixed point set. Totally real subvarieties of smooth varieties
are always smooth (the fixed point set of an involution is always smooth).
From the definition it is easy to see that
$TM{\left|{}_{{\phantom{|}\\!\\!}_{N}}\right.}=TN\otimes_{\mathbb{R}}{\mathbb{C}}$.
Therefore, a tangent space $T_{x}N$ to a totally real submanifold contains no
positive-dimensional complex subspaces, and it is not contained in a proper
complex subspace of $T_{x}M$.

This gives the following claim.

Claim 2.6: Let $M$ be a complex manifold, and $N\subset M$ a totally real
submanifold. Then $N$ contains no positive-dimensional complex subvarieties of
$M$, and is contained in no proper subvariety of $M$.

Proposition 2.7:  Let $V$ be a vector space equipped with a scalar product of
signature $(a,b)$, $a>2$, and
$\operatorname{Gr}_{++}:=\frac{SO^{+}(a,b)}{SO^{+}(a-2,b)\times SO(2)}$ the
positive oriented Grassmannian, equipped with the complex structure as in 2.3.
Fix a vector $v\in V$,= with $(v,v)>0$, and let
$\operatorname{Gr}_{++}(v)\subset\operatorname{Gr}_{++}$ be the set of all
positive, oriented 2-planes containing $v$. Then $\operatorname{Gr}_{++}(v)$
is a fixed point set of an antiholomorphic involution of
$\operatorname{Gr}_{++}$. This means that $\operatorname{Gr}_{++}(v)$ is
totally real, contains no positive-dimensional complex subvarieties of
$\operatorname{Gr}_{++}$, and is contained in no proper complex subvariety of
$\operatorname{Gr}_{++}$.

Proof: Let $\iota_{v}:\;V{\>\longrightarrow\>}V$ be the reflection

$x\mapsto x-2\frac{(x,v)}{(v,v)}v.,$

and
$\gamma_{v}:\;\operatorname{Gr}_{++}{\>\longrightarrow\>}\operatorname{Gr}_{++}$
be the composition of the map induced by $\iota_{v}$ and the orientation
reversal. Clearly, $\gamma_{v}$ fixes a 2-plane $W\subset V$ if and only if
$W$ contains $v$. Since the complex structure on $\operatorname{Gr}_{++}$ is
replaced by its opposite when the orientation changes, $\gamma_{v}$ is an
anti-holomorphic map.

##  3 Mapping class group action on the Teichmüller space

###  3.1 Mapping class group action and non-Hausdorff points

The period space $\operatorname{{\mathbb{P}}\sf er}$ is obtained by taking the
Teichmüller space $\operatorname{\sf Teich}$ and gluing together all non-
separable points. It is not obvious how to relate closed or dense orbits on
$\operatorname{{\mathbb{P}}\sf er}$ to that on $\operatorname{\sf Teich}$. The
answer to this problem, given in [V2], remains valid. Let us remind how it is
done.

Theorem 3.1:  Let $\operatorname{\sf Per}:\;\operatorname{\sf
Teich}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf er}$ be the period
map, and $\Gamma$ the mapping class group. Assume that $b_{2}(M)\geqslant 5$.
Then the period map commutes with taking closures: $\operatorname{\sf
Per}(\overline{\Gamma I})=\overline{\Gamma\operatorname{\sf Per}(I)}$, for all
$I\in\operatorname{\sf Teich}$.

The proof of 3.1 takes the rest of this section.

Recall that the positive cone $\operatorname{Pos}([I])$ is the connected
component of the set of all real (1,1)-classes $v\in H^{1,1}([I])$ satisfying
$q(v,v)>0$, where $q$ is the Bogomolov-Beauville-Fujiki form. There are two
connected components, and we choose one which contains the Kähler classes. A
subset $K\subset\operatorname{Pos}([I])$ is called Kähler chamber if it is a
Kähler cone for some $I\in\operatorname{\sf Teich}$ satisfying
$\operatorname{\sf Per}(I)=[I]$. The following fundamental result is due to
Eyal Markman.

Proposition 3.2:  Different Kähler chambers of $[I]$ do not intersect, and
$\operatorname{Pos}([I])$ is a closure of their union. Moreover, there is a
bijective correspondence between points of $\operatorname{\sf Per}^{-1}([I])$
in the Teichmüller component of $I$ and the set of Kähler chambers of $[I]$.

Proof: [Mar, Proposition 5.14].

Consider the set $\operatorname{\sf Teich}_{K}$ of pairs
$\left(I\in\operatorname{\sf Teich},\omega\in\operatorname{Kah}(M,I)\right)$,
where $\operatorname{Kah}(M,I)$ denotes the Kähler cone, and
$q(\omega,\omega)=1$. Let $\operatorname{\sf Per}_{K}$ be the set of all pairs

$\\{\left([I]\in\operatorname{{\mathbb{P}}\sf
er},\omega\in\operatorname{Pos}([I])\right)\ \ |\ \ q(\omega,\omega)=1\\}.$

Consider the period map $\operatorname{\sf Per}_{K}:\;\operatorname{\sf
Teich}_{K}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf er}_{K}$ mapping
$(I,\omega)$ to $(\operatorname{\sf Per}(I),\omega)$. By 3.1,
$\operatorname{\sf Per}_{K}$ is injective with dense image. This means that
points of $\operatorname{\sf Teich}$ can be identified with the space
${\mathfrak{P}}$ pairs $(I,k)$, where $I\in\operatorname{{\mathbb{P}}\sf er}$
and $k\subset H^{1,1}(M,I)$ is one of the Kähler chambers of
$I\in\operatorname{{\mathbb{P}}\sf er}$. The topology on this space of pairs
is the quotient topology induced by the map
$\tau:\;\operatorname{{\mathbb{P}}\sf
er}_{K}{\>\longrightarrow\>}{\mathfrak{P}}$.

Taking in account this picture, the closure of the monodromy orbit of
$I\in\operatorname{\sf Teich}$ can be understood as the closure of
$\Gamma\cdot(\operatorname{\sf Per}(I),k)$ in $\operatorname{\sf
Teich}_{K}\subset\operatorname{{\mathbb{P}}\sf er}_{K}$, where $k$ is the
Kähler chamber $I\in\operatorname{\sf Teich}$, and $\operatorname{\sf
Per}(I),k$ is considered as a subset in $\operatorname{{\mathbb{P}}\sf
er}_{K}$.

To prove that $\overline{\Gamma\cdot(\operatorname{\sf Per}(I),k)}$ contains
the positive cone
$(I,\operatorname{Pos}(I))\subset\operatorname{{\mathbb{P}}\sf er}_{K}$, we
use the Ratner’s theorem again. We find an orbit $R$ of a one-parametric group
$H_{0}\subset SO(H^{2}(M,{\mathbb{R}}))$ generated by unipotents and fixing
$I\in\operatorname{{\mathbb{P}}\sf er}$ with
$R\subset\operatorname{{\mathbb{P}}\sf er}_{K}$ lying within
$(\operatorname{\sf Per}(I),k)$, and show (using Ratner’s theorem) that
$\Gamma\cdot R$ is dense in $\operatorname{{\mathbb{P}}\sf er}_{K}$.

The same strategy was used in [V2], however, the one-parametric group that we
chose in [V2] was in fact not generated by unipotents. Here we fill this gap
and give a corrected version of the proof.

###  3.2 Round segments on the boundary of the Kähler cone

In this subsection we shall speak of “round bits” on the boundary of the
convex cone. The geometric intuition behind this terminology is that the
Kähler cone is polyhedral at some points, and strictly convex with smooth
boundary at other points; the points of the boundary where it’s smooth and
strictly convex constitute “round bits”.

Definition 3.3: Let $(M,I)$ be a hyperkähler manifold, and $W\subset
H^{1,1}_{I}(M,{\mathbb{R}})$ a 3-dimensional real subspace of signatue
$(1,2)$. We say that the Kähler cone boundary has round bits on $W$ if the
projectivization of the closure
${\mathbb{P}}\overline{\operatorname{Kah}(M,I)\cap W}$ contains an open subset
of the boundary of the disk ${\mathbb{P}}\operatorname{Pos}(W)$.111Here, as
elsewhere, $\operatorname{Pos}(W)$ denotes the positive cone, which is one of
two components of the set $\\{w\in W\ \ |\ \ q(w,w)>0\\}$. Notice that
${\mathbb{P}}\operatorname{Pos}(W)$ is a 2-dimensional disk in
${\mathbb{P}}(W)={\mathbb{R}}P^{2}$, identified with one of the standard
models of the hyperbolic plane.

The following proposition is used to construct a one-parametric unipotent
orbit which lies in $\operatorname{Kah}(M,I)$. The idea is to construct a
horocycle in the hyperbolic segment
${\mathbb{P}}\operatorname{Pos}(W)\subset{\mathbb{P}}\operatorname{Kah}(M,I)=(I,k)\subset\operatorname{{\mathbb{P}}\sf
er}_{K}$. However, a horocycle exists only in subsets of
$\overline{\operatorname{Pos}(W)}$ which contain open parts of the boundary.

Proposition 3.4:  Let $(M,I)$ be a hyperkähler manifold such that
$b_{2}(M)\geqslant 5$ and the Picard rank of $(M,I)$ is non-maximal. Denote by
${\mathfrak{T}}$ the orthogonal complement to $H^{1,1}_{I}(M,{\mathbb{R}})\cap
H^{2}(M,{\mathbb{Q}})$ in $H^{1,1}_{I}(M,{\mathbb{R}})$; since the Picard rank
of $(M,I)$ is non-maximal, the space ${\mathfrak{T}}$ has positive dimension.
Then for each 3-dimensional space $W\subset H^{1,1}_{I}(M,{\mathbb{R}})$ of
signature $(1,2)$ such that ${\mathfrak{T}}\cap W\neq 0$, the Kähler cone
boundary has round bits on $W$.

Proof: By [AV1, Theorem 1.19], the Kähler cone $\operatorname{Kah}(M,I)$ is a
connected component of $\operatorname{Pos}(M,I)\backslash S^{\bot}$, where
$S^{\bot}=\bigcup_{i}s_{i}^{\bot}$ is a countable union of hyperplanes
obtained as orthogonal complement to the so-called MBM classes $s_{i}\in
H^{1,1}_{I}(M,{\mathbb{Z}})$. Clearly, all $s_{i}$ are orthogonal to
${\mathfrak{T}}$. Therefore, all hyperplanes $s_{i}^{\bot}$ intersect the
subspace $P:={\mathfrak{T}}\cap W\subset W$. The corresponding partition of
the disk ${\mathbb{P}}\operatorname{Pos}(W)$ onto pieces by $S^{\bot}$ depends
on the signature of $P$.

We may assume that $P$ is one-dimensional; if it’s 2-dimensional, one has
$S^{\bot}_{i}\cap W=P$, and the disk is partitioned onto at most two pieces,
hence each piece contains a round part of the boundary.

Notice that the set $s_{i}^{\bot}$ of walls of the Kähler chambers is locally
finite ([HT, Proposition 10]). On the disk ${\mathbb{P}}\operatorname{Pos}(W)$
these lines trace geodesics which intersect in one point of the disk if
${\mathfrak{T}}$ has positive signature, and intersect in “imaginary”
(negative) point outside of the disk if the signature is negative, We obtain
that the hyperplanes $s_{i}^{\bot}$ partition the disk
${\mathbb{P}}\operatorname{Pos}(W)$ as follows.

$\mathfrak{T}$ has positive signature $\mathfrak{T}$ has negative signature

In both cases, as seen from the picture, all connected components of
${\mathbb{P}}\operatorname{Pos}(W)\backslash S^{\bot}$ have round bits on the
boundary of the disk.

###  3.3 Horocycles in the Kähler cone

Let $\operatorname{{\mathbb{P}}\sf er}_{K}$ the period space of Kähler classes
(Subsection 3.1), and $\Gamma$ the monodromy group acting on
$\operatorname{{\mathbb{P}}\sf er}_{K}$. We find a one-parametric group
$H_{0}\subset SO^{+}(H^{2}(M,{\mathbb{R}}))$ generated by unipotents such that
for some $x\in\operatorname{Kah}(M,I)$, the orbit $H_{0}\cdot x$ is contained
in $\operatorname{Kah}(M,I)$. We apply 3.2 and find $W\subset
H^{1,1}(M,{\mathbb{R}})$ such that the Kähler cone boundary of $(M,I)$ has
round bits on $W$. The group $H_{0}$ acts on the Poincare disk
$\Delta:={\mathbb{P}}\operatorname{Pos}(W)$ by parabolic isometries, and its
orbits are horocycles. From the picture below it is clear that there are
uncountably many horocycle subgroups with orbits in
$\operatorname{Kah}(M,I)\cap W$ whenever the Kähler cone boundary of $(M,I)$
has round bits on $W$.

Horocycles on a Poincare disc

The following proposition, applied together with Ratner’s theorem, finishes
the proof of 3.1, as indicated in Subsection 3.1.

By “general” we always mean “not contained in a union of countable number of
Zariski closed subsets of positive codimension”.

Proposition 3.5:  Let $V_{\mathbb{Q}}$ be a rational vector space of signature
$(3,d)$, $d\geqslant 2$, $V:=V_{\mathbb{Q}}\otimes_{\mathbb{Q}}{\mathbb{R}}$,
and $\Gamma\subset SO^{+}(V)$ an arithmetic lattice. Consider a subspace
$V_{1}\subset V$ of signature $(1,d)$, let $P\subset V_{1}$ be a 1-dimensional
subspace, $W\supset P$ a general 3-dimensional subspace of $V_{1}$ of
signature $(1,2)$ and containing $P$, and $H_{0}\subset SO(W)\subset SO(V)$ a
general one-parametric unipotent subgroup. Then the smallest rational Lie
subgroup containing $H_{0}$ also contains $SO^{+}(V_{1})$.

Proof: Denote by $\operatorname{Cl}(H_{0})$ the smallest rational Lie subgroup
containing $H_{0}$, and let
${\mathfrak{I}}_{H_{0}}\subset\coprod{\mathbb{P}}V^{\otimes i}$ be the set of
all rational projective invariants of $H_{0}$. By Chevalley’s theorem (see
e.g. [Mor]), $\operatorname{Cl}(H_{0})$ is the biggest Lie group fixing all
points $x\in{\mathfrak{I}}_{H_{0}}$. Since the set ${\mathfrak{I}}_{H_{0}}$ is
countable and semicontinuous in $H_{0}$, the group $\operatorname{Cl}(H_{0})$
for generic $H_{0}$ is independent on $H_{0}$ and contains all unipotent
sugroups of $SO^{+}(W)$, for all $W$ of appropriate signature satisfying
$P\subset W\subset V_{1}$. This implies $\operatorname{Cl}(H_{0})\supset
SO^{+}(V_{1})$.

From 3.3, 3.1 follows immediately. Indeed, apply 3.3 to
$V=H^{2}(M,{\mathbb{R}})$, $V_{1}=H^{1,1}_{I}(M,{\mathbb{R}})$, and
$P\subset{\mathfrak{T}}$, and choose generic $H_{0}\subset SO(W)$ which has an
orbit in $\operatorname{Kah}(M,I)$. By Ratner’s theorem and 3.3, the closure
of this orbit in $\operatorname{{\mathbb{P}}\sf er}_{K}/\Gamma$ contains an
orbit of $SO^{+}(V_{1})$ which contains the set of unit vectors in the
positive cone of $I$ considered as a subset of $\operatorname{{\mathbb{P}}\sf
er}_{K}$. However, the closure of this set in $\operatorname{{\mathbb{P}}\sf
er}_{K}/\Gamma$ is mapped surjectively to the closure of $I$ in
$\operatorname{{\mathbb{P}}\sf er}/\Gamma$, because
$\operatorname{{\mathbb{P}}\sf er}_{K}$ is fibered over
$\operatorname{{\mathbb{P}}\sf er}$ with fiber $\operatorname{Pos}(M,I)$, and
this fibration is $\Gamma$-equivariant.

##  4 Applications to hyperkähler geometry: Hyperbolicity and Kobayashi
pseudometric

Let $\Gamma$ be the mapping class group of a complex manifold. Recall that a
complex structure is called ergodic if its $\Gamma$-orbit is dense in its
connected component of the Teichmüller space of complex structures.

In [V2], we proved that a hyperkähler manifold is not Kobayashi hyperbolic. To
prove this, one needs to produce a deformation which is not Kobayashi
hyperbolic and has ergodic complex structure.

As shown by F. Campana in [Cam], for any hyperkähler structure, the
corresponding twistor deformation has at least one non-hyperbolic fiber. The
twistor deformation can be interpreted in terms of the period space as
follows. Let $W\subset H^{2}(M,{\mathbb{R}})$ be the 3-dimensional space
generated by three Kähler forms $\omega_{I},\omega_{J},\omega_{K}$, associated
with the quaternionic triple of complex structures. Consider the space $S_{W}$
of oriented 2-planes in $W$. We can consider $S_{W}$ as a subset in
$\operatorname{Gr}_{++}$. This subset is precisely the set of all complex
structures in the twistor deformation associated with $I,J,K$.

Let now $W_{0}\subset W$ be the 2-plane corresponding to the non-hyperbolic
complex structure constructed in [Cam]. If we chose $W$ such that it contains
no rational vectors, $W_{0}\cap H^{2}(M,{\mathbb{Q}})=0$, hence the associated
$\Gamma$-orbit is dense in $\operatorname{{\mathbb{P}}\sf er}$ (2.3). The
corresponding complex structure is ergodic. This proves the non-hyperbolicity
of hyperkähler manifolds.

In [KLV, Theorem 1.1], it was shown that a hyperkähler manifold $M$ with
$b_{2}(M)\geqslant 7$ has vanishing Kobayashi pseudometric if the “SYZ
conjecture” holds. SYZ conjecture, which is formulated further on in this
paper, is the same as Kawamata’s abundance conjecture for hyperkähler
manifolds. It is known for the Hilbert scheme of K3 surfaces and all its
deformations ([BM]) and for the generalized Kummer varieties ([Y]).

By a theorem of Matsushita (see [Mat1]), any holomorphic map
$\pi:\;M{\>\longrightarrow\>}B$ is a Lagrangian fibration if $0<\dim B<\dim
M$. The space $B$ is known to be a Fano manifold, and it has the same rational
cohomology as ${\mathbb{C}}P^{n}$ if it is normal ([Mat2]). The pullback
$\eta:=\pi^{*}\omega$ of an ample class $\omega\in H^{2}(B,{\mathbb{Z}})$ is
an integer class on the boundary of the Kähler cone of $M$ which satisfies
$\int_{M}\eta^{\dim_{\mathbb{C}}M}=0$. Such cohomology classes are known as
parabolic111In more generality, parabolic classes can be defined as nef
classes of volume 0., and SYZ conjecture claims that, conversely, any
parabolic class $\eta\in H^{2}(M,{\mathbb{Z}})$ is obtained this way.

Consider a manifold $M$ which has two transversal Lagrangian fibrations
(possibly rational). Since the fibers of these fibrations are tori, the
Kobayashi pseudometric on $M$ vanishes.

To prove that any manifold $(M,I)$ has vanishing Kobayashi pseudometric, we
need to show that in the closure $\overline{\Gamma I}$ of its $\Gamma$-orbit
there exists a manifold with vanishing Kobayashi pseudometric. Since the
diameter of the Kobayashi pseudometric is upper semicontinuous ([KLV]), this
would imply that diameter of Kobayashi pseudometric on $(M,I)$ also vanishes.

Meyer’s theorem states that any integer lattice which is not sign-definite and
of rank $\geqslant 5$ represents 0, that is, contains an integer (1,1)-class
with square 0. Applying this result and global Torelli theorem to a
hyperkähler manifold with $b_{2}(M)\geqslant 5$, we can find a complex
structure with any Picard rank $\leqslant b_{2}(M)-2$ and a vector with square
0, which would lie in a boundary of the Kähler cone for some complex structure
([AV2]), and give a Lagrangian fibration if SYZ conjecture is true.

Using the same argument for two non-proportional nef vectors, we would get two
transversal Lagrangian fibrations. If we want to have a point with the two
Lagrangian fibrations on a positive-dimensional orbit
$\operatorname{Gr}_{++}(v)$ (2.3), we would need to find two Lagrangian
fibration (that is, two nef classes) on a manifold with
$\operatorname{Re}(H^{2,0})\ni v$, which is done using the same argument. This
argument takes care of [KLV, Theorem 1.1].

A special case of this result, [KLV, Corollary 2.2], says that any non-
algebraic K3 surface has vanishing Kobayashi metrics (for algebraic K3 it was
already known). The idea is to show that an algebraic complex structure can be
found in the closure of the mapping class orbit of a non-algebraic complex
structure and use the semicontinuity of the diameter of Kobayashi metric. This
argument remains true. However, we should take care about an extra orbit and
notice that the closure of this orbit is fixed point set of a rational
anticomplex involution on the period space. It is easy to see that this fixed
point set intersects the divisor $\operatorname{\sf
Teich}_{g}\subset\operatorname{\sf Teich}$ representing polarized K3 of any
given genus. This implies that there is a dense set of algebraic complex
structures in any intermediate orbit.

The rest of results of [KLV] remain the same, with the same proofs.

Acknowledgements: The gap in the classification of orbits in [V2] was found
when I collaborated with Nessim Sibony on a different project extending the
classification of [V2] to the space of complex structures equipped with a
parabolic nef class. I am grateful to Nessim for his questions and many
interesting discussions. Apologies to those who were misled by the omission of
the elusive orbit. Many thanks to Ekaterina Amerik and Ljudmila Kamenova for
their remarks and questions about the preliminary version of this paper.
Section 3 is based on a long series of talks with Ekaterina Amerik, who is
responsible for finding many bugs in earier versions, and many ideas of the
proof and its presentation.

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Misha Verbitsky
Instituto Nacional de Matemática Pura e Aplicada
Estrada Dona Castorina, 110
Jardim Botânico, CEP 22460-320
Rio de Janeiro, RJ - Brasil
also:
Laboratory of Algebraic Geometry,
Faculty of Mathematics, National Research University HSE,
7 Vavilova Str. Moscow, Russia,
<EMAIL_ADDRESS><EMAIL_ADDRESS>