Hermitian-symplectic and
Kähler structures on degenerate twistor deformations

Andrey Soldatenkov, Misha Verbitsky111Partially supported by FAPERJ
SEI-260003/000410/2023 and CNPq - Process 310952/2021-2.
Keywords: hyperkähler manifold, twistor construction 2010 Mathematics Subject
Classification: 53C26

Abstract Let $(M,\Omega)$ be a holomorphically symplectic manifold equipped
with a holomorphic Lagrangian fibration $\pi:M\to B$, and $\eta$ a closed
$(1,1)$-form on $B$. Then $\Omega+\pi^{*}\eta$ is a holomorphically symplectic
form on a complex manifold which is called the degenerate twistor deformation
of $M$. We prove that degenerate twistor deformations of compact
holomorphically symplectic Kähler manifolds are also Kähler. First, we prove
that degenerate twistor deformations are Hermitian symplectic, that is, tamed
by a symplectic form; this is shown using positive currents and an argument
based on the Hahn–Banach theorem, originally due to Sullivan. Then we apply a
version of Huybrechts’s theorem showing that two non-separated points in the
Teichmüller space of holomorphically symplectic manifolds correspond to
bimeromorphic manifolds if they are Hermitian symplectic.

###### Contents

  1. 1 Introduction
    1. 1.1 Degenerate twistor deformations and the Tate–Shafarevich twist
    2. 1.2 Non-Kähler deformations of hyperkähler manifolds
    3. 1.3 Hermitian symplectic geometry and Streets–Tian conjecture
  2. 2 Preliminaries
    1. 2.1 Hyperkähler manifolds
    2. 2.2 Degenerate twistor deformations
    3. 2.3 Weakly positive and strongly positive forms
    4. 2.4 Positive currents
  3. 3 Hermitian symplectic structures and degenerate twistor deformations
  4. 4 $d$-Commendable manifolds
    1. 4.1 Barlet spaces on commendable manifolds
    2. 4.2 Non-separated points in the Teichmüller space
    3. 4.3 Degenerate twistor deformations are Kähler
  5. 5 Appendix: Hermitian symplectic forms and Fujiki class C

##  1 Introduction

###  1.1 Degenerate twistor deformations and the Tate–Shafarevich twist

Let $(M,I,\Omega)$ be a holomorphically symplectic manifold, that is, an
almost complex manifold equipped with a closed, non-degenerate $(2,0)$-form.
Clearly, the almost complex structure $I$ is uniquely determined by the
differential form $\Omega$. It is not hard to see that the integrability of
the almost complex structure $I$ follows from closedness of $\Omega$ ([V3]).
This allows one to define the complex structure in terms of $\Omega$, giving a
motivation for the notion of C-symplectic form (2.2).

The notion of degenerate twistor deformation, introduced in [V3], and further
explored in [BDV] and [SV], is one of the applications of this approach. Given
a holomorphic Lagrangian fibration $\pi:\;M{\>\longrightarrow\>}B$ on a
holomorphic symplectic manifold $(M,\Omega)$, we can replace $\Omega$ by
$\Omega+t\pi^{*}\eta$, where $\eta$ is a closed $(2,0)+(1,1)$-form on $B$, and
$t\in{\mathbb{C}}$. The new form $\Omega_{t}:=\Omega+t\pi^{*}\eta$ is
C-symplectic, that is, holomorphic symplectic with respect to a complex
structure $I_{t}$ (2.2). This defines a holomorphic family of complex
manifolds $(M,I_{t}),t\in{\mathbb{C}}$, which is called the degenerate twistor
deformation.

It shares many of its properties with twistor deformations (the ones which
come from a hyperkähler rotation); indeed, the degenerate twistor deformation
can be obtained as a limit of twistor deformations ([V3, Section 1.2]).

A version of this construction is known as the “Tate–Shafarevich twist”. For
elliptic surfaces, it was explored at some length in [FM], and for higher
dimensions in [Mar] and [AR]. In [AR] it was shown that the Tate–Shafarevich
twist and the degenerate twistor deformation coincide if the Lagrangian
fibration has no multiple fibers in codimension 1.

In the present paper, we prove that the degenerate twistor deformation is
always Kähler (4.3), answering the question which was originally posed by E.
Markman.

Theorem 1.1: Let $M$ be a compact holomorphically symplectic manifold of
Kähler type, $\pi:\;M{\>\longrightarrow\>}B$ a Lagrangian fibration, and
$\tau:\;{\cal M}{\>\longrightarrow\>}{\mathbb{C}}$ the corresponding
degenerate twistor deformation. Then all fibers of $\tau$ admit a Kähler
metric.

Proof: 4.3.

In [Mar], Markman has studied the Tate–Shafarevich twist of a Lagrangian
fibration on a Hilbert scheme of a K3 surface. Markman calls a compact
holomorphically symplectic manifold $(M,\Omega)$ “M-special” if the space
$\langle[\operatorname{Re}\Omega],[\operatorname{Im}\Omega]\rangle\subset
H^{2}(M,{\mathbb{R}})$ does not contain non-zero rational cohomology classes.
He has shown that the Tate–Shafarevich twist of a Lagrangian fibration on a
Hilbert scheme of a K3 surface is Kähler, if it is not M-special. In [AR],
this result was generalized to all Lagrangian fibrations without multiple
fibers in codimension 1. In the present paper we give an independent proof
that works for arbitrary compact hyperkähler manifolds without the assumption
of non-M-speciality.

###  1.2 Non-Kähler deformations of hyperkähler manifolds

The earliest publication on Kählerness of holomorphically symplectic manifolds
was a preprint of Andrey Todorov [T]. Using a Harvey-Lawson-type Hahn-Banach
argument, Todorov tried to show that a simply connected, compact,
holomorphically symplectic manifold with $H^{2,0}(M)={\mathbb{C}}$ is Kähler.

The earliest counterexamples to this statement were obtained in 1994 by D.
Guan, [Gu1, Gu2, Gu3, Bo]. These are manifolds, obtained from the Kodaira
surface (non-Kahler, holomorphically symplectic complex surface) by a
procedure reminding of the one which is used to construct the generalized
Kummer variety from a 2-dimensional torus. This manifold is non-Kähler because
it contains a complex surface isomorphic to a blow-up of Kodaira surface ([Bo,
Corollary 4.10]).

Another example of a non-Kähler holomorphically symplectic manifold comes from
birational geometry.111We benefited from a Mathoverflow discussion at
http://mathoverflow.net/questions/302649/are-there-non-projective-but-
algebraic-hyperkahler-varieties and the information provided by the user
YangMills. Recall that Mukai flop is a bimeromorphic transform which takes a
Lagrangian ${\mathbb{C}}P^{n}$ in a holomorphically symplectic manifold $M$,
and replaces it with another one, obtaining another holomorphically symplectic
manifold (say, $M^{\prime}$). The second homology groups of these manifolds
are isomorphic, however, the homology class of a rational curve (a line
contained in the ${\mathbb{C}}P^{n}$) on $M$ is opposite to that on
$M^{\prime}$. If we start from a holomorphically symplectic manifold
containing two disjoint Lagrangian planes with homologous lines, and do a flop
in only one of these, we will obtain a holomorphically symplectic manifold
which contains a complex curve (the union of two lines) homologous to zero.
This manifold is Fujiki class C (that is, bimeromorphic to Kähler). For
explicit examples of non-Kähler Fujiki class C holomorphic symplectic
manifolds and a reference, see [GHJ, Example 21.9].

In the sequel, hyperkähler means “compact holomorphically symplectic manifold
of Kähler type”. In [Hu1], D. Huybrechts proved that two bimeromorphic
hyperkähler manifolds corresponds to non-separable (in the sense of being non-
Hausdorff) points of the corresponding Teichmüller space. Conversely, given
two non-separable points $I_{1},I_{2}$ in the Teichmüller space, the
hyperkähler manifolds $(M,I_{1})$ and $(M,I_{2})$ are bimeromorphic. Curiously
enough, this argument uses the Kählerness in an essential way.

If one drops the Kählerness assumption (or a weaker assumption of existence of
the Hodge decomposition on $H^{2}(M)$ as in [Per]; see the discussion of this
work below), the proof is still unknown, though we may dare to make a
conjecture.

Conjecture 1.2:  Let $(M_{t},\Omega_{t})$ be a smooth family of compact
holomorphically symplectic manifolds over a disk $\Delta$. Assume that all
fibers except the central one are hyperkähler. Then the central fiber
$(M_{0},\Omega_{0})$ is of Fujiki class C.

In 4.2, we prove this conjecture when the central fiber $M_{0}$ is Hermitian
symplectic. In [Per], A. Perego proved 1.2 when $M_{0}$ satisfies
$b_{2}(M_{0})=h^{2,0}(M_{0})+h^{1,1}(M_{0})+h^{0,2}(M_{0})$.

Let us indicate where Huybrechts’s proof uses the Kählerness. Consider two
deformations $M_{t}$ and $M_{t}^{\prime}$ of a hyperkähler manifold, with
$t\in\Delta$. Assume that there is an isomorphism $M_{t}\cong M^{\prime}_{t}$
for all $t\neq 0$. The graph of this isomorphism defines a family of
subvarieties $\Gamma_{t}$ in $M_{t}\times M^{\prime}_{t},t\neq 0$. Using
Bishop’s compactness theorem, Huybrechts proves that the family $\Gamma_{t}$
can be extended to $t=0$, and the corresponding cycle $\Gamma_{0}$ contains a
component that is the graph of a bimeromorphic map $M_{0}\dashrightarrow
M^{\prime}_{0}$. When $M_{0}$ is non-Kähler, Bishop’s compactness fails and
the argument becomes invalid. However, Bishop’s compactness still holds for
Hermitian symplectic manifolds (4.1), and this leads to 4.2.

1.2 is a special case of a more ambitious conjecture proposed in [LL].

Conjecture 1.3:  Let $M_{t}$ be a smooth family of compact complex manifolds
over a disk $\Delta$. Assume that all fibers except the central one are
Kähler. Then the central fiber $M_{0}$ is Fujiki class C.

In [LL], this conjecture is proven under two extra assumptions on a central
fiber: (a) $M_{0}$ admits a Hermitian form $\omega$ satisfying
$dd^{c}\omega=0$ and $\partial\omega\wedge\overline{\partial}\omega=0$, and
(b) $h^{0,2}(M_{0})=h^{0,2}(M_{t})$ for $t\neq 0$.

In [Po], this conjecture is proven when $M_{t}$, $t\neq 0$ are projective, and
$h^{0,1}(M_{0})=h^{0,1}(M_{t})$ for $t\neq 0$; in this case, the central fiber
is Moishezon.

###  1.3 Hermitian symplectic geometry and Streets–Tian conjecture

The original impulse to this paper is an observation due to Peternell: any
non-projective Moishezon manifold admits a positive, exact
$(n-1,n-1)$-current, see [Pe]. We were trying to use this observation to prove
that a degenerate twistor deformation of a hyperkähler manifold cannot be non-
projective Moishezon, in hope to extend it further to Fujiki class C, and then
try to prove 1.2 in this particular situation.

In 3 we accomplish the first task, indeed, showing that a degenerate twistor
deformation of a hyperkähler manifold cannot support a positive, exact
$(n-1,n-1)$-current.

Let us illustrate this in a simpler situation, showing that a degenerate
twistor deformation cannot admit a complex curve $C$ which is homologous to
zero.

Let $\pi:\;M{\>\longrightarrow\>}B$ be a Lagrangian fibration on a hyperkähler
manifold $(M,\Omega)$. Then $B$ is projective; let $\omega\in
H^{2}(B,{\mathbb{R}})$ be its Kähler class. Since $\int_{C}\pi^{*}\omega=0$,
$C$ is contained in the fibers of $\pi$. However, the degenerate twistor
deformation does not change the complex structures on the fibers, hence $C$ is
a complex curve on the original hyperkähler manifold, which is impossible,
because a subvariety of a Kähler manifold cannot be homologous to zero.

In 3 we modify this argument showing that it remains valid for any positive,
exact $(n-1,n-1)$-current in place of the current of integration of $C$.

A theorem of Sullivan [Su] implies that a complex manifold admits no positive,
exact $(n-1,n-1)$-currents if and only if it is Hermitian symplectic, that is,
admits a taming symplectic structure (3).

Streets and Tian have conjectured that Hermitian symplectic manifolds are
Kähler, see [ST]. This conjecture was treated at great length, and solved for
some classes of complex manifolds (for complex surfaces in [LZ, Theorem 1.2],
for twistor spaces in [V2]), but in full generality it remains very difficult.
In [EFV, FV] the Streets–Tian conjecture was proven for complex nilmanifolds.

The Streets–Tian conjecture makes sense for almost complex manifolds as well;
in this context, it is known as the “Donaldson’s tamed to compatible
conjecture”. In [Do, Question 2], Donaldson asked whether an almost complex
4-manifold admitting a taming symplectic form must also admit a compatible
symplectic form.

Using 3, we reduce the proof of Kählerness of degenerate twistor deformations
to the Streets–Tian conjecture, and in Subsection 4.2, we prove a special case
of this conjecture, finishing the proof of Kählerness.

Theorem 1.4: Let $\operatorname{\sf Teich}$ be the Teichmuller space of
holomorphically symplectic complex structures, and $I\in\operatorname{\sf
Teich}$ a holomorphically symplectic complex structure obtained as a limit
$I=\lim_{i}I_{i}$, where $I_{i}\in\operatorname{\sf Teich}$ are
holomorphically symplectic complex structures of Kähler type. Assume that
$(M,I)$ is Hermitian symplectic. Then $(M,I)$ is also of Kähler type.

Proof: By 4.2, $(M,I)$ is of Fujiki class C. By [Ch, Theorem 0.2], any
Hermitian symplectic Fujiki C manifold is of Kähler type.

##  2 Preliminaries

###  2.1 Hyperkähler manifolds

We start by recalling the basic facts of the theory of hyperkähler manifolds.
For a detailed exposition of this theory see [Hu1].

We will denote by $\mathbb{H}$ the algebra of quaternions and by $Sp(n)$ the
group of quaternionic-linear transformations of $\mathbb{H}^{n}$. Let $M$ be a
compact $C^{\infty}$ manifold. A Riemannian metric $g$ on $M$ is called
hyperkähler if the holonomy group of its Levi–Civita connection $\nabla^{g}$
is contained in $Sp(n)$. The elements of $Sp(n)$ commute with a two-
dimensional family of complex structures defined via multiplication by
imaginary unit quaternions. Hence by the holonomy principle we get a family of
$\nabla^{g}$-parallel complex structures on $M$ called the twistor family.

In this paper we will always assume that hyperkähler metrics are of maximal
holonomy, i.e. the holonomy group of $\nabla^{g}$ is isomorphic to $Sp(n)$.
Given a complex structure $I$ on $M$, we will say that $I$ is of hyperkähler
type if there exists a hyperkähler metric of maximal holonomy $g$ on $M$ such
that $\nabla^{g}I=0$. If $I$ is of hyperkähler type and $M$ is simply
connected, the manifold $(M,I)$ is usually called irreducible holomorphic
symplectic (IHS).

Given a hyperkähler metric $g$ on $M$ one can find $\nabla^{g}$-parallel
complex structures $I$, $J$ and $K$ such that $K=IJ=-JI$. Denote by
$\omega_{I}$, $\omega_{J}$ and $\omega_{K}$ the corresponding Kähler forms.
The 2-form $\Omega=\omega_{J}+\sqrt{-1}\omega_{K}$ is a holomorphic symplectic
form on $(M,I)$.

Recall the following facts about the cohomology of a hyperkähler manifold $M$
of maximal holonomy (see [Bea] and [Hu1] for the details). The vector space
$H^{2}(M,\mathbb{Q})$ carries a non-degenerate quadratic form $q$ called the
Beauville–Bogomolov–Fujiki (BBF) form. It has the following property: there
exists a positive constant $c_{M}\in\mathbb{Q}$ such that for all $a\in
H^{2}(M,\mathbb{Q})$ the Fujiki relation holds:

$\displaystyle\int_{M}a^{2n}=c_{M}q(a)^{n}.$ (2.1)

We will normalize $q$ in such a way that it is primitive and integral on
$H^{2}(M,\mathbb{Z})$. The BBF form has signature $(3,b_{2}(M)-3)$.

Consider the collection $\operatorname{\sf Comp}(M)$ of complex structures of
hyperkähler type on $M$ and the group $\operatorname{\sf Diff}^{\circ}(X)$ of
diffeomorphisms of $M$ isotopic to the identity. The quotient
$\operatorname{\sf Teich}(M)=\operatorname{\sf Comp}(M)/\operatorname{\sf
Diff}^{\circ}(M)$ carries the structure of a non-Hausdorff complex manifold
and is called the Teichmüller space of $M$. The globall Torelli theorem for
hyperkähler manifolds [V1] describes the Teichmüller space in terms of the
cohomology of $M$. Define the period domain:

$\mathcal{D}=\\{x\in\mathbb{P}H^{2}(M,\mathbb{C})\enskip|\enskip
q(x)=0,\,q(x,\overline{x})>0\\}$

If $I$ is a complex structure of hyperkähler type then the space of
$I$-holomorphic two-forms on $M$ is spanned by the symplectic form $\Omega$,
and the corresponding point $[\Omega]\in\mathcal{D}$ is called the period of
$I$. This defines the period map $\operatorname{\sf
Per}\colon\operatorname{\sf Teich}(M)\to\mathcal{D}$. The global Torelli
theorem [V1] claims that the restriction of $\operatorname{\sf Per}$ to every
connected component of $\operatorname{\sf Teich}$ is the Hausdorff reduction
map, i.e. after identifying the non-separated points of $\operatorname{\sf
Teich}$ it induces an isomorphism of complex manifolds.

###  2.2 Degenerate twistor deformations

In this subsection, we briefly recall the definition of a degenerate twistor
deformation associated with a holomorphic Lagrangian fibration on a
hyperkähler manifold, following [BDV] (see also [V3] and [SV]). Holomorphic
Lagrangian fibrations are ubiquitous in hyperkähler geometry, as shown by the
following foundational theorem.

Theorem 2.1: (D. Matsushita) Let $\pi\colon M\to B$ be a surjective
holomorphic map from a compact hyperkähler manifold $M$ of maximal holonomy to
a complex variety $B$ with $0<\dim B<\dim M$. Then $\dim B={1\over 2}\dim M$,
and the fibers of $\pi$ are holomorphic Lagrangian.

Proof: [Mat].

A map as in the above theorem is called a holomorphic Lagrangian fibration.
The base $B$ is conjectured to be ${\mathbb{C}}P^{n}$, and this is the case in
all known examples; when $B$ is smooth, it is biholomorphic to
${\mathbb{C}}P^{n}$, as shown in [Hw].

Definition 2.2:  Let $M$ be a real $4n$-dimensional $C^{\infty}$-manifold. A
C-symplectic form on $M$ is a smooth section $\Omega$ of the complexified
bundle of 2-forms $\Lambda^{2}M\otimes\mathbb{C}$ such that: $d\Omega=0$,
$\Omega^{n+1}=0$ and the form
$\Omega^{n}\wedge\overline{\Omega}\mathstrut^{n}$ is pointwise non-vanishing.

Let $\Omega$ be a C-symplectic form on $M$. For a point $x\in M$ let
$T^{0,1}_{x}M\subset(T_{x}M)\otimes\mathbb{C}$ be the kernel of $\Omega$, i.e.
the subspace consisting of complex tangent vectors
$v\in(T_{x}M)\otimes\mathbb{C}$ such that the contraction
$v\,\lrcorner\,\Omega$ vanishes.

One verifies (see [BDV], [SV]) that the kernel of $\Omega$ has rank $2n$ at
all points of $M$, hence $T^{0,1}M\subset TM\otimes\mathbb{C}$ is a subbundle.
Defining $T^{1,0}M$ as the complex conjugate of $T^{0,1}M$, one gets the
decomposition $TM\otimes\mathbb{C}=T^{1,0}M\oplus T^{0,1}M$. Using the
closedness of the form $\Omega$, one checks that the subbundle $T^{1,0}M$ is
closed under the Lie bracket. This defines an integrable almost complex
structure on $M$, and therefore by the Newlander–Nirenberg theorem any
manifold with a C-symplectic form admits an intrinsically defined complex
structure.

Theorem 2.3:  Let $M$ be a compact complex manifold with a holomorphic
symplectic form $\Omega$, and let $\pi\colon M\to B$ be a holomorphic
Lagrangian fibration. Consider a closed two-form
$\eta\in\Lambda^{2}B\otimes\mathbb{C}$ such that $\eta^{0,2}=0$. Then the two-
form $\Omega_{t}:=\Omega+t\pi^{*}\eta$ is C-symplectic for any
$t\in\mathbb{C}$. If we denote by $I_{t}$ the complex structure on $M$
corresponding to $\Omega_{t}$, then for any $t\in\mathbb{C}$ the map $\pi$ is
holomorphic with respect to $I_{t}$ and the fixed complex structure on the
base $B$.

Proof: [BDV].

The following property of the complex structures $I_{t}$ from the above
theorem appears implicitly in [BDV] and [SV]. Let us record it for the later
use.

Lemma 2.4:  In the setting of 2.2 denote by $F$ one of the fibres of $\pi$ and
let $x\in F$. Let $V=T_{x}F\otimes\mathbb{C}$ be the complexified tangent
space of $F$ at the point $x$. Then the subspace $V_{t}^{0,1}=V\cap
T^{0,1}_{I_{t}}M$ does not depend on $t\in\mathbb{C}$.

Proof: Consider $v\in V$. Then $v\in V^{0,1}_{t}$ if and only if
$v\,\lrcorner\,\Omega_{t}=0$. Note that $V=\ker(d\pi)_{x}$, therefore
$v\,\lrcorner\,\pi^{*}\eta=0$ and
$v\,\lrcorner\,\Omega_{t}=v\,\lrcorner\,\Omega$. Hence
$V^{0,1}_{t}=V^{0,1}_{0}$, so $V^{0,1}_{t}$ is a fixed subspace of $V$ not
depending on $t$.

Assume now that $M$ is a compact simply connected hyperkähler manifold of
maximal holonomy, and $\pi\colon M\to B$ a holomorphic Lagrangian fibration
over a projective base $B$ (the projectivity of $B$ follows from the other
assumptions, see [AC, footnote on page 53]). Let $\eta\in\Lambda^{1,1}B$ be
the restriction of the Fubini–Study form under some projective embedding of
the base, $\Omega_{t}=\Omega+t\pi^{*}\eta$ the C-symplectic form and $I_{t}$
the corresponding complex structure, where $t\in{\mathbb{C}}$. The manifolds
$\mathcal{M}_{t}=(M,I_{t})$ are fibres of the the degenerate twistor
deformation of $M$, which is a smooth holomorphic family
$f\colon\mathcal{M}\to\mathbb{A}^{1}$ over the affine line, see [V3]. This
deformation has a number-theoretic interpretation (“the Shafarevich–Tate
family”), see [Mar, AR].

The periods of the manifolds $\mathcal{M}_{t}$ form an affine line in the
period domain. More precisely, consider the subspace $W\subset
H^{2}(M,\mathbb{C})$ spanned by $\Omega$, $\overline{\Omega}$ and
$\pi^{*}\eta$. Then the restriction of the BBF form $q|_{W}$ is a degerate
positive-semidefinite quadratic form with one-dimensional kernel spanned by
$\pi^{*}\eta$. The corresponding quadric $Q_{W}$ is the union of two
projective lines intersecting at the point $[\pi^{*}\eta]$. The complement to
that point in $Q_{W}$ is the disjoint union of two affine lines, one of them
passing through $[\Omega]$ and the other through $[\overline{\Omega}]$. The
first of the two affine lines is formed exactly by the periods of the
manifolds $\mathcal{M}_{t}$, $t\in\mathbb{C}$ of the degenerate twistor
family. The other affine line corresponds to the complex-conjugate degenerate
twistor family which one obtains replacing $\Omega$ by $\overline{\Omega}$.

###  2.3 Weakly positive and strongly positive forms

When we speak about “weakly” and “strongly positive” forms or currents, we use
the French convention about positivity: “positive” always means $\geqslant 0$.
We follow [De, Ha].

Definition 2.5: Let $(V,I)$ be a $2n$-dimensional real vector space eqipped
with a complex structure operator $I$, $I^{2}=-\operatorname{Id}$. A weakly
positive $(p,p)$-form on $V$ is a real $(p,p)$-form
$\eta\in\Lambda^{p,p}_{\mathbb{R}}(V^{*})$ which satisfies

$\eta(x_{1},Ix_{1},x_{2},Ix_{2},\ldots,x_{p},Ix_{p})\geqslant 0$

for all $x_{1},\ldots,x_{p}\in V$.

Remark 2.6: Clearly, the set of weakly positive $(p,p)$-forms is a convex
cone.

Definition 2.7: The cone of strongly positive polyvectors is the cone in
$\Lambda^{p,p}_{\mathbb{R}}(V)$ generated by the monomials $x_{1}\wedge
Ix_{1}\wedge x_{2}\wedge Ix_{2}\wedge\ldots\wedge x_{p}\wedge Ix_{p}$ with
positive coefficients. Replacing $V$ with $V^{*}$, we also define weakly
positive $(p,p)$-polyvectors and strongly positive $(p,p)$-forms.

Remark 2.8: By definition, the cone of strongly positive polyvectors is dual
to the cone of weakly positive forms under the natural pairing
$\Lambda^{p,p}_{\mathbb{R}}(V^{*})\times\Lambda^{p,p}_{\mathbb{R}}(V){\>\longrightarrow\>}{\mathbb{R}}$.

Remark 2.9: Fix a volume element $\operatorname{Vol}\in\Lambda^{2n}(V^{*})$
positive with respect to $I$. For any $v\in\Lambda^{m}(V)$, consider the
contraction $v\,\lrcorner\,\operatorname{Vol}\in\Lambda^{2n-m}(V^{*})$. For
any monomial $\eta\in\Lambda^{m}(V)$, $\eta\,\lrcorner\,\operatorname{Vol}$ is
the complementary monomial. Therefore, the contraction takes the strongly
positive cone in $\Lambda^{p,p}_{\mathbb{R}}(V)$ to the strongly positive cone
in $\Lambda^{n-p,n-p}_{\mathbb{R}}(V^{*})$. Using this remark, it is not hard
to prove the following basic claim.

Proposition 2.10:  Consider the natural non-degenerate pairing

$\Lambda^{p,p}(V^{*})\times\Lambda^{n-p,n-p}(V^{*})\to{\mathbb{R}},\quad(\alpha,\beta)\mapsto\frac{\alpha\wedge\beta}{\operatorname{Vol}}.$

Under this pairing the cone of strongly positive forms is dual to the cone of
weakly positive forms. Moreover, if $\alpha\in\Lambda^{p,p}(V^{*})$ is weakly
positive and $\beta\in\Lambda^{q,q}(V^{*})$ is strongly positive, then
$\alpha\wedge\beta$ is weakly positive.

Proof: [De].

Lemma 2.11:  Let $\eta$ be a $(p,0)$-form. Then the form
$(-1)^{\frac{p(p-1)}{2}}(\sqrt{-1}\>)^{p}\eta\wedge\overline{\eta}$ is weakly
positive.

Proof: Clearly, a $(p,p)$-form $\alpha\in\Lambda^{p,p}(V^{*})$ is weakly
positive iff its restriction to any $p$-dimensional $I$-invariant subspace
$W\subset V$ is a non-negatively oriented volume form. However, $\eta|_{W}$ is
a complex linear volume form, hence it can be written as $uz_{1}\wedge
z_{2}\wedge\ldots\wedge z_{p}$, where $z_{i}\in\Lambda^{1,0}(W^{*})$ is a
basis and $u\in\mathbb{C}$. Write $z_{i}:=x_{i}+\sqrt{-1}\>Ix_{i}$, where
$x_{i}\in W^{*}$. Then $z_{i}\wedge\overline{z}_{i}=-2\sqrt{-1}\>x_{i}\wedge
Ix_{i}$ which gives

$uz_{1}\wedge z_{2}\wedge...\wedge
z_{p}\wedge\overline{u}\overline{z}_{1}\wedge\overline{z}_{2}\wedge...\wedge\overline{z}_{p}=(-1)^{\frac{p(p-1)}{2}}(-2\sqrt{-1}\>)^{p}|u|^{2}\prod_{i=1}^{p}x_{i}\wedge
Ix_{i}.$

The volume form $\prod_{i=1}^{p}x_{i}\wedge Ix_{i}$ is positively oriented.
This implies that
$(-1)^{\frac{p(p-1)}{2}}(\sqrt{-1}\>)^{p}\eta\wedge\overline{\eta}$ is weakly
positive.

###  2.4 Positive currents

Let $M$ be a compact complex manifold of complex dimension $n$. Denote by
$\mathcal{E}^{k}(M)$ the space of complex $k$-forms of class $C^{\infty}$ on
$M$. Fixing an arbitrary Hermitian metric and an affine connection on $M$, one
may define the $C^{d}$-norm of $k$-forms, for arbitrary $d\geqslant 0$. It is
well known that this family of norms makes the space of smooth $k$-forms
$\mathcal{E}^{k}(M)$ a Fréchet topological vector space, the topology being
independent on the Hermitian metric and the connection. The space
$\mathcal{E}^{k}(M)$ has the Hodge decomposition

$\mathcal{E}^{k}(M)=\bigoplus_{p+q=k}\mathcal{E}^{p,q}(M),$

where $\mathcal{E}^{p,q}(M)$ are the closed subspaces of differential forms of
type $(p,q)$. We will denote by $\mathcal{E}^{k}(M,\mathbb{R})$ the space of
real $k$-forms with the induced topology.

Denote by $\mathcal{E}^{\prime}_{k}(M)$ the topological dual of
$\mathcal{E}^{k}(M)$, i.e. the space of continuous linear forms on
$\mathcal{E}^{k}(M)$. We will always work with the weak topology on
$\mathcal{E}^{\prime}_{k}(M)$, i.e. the coarsest topology for which all the
evaluation maps
$\mathrm{ev}_{\alpha}\colon\mathcal{E}^{\prime}_{k}(M)\to\mathbb{C}$,
$T\mapsto T(\alpha)$ for arbitrary $\alpha\in\mathcal{E}^{k}(M)$ are
continuous. The topological vector space $\mathcal{E}^{\prime}_{k}(M)$ is
called the space of currents of dimension $k$. The pairing
$\mathcal{E}^{2n-k}(M)\times\mathcal{E}^{k}(M)\to\mathbb{C}$ given by the
wedge product and integration over $M$ induces a natural continuous embedding
$\mathcal{E}^{2n-k}(M)\hookrightarrow\mathcal{E}^{\prime}_{k}(M)$. For this
reason the elements of $\mathcal{E}^{\prime}_{k}(M)$ are also called currents
of degree $2n-k$.

The Hodge decomposition of $k$-forms induces the Hodge decomposition of
currents. We denote by $\mathcal{E}^{\prime}_{p,q}(M)$ the dual of
$\mathcal{E}^{p,q}(M)$ and call it the space of currents of bidimension
$(p,q)$ or of bidegree $(n-p,n-q)$. The space of real currents will be denoted
by $\mathcal{E}^{\prime}_{k}(M,\mathbb{R})$ and
$\mathcal{E}^{\prime}_{p,p}(M,\mathbb{R})=\mathcal{E}^{\prime}_{p,p}(M)\cap\mathcal{E}^{\prime}_{2p}(M,\mathbb{R})$.

The de Rham differential $d\colon\mathcal{E}^{k}(M)\to\mathcal{E}^{k+1}(M)$ is
continuous, and by the well known theory of elliptic complexes its image is
closed in Fréchet topology. It follows from the Banach open mapping theorem
and the Hahn–Banach theorem that the adjoint map
$d\colon\mathcal{E}^{\prime}_{k+1}(M)\to\mathcal{E}^{\prime}_{k}(M)$ also has
closed image, i.e. the space of exact currents is closed in the weak topology.

Recall that a real current $T\in\mathcal{E}^{\prime}_{1,1}(M,\mathbb{R})$ is
called positive if for any $\eta\in\mathcal{E}^{1,0}(M)$ we have
$\sqrt{-1}T(\eta\wedge\overline{\eta})\geqslant 0$. Let us denote the set of
positive currents by $\mathcal{E}^{+}_{1,1}(M)$. This set is clearly a closed
convex cone in $\mathcal{E}^{\prime}_{1,1}(M,\mathbb{R})$. The positive
currents admit the following description, see e.g. [HL, §3]. Given a positive
current $T\in\mathcal{E}^{+}_{1,1}(M)$ there exists a positive Borel measure
$\|T\|$ called the total variation of $T$ and a $\|T\|$-measurable section
$\tau$ of the bundle $\Lambda^{1,1}TM$, i.e. a $\|T\|$-measurable bivector
field of Hodge type $(1,1)$, such that $T=\tau\|T\|$. The latter equality
means that the action of $T$ on $\alpha\in\mathcal{E}^{1,1}(M)$ is given by

$T(\alpha)=\int_{M}\langle\tau,\alpha\rangle d\|T\|,$

where $\langle\tau,\alpha\rangle$ denotes the pairing between bivectors and
two-forms. The bivector field $\tau$ is positive in the following sense: for
any $\eta\in\mathcal{E}^{1,0}(M)$ the function
$\langle\tau,\sqrt{-1}\eta\wedge\overline{\eta}\rangle$ is non-negative
$\|T\|$-almost everywhere on $M$.

Proposition 2.12:  Let $\pi\colon M\to X$ be a proper holomorphic map, and
$T\in\mathcal{E}^{+}_{1,1}(M)$ a positive current such that
$T(\pi^{*}\alpha)=0$ for any positive $(1,1)$-form $\alpha$ on $X$. Then in
the above decomposition $T=\tau\|T\|$ the bivector field $\tau$ is almost
everywhere tangent to the fibers of $\pi$, i.e. $\tau\in\ker{d\pi}$ outside a
subset of $\|T\|$-measure zero.

Proof: The differential $d\pi:\;T_{x}M{\>\longrightarrow\>}T_{\pi(x)}X$ can be
interpreted as a map from $TM$ to $\pi^{*}(TX)$. Then $d\pi(\tau)$ is a
measurable section of the bundle $\pi^{*}\Lambda^{1,1}TX$. By our assumptions,
for any $\eta\in\mathcal{E}^{1,0}(X)$ we have

$\sqrt{-1}T(\pi^{*}\eta\wedge\pi^{*}\overline{\eta})=\int_{M}\langle
d\pi(\tau),\sqrt{-1}\eta\wedge\overline{\eta}\rangle d\|T\|=0,$

and since $d\pi(\tau)$ is positive, the function $\langle
d\pi(\tau),\sqrt{-1}\eta\wedge\overline{\eta}\rangle$ is non-negative almost
everywhere on $M$, therefore it must vanish almost everywhere. Since this is
true for arbitrary $\eta$ as above, we conclude that $d\pi(\tau)$ vanishes
almost everywhere, which concludes the proof.

##  3 Hermitian symplectic structures and degenerate twistor deformations

In the sequel, we use the following version of the Hahn–Banach theorem,
sometimes called the “Hahn–Banach separation theorem”.

Theorem 3.1:  (Hahn–Banach theorem, [Bou], or [Sch, Chapter II, Theorem 3.1])
Let $V$ be a topological vector space, $A\subset V$ an open convex subset of
$V$, and $W$ a subspace of $V$ satisfying $W\cap A=\emptyset$. Then there
exists a continuous linear functional $\theta$ on $V$, such that
$\theta|_{A}>0$ and $\theta|_{W}=0$.

Recall the following definitions.

Definition 3.2: Let $(M,I)$ be a complex manifold and $\omega\in\Lambda^{2}M$
a real symplectic form. The complex structure $I$ is tamed by $\omega$ if for
any non-zero tangent vector $v$ at an arbitrary point of $M$ we have
$\omega(v,Iv)>0$. In this case we call $\omega$ a Hermitian symplectic
structure. Equivalently, a symplectic form $\omega$ is Hermitian symplectic if
its $(1,1)$-part $\omega^{1,1}$ is a Hermitian form.

The following characterization of compact complex manifolds admitting
Hermitian symplectic structures is originally due to Sullivan [Su], see [Ch],
[DP] and [HL] for related results and a broader context.

Theorem 3.3:  Let $M$ be a compact complex manifold. The manifold $M$ does not
admit a Hermitian symplectic structure if and only if if admits a non-zero
positive exact $(n-1,n-1)$-current.

Proof: We sketch the well known proof for the sake of completeness and because
we refer to these arguments elsewhere. We use the Hahn–Banach separation
theorem (3), applied to $V$, $A$ and $W$ defined below.

Let $V=\Lambda^{2}(M,{\mathbb{R}})$ be the Fréchet space of smooth two-forms
equipped with the $C^{\infty}$-topology, $A$ the cone of two-forms with
strictly positive $(1,1)$-part, and $W\subset V$ the space of closed two-
forms. The intersection $A\cap W$ is the set of all Hermitian symplectic
forms. If this set is empty, we use 3 to find a non-zero functional $\theta\in
V^{*}$ which is positive on $A$ (this means that $\theta$ is a positive
$(n-1,n-1)$-current) and vanising on $W$. The latter condition means that
$\theta$ is exact. Indeed, $\langle\theta,du\rangle=0$ for all
$u\in\Lambda^{1}M$ implies that $\theta$ is closed. It is exact, because
otherwise it is non-zero on $W$ by Poincaré duality (the de Rham cohomology of
currents is equal to the de Rham cohomology of differential forms, [De]).

Theorem 3.4:  Let $\pi\colon M\to B$ be a holomorphic Lagrangian fibration on
a compact simply connected hyperkähler manifold of maximal holonomy, and
$\mathcal{M}_{t}=(M,I_{t})$, $t\in{\mathbb{C}}$ the corresponding degenerate
twistor deformation. Then $\mathcal{M}_{t}$ admits an Hermitian symplectic
structure for all $t\in{\mathbb{C}}$.

Proof:  We use 3. Let $T\in\mathcal{E}^{+}_{1,1}(\mathcal{M}_{t})$ be an exact
positive current of type $(1,1)$ on $\mathcal{M}_{t}$. We need to prove that
$T=0$.

As explained above, $T$ admits a representation $T=\tau\mu$, where $\mu=\|T\|$
is a positive Borel measure and $\tau$ is a $\mu$-measurable bivector field on
$\mathcal{M}_{t}$. Let us check that the conditions of 2.4 are satisfied. Fix
a Kähler form $\eta\in\Lambda^{1,1}B$. For any positive form
$\alpha\in\Lambda^{1,1}B$ we have $0\leqslant\alpha\leqslant C\eta$ for some
constant $C>0$. Then $0\leqslant T(\pi^{*}\alpha)\leqslant T(\pi^{*}\eta)$,
but $\pi^{*}\eta$ is closed, and since $T$ is exact $T(\pi^{*}\eta)=0$.
Therefore $T(\pi^{*}\alpha)=0$ and we conclude by 2.4 that the bivector field
$\tau$ is tangent to the fibers of $\pi$ (after, possibly, changing $\tau$ on
a subset of $\mu$-measure zero).

Fix a point $x\in M$, let $V=T_{x}M\otimes\mathbb{C}$ and let
$W=\ker(d\pi)_{x}\subset V$ be the complexified tangent space to the fiber of
$\pi$ through $x$. The bivector $\tau(x)$ is a linear combination with
positive coefficients of bivectors of the form $-\sqrt{-1}u\wedge\overline{u}$
for $u\in V^{1,0}_{t}$. But since $\tau$ is tangent to the fibers of $\pi$, we
have $u\in W^{1,0}_{t}$ for all bivectors appearing in that linear
combination. Now, we know by 2.2 that the subspace $W^{1,0}_{t}\subset W$ does
not depend on $t$. We conclude that the bivector field $\tau$ is of type
$(1,1)$ and positive for all complex structures $I_{t}$ in the degenerate
twistor family. Therefore $T$ is an exact positive current of bidimension
$(1,1)$ on $\mathcal{M}_{0}=M$. The latter being a Kähler manifold, the
current $T$ must vanish by 3.

##  4 $d$-Commendable manifolds

Definition 4.1: Let $M$ be a compact complex manifold, $h\in\Lambda^{1,1}M$ a
Hermitian form and $0<d<\dim_{\mathbb{C}}(M)$. We say that $M$ is
$d$-commendable if there exists $\eta\in\Lambda^{2d}M$ such that $d\eta=0$ and
the form $\eta^{d,d}-\varepsilon h^{d}$ is weakly positive for some
$\varepsilon>0$.111In the sequel, we will denote this condition as
$\eta^{d,d}\geqslant\varepsilon h^{d}$ . Note that 1-commendable is the same
as Hermitian symplectic.

Proposition 4.2:  Let $M$ be a complex manifold with a Hermitian symplectic
structure $\omega$. Then $M$ is $d$-commendable, for all $d>0$.

Proof:  For $0<d<\dim_{\mathbb{C}}(M)$ let $\eta=\omega^{d}$. Then $\eta$
closed and $\eta^{d,d}$ is a positive linear combination of the forms
$(\omega^{2,0}\wedge\overline{\omega^{2,0}})^{k}\wedge(\omega^{1,1})^{d-2k}$,
for $k=0,\ldots,[d/2]$. The form
$(\omega^{2,0})^{k}\wedge(\overline{\omega^{2,0}})^{k}$ is weakly positive by
2.3, and $(\omega^{1,1})^{d-2k}$ is strongly positive for $2k<d$. By 2.3 the
forms
$(\omega^{2,0}\wedge\overline{\omega^{2,0}})^{k}\wedge(\omega^{1,1})^{d-2k}$
are weakly positive. Moreover, $(\omega^{1,1})^{d}\geqslant Ch^{d}$ for some
constant $C>0$, because $\omega^{1,1}$ is a Hermitian form. Therefore
$\eta^{d,d}\geqslant Ch^{d}$.

###  4.1 Barlet spaces on commendable manifolds

The purpose of introducing the notion of $d$-commendability is that it can
serve as a replacement of the condition of being Kähler in some situations.
More precisely, we will observe that the Barlet spaces of $d$-dimensional
cycles of $d$-commendable manifolds behave in some ways similar to the spaces
of cycles of Kähler manifolds. Let us start with the following claim.

Proposition 4.3:  Let $M$ be a $d$-commendable compact complex manifold and
$\mathcal{C}_{d}$ a connected component of the Barlet space of dimension $d$
cycles on $M$. Then $\mathcal{C}_{d}$ is compact.

Proof:  Let $h\in\Lambda^{1,1}M$ be a Hermitian metric on $M$ and $\eta$ a
closed $2d$-form with $\eta^{d,d}\geqslant Ch^{d}$. By the compactness
criterion of Barlet, based on Bishop’s theorem ([Mag], [BM, Theorem 4.2.69]),
we need to show that the volume of all cycles in $\mathcal{C}_{d}$ with
respect to $h$ is bounded. Assume that $[Z]\in\mathcal{C}_{d}$ is a cycle.
Then

$\int_{Z}h^{d}\leqslant(1/C)\int_{Z}\eta^{d,d}=(1/C)\int_{Z}\eta.$

Since $d\eta=0$, the latter integral depends only on the cohomology class of
$[Z]$ which is constant for all cycles in the connected component
$\mathcal{C}_{d}$. The claim follows.

In what follows we will need a relative version of the above statement. We
will consider the following setting. Let $\pi\colon\mathcal{M}\to B$ be a
proper smooth holomorphic map between complex manifolds. Let
$\mathcal{C}_{d}(\mathcal{M}/B)$ be a connected component of the Barlet space
of $d$-dimensional cycles contained in the fibers of $\pi$, see [BM, section
4.8.2]. Then the map $p\colon\mathcal{C}_{d}(\mathcal{M}/B)\to B$ that sends a
cycle to the point $t\in B$ over which this cycle is supported is holomorphic
(loc. cit.) The following proposition is a generalization of [BM, Corollary
4.8.].

Proposition 4.4:  In the above setting assume that there exists a differential
form $\eta\in\Lambda^{2d}(\mathcal{M})$ such that for any $t\in B$ the fiber
$\mathcal{M}_{t}$ and the restriction $\eta_{t}=\eta|_{\mathcal{M}_{t}}$
satisfy the conditions of $d$-commendability. Then the support morphism
$p\colon\mathcal{C}_{d}(\mathcal{M}/B)\to B$ is proper, in particular its
image is an analytic subvariety of $B$.

Proof:  To prove properness we restrict to a compact subset $K\subset B$. The
cycles in the preimage $p^{-1}(K)$ are supported in the compact subset
$\pi^{-1}(K)$ of $\mathcal{M}$. So by the compactness criterion [BM, Theorem
4.2.69] we need to prove that the volume of the cycles
$[Z]\in\mathcal{C}_{d}(\mathcal{M}/B)$ in $p^{-1}(K)$ is bounded above. Let us
fix a Hermitian metric $h$ on $\mathcal{M}$.

We can assume that $K$ is sufficiently small and find a constant $C>0$ such
that for any $t\in K$ we have $\eta_{t}^{d,d}\geqslant Ch_{t}^{d}$, where the
subscript $t$ denotes the restriction to the fiber $\mathcal{M}_{t}$. Now,
given a cycle $[Z]\in p^{-1}(K)$ let $t\in K$ be the point over which $[Z]$ is
supported. We estimate the volume of $[Z]$:

$\int_{Z}h^{d}=\int_{Z}h^{d}_{t}\leqslant(1/C)\int_{Z}\eta_{t}^{d,d}=(1/C)\int_{Z}\eta_{t}.$

The homology class of $[Z]$ is constant, and the restriction of $\eta$ to any
fiber $\mathcal{M}_{t}$ is closed, so the integral $\int_{Z}\eta_{t}$ depends
only on the point $t\in K$ over which $[Z]$ is supported; let us denote the
value of the integral by $\varphi(t)$. Since $\eta$ is a smooth differential
form on $\mathcal{M}$, $\varphi$ is a continuous function. Since $K$ is
compact, $\varphi$ is bounded from above on $K$. This gives the required upper
bound for the volume of the cycles $[Z]\in p^{-1}(K)$.

###  4.2 Non-separated points in the Teichmüller space of Hermitian symplectic
holomorphically symplectic manifolds

Let $M$ be a simply connected compact hyperkähler manifold of maximal holonomy
and $\pi\colon M\to B$ a holomorphic Lagrangian fibration. Let
$\mathcal{M}\to\mathbb{A}^{1}$ be the corresponding degenerate twistor family.
Denote by $U\subset\mathbb{A}^{1}$ the open subset of points of the base such
that for $t\in U$ the fiber $\mathcal{M}_{t}$ is Kähler.

The argument used to prove 4.2 below is a version of the one used in [Hu1] to
establish the bimeromorphic correspondence between non-separated fibers of a
twistor deformation. In [Hu1], both fibers are assumed to be Kähler. We prove
the same statement when the fibers are Hermitian symplectic.

By the global Torelli theorem ([V1]) the hyperkähler manifold
$\mathcal{M}_{t}$ admits a hyperkähler deformation $X$ such that the period of
$X$ corresponds to the point $t_{0}$ and $X$ lies in the same connected
component of the Teichmüller space as $\mathcal{M}_{t}$ ([V3, Section 1.2]).
Let $\mathcal{X}\to\Delta$ denote the restriction of the universal deformation
of $X$ to $\Delta$.

Proposition 4.5:  Consider a family $\mathcal{M}\to\mathbb{A}^{1}$ of
holomorphically symplectic manifolds, and let $U\subset\mathbb{A}^{1}$ be the
set of all points corresponding to Kähler fibers. Assume the the fiber over a
limit point $t_{0}\in\partial U$ is Hermitian symplectic. Then there exists a
hyperkähler deformation $X$ of $M$ that is bimeromorphic to
$\mathcal{M}_{t_{0}}$

Proof:  Let $\Delta\subset\mathbb{A}^{1}$ be a small disc around $t_{0}$. Note
that $\Delta\cap U\neq\emptyset$, so we may find a Kähler fiber
$\mathcal{M}_{t}$ with $t\in\Delta$.

Let $\omega$ be a Hermitian symplectic form on $\mathcal{M}_{t_{0}}$, and
$\omega^{\prime}$ a Kähler form on $X$.

Note that $\omega+\omega^{\prime}$ is a Hermitian symplectic form on
$X\times\mathcal{M}_{t_{0}}$. By continuity, after possibly shrinking
$\Delta$, we may assume that the restriction of $\omega+\omega^{\prime}$ to
any fiber $\mathcal{X}_{t}\times\mathcal{M}_{t}$, $t\in\Delta$ makes this
fiber Hermitian symplectic.

The idea now is to take the graph $\Gamma_{t}$ of the bimeromorphism
$\mathcal{X}_{t}\dashrightarrow\mathcal{M}_{t}$ and use 4.1 to show that there
is a subsequence $\\{\Gamma_{t_{i}}\\}$ converging to a subvariety
$\Gamma_{t_{0}}$ in $\mathcal{X}_{t_{0}}\times\mathcal{M}_{t_{0}}$. Then we
use a topological argument to show that one of the irreducivle components of
$\Gamma_{t_{0}}$ is a graph of a bimeromorphism
$\mathcal{X}_{t_{0}}\dashrightarrow\mathcal{M}_{t_{0}}$.

Let us consider the open subset $\Delta\cap U$. It is non-empty because
$t_{0}$ is a limit point of $U$. The fibers of the family $\mathcal{M}$ over
this open subset are Kähler by the definition of $U$. The points of the
Teichmüller space corresponding to $\mathcal{M}_{t}$ and $\mathcal{X}_{t}$,
$t\in\Delta\cap U$ are either equal or non-separated by construction, hence by
the theorem of Huybrechts [Hu1, Theorem 4.3] the manifolds $\mathcal{M}_{t}$
and $\mathcal{X}_{t}$ are bimeromorphic for any $t\in\Delta\cap U$.

Consider the complex manifold
$\mathcal{Y}=\mathcal{X}\times_{\Delta}\mathcal{M}$. For $t\in\Delta\cap U$
the graph of a bimeromorphic map between $\mathcal{X}_{t}$ and
$\mathcal{M}_{t}$ is a subvariety $\Gamma_{t}\subset\mathcal{Y}$. Let
$\mathcal{C}_{\dim_{\mathbb{C}}X}(\mathcal{Y}/\Delta)$ be an irreducible
component of the relative Barlet space that contains a point corresponding to
one of these graphs. By 4.1 the image of
$\mathcal{C}_{\dim_{\mathbb{C}}X}(\mathcal{Y}/\Delta)$ in $\Delta$ is a closed
subvariety, hence it is either a countable set of points, or the whole disc
$\Delta$. Since the subset $\Delta\cap U$ is uncountable, one of the connected
components of the Barlet space must map surjectively onto $\Delta$.

We conclude that there exists a sequence of cycles $[\Gamma_{t_{i}}]$ in the
fibers $\mathcal{Y}_{t_{i}}$, where $t_{i}\in\Delta\cap U$, $t_{i}\to t_{0}$
for $i\to\infty$, giving the above bimeromorphic maps, contained in the same
irreducible component of the Barlet space and converging to a limit cycle
$[\Gamma_{t_{0}}]$ in the fiber $X\times\mathcal{M}_{t_{0}}$ over $t_{0}$.
Then the homology classes of $[\Gamma_{t_{i}}]$ do not depend on $i$. Denote
by $\Omega^{\prime}$ a holomorphic symplectic form on $X$. The bimeromorphic
maps induced by $\Gamma_{t_{i}}$ map a holomorphic symplectic form on
$\mathcal{M}_{t_{i}}$ to a holomorphic symplectic form on
$\mathcal{X}_{t_{i}}$. Therefore by continuity $[\Gamma_{t_{0}}]$ seen as a
correspondence acting on cohomology maps the cohomology class
$[\Omega_{t_{0}}]$ on $\mathcal{M}_{t_{0}}$ to the cohomology class
$[\Omega^{\prime}]$ on $X$. We can therefore apply 4.2 below to conclude that
the cycle $[\Gamma_{t_{0}}]$ contains exactly one irreducible component that
is the graph of a bimeromorphic map between $\mathcal{M}_{t_{0}}$ and $X$.

The proof of the following proposition is a version of [Hu1, proof of Theorem
4.3]. We use it to establish the bimeromorphic equivalence of two
holomorphically symplectic manifolds, the only difference from loc. cit. being
that we do not assume one of the manifolds to be Kähler.

We consider the following setting. Assume that $M$ is a compact complex
manifold with a holomorphic symplectic form $\Omega$ and $X$ is a compact
hyperkähler manifold of maximal holonomy with a holomorphic symplectic form
$\Omega^{\prime}$. Assume that $\dim_{\mathbb{C}}(M)=\dim_{\mathbb{C}}(X)=2n$
and that $[Z]$ is a $2n$-cycle on $X\times M$. Assume that $[Z]$ is of degree
one both over $X$ and over $M$. Assume moreover that $[Z]_{*}\colon
H^{2}(M,\mathbb{C})\to H^{2}(X,\mathbb{C})$ is an isomorphism that sends
$[\Omega]$ to $[\Omega^{\prime}]$. We can decompose the cycle into a finite
sum $[Z]=\sum a_{i}[Z_{i}]$, where $a_{i}>0$ and $Z_{i}\subset X\times M$ is
an irreducible subvariety.

Proposition 4.6:  In the above decomposition of $[Z]$ there is exactly one
irreducible component dominating both $M$ and $X$ and this component is the
graph of a bimeromorphic isomorphism between $M$ and $X$.

Proof:  The degree of $[Z]=\sum a_{i}[Z_{i}]$ over $X$ is equal to $\sum
a_{i}\mathrm{deg}_{X}(Z_{i})$, and since all $a_{i}$ are positive, there
exists $i_{0}$ such that $\mathrm{deg}_{X}(Z_{i_{0}})=1$, $a_{i_{0}}=1$ and
$\mathrm{deg}_{X}(Z_{i})=0$ for $i\neq i_{0}$. Analogously, there exists
$i_{1}$, such that $\mathrm{deg}_{M}(Z_{i_{1}})=1$, $a_{i_{1}}=1$ and
$\mathrm{deg}_{M}(Z_{i})=0$ for $i\neq i_{1}$. We need to prove that
$i_{0}=i_{1}$: in this case the subvariety $Z_{i_{0}}=Z_{i_{1}}$ is the graph
of a bimeromorphic map between $X$ and $M$.

Denote by $p\colon X\times M\to X$ and $q\colon X\times M\to M$ the two
projections. Let us consider the action of the correspondence given by $[Z]$
on the cohomology class of the symplectic form $\Omega$. Recall that by our
assumptions $[Z]_{*}[\Omega]=[\Omega^{\prime}]$. For a connected component
$Z_{i}$ of the cycle $Z$ denote by $\widetilde{Z}_{i}$ its resolution of
singularities and by $\widetilde{p}_{i}\colon\widetilde{Z}_{i}\to X$ and
$\widetilde{q}_{i}\colon\widetilde{Z}_{i}\to M$ the induced maps. Then
$[Z_{i}]_{*}[\Omega]$ is the cohomology classs of the current
$\widetilde{p}_{i*}(\widetilde{q}_{i}^{*}\Omega)$. For $i\neq i_{0}$ the map
$\widetilde{p}_{i}$ is not dominant, and since $\widetilde{q}_{i}^{*}\Omega$
is a holomorphic two-form, the current
$\widetilde{p}_{i*}(\widetilde{q}_{i}^{*}\Omega)$ vanishes. Indeed, a closed
(2,0)-current is holomorphic, hence smooth by elliptic reguarity, but the
support of $\widetilde{p}_{i*}(\widetilde{q}_{i}^{*}\Omega)$ is contained in a
proper subvariety. Now consider the case $i=i_{0}$. Then
$\widetilde{p}_{i_{0}*}(\widetilde{q}_{i_{0}}^{*}\Omega)$ is a holomorphic
two-form on $X$, therefore it must be proportional to $\Omega^{\prime}$,
because $X$ is hyperkähler of maximal holonomy. It follows that
$\widetilde{q}_{i_{0}}^{*}\Omega$ is generically a symplectic form, so
$\widetilde{q}_{i_{0}}$ must be dominant, hence $i_{0}=i_{1}$.

###  4.3 Degenerate twistor deformations are Kähler

We finish the paper by proving our main result.

Theorem 4.7:  Let $M$ be a simply connected compact hyperkähler manifold of
maximal holonomy and $\pi\colon M\to B$ a holomorphic Lagrangian fibration.
Let $\mathcal{M}\to\mathbb{A}^{1}$ be the corresponding degenerate twistor
family. Then all fibers $\mathcal{M}_{t}$, $t\in\mathbb{A}^{1}$ are Kähler.

Proof:  Denote by $U\subset\mathbb{A}^{1}$ the open subset of points of the
base such that for $t\in U$ the fiber $\mathcal{M}_{t}$ is Kähler. Assume that
$U\neq\mathbb{A}^{1}$ and let $x\in\partial U$. Then by 4.2, the fiber
$\mathcal{M}_{t}$ is of Fujiki class C. Moreover, by 3 the manifold
$\mathcal{M}_{t}$ admits a Hermitian symplectic form $\omega$. Then
$\omega^{1,1}$ is a Hermitian metric and $dd^{c}(\omega^{1,1})=0$, therefore
$\mathcal{M}_{t}$ is Kähler by [Ch, Theorem 0.2] 222We thank the participants
of the mathoverflow discussion for this reference or 5. It follows that
$U=\mathbb{A}^{1}$.

##  5 Appendix: Hermitian symplectic forms and Fujiki class C symplectic
manifolds

For the sake of completeness, we give a self-contained proof of [Ch, Theorem
0.2] for holomophically symplectic manifolds.

Theorem 5.1:  Assume that a compact holomorphically symplectic manifold $M$ is
bimeromorphic to a hyperkähler manifold of maximal holonomy and that
$\omega\in\Lambda^{2}M$ is a Hermitian symplectic form. Then $M$ is Kähler.

Proof:  Denote by $\Omega$ the holomorphic symplectic form on $M$ and let
$f\colon M\dashrightarrow M^{\prime}$ be a bimeromorphic map to a hyperkähler
manifold of maximal holonomy $M^{\prime}$. Then $\Omega^{\prime}=f_{*}\Omega$
is a holomorphic symplectic form on $M^{\prime}$. Since $M$ is of Fujiki class
C, we may apply Hodge theory to it. Since $f$ maps a holomorphic volume form
on $M$ to a holomorphic volume form on $M^{\prime}$, the indeterminacy locus
of $f$ has codimension at least two. Hence $f_{*}\colon H^{2}(M,\mathbb{C})\to
H^{2}(M^{\prime},\mathbb{C})$ is an isomorphism of Hodge structures. We will
denote by $q$ the BBF form on $H^{2}(M^{\prime},\mathbb{C})$.

Let us denote $a=[\omega]^{1,1}$, the $(1,1)$-part of the cohomology class of
the Hermitian symplectic form on $M$. Assume that
$\omega^{\prime}\in\Lambda^{1,1}M^{\prime}$ is a Kähler form and
$b=[\omega^{\prime}]\in H^{2}(M^{\prime},\mathbb{C})$ is its cohomology class.
Let us observe that $q(f_{*}a,b)>0$. Indeed, resolving the indeterminacy of
$f$ we can construct a manifold $X$ and two bimeromorphic maps $p_{1}\colon
X\to M$, $p_{2}\colon X\to M^{\prime}$ such that $f=p_{2}\circ p_{1}^{-1}$.
Then using the Fujiki relations for $q$ we get:

$q(f_{*}a,b)=\frac{q(b)}{\int_{M^{\prime}}b^{2n}}\int_{M^{\prime}}(f_{*}a)b^{2n-1}=\frac{q(b)}{\int_{M^{\prime}}b^{2n}}\int_{X}p_{1}^{*}(\omega^{1,1})\wedge(p_{2}^{*}\omega^{\prime})^{2n-1}>0,$

the latter inequality being valid because both $\omega^{1,1}$ and
$\omega^{\prime}$ are strictly postitive forms. We conclude that the class
$f_{*}a$ lies in the positive cone of $M^{\prime}$. It is clear from the
definition that $\omega$ will stay Hermitian symplectic under small
perturbations. Therefore, after possibly modifying $\omega$, we may assume
that $f_{*}a$ lies in one of the Kähler chambers of $M^{\prime}$. Replacing
$M^{\prime}$ by another bimeromorphic hyperkähler model we may assume that
$f_{*}a$ is Kähler. The claim now follows from [Hu2, Proposition 2.1] which
shows that the bimeromorphic map $f$ taking a Kähler class to a Kähler class
is regular, hence defines an isomorphism of $M$ and $M^{\prime}$.

Acknowledgements: We are grateful to Ekaterina Amerik, Alexey Golota, Olivier
Martin and Eyal Markman for interesting discussions of the subject.

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Andrey Soldatenkov
<EMAIL_ADDRESS>
Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Estrada Dona Castorina, 110
Jardim Botânico, CEP 22460-320
Rio de Janeiro, RJ - Brasil

Misha Verbitsky
Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Estrada Dona Castorina, 110
Jardim Botânico, CEP 22460-320
Rio de Janeiro, RJ - Brasil
<EMAIL_ADDRESS>


# Quasiperiodic graphs at the onset of chaos

B. Luque1, M. Cordero-Gracia1, M. Gómez1, and A. Robledo2 1 Dept. Matemática
Aplicada y Estadística. ETSI Aeronáuticos, Universidad Politécnica de Madrid,
Spain.
2 Instituto de Física y Centro de Ciencias de la Complejidad, Universidad
Nacional Autónoma de México, Mexico.

###### Abstract

We examine the connectivity fluctuations across networks obtained when the
horizontal visibility (HV) algorithm is used on trajectories generated by
nonlinear circle maps at the quasiperiodic transition to chaos. The resultant
HV graph is highly anomalous as the degrees fluctuate at all scales with
amplitude that increases with the size of the network. We determine families
of Pesin-like identities between entropy growth rates and generalized graph-
theoretical Lyapunov exponents. An irrational winding number with pure
periodic continued fraction characterizes each family. We illustrate our
results for the so-called golden, silver and bronze numbers.

###### pacs:

05.45.Ac, 05.90.+m, 05.10.Cc

##  I Introduction

The onset of chaos is a prime dynamical phenomenon that has attracted
continued attention motivated by the aim to both expand its understanding and
to explore its manifestations in many fields of study strogatz1 . From a
theoretical viewpoint, chaotic attractors generated by low-dimensional
dissipative maps have ergodic and mixing properties and, not surprisingly,
they can be described by a thermodynamic formalism compatible with Boltzmann-
Gibbs (BG) statistics dorfman1 . But at the transition to chaos, the infinite-
period accumulation point of periodic attractors, these two properties are
lost and this suggests the possibility of exploring the limit of validity of
the BG structure in a precise but simple enough setting. The horizontal
visibility (HV) algorithm luque1 ; luque2  that transforms time series into
networks has offered luque3 ; luque4 ; luque5 ; luque6 ; luque7  a view of
chaos and its genesis in low-dimensional maps from an unusual perspective
favorable for the appreciation and understanding of basic features. Here we
present the scaling and entropic properties associated with the connectivity
of HV networks obtained from trajectories at the quasiperiodic onset of chaos
of circle maps hilborn1  and show that this is an unusual but effective
setting to observe the universal properties of this phenomenon.

The three well-known routes to chaos in low-dimensional dissipative systems,
period-doubling, intermittency and quasiperiodicity, have been analyzed
recently luque3 ; luque4 ; luque5 ; luque6 ; luque7  via the HV formalism, and
complete sets of graphs, that encode the dynamics of all trajectories within
the attractors along these routes, have been determined. These graphs display
structural and entropic properties through which a distinct characterization
of the families of time series spawned by these deterministic systems is
obtained. The quantitative basis for these results is provided by the
corresponding analytical expressions for the degree distributions. The graph
at the transition to chaos has been studied only for the period-doubling route
for which connectivity expansion an entropy growth rates have been determined
and found to be linked by Pesin-like identities luque5 . Here we present
results for the transition to chaos for the quasiperiodic route that expand on
this findings and suggest that structural and entropic properties of such
networks are linked by Pesin-like equalities that use generalizations of the
ordinary Lyapunov and BG entropy expressions.

We refer to Pesin-like identities as those that were first found to occur at
the period-doubling transitions to chaos that link generalized Lyapunov
exponents to entropy growth rates at finite, but all, iteration times
baldovin1 ; mayoral1 . Recently luque5  these identities were retrieved in a
network context via the HV method. Pesin-like identities differ from the
genuine Pesin identity, the single positive Lyapunov exponent version of the
Pesin theorem pesin1 , for chaotic attractors in one-dimensional iterated
maps. The Pesin identity links asymptotic quantities that are invariant under
coordinate transformations, whereas the finite-time Pesin-like identities,
that appear for vanishing ordinary Lyapunov exponent are coordinate dependent.
However, in the case of period doubling it has been seen that the identities
remain valid when different coordinate systems are used to determine them, as
in Refs. luque5  and baldovin1 .

The rest of this paper is as follows: We first recall the HV algorithm luque1
; luque2  that converts a time series into a network and focus on the
quasiperiodic graphs luque6  as the specific family of HV graphs generated by
the standard circle map. We then expose the universal scale-invariant
structure of the graphs that arise at the infinite period accumulation points
by focusing on the golden ratio route. We describe the diagonal structure of
these graphs when represented by the exponential of the connectivity, and
introduce a generalized graph-theoretical Lyapunov exponent appropriate for
the subexponential growth of connectivity fluctuations. Subsequently, we show
how the collapse of the diagonal structure into a single one represents the
scale-invariant property that governs the degree fluctuations. Following this,
we analyze the network expression for the entropy rate of growth and find a
spectrum of Pesin-like identities. Finally, we show that all the previous
results can be generalized by considering winding numbers given by any
quadratic irrational. We discuss our results.

Figure 1: Six levels of the Farey tree and the periodic motifs of the graphs
associated with the corresponding rational fractions $p/q$ taken as dressed
winding numbers $\omega$ in the circle map (for space reasons only two of
these are shown at the sixth level). (a) In order to show how graph
inflationary process works, we have highlighted an example using different
grey tones on the left side. See Ref. luque6  for details. (b) First five
steps along the Golden ratio route, $b=1$ (thick solid line); (c) First three
steps along the Silver ratio route, $b=2$ (thick dashed line).

##  II Quasiperiodic graphs at the golden ratio onset of chaos

The idea of extracting graphs from time series is hardly new and over the past
years several approaches have been proposed and are currently developed
Crutchfield ; zhang06 ; kyriakopoulos07 ; xu08 ; donner10 ; donner11 ;
donner11-2 ; campanharo11 . The HV approach is chosen here because of both its
simplicity of implementation and its capability to produce analytical results
in closed form for quantities that are generally difficult to determine. As we
see below this is corroborated for the present enterprise. For the circle map
it has been possible to determine previously the relevant dynamical quantities
at the transition to chaos only for the golden route robledo1 . In contrast,
in the present study it has been possible to generalize this result
effortlessly for an infinite number of routes to chaos associated with all the
quadratic irrational numbers.

The horizontal visibility (HV) algorithm is a general method to convert time
series data into a graph luque1 ; luque2  and is minimally stated as follows:
assign a node $i$ to each datum $\theta_{i}$ of the time series
$\\{\theta_{i}\\}_{i=1,2,...}$ of real data, and then connect any pair of
nodes $i$, $j$ if their associated data fulfill the criterion $\theta_{i}$ ,
$\theta_{j}$ $>$ $\theta_{n}$ for all $n$ such that $i<n<j$. We note that the
HV algorithm is related to the permutation entropy scheme Bandt  in which the
problem of the partition of symbols of a time series is sorted out by simple
comparison of nearest-neighbor values within the series. The HV method
addresses in a similar way this problem, but in addition it makes use of
comparisons of values between neighbors that can be separated by long
distances, and consequently it stores additional information of the series in
the structure of the resulting HV graph.

The HV method has been applied luque6  to trajectories generated by the
standard circle map Landau ; Ruelle ; Shenker ; Kadanoff ; Rand ; Rand2 ;
hilborn1  given by

$\theta_{t+1}=f_{\Omega,K}(\theta_{t})=\theta_{t}+\Omega-\frac{K}{2\pi}\sin(2\pi\theta_{t}),\;\textrm{mod}\;1,$
(1)

representative of the general class of nonlinear circle maps:
$\theta_{t+1}=f_{\Omega,K}(\theta_{t})=\theta_{t}+\Omega+K\cdot
g(\theta_{t}),\;\textrm{mod}\;1$, where $g(\theta)$ is a periodic function
that fulfills $g(\theta+1)=g(\theta)$. This family of maps exhibit universal
properties that are preserved by the HV algorithm luque6  so that without loss
of generality we explain below our findings in terms of the standard circle
map, where $\theta_{t}$, $0\leq\theta_{t}<1$, is the dynamical variable, the
control parameter $\Omega$ is called _bare winding number_ , and $K$ is a
measure of the strength of the nonlinearity. The _dressed winding number_ for
the map is defined as the limit of the ratio:
$\omega\equiv\lim_{t\rightarrow\infty}(\theta_{t}-\theta_{0})/t$. For $K\leq
1$ trajectories are periodic (locked motion) when the corresponding dressed
winding number $\omega(\Omega)$ is a rational number $p/q$ and quasiperiodic
when it is irrational. For $K=1$ (critical circle map) locked motion covers
the entire interval of $\Omega$ leaving only a multifractal subset of $\Omega$
unlocked.

The periodic time series of period $q$ that constitutes the trajectory within
an attractor with $\omega(\Omega)=p/q$ is represented in the HV graph by the
repeated concatenation of a motif, a number of which are shown in Fig. 1. The
display of these motifs in the Farey tree in Fig. 1 helps visualize the
inflationary process that takes place when the HV network grows at the onset
of chaos luque6 . For illustrative purposes in Fig. 1 we show the periodic
motifs of the HV graphs that are associated with the irreducible rational
numbers $p/q\in[0,1]$, and we place them on the Farey tree hilborn1  along
which routes to chaos take place. A well-studied case is the sequence of
rational approximations of $\omega_{\infty}=\phi^{-1}=(\surd
5-1)/2=0.618034...$ , the reciprocal of the golden ratio, which yields winding
numbers $\\{\omega_{n}=F_{n-1}/F_{n}\\}_{n=1,2,3...}$ where $F_{n}$ is the
Fibonacci number generated by the recurrence $F_{n}=F_{n-1}+F_{n-2}$ with
$F_{0}=1$ and $F_{1}=1$. The first few steps of this route can be seen in Fig.
1(b).

The trajectories generated by the map with initial condition $\theta_{0}=1$ at
the golden ratio onset of chaos define a multifractal attractor that forms a
striped pattern of positions when plotted in logarithmic scales, i.e.
$\ln\theta_{t}$ vs $\ln t$. See Fig. 3 in Ref. robledo1 . This attractor
corresponds to the accumulation point
$\Omega_{\infty}=\lim_{n\rightarrow\infty}\Omega_{n}$ of bare winding numbers
$\Omega_{n}$ that characterize superstable trajectories of periods $F_{n}$,
$n=1,2,3,...$, $\Omega_{\infty}=0.606661...$ robledo1 . A sample of this time
series is shown in the top panel of Fig. 2. In the bottom panel of the same
figure we plot, in logarithmic scales, the outcome of the HV method with use
of the variable $\exp k(N)$, where $k(N)$ is the degree of node $N$ in the
graph generated by the time series $\theta_{t}$ (that is, $N\equiv
t=1,2,3,...$). Notice that the distinctive striped pattern of the attractor
robledo1  is present in the figure, although in a simplified manner where the
fine structure is replaced by single lines of constant degree. The HV
algorithm transforms the multifractal attractor into a discrete set of
connectivities.

##  III Diagonal structure of the connectivity fluctuations

It is clear from the bottom panel of Fig. 2 that the degree $k(N)$, and also
$\exp k(N)$, fluctuates when $N$ is increased step by step via a deterministic
pattern of ever increasing amplitude. Notice also in the same panel the
diagonal lines that are drawn to connect sequences of node-connectivity
($N,k$) values; there is a main diagonal followed by two other diagonals close
to each other. These ($N,k$) sequences fall asymptotically along parallel
straight lines, that begin after the initial steps from the lowest values of
the degree, $k=2$ or $k=3$, skip the absent $k=4$, and reach the values $k=5$
or $k=6$, and therefore the sequences obey a power law with the same exponent.
There are many more sequences along same-slope diagonals, not highlighted in
the figure, arranged in close groups and that trace all other possible
connectivities $k(N)$. See also Fig. 3 in Ref. robledo1 . It is by examining
the dependence of $k(N)$ along each member of this family of diagonals that
the scaling and entropic properties of the network are determined.

Figure 2:  Top: Positions $\theta_{t}$ as a function of ${\small t}$ for the
first ${\small 55}$ data for the orbit with initial condition
${\small\theta}_{0}{\small=1}$ at the golden ratio onset of chaos (see text)
of the critical circle map ${\small K=1}$. The data highlighted are associated
with specific subsequences of nodes (see text). Bottom: Log-log plot of
${\small\exp k(N)}$ as a function of the node $N$ for the HV graph generated
from same time series as as for the upper panel but for $3\times 10^{2}$
iterations, where ${\small N=t}$. The distinctive band pattern of the
attractor manifests through a pattern of single lines of constant degree. The
node positions of some node subsequences along diagonals is highlighted as
guide lines to the eye. The inset shows the collapse of all nodes in the graph
into a single diagonal (see text).

Thus, the ($N,k$) pairs in the graph define a structure in diagonals
$d=1,2,3,...$, and on each diagonal $d$ we label the particular nodes that lie
on it as $n=0,1,2,...$ Thus, $N(n;d)$ indicates the node/time for the $n$-th
position on diagonal $d$. For example, in the first and main diagonal $d=1$ in
Fig. 2 we have $N(0;1)=1=F_{1}$, $N(1;1)=3=F_{3}$, $N(2;1)=8=F_{5}$,
$N(3;1)=21=F_{7}$,… As it can be seen in the top panel of Fig. 2, the matching
positions $\theta_{t}$, $t=F_{2n+1}$ (highlighted) grow monotonically when
removed from the rest of the time series, and according to the HV algorithm
this implies increasing values for the degrees of their corresponding nodes.
For $d=2$ (the second diagonal in Fig. 2) $N(0;2)=2$, $N(1;2)=6$, $N(2;2)=16$,
$N(3;2)=42$,… All the nodes $N(n;d)$ can be expressed via the recurrence
formula

$\displaystyle N(0;d)$ $\displaystyle=$
$\displaystyle\textrm{mex}\\{N(n;i):1\leq i<d,n\geq 0\\},$ $\displaystyle
N(1;d)$ $\displaystyle=$ $\displaystyle 2N(0;d)+d,$ $\displaystyle N(n;d)$
$\displaystyle=$ $\displaystyle 3N(n-1;d)-N(n-2;d),$ (2)

with $d=1,2,...$ and $n=0,1,2,..$., where the term mex stands for
MinimumEXclude value conway1  that in this case it means the smallest value of
$N$ that has not appeared in the previous diagonals. In fraenkel1  it is
demonstrated that every integer $N$ appears only once under the above
recurrence and this exotic enumeration occurs in a natural way in the golden
ratio route. In fact, all the time labels $n$ along the diagonals $d=1,2,...$
can be expressed as Fibonacci numbers
$F_{n}^{(d)}=F_{n-1}^{(d)}+F_{n-2}^{(d)}$ with different initial conditions
for each one of them,

$\displaystyle F_{0}^{(d)}$ $\displaystyle=$ $\displaystyle d,\
F_{1}^{(d)}=N(0;d),$ $\displaystyle N(n;d)$ $\displaystyle=$ $\displaystyle
F_{2n+1}^{(d)}.$ (3)

This recurrence is the consequence of the inflationary process that takes
place in the generation of graphs via the golden ratio route luque6 . Notice
that this route goes through successive approximants of the continued fraction
$[1,1,1,...]$ (see Fig. 1b). These approximants permanently alternate from
larger to smaller to larger values around the golden number, such that an
approximant graph is generated by concatenation of the two preceding
approximant graphs alternating the order of concatenation at each stage. This
can be seen explicitly in Fig. 3.

Figure 3:  First substructures of the quasiperiodic graph associated with the
golden ratio route to chaos. The resulting patterns follow from the universal
order with which an orbit visits the positions of the attractor. The
quasiperiodic graph associated with the time series generated at the onset of
chaos ($n\rightarrow\infty$) is the result of an infinite application of the
inflationary process by which a graph at period $F_{2n+1}$ is generated out of
graphs at periods $F_{2n}$ and $F_{2n-1}$ luque6 . The first few node/time
steps along the first diagonal ($d=1$) are highlighted.

The recurrence formula in Eq. (2) can be solved leading to an explicit
expression convenient for our purposes. First, it can be demonstrated
fraenkel1  that

$\displaystyle N(0;d)$ $\displaystyle=$
$\displaystyle\lfloor(d-1)\phi\rfloor+1,$ $\displaystyle N(n;d)$
$\displaystyle=$ $\displaystyle\lfloor N(n-1;d)\phi^{2}\rfloor+1.$ (4)

Then, use of the approximation $N(n;d)\approx N(n-1;d)\phi^{2}$ and of the
definition $C_{d}\equiv
N(1;d)=\big{\lfloor}(\lfloor(d-1)\phi\rfloor+1)\phi^{2}\big{\rfloor}+1$ yields
the solution

$N(n;d)=C_{d}\phi^{2n-2},\ n\geq 1.$ (5)

This equation captures the values $N(n;d)$ along the diagonals starting always
from $n=1$, that, as we can observe in the bottom panel of Fig. 2, are the
nodes with connectivities $k=5$ or $k=6$. Furthermore, all the (parallel
straight- line) diagonals can be collapsed into a single one by first
redefining the connectivities in each of them such that the degree is zero in
the initial position $n=1$. To do this it is only necessary to subtract $5$ or
$6$ according to the given diagonal, with the outcome that
$\widetilde{k}=2n-2$ with $n=1,2,...$ To get the collapse it is sufficient to
introduce the change of variable $\widetilde{N}(n;d)=N(n;d)/C_{d}$ so that
$\widetilde{N}(n;d)=\phi^{2n-2}$. We can see the result in the inset in the
bottom panel of Fig. 2. To keep notation simple we make use of this variable
and write $k$ instead of $\widetilde{k}$ from now on.

Figure 4: Log-log plot of the distance between two nearby trajectories
$l_{t}=|\theta_{t}-\theta_{t}^{\prime}|$ close to
${\small\theta}_{0}{\small=1}$, where ${\small l}_{0}{\small=10}^{-4}$,
measured at times ${\small t=N(n;d)}$, ${\small n=0,1,2,...}$, along the main
diagonal ${\small d=1}$ at the transition to chaos for the golden, silver and
bronze routes (see text).

##  IV Generalized Lyapunov exponents at the accumulation point of the golden
ratio route to chaos

We define now a connectivity expansion rate for the graph under study. The
formal network analog of the sensitivity to initial conditions in the map is
luque5

$\xi(N(n;d))\equiv\frac{\exp(k(n))}{\exp(k(1))}=\exp(k(n)),$ (6)

since $k(1)=k(N(1;d))=0$. That is, we compare the expansion $\exp(k(n))$ with
the minimal $\exp(k(1))=1$ occurring always at nodes at positions $N(1;d)$.

From Eq. (5) we have

$k(N(n;d))=2n-2=\ln\left(\frac{N}{C_{d}}\right)^{\frac{1}{\ln\phi}},$ (7)

or

$\xi(N(n;d))=\left(\frac{N}{C_{d}}\right)^{\frac{1}{\ln\phi}}.$ (8)

The standard network Lyapunov exponent is defined as

$\lambda\equiv\lim_{N\rightarrow\infty}\frac{1}{N}\ln\xi(N),$ (9)

but since Eq. (8) indicates that the bounds of the fluctuations of $\xi(N)$
grow with $N$ slower than $\exp N$ we have $\lambda=0$, in agreement to the
ordinary Lyapunov exponent at the onset of chaos.

To get a suitable expansion rate that grows linearly with the size of the
network, we deform the ordinary logarithm in $\ln{\xi(N)}=k(N)$ into
$\ln_{q}\xi(N)$ by an amount $q>1$ such that $\ln_{q}\xi(N)$ depends linearly
in $N$, where $\ln_{q}x\equiv(x^{1-q}-1)/(1-q)$ and $\ln x$ is restored in the
limit $q\rightarrow 1$ Koelink ; robledo2 . And through this deformation we
define the generalized graph-theoretical Lyapunov exponent as

$\lambda_{q}\equiv\frac{1}{\Delta N}\ln_{q}\xi(N),$ (10)

where $\Delta N=N(n;d)-C_{d}$ is the node distance or iteration time duration
between an initial node $N(1;d)$ where $d$ is fixed and $N(n;d)$ is the final
node position. From Eq. (8) we obtain

$\lambda_{q}(d)=\frac{1}{N-C_{d}}\frac{\left(\frac{N}{C_{d}}\right)^{\frac{1-q}{\ln\phi}}-1}{1-q}=\frac{1}{C_{d}\ln\phi},$
(11)

where the degree of deformation $q$ is found to be $q=1-\ln\phi$. This way we
have determined a spectrum of generalized Lyapunov exponents $\lambda_{q}(d)$,
one for each diagonal $d=1,2,...$ in Fig.2. The largest value is for the main
diagonal, $\lambda_{q}(1)=(C_{1}\ln\phi)^{-1}$, and the others gradually
decrease as $d\rightarrow\infty$.

##  V $q$-deformed entropy expression and Pesin-like identities

Having obtained the family of generalized Lyapunov exponents $\lambda_{q}(d)$
from a suitable expansion rate $\ln_{q}\xi(N)$, we proceed to analyze the
entropic properties of the network. At the transition to chaos for the golden
ratio the HV method creates a single network that represents many different
trajectories. Trajectories initiated at different positions of the attractor
produce networks related to each other by a node translation equal to the
number of iterations needed from one initial position $\theta_{0}^{(1)}$ to
reach the second $\theta_{0}^{(2)}$. The two positions appear in the
trajectory initiated at $\theta_{0}=0$ at times $t_{1}$ and $t_{2}$,
$\theta_{0}^{(1)}=\theta_{t_{1}}$ and $\theta_{0}^{(2)}=\theta_{t_{2}}$, and
the node translation is $\delta N=t_{2}-t_{1}>0$. This shift property can be
visualized in Fig. 2, and is implicated in the derivation of Eq. (10) for
$\lambda_{q}(d)$. But also, trajectories initiated at positions off the
attractor, but sufficiently close to a position of this set generate the same
network, as the HV method distinguishes differences in trajectory positions
only when they surpass threshold values. There is a basic property of
trajectories at the onset of chaos that combines with the previous remark and
that can be used to describe the rate of entropy growth of the network with
its size. This property is that for a small interval of length $l_{0}$ with
$\mathcal{N}$ uniformly-distributed initial conditions around, say,
$\theta_{0}=0$, all trajectories behave similarly, remain uniformly-
distributed at later times and follow the concerted pattern shown in Fig. 3 in
Ref. robledo1 . Studies of entropy growth associated with an initial
distribution of positions with iteration time $t$ of several chaotic maps
latora1  have established that a linear growth occurs during an intermediate
stage in the evolution of the entropy, after an initial transient dependent on
the initial distribution and before an asymptotic approach to a constant
equilibrium value. In relation to this it was found, both at the period-
doubling baldovin1 ; mayoral1  and at the quasiperiodic golden ratio robledo1
transitions to chaos, that (i) there is no initial transient if the initial
distribution is uniform and defined around a small interval of an attractor
position, and (ii) the distribution remains uniform for an extended period of
time due to the subexponential dynamics. In Fig. 4 we demonstrate this
property by presenting the time evolution of the distance between to nearby
trajectories, say the endpoints of the interval of length $l_{t}$ containing
the $\mathcal{N}$ uniformly-distributed positions at time $t$, for the golden
ratio transition to chaos, and also for other quasiperiodic transitions to
chaos along other routes discussed below. But the time evolution of the
trajectory distances in Fig. 4 can also be that between any pair of adjacent
positions in the initial uniform distribution and therefore the trajectories
distribution remains uniform after continued iterations.

We denote the above-referred distribution by $\pi(t)=1/W(t)$ where
$W(0)=l_{0}/\mathcal{N}$ is the number of cells that cover the initial
interval $l_{0}$. As stated, all such trajectories give rise to the same HV
graph, and at iteration times, say, of the form $t=N(n;d)$, $n=1,2,3,\dots$,
the HV criterion assigns $k=2n-2$ links to the common node $N(n;d)$. The
distribution $\pi$ is defined in the map but we can look at its
$n$-dependence, $\pi(N(n;d))$, if the scaling properties of the network retain
the scaling property of $\pi$ in the map. We can corroborate this and also
that the entropic properties derived from this distribution are connected to
the network Lyapunov exponents described in the previous section. The scaling
property of the network that yields the collapse of the diagonals in Fig. 2
described above implies that the uniform distributions $\pi$ for the
consecutive node-connectivity pairs ($N(n;d),2n-2$) and ($N(n+1;d),2(n+1)-2$)
along the same diagonal $d$ scale with the same factors and this leads us to
conclude that the $n$-dependence for these distributions is

$\pi(N)=W_{n}^{-1}=\exp(-2n+2).$ (12)

But since

$W_{n}=\exp(2n-2)=\left(\frac{N}{C_{d}}\right)^{\frac{1}{\ln\phi}},$ (13)

the ordinary entropy associated with $\pi$ grows logarithmically with the
number of nodes $N$, $S_{1}\left[\pi(N)\right]=\ln W_{j}\sim\ln N$. However,
the $q$-deformed entropy

$S_{q}\left[\pi(N)\right]\equiv\ln_{q}W_{n}=\frac{1}{1-q}\left[W_{n}^{1-q}-1\right],$
(14)

where the amount of deformation $q$ of the logarithm has the same value as
before, grows linearly with $N$, as $W_{n}$ can be rewritten as

$W_{n}=\exp_{q}[\lambda_{q}\Delta N],$ (15)

with $q=1-\ln\phi$ and $\lambda_{q}(d)=(C_{d}\ln\phi)^{-1}$. Therefore, if we
define the entropy growth rate

$h_{q}\left[\pi(N)\right]\equiv\frac{1}{\Delta N}S_{q}\left[\pi(N)\right]$
(16)

we obtain

$h_{q}\left[\pi(N)\right]=\lambda_{q}(d),$ (17)

a Pesin-like identity at the onset of chaos (effectively one identity for each
subsequence of node numbers , $n=1,2,3,\dots$, given each by a value of
$d=1,2,3,...$).

Figure 5:  Top: Positions $\theta_{t}$ as a function of ${\small t}$ for the
first ${\small 70}$ data for the orbit with initial condition
${\small\theta}_{0}{\small=1}$ at the silver number onset of chaos (see text)
of the critical circle map ${\small K=1}$. The data highlighted are associated
with specific subsequences of nodes (see text). Bottom: Log-log plot of
${\small\exp k(N)}$ as a function of the node ${\small N}$ for the HV graph
generated from same time series as as for the upper panel but for $3\times
10^{2}$ iterations, where ${\small N=t}$. The distinctive band pattern of the
attractor manifests through a pattern of single lines of constant degree. The
node positions of some node subsequences along diagonals is highlighted as
guide lines to the eye.

##  VI Quasiperiodic graphs at the onset of chaos for quadratic irrationals

We can generalize the above results for every quadratic irrational in $[0,1]$
with pure periodic continued fraction representation:
$\phi_{b}^{-1}=[b,b,b,...]=[\overline{b}]$ ($b=1$ , $2$, $3$, correspond to
the golden, silver and bronze routes, respectively). These irrationals are the
solutions of the equation $x^{2}-bx-1=0$, where $b$ is a natural number. The
dressed winding number is now
$\omega_{\infty}=\lim_{n\rightarrow\infty}[1-(F_{n-1}/F_{n})]=\phi_{b}^{-1}$with
$F_{n}=bF_{n-1}+F_{n-2}$, $F_{0}=0$, $F_{1}=1$ and the route to chaos is the
infinite sequence of attractors with periods $F_{n}$, $n=1,2,3,...$(Notice now
$F_{n}$ is only a Fibonacci number when $b=1$). The first few steps of the
silver route $b=2$ can be seen in Fig. 1(c), whereas Fig. 5 shows results for
the attractor at the onset of chaos via this route. Similarly to Fig. 2 for
$b=1$, in the top panel of Fig. 5 is the time series for the first $70$
iteration times, while in the bottom panel of the same figure we plot, in
logarithmic scales, the outcome of the HV method with use of the variable
$\exp k(N)$. As it can be observed the networks for the two cases are
qualitatively similar, although there are differences, mainly the absence of
even connectivities when $k>5$.

This absence can be verified by inspection of the degree distribution
$P_{\infty}(k)$ for the graphs at the $\omega_{\infty}=\phi_{b}^{-1}$
accumulation points luque6

$P_{\infty}(k)=\left\\{\begin{array}[]{ll}\phi_{b}^{-1}&k=2\\\
1-2\phi_{b}^{-1}&k=3\\\
(1-\phi_{b}^{-1})\phi_{b}^{(3-k)/b}&k=bn+3,\;n\in\mathbb{N}\\\
0&\mathrm{otherwise,}\end{array}\right.$ (18)

where we can see explicitly which values of $k$ are not present for a given
value of $b$. This and other connectivity properties can be worked out from
the inflation process of the graphs. See Fig. 6.

Figure 6:  First substructures of the quasiperiodic graph associated to (a)
the silver number ${\small b=2}$ and (b) the bronze number ${\small b=3}$
routes to chaos. The resulting patterns follow from the universal order with
which an orbit visits the positions of the attractors. The quasiperiodic graph
associated with the time series generated at the onset of chaos
($n\rightarrow\infty$) is the result of an infinite application of the
inflationary process by which a graph at period $F_{2n}$ is generated out of
graphs at periods $F_{2n-2}$ and $F_{2n-1}$ luque6 .

We will center our attention on the first diagonal $d=1$. For every $b$, the
node positions on the first diagonal, $n=1,2,3,...$, are

$N(n;1)=F_{2n},$ (19)

that with the use of the generalized Binet formula

$F_{n}=\frac{1}{\sqrt{b^{2}+4}}\left[\phi_{b}^{n}-\left(\frac{-1}{\phi_{b}}\right)^{n}\right]\approx\frac{\phi_{b}^{n}}{\sqrt{b^{2}+4}},$

can be written as

$N(n;1)\approx\frac{1}{\sqrt{b^{2}+4}}\phi_{b}^{2n}=\frac{1}{\sqrt{b^{2}+4}}\phi_{b}^{2}\phi_{b}^{2n-2}=C_{b}\phi_{b}^{2n-4},$
(20)

where the position $n=1$ is

$N(1;1)=F_{2}\approx\frac{1}{\sqrt{b^{2}+4}}\phi_{b}^{2}\equiv C_{b}.$ (21)

We note that the connectivity of the first node is $k(n=1)=b+3$ and in general
$k(n)=b+3+2b(n-1)$, $n\geq 2$. As before we redefine the connectivities such
that the degree is zero at the initial position $n=1$, $k(n)=2b(n-1)$,
$n=1,2,3,...$ Following the same procedure as in Section 4, from Eq. (20) we
have

$k(N(n;1))=2b(n-1)=\ln\left(\frac{N}{C_{b}}\right)^{\frac{1}{\ln\phi_{b}}},$
(22)

and use of it in the sensitivity $\xi(N(n;1))\equiv\exp(k(n))$ yields

$\xi(N(n;1))=\left(\frac{N}{C_{b}}\right)^{\frac{1}{\ln\phi_{b}}}.$ (23)

Since all the features required for the $q$-deformation described in Section 4
are present for general $b$, we obtain for the generalized Lyapunov exponent
the expression

$\lambda_{q}^{(b)}(1)=\frac{1}{N-C_{b}}\frac{\left(\frac{N}{C_{b}}\right)^{\frac{1-q}{\ln\phi_{b}}}-1}{1-q}=\frac{1}{C_{b}\ln\phi_{b}},$
(24)

where $q=1-\ln\phi_{b}$. Likewise, the contents of Section 5 can also be
reproduced for general $b$ with the result that

$h_{q}\left[\pi(N)\right]=\lambda_{q}^{(b)}(1).$ (25)

##  VII Summary and discussion

At the quasiperiodic onset of chaos the HV method leads to a self-similar
network with a structure illustrated by the related periodic networks obtained
from the sequence of attractors of finite periods along the route to chaos
luque6 . Under the HV algorithm many nearby trajectory positions lead to the
same network, since only when the values of trajectory positions cross a
threshold the corresponding node increases its degree with new links. (See the
succinct definition of the algorithm and the top panel in Fig. 2). Therefore
trajectories off the attractor but close to it transform into the same network
structure. As we have seen the fluctuations of the degree capture the
anomalous but basic behavior of the fluctuations of the sensitivity to initial
conditions at the transition to chaos robledo1 . The graph-theoretical
analogue of the sensitivity was identified as $\exp(k)$ while the amplitude of
the variations of $k$ grows logarithmically with the number of nodes $N$.
These deterministic fluctuations are described by a discrete spectrum of
generalized graph-theoretical Lyapunov exponents that are shown to relate to
an equivalent spectrum of generalized entropy growth rates, yielding a set of
Pesin-like identities. This behavior is similar to what was observed for the
case of the more straightforward period-doubling accumulation point luque5 .
The definitions of these quantities involve a scalar deformation of the
ordinary logarithmic function that ensures their linear growth with the number
of nodes. Therefore the entropy expression involved is extensive and of the
Tsallis type with a precisely fixed value of the deformation index $q$,
$q=1-\ln\phi_{b}$, where $\phi_{b}$ is the inverse of the irrational (dressed)
winding number.

We have considered special families of time series and converted each into a
network, each family consists of the trajectories associated with an attractor
at the quasiperiodic transition to chaos of circle maps. The attractors
studied are defined by a winding number given by a quadratic irrational or,
equivalently, by a pure periodic continued fraction. Each winding number
singles out a specific route to chaos. Amongst these we described in some
detail the so-called golden route, but also we have shown results for those
known as the silver and bronze routes luque6 . See Figs. 1 and 4. The HV
algorithm proved to be capable of generating a single network that contains
the scaling and entropic properties of the trajectories associated with each
attractor. The results presented here are of the same kind as those obtained
for the period-doubling route to chaos luque5  suggesting that the HV networks
associated with the onset of chaos are useful for describing the universal
properties at these special systems. The Pesin identity is a reflection of a
basic connection between BG statistical mechanics and chaos so that our
results provide elements for an analogous connection for the case of
nonergodic and nonmixing dynamics at vanishing ordinary Lyapunov exponent.

Acknowledgements. We acknowledge financial support by the Comunidad de Madrid
(Spain) through Project No. S2009ESP-1691 (B.L.), support from CONACyT & DGAPA
(PAPIIT IN100311)-UNAM (Mexican agencies) (A.R.).

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