# On Automorphisms of Complex $b^{k}$-Manifolds

Tatyana Barron and Michael Francis

###### Abstract

The $b$-calculus of Melrose is a tool for studying structures on a smooth
manifold with a first order degeneracy at a given hypersurface. In this
framework, Mendoza defined complex $b$-manifolds. In the spirit of work of
Scott, we extend Mendoza’s definition to the case of higher-order
degeneracies, introducing the notion of a complex $b^{k}$-manifold for $k$ a
positive integer. We then investigate the local and global automorphisms of
complex $b^{k}$-manifolds. We also propose $b^{k}$-analogues for some
classical spaces of holomorphic functions.

Mathematics Subject Classification (2010): Primary 32Q99; Secondary 32M18

Keywords: $b$-geometry, complex $b$-manifold, automorphism group

##  1 Introduction

Throughout, $M$ is a smooth manifold and $Z\subseteq M$ is a closed
hypersurface. We write $C^{\infty}(M)$ for the ring of smooth,
$\mathbb{R}$-valued functions on $M$ and $\mathfrak{X}(M)$ for the
$C^{\infty}(M)$-module of (smooth) vector fields on $M$.

Melrose [11] introduced $b$-calculus as an organizational framework for the
study of differential operators on $M$ with a first order degeneracy along
$Z$. His formalism has significant applications to index theory on open
manifolds.

It is interesting to study “$b$ versions” of classical geometries in which the
defining data of a given geometry suffers a degeneracy along $Z$. Various
authors have considered $b$-symplectic manifolds, also called log-symplectic
manifolds ([5], [7], [6], [8], [10]). Mendoza [12] defined complex
$b$-manifolds and studied the cohomology of the $b$-Dolbeault complex.

Scott [14] generalized $b$-calculus by introducing $b^{k}$-manifolds, where
$k$ encodes the order of degeneracy along the hypersurface $Z\subseteq M$.
From the point of view of Scott’s theory, ordinary $b$-geometry is the case
$k=1$.

In this article, we extend Mendoza’s definition in the spirit of Scott’s work
by defining what is a complex $b^{k}$-manifold for $k>1$. We then restrict
attention to the (real) two-dimensional case and investigate the local and
global automorphisms of complex $b^{k}$-manifolds. We briefly discuss
candidates for function spaces one can attach to a complex $b^{k}$-manifold.

##  2 Almost-injective Lie algebroids

Let $M$ be a smooth manifold. The categories of smooth vector bundles on $M$
and finitely-generated, projective $C^{\infty}(M)$-modules are equivalent by
Serre-Swan duality [15]. Under this equivalence, injective maps of
$C^{\infty}(M)$-modules correspond exactly to vector bundle maps that are
injective over a dense set. One may therefore regard the following data as
equivalent:

  * •

a projective $C^{\infty}(M)$-module $\mathcal{F}$ of vector fields on $M$,

  * •

a vector bundle $A$ over $M$ together with a bundle map $\rho:A\to TM$ that is
injective over a dense (open) subset of $M$.

One may also deal with the corresponding sheaf of local vector fields:

$\displaystyle\mathcal{F}_{U}$ $\displaystyle\coloneq
C^{\infty}(U)\\{X|_{U}:X\in\mathcal{F}\\}=\rho(C^{\infty}(U;A)),\text{ for
$U\subseteq M$ open.}$ (2.1)

If $\mathcal{F}$ is closed under Lie bracket, then $A$ is naturally a Lie
algebroid with bracket coming from the identification
$C^{\infty}(M;A)\cong\mathcal{F}$. Lie algebroids whose anchor map is
injective over a dense set are called almost-injective and appear in the study
of Stefan-Sussmann singular foliations. See, for example, [3] and 5.2.5 in
[2]. In this language, the following data are equivalent:

  * •

an involutive, projective $C^{\infty}(M)$-module $\mathcal{F}$ of vector
fields on $M$,

  * •

an almost-injective Lie algebroid $A=(A,\rho,[\cdot,\cdot])$ over $M$.

##  3 The $b$-tangent bundle

Two superficially different approaches to $b$-calculus are in use. The
original approach of [11] uses a manifold with boundary. It is also common to
work on a manifold without boundary that is equipped with a preferred
hypersurface [1]. We shall adopt the second approach here.

######  Definition 3.1.

A $\mathbf{b}$-manifold is a smooth manifold $M$ together with a closed
hypersurface $Z\subseteq M$ that we refer to as the singular locus. A
$\mathbf{b}$-vector field is a vector field on $M$ tangent to $Z$. The
$C^{\infty}(M)$-module of $b$-vector fields is denoted
${{}^{b}\mathfrak{X}}(M)$.

######  Example 3.2.

If $M=\mathbb{R}^{2}$ and $Z=\\{0\\}\times\mathbb{R}$, then
${{}^{b}\mathfrak{X}(\mathbb{R}^{2})}=\langle
x\partial_{x},\partial_{y}\rangle$, the $C^{\infty}(\mathbb{R}^{2})$-module
generated by $x\partial_{x}$ and $\partial_{y}$.

Above, ${{}^{b}\mathfrak{X}(\mathbb{R}^{2})}$ is a free
$C^{\infty}(\mathbb{R}^{2})$-module. In general, for any $b$-manifold $M$,
${{}^{b}\mathfrak{X}}(M)$ is a projective $C^{\infty}(M)$-module closed under
Lie bracket. Thus, referring to Section 2, the following definition makes
sense.

######  Definition 3.3.

The $\mathbf{b}$-tangent bundle of a $b$-manifold $M$ is the almost injective
Lie algebroid ${{}^{b}TM}$ associated to the involutive, projective module
${}^{b}\mathfrak{X}(M)$.

##  4 $b^{k}$-Tangent bundles

Fix an integer $k\geq 2$. By analogy with $b$-vector fields ($k=1$), a
$b^{k}$-vector field on a manifold $M$ with given closed hypersurface $Z$
ought to be a vector field that is “tangent to $Z$ to $k$th order”. It is
clear how to best proceed in the model Euclidean example where
$M=\mathbb{R}^{n}$, $Z=\\{0\\}\times\mathbb{R}^{n-1}$.

######  Definition 4.1.

Fix a positive integer $k$. The standard module of $\mathbf{b^{k}}$-vector
fields on $\mathbb{R}^{n}$ is $\langle
x_{1}^{k}\partial_{x_{1}},\partial_{x_{2}},\ldots,\partial_{x_{n}}\rangle$.
That is, the $C^{\infty}(\mathbb{R}^{n})$-module of vector fields on
$\mathbb{R}^{n}$ whose first component vanishes to $k$th order on
$\\{0\\}\times\mathbb{R}^{n-1}$.

Unfortunately, the naive notion of “$k$th order tangency” used above is not
compatible with smooth coordinate changes.

######  Example 4.2.

The diffeomorphism $\theta:\mathbb{R}^{2}\to\mathbb{R}^{2}$,
$\theta(x,y)=(e^{y}x,y)$ satisfies
$\theta_{*}(\partial_{y})=x\partial_{x}+\partial_{y}$. Thus, although $\theta$
preserves the $y$-axis, pushforward by $\theta$ does not preserve the standard
module of $b^{2}$-vector fields.

The above example shows it does not make sense to talk about _the_ module of
$b^{k}$-vector fields. Instead, we prescribe as external data a projective
module of vector fields on $M$ that is tangent to $Z$ and locally
$C^{\infty}$-conjugate to the model example of Definition 4.1.

######  Definition 4.3.

Fix a positive integer $k$. A module of $\mathbf{b^{k}}$-vector fields for an
$n$-dimensional, smooth manifold $M$ with a given closed hypersurface
$Z\subseteq M$ is a projective $C^{\infty}(M)$-module
$\mathcal{F}\subseteq\mathfrak{X}(M)$ such that every $p\in M$ admits an open
neighbourhood $U$ and a diffeomorphism $\theta:U\to\mathbb{R}^{n}$ satisfying:

$\theta_{*}(\mathcal{F}_{U})=\begin{cases}\langle
x_{1}^{k}\partial_{x_{1}},\partial_{x_{2}},\ldots,\partial_{x_{n}}\rangle&\text{
if }p\in Z\phantom{.}\\\ \mathfrak{X}(\mathbb{R}^{n})&\text{ if }p\notin Z\\\
\end{cases}$

(see (2.1) for notation). Such an $\mathcal{F}$ is involutive; its
corresponding almost-injective Lie algebroid is called the
$\mathbf{b^{k}}$-tangent bundle and denoted ${{}^{b^{k}}TM}$.

######  Remark 4.4.

It is tempting to call a pair $(M,\mathcal{F})$ consisting of a manifold with
a given module of $b^{k}$-vector fields a _$b^{k}$ -manifold_. However, we
shall not do this because Scott [14] has already defined a $b^{k}$-manifold to
be a triple $(M,Z,j_{Z})$, where $j_{Z}$ is the $(k-1)$-jet along $Z$ of a
defining function for $Z$ (plus an orientation on $M$ and $Z$). Such a jet
induces a module of $b^{k}$-vector fields in our sense, namely (Definition 2.5
in [14]) the collection $\mathcal{F}$ of vector fields $X$ such that $Xf$
vanishes to $k$th order on $Z$, for any defining function $f$ representing
$j_{Z}$. On the other hand, the assignment $j_{Z}\mapsto\mathcal{F}$ is
neither injective nor surjective. Regarding injectivity, taking $k=2$,
$M=\mathbb{R}^{2}$, $Z=\\{0\\}\times\mathbb{R}$, both $j_{Z}=x$ and $j_{Z}=2x$
induce $\mathcal{F}=\langle x^{2}\partial_{x},\partial_{y}\rangle$. Regarding
surjectivity, the module of $b^{2}$-vector fields $\langle
x^{2}\partial_{x},\partial_{y}+x\partial_{x}\rangle$ for $M=\mathbb{R}\times
S^{1}$, $Z=\\{0\\}\times S^{1}$, for which local trivializations can be
obtained from $\theta(x,y)=(e^{y}x,y)$, cannot arise from the jet of a
defining function. This can be seen from Chapter 5 of [4] where a holonomy
invariant is attached to a module of $b^{k}$-vector fields (there called a
_transversely order $k$ singular foliation_).

##  5 $b$-Complex and $b^{k}$-complex manifolds

Recall that a complex structure on $M$ is given by a subbundle ${T^{0,1}M}$ of
the complexified tangent bundle satisfying: (i)
$\mathbb{C}TM=\overline{{T^{0,1}M}}\oplus{T^{0,1}M}$, and (ii) $T^{0,1}M$ is
involutive. This bundle-theoretic definition is equivalent to the definition
via holomorphic charts by the Newlander-Nirenberg theorem: every point in $M$
admits local coordinates $(x_{1},y_{1},\ldots,x_{n},y_{n})$ in which the
complex derivatives
$\partial_{\overline{z}}=\frac{1}{2}(\partial_{x_{j}}+i\partial_{y_{j}})$,
$j=1,\ldots,n$ form a local frame for $T^{0,1}M$. To define complex
$b^{k}$-manifolds, we simply replace $TM$ by ${{}^{b^{k}}TM}$. The case $k=1$
was given in [12].

######  Definition 5.1.

Fix a positive integer $k$. A complex $\mathbf{b^{k}}$-structure for a smooth
manifold $M$ with a given closed hypersurface $Z\subseteq M$ and module of
$b^{k}$-vector fields $\mathcal{F}$ is a (complex) subbundle
${{}^{b^{k}}T^{0,1}M}$ of the complexified $b^{k}$-tangent bundle satisfying:
(i)
$\mathbb{C}({{}^{b^{k}}TM})=\overline{{}^{b^{k}}T^{0,1}M}\oplus{{}^{b^{k}}T^{0,1}M}$,
and (ii) ${{}^{b^{k}}T^{0,1}M}$ is involutive. It is equivalent to work with
the corresponding bracket-invariant $C^{\infty}(M,\mathbb{C})$-module of
complex vector fields $\mathcal{F}^{0,1}\subseteq\mathbb{C}\mathcal{F}$ and,
actually, it is the pair $(M,\mathcal{F}^{0,1})$ that we call a complex
$\mathbf{b^{k}}$-manifold.

######  Remark 5.2.

The module $\mathcal{F}^{0,1}$ recovers all the other data at hand. For
example, taking real parts recovers $\mathcal{F}$, and hence ${{}^{b^{k}}TM}$.
The singular set for the anchor map is the hypersurface $Z$. We define complex
$b^{k}$-manifolds as pairs $(M,\mathcal{F}^{0,1})$ in order to make the
definition below simple.

######  Definition 5.3.

An isomorphism $\theta:(M,\mathcal{E}^{0,1})\to(N,\mathcal{F}^{0,1})$ of two
complex $b^{k}$-manifolds is a diffeomorphism $\theta:M\to N$ satisfying
$\theta_{*}(\mathcal{E}^{0,1})=\mathcal{F}^{0,1}$.

######  Remark 5.4.

The $b^{k}$-tangent bundle of a complex $b^{k}$-manifold $M$ is isomorphic to
the usual tangent bundle away from the singular locus $Z$, so $M\setminus Z$
is an ordinary complex manifold. Moreover, complex $b^{k}$-manifold
isomorphisms restrict to complex manifold isomorphisms away from the singular
loci.

######  Remark 5.5.

Nothing prevents one from speaking about involutive almost complex structures
on arbitrary Lie algebroids. This is done in [9] and [13], for example. On the
other hand, at this level of generality, it is not clear what the formal
integrability condition of involutivity might imply about the existence of a
special local normal forms in the spirit of Newlander-Nirenberg. The case of
complex $b$-manifolds is discussed in the next section.

##  6 Remark on the $b$-Newlander-Nirenberg theorem

Ideally, complex $b^{k}$-manifolds should have a single local model depending
only on $k$ and the dimension, thus allowing for an equivalent definition in
terms of appropriate “$b^{k}$-holomorphic charts”. This holds for ordinary
complex manifolds (if one likes, the case $k=0$) by the Newlander-Nirenberg
theorem. The $k=1$ case was partially resolved in Section 5 of [12]; complex
$b$-manifolds do not have “formal local invariants” at the boundary. Recently
(results to appear elsewhere), the authors have established the full “no local
invariants” result in the $k=1$ case and expect the same to hold for $k\geq
2$. For this reason, we focus on a single model case in our study of
automorphisms below.

######  Definition 6.1.

The standard $\mathbf{b^{k}}$-complex plane is the complex $b$-manifold
${{}^{b^{k}}\mathbb{R}^{2}}\coloneq(\mathbb{R}^{2},\langle L_{k}\rangle)$,
where $\langle L_{k}\rangle$ is the
$C^{\infty}(\mathbb{R}^{2},\mathbb{C})$-module singly-generated by

$\displaystyle L_{k}\coloneq x^{k}\partial_{x}+i\partial_{y}.$ (6.1)

As stated above, our justification for focusing on these model cases is the
following result whose proof will appear elsewhere.

######  Theorem 6.2.

Let $M$ be a two-dimensional complex $b$-manifold with singular locus $Z$.
Then, for every $p\in Z$, there is a neighbourhood $U\subseteq M$ of $p$ and
an open set $V\subseteq{{}^{b^{k}}\mathbb{R}^{2}}$ such that $U\cong V$ as
complex $b$-manifolds.

##  7 Automorphisms of complex $b^{k}$-manifolds

We describe the global automorphisms of the standard $b^{k}$-complex planes by
relating complex $b^{k}$-manifold automorphisms to standard complex
automorphisms (see Remark 5.4). We first note the following simple fact.

######  Proposition 7.1.

Fix a positive integer $k$. Then,

$\operatorname{Aut}({{}^{b^{k}}\mathbb{R}^{2}})=\\{\theta\in\operatorname{Diff}(\mathbb{R}^{2}):\theta_{*}(L_{k})\sim
L_{k}\\}.$

Here, $L_{k}=x^{k}\partial_{x}+i\partial_{y}$ and $\sim$ denotes equality
modulo multiplication by a nowhere-vanishing smooth function. Every
automorphism $\theta$ of ${{}^{b^{k}}\mathbb{R}^{2}}$ satisfies
$\theta_{*}(Z)=Z$ where $Z=\\{0\\}\times\mathbb{R}$.

###### Proof.

From Definition 5.3, an isomorphism
${{}^{b^{k}}\mathbb{R}^{2}}\to{{}^{b^{k}}\mathbb{R}^{2}}$ is a diffeomorphism
preserving the $C^{\infty}(\mathbb{R}^{2},\mathbb{C})$-module singly-generated
by $L_{k}$, hence the first statement. For the second statement, see Remark
5.2. ∎

######  Example 7.2.

For every positive integer $k$, $(x,y)\mapsto(-x,(-1)^{k+1}y)$ is an order-two
automorphism of ${{}^{b^{k}}\mathbb{R}^{2}}$ that maps the (open) left half-
plane $\mathbb{R}^{2}_{-}$ onto the right half-plane $\mathbb{R}^{2}_{+}$.

######  Definition 7.3.

We write $\operatorname{Aut}_{+}({{}^{b^{k}}\mathbb{R}^{2}})$ for the normal
subgroup of $\operatorname{Aut}({{}^{b^{k}}\mathbb{R}^{2}})$ consisting of
automorphisms which preserve the right half-plane $\mathbb{R}^{2}_{+}$.

As the full automorphism group
$\operatorname{Aut}({{}^{b^{k}}\mathbb{R}^{2}})$ is the semidirect product
product of $\operatorname{Aut}_{+}({{}^{b^{k}}\mathbb{R}^{2}})$ by the order-
two automorphism of Example 7.2, we are content to obtain a description of
$\operatorname{Aut}_{+}({{}^{b^{k}}\mathbb{R}^{2}})$.

######  Theorem 7.4.

The group $\operatorname{Aut}_{+}({{}^{b}\mathbb{R}^{2}})$ is isomorphic to
$\mathbb{R}\times\mathbb{R}$. More precisely, it is generated by the following
1-parameter subgroups:

vertical translations: $\displaystyle(x,y)$ $\displaystyle\mapsto(x,y+t)$
$\displaystyle t\in\mathbb{R}$ horizontal scalings: $\displaystyle(x,y)$
$\displaystyle\mapsto(e^{t}x,y)$ $\displaystyle t\in\mathbb{R}.$

###### Proof.

Let $\mathbb{C}$ have its usual complex structure. The diffeomorphism

$\displaystyle\theta:\mathbb{R}^{2}_{+}\to\mathbb{R}^{2}$
$\displaystyle\theta(x,y)=\log x+iy$

satisfies
$\theta_{*}(x\partial_{x}+i\partial_{y})=\partial_{x}+i\partial_{y}$; it is an
isomorphism from $(\mathbb{R}^{2}_{+},\langle L_{1}\rangle)$ to
$\mathbb{C}=(\mathbb{R}^{2},\langle L_{0}\rangle)$. From Remark 5.4, the
restriction of any $\alpha\in\operatorname{Aut}_{+}({{}^{b}\mathbb{R}^{2}})$
to $\mathbb{R}^{2}_{+}$ pushes forward through $\theta$ to an automorphism of
the ordinary complex structure of $\mathbb{R}^{2}=\mathbb{C}$. The
automorphisms of $\mathbb{C}$ are the group $\mathbb{C}^{*}\ltimes\mathbb{C}$
of affine transformations. Pulled back through $\theta$, the translation
automorphisms of $\mathbb{C}$ become the automorphisms
$(x,y)\mapsto(x+s,e^{t}y)$ of $(\mathbb{R}^{2}_{+},\langle L_{1}\rangle)$, for
$s,t\in\mathbb{R}$. These extend (by the same formula) to automorphisms of
${{}^{b}\mathbb{R}^{2}}$. Moreover, these extensions can be seen to be unique
by considering the analogous identification of the left half-plane
$(\mathbb{R}^{2}_{-},\langle L_{1}\rangle)$ with $\mathbb{C}$.

On the other hand, the scaling symmetries of $\mathbb{C}$ pull back through
$\theta$ to the automorphisms $(x,y)\mapsto(e^{s\log x-ty},t\log x+sy)$ of
$(\mathbb{R}^{2}_{+},\langle L_{1}\rangle)$, for $s+it\in\mathbb{C}^{*}$. Such
maps do not extend continuously to $\mathbb{R}^{2}$ unless $t=0$ and $s>0$. In
the latter case, one obtains the automorphisms $(x,y)\mapsto(sx,y^{s})$, $s>0$
of $(\mathbb{R}^{2}_{+},\langle L_{1}\rangle)$ which can be extended
continuously to $\mathbb{R}^{2}$, but not to diffeomorphisms (excepting the
trivial case $s=1$). ∎

######  Theorem 7.5.

If $k\geq 2$ is an integer, the group
$\operatorname{Aut}_{+}({{}^{b^{k}}\mathbb{R}^{2}})$ is isomorphic to the
“$ax+b$ group” $\mathbb{R}_{+}\ltimes\mathbb{R}$. More precisely, it is
generated by the following 1-parameter subgroups:

vertical translations: $\displaystyle(x,y)$ $\displaystyle\mapsto(x,y+t)$
$\displaystyle t\in\mathbb{R}$ hyperbolic transformations:
$\displaystyle(x,y)$ $\displaystyle\mapsto(e^{-\frac{t}{k-1}}x,e^{t}y)$
$\displaystyle t\in\mathbb{R}.$

###### Proof.

Let $\mathbb{H}\subseteq\mathbb{C}$ denote the upper half complex plane
$\operatorname{Im}(z)>0$ with its standard complex structure. The
diffeomorphism

$\displaystyle\theta:\mathbb{R}^{2}_{+}\to\mathbb{H}$
$\displaystyle\theta(x,y)=y+\tfrac{1}{(k-1)x^{k-1}}i$

satisfies
$\theta_{*}(x^{k}\partial_{x}+i\partial_{y})=-\partial_{y}+i\partial_{x}\sim\partial_{x}+i\partial_{y}$;
it is an isomorphism from $(\mathbb{R}^{2}_{+},\langle L_{k}\rangle)$ to
$\mathbb{H}=(\mathbb{H},\langle L_{0}\rangle)$. Referring again to Remark 5.4,
every element of $\operatorname{Aut}_{+}(\mathbb{R}^{2},\langle L_{k}\rangle)$
determines, by conjugation through $\theta$, an automorphism of $\mathbb{H}$.
The usual group of complex automorphisms of $\mathbb{H}$ is
$\operatorname{PSL}(2,\mathbb{R})$, acting as fractional linear
transformations. Using the KAN decomposition for
$\operatorname{SL}(2,\mathbb{R})$, $\operatorname{Aut}(\mathbb{H})$ is the
product of the following three one-parameter subgroups:

$\displaystyle K$ $\displaystyle=\\{z\mapsto\tfrac{z\cos t-\sin t}{z\sin
t+\cos t}:t\in\mathbb{R}\\}$ (elliptic) $\displaystyle A$
$\displaystyle=\\{z\mapsto e^{t}z:t\in\mathbb{R}\\}$ (hyperbolic)
$\displaystyle N$ $\displaystyle=\\{z\mapsto z+t:t\in\mathbb{R}\\}$
$\displaystyle\text{(parabolic)}.$

Pulled back through $\theta$, the latter two groups of automorphisms of
$\mathbb{H}$ become the following groups of automorphisms
$(\mathbb{R}^{2}_{+},\langle L_{k}\rangle)$:

$\displaystyle
A^{\prime}=\\{(x,y)\mapsto(e^{-\frac{t}{k-1}}x,e^{t}y):t\in\mathbb{R}\\},$
$\displaystyle N^{\prime}=\\{(x,y)\mapsto(x,y+t):t\in\mathbb{R}\\}.$

These groups of automorphisms extend (by the same formulas) to automorphisms
of ${{}^{b^{k}}\mathbb{R}^{2}}$. Their extensions are also unique (by
considering their action on the left half-plane as well). On the other hand,
the automorphisms of $(\mathbb{R}^{2}_{+},\langle L_{k}\rangle)$ arising as
the pullback through $\theta$ of $z\mapsto\tfrac{z\cos t-\sin t}{z\sin t+\cos
t}$ for $t$ not an integer multiple of $\pi$ do not extend continuously to
$\mathbb{R}^{2}$. Indeed, an elementary computation shows that, if the
imaginary part of $z$ goes to $+\infty$ and the real part of $z$ is held
fixed, then $\tfrac{z\cos t-\sin t}{z\sin t+\cos t}\to\cot(t)$. It follows, if
$\alpha_{t}$ is the corresponding automorphism of $(\mathbb{R}^{2},\langle
L_{k}\rangle)$, that $\alpha_{t}(x,y)$ approaches $(+\infty,\cot(t))$ as $x\to
0^{+}$ and $y$ is held fixed. In particular, $\alpha_{t}$ does not admit a
continuous extension to $\mathbb{R}^{2}$. ∎

######  Remark 7.6.

We note that the action of
$\operatorname{Aut}_{+}({{}^{b^{k}}\mathbb{R}^{2}})$ on the singular locus
$Z=\\{0\\}\times\mathbb{R}$ is faithful for $k\geq 2$, but not for $k=1$.

We conclude this section with an example of a family of local automorphisms of
${{}^{b}\mathbb{R}^{2}}$ that do not extend to a global ones.

######  Example 7.7.

Let $\Omega\subseteq{{}^{b}\mathbb{R}^{2}}$ be the strip
$-\frac{\pi}{2}<y<\frac{\pi}{2}$. The function $u:\mathbb{R}^{2}\to\mathbb{C}$
given by $u(x,y)=xe^{iy}$ maps $\Omega\setminus(\\{0\\}\times\mathbb{R})$
diffeomorphically onto $\mathbb{C}\setminus\mathbb{R}$ and satisfies
$u_{*}(x\partial_{x}+i\partial_{y})=\overline{z}\partial_{\overline{z}}$.
Thus,
$u_{*}(xe^{iy}(x\partial_{x}+i\partial_{y}))=\overline{z}^{2}\partial_{\overline{z}}$
and, after taking real parts,

$u_{*}(x^{2}\cos x\partial_{x}+x\sin
y\partial_{y})=(x^{2}-y^{2})\partial_{x}+2xy\partial_{y}.$

The vector feld on the right is the generator of the 1-parameter group of
Mobius transformations $z\mapsto\frac{z}{1-tz}$, $t\in\mathbb{R}$ which is
complete when considered as a flow on $\mathbb{C}\setminus\mathbb{R}$ (or, of
course, the Riemann sphere). Pulling back through $u$, one checks that the
flow of $x^{2}\cos x\partial_{x}+x\sin y\partial_{y}$ defines a 1-parameter
group of complex $b$-manifold automorphisms of $(\Omega,\langle
L_{1}\rangle)$.

##  8 Function spaces

A $\mathbf{b}$-holomorphic function on ${{}^{b}\mathbb{R}^{2}}$ is a smooth
function $f:\mathbb{R}^{2}\to\mathbb{C}$ (perhaps only locally defined)
satisfying $L_{1}f=0$ where $L_{1}=x\partial_{x}+i\partial_{y}$.

######  Example 8.1.

The function $u:{{}^{b}\mathbb{R}^{2}}\to\mathbb{C}$, $u(x,y)=xe^{iy}$ is
$b$-holomorphic. More generally, $h\circ u$ is $b$-holomorphic near $(0,0)$
for any holomorphic function $h$ defined near $0$. There exist nonanalytic
$b$-holomorphic functions defined near $(0,0)$ as well;
$f(x,y)=\exp(\frac{-1}{u(x,y)})$ for $x>0$, $f(x,y)=0$ for $x\leq 0$ defines a
$b$-holomorphic function on
$\\{(x,y)\in{{}^{b}\mathbb{R}^{2}}:-\frac{\pi}{4}<y<\frac{\pi}{4}\\}$.

It is interesting to contemplate potential “$b$ analogues” for classical
spaces of holomorphic functions such as Segal-Bargmann space, (weighted)
Bergman space, Hardy space, etc. Let us define ad hoc $\mathbf{b}$-Segal-
Bargmann space to be the space of $b$-holomorphic functions on
${{}^{b}\mathbb{R}^{2}}$ whose restriction to the right half-plane
$\mathbb{R}^{2}_{+}$ pushes forward through the isomorphism $(x,y)\mapsto(\log
x,y):(\mathbb{R}^{2}_{+},\langle L_{1}\rangle)\to\mathbb{C}$ used earlier to
an element of the classical Segal-Bargmann space (entire functions $f$ on
$\mathbb{C}$ satisfying $\int_{\mathbb{C}}|f(z)|^{2}e^{-|z|^{2}}\ dA<\infty$).

######  Example 8.2.

The function $u(x,y)=xe^{iy}$ belongs to the $b$-Segal-Bargman space (it
pushes forward to the exponential function). Considering powers of $u$, we
conclude the $b$-Segal-Bargmann space is infinite-dimensional.

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