The Newlander-Nirenberg theorem for
complex $b$ -manifolds

Tatyana Barron and Michael Francis

(October 12, 2023)

Abstract

Melrose defined the $b$ -tangent bundle of a smooth manifold $M$ with boundary as the vector bundle whose sections are vector fields on $M$ tangent to the boundary. Mendoza defined a complex $b$ -manifold as a manifold with boundary together with an involutive splitting of the complexified $b$ -tangent bundle into complex conjugate factors. In this article, we prove complex $b$ -manifolds have a single local model depending only on dimension. This can be thought of as the Newlander-Nirenberg theorem for complex $b$ -manifolds: there are no “local invariants”. Our proof uses Mendoza’s existing result that complex $b$ -manifolds do not have “formal local invariants” and a singular coordinate change trick to leverage the classical Newlander-Nirenberg theorem.

1 Introduction

Throughout, we conflate $\mathbb{R}^{2}=\mathbb{C}$ and use coordinates $z=(x,y)=x+iy$ interchangeably. The word “smooth” means “infinitely real-differentiable”, not “holomorphic”.

Recall that a complex manifold may equivalently defined either by a holomorphic atlas or by an integrable almost-complex structure. The equivalence of these two definitions is given by the Newlander-Nirenberg theorem [8]. In more detail, a complex structure on a smooth manifold $M$ is an involutive subbundle $T^{0,1}M$ of the complexified tangent bundle such that $\mathbb{C}TM=T^{1,0}M\oplus T^{0,1}M$ where $T^{1,0}M\coloneq\overline{T^{0,1}M}$ . The Newlander-Nirenberg theorem implies that every point belongs to a coordinate chart $(x_{1},y_{1},\ldots,x_{n},y_{n})=(z_{1},\ldots,z_{n})$ in which $T^{0,1}M$ is spanned by

\displaystyle\partial_{\overline{z}_{j}}\coloneq\tfrac{1}{2}(\partial_{x_{j}}+i\partial_{y_{j}})

\displaystyle j=1,\ldots,n.

Accordingly, $T^{1,0}M$ is spanned by

\displaystyle\partial_{z_{j}}\coloneq\tfrac{1}{2}(\partial_{x_{j}}-i\partial_{y_{j}})

\displaystyle j=1,\ldots,n.

In this article, a $b$ -manifold is a smooth manifold $M$ equipped with a closed hypersurface $Z\subseteq M$ referred to as the singular locus. The $b$ -tangent bundle is the vector bundle ${{}^{b}TM}$ over $M$ whose smooth sections are the vector fields on $M$ that are tangent along $Z$ . These terminologies stem from the $b$ -geometry of Melrose [6], also called log-geometry.

Mendoza [7] defined a complex $b$ -structure on a $b$ -manifold $M$ to be an involutive subbundle ${{}^{b}T^{0,1}}M$ of the complexified $b$ -tangent bundle such that $\mathbb{C}{{}^{b}TM}={{}^{b}T^{1,0}}M\oplus{{}^{b}T^{0,1}}M$ , where ${{}^{b}T^{1,0}}M\coloneq\overline{{{}^{b}T^{0,1}}M}$ . Thus, complex $b$ -structures are defined exactly analogously to complex structures, replacing the tangent bundle by the $b$ -tangent bundle.

To complete the analogy, one would like there to be a Newlander-Nirenberg type result to the effect that all complex $b$ -structures are locally isomorphic to a standard model depending only dimension. Such a result would empower one to give a second (equivalent) definition of a complex $b$ -manifold in terms of an appropriately defined $b$ -holomorphic atlas. A logical candidate model for a complex $b$ -manifolds is $M=\mathbb{R}^{2n+2}=\mathbb{C}^{n+1}$ , $n\geq 0$ equipped with coordinates $(x_{0},y_{0},\ldots,x_{n},y_{n})=(z_{0},\ldots,z_{n})$ with ${{}^{b}T^{0,1}}M$ spanned by

	$\displaystyle{{}^{b}\partial}_{\overline{z}_{0}}$	$\displaystyle\coloneq\tfrac{1}{2}(x_{0}\partial_{x_{0}}+i\partial_{y_{0}})$
	$\displaystyle\partial_{\overline{z}_{j}}$	$\displaystyle\coloneq\tfrac{1}{2}(\partial_{x_{j}}+i\partial_{y_{j}})$	$\displaystyle j=1,\ldots,n.$

Accordingly, ${{}^{b}T^{1,0}}M$ is spanned by

	$\displaystyle{{}^{b}\partial}_{z_{0}}$	$\displaystyle\coloneq\tfrac{1}{2}(x_{0}\partial_{x_{0}}-i\partial_{y_{0}})$
	$\displaystyle\partial_{z_{j}}$	$\displaystyle\coloneq\tfrac{1}{2}(\partial_{x_{j}}-i\partial_{y_{j}})$	$\displaystyle j=1,\ldots,n.$

Our goal in this article is to show that, locally near points on the singular locus, every complex $b$ -manifold of dimension $2n+2$ is indeed isomorphic to the model example above. Mendoza [7] already took up this problem and partially settled it, up to deformation by terms whose Taylor expansions vanish along the singular locus. Indeed, Mendoza’s result, which we state below, will play a crucial role.

Theorem 1.1 ([7], Proposition 5.1).

With $n\geq 0$ , let $M$ be a $(2n+2)$ -dimensional complex $b$ -manifold with singular locus $Z$ . Then, every $p\in Z$ has an open neighbourhood $U$ on which there are smooth local coordinates $(x_{0},y_{0},\ldots,x_{n},y_{n})=(z_{0},\ldots,z_{n})$ centered at $p$ with $x_{0}$ vanishing on $Z$ and smooth, complex vector fields $\Gamma_{0},\ldots,\Gamma_{n}$ on $U$ vanishing to infinite order on $Z$ such that

	$\displaystyle L_{0}$	$\displaystyle\coloneq{{}^{b}\partial}_{\overline{z}_{0}}+\Gamma_{0}$
	$\displaystyle L_{j}$	$\displaystyle\coloneq\partial_{\overline{z}_{j}}+\Gamma_{j}$	$\displaystyle j=1,\ldots,n$

is a frame for ${{}^{b}T^{0,1}}M$ over $U$ . Moreover, each of $\Gamma_{0},\ldots,\Gamma_{n}$ is a linear combination of ${{}^{b}\partial}_{z_{0}},\partial_{z_{1}},\ldots,\partial_{z_{n}}$ whose coefficients are smooth, complex-valued functions on $U$ vanishing to infinite order on $Z$ .

The contribution of the present article will be to improve the above result by showing that the deformation terms $\Gamma_{0},\ldots,\Gamma_{n}$ can be got rid of. Thus, our main result is the following.

Theorem 1.2.

	$\displaystyle{{}^{b}\partial}_{\overline{z}_{0}}$	$\displaystyle\coloneq\tfrac{1}{2}(x_{0}\partial_{x_{0}}+i\partial_{y_{0}})$
	$\displaystyle\partial_{\overline{z}_{j}}$	$\displaystyle\coloneq\tfrac{1}{2}(\partial_{x_{j}}+i\partial_{y_{j}})$	$\displaystyle j=1,\ldots,n$

is a frame for ${{}^{b}T^{0,1}}M$ over $U$ .

Readers familiar with the techniques needed to prove the classical Newlander-Nirenberg theorem will be unsurprised that nonellipticity of the operator ${{}^{b}\partial}_{\overline{z}}=\frac{1}{2}(x\partial_{x}+i\partial_{y})$ is at the heart of the matter. The difficulty is that, without ellipticity, we do not automatically have good existence theory for solutions to deformations of this operator. The main insight of this article is that it is possible to get around this difficulty by leveraging the polar coordinate change $g:\mathbb{R}^{2}\to\mathbb{R}^{2}$ , $g(x,y)=xe^{iy}$ (negative radii permitted). Notice that ${{}^{b}\partial}_{\overline{z}}g=0$ , i.e. $g$ is “ $b$ -holomorphic”.

Let us summarize the contents of this paper. In Section 2, we collect definitions and basic results on complex $b$ -manifolds. In Section 3, we prove in the two-dimensional case that nontrivial $b$ -holomorphic functions exist locally. In Section 4, we use the latter existence result to prove the main result Theorem 1.2 in the two-dimensional case. Sections 5 and 6 extend the results of Sections 3 and 4 to the general case. Finally, in Section 7, we situate the singular coordinate change $g(x,y)=xe^{iy}$ in a more general context which will be relevant for planned future work on complex $b^{k}$ -manifolds (see [1] for the definition of a complex $b^{k}$ -manifold).

2 Background

The language of $b$ -geometry, also known as log-geometry, was introduced by Melrose [6]. One encounters two superficially different formulations of $b$ -geometry in the literature. In the original approach, one works on a manifold with boundary. Other authors instead use a manifold without boundary that is equipped with a given hypersurface [2]. We follow the latter approach.

Definition 2.1.

A $\mathbf{b}$ -manifold is a smooth manifold $M$ together with a closed hypersurface $Z\subseteq M$ which we refer to as the singular locus. An isomorphism of $b$ -manifolds is a diffeomorphism that preserves the singular loci.

Note ordinary smooth manifolds may be considered as $b$ -manifolds, taking $Z=\varnothing$ .

Definition 2.2.

A $\mathbf{b}$ -vector field on a $b$ -manifold $M=(M,Z)$ is a smooth vector field on $M$ that is tangent to $Z$ . The collection of $b$ -vector fields on $M$ is denoted ${{}^{b}\mathfrak{X}}(M)\subseteq\mathfrak{X}(M)$ .

Example 2.3.

Consider the $b$ -manifold $M=\mathbb{R}^{n+1}$ with coordinates $(x_{0},x_{1},\ldots,x_{n})$ and singular locus $Z=\{0\}\times\mathbb{R}^{n}$ . Then, ${{}^{b}\mathfrak{X}}(M)=\langle x_{0}\partial_{x_{0}},\partial_{x_{1}},\ldots,\partial_{x_{n}}\rangle$ , the free $C^{\infty}(M)$ -module generated by $x_{0}\partial_{x_{0}},\partial_{x_{1}},\ldots,\partial_{x_{n}}$ .

The example above completely captures the local structure of $b$ -manifolds. Accordingly, for any $b$ -manifold $M$ , ${{}^{b}\mathfrak{X}}(M)$ is a projective $C^{\infty}(M)$ -module closed under Lie bracket. Applying Serre-Swan duality to the inclusion ${{}^{b}\mathfrak{X}}(M)\to\mathfrak{X}(M)$ , one has a corresponding Lie algebroid ${{}^{b}TM}$ whose anchor map, denoted $\rho:{{}^{b}TM}\to TM$ , induces (abusing notation) an identification $C^{\infty}(M;{{}^{b}TM})={{}^{b}\mathfrak{X}}(M)$ .

Definition 2.4.

For a $b$ -manifold $M$ , the Lie algebroid ${{}^{b}TM}=({{}^{b}TM},\rho,[\cdot,\cdot])$ satisfying $C^{\infty}(M;{{}^{b}TM})={{}^{b}\mathfrak{X}}(M)$ described above is called the $\mathbf{b}$ -tangent bundle of $M$ .

In Example 2.3, $x_{0}\partial_{x_{0}},\partial_{x_{1}},\ldots,\partial_{x_{n}}$ is a global frame for ${{}^{b}TM}$ . The anchor map $\rho$ descends from evaluation of vector fields. Thus, over $M\setminus Z$ , $\rho$ is an isomorphism and, over $Z$ , the kernel of $\rho$ is the line bundle spanned by $x_{0}\partial_{x_{0}}$ . In general, for any $b$ -manifold $M$ , the bundles ${{}^{b}TM}$ and $TM$ are canonically isomorphic over $M\setminus Z$ via $\rho$ .

If $\theta:M_{1}\to M_{2}$ is an isomorphism of $b$ -manifolds, it is clear that $\theta$ preserves the associated modules of $b$ -vector fields. By Serre-Swan duality, $\theta$ induces a Lie algebroid isomorphism

\displaystyle{{}^{b}\theta}_{*}:{{}^{b}TM_{1}}\to{{}^{b}TM_{2}}.

(2.1)

Over $M_{i}\setminus Z_{i}$ , if we apply the aforementioned natural identifications of the $b$ -tangent bundle and usual tangent bundle, ${{}^{b}\theta_{*}}$ coincides with $\theta_{*}:TM_{1}\to TM_{2}$ , the usual induced isomorphism given by pushforward of tangent vectors.

Definition 2.5.

The $\mathbf{b}$ -cotangent bundle ${{}^{b}TM^{*}}$ of a $b$ -manifold $M$ is the dual of the $b$ -tangent bundle ${{}^{b}TM}$ .

The dual $\rho^{*}:TM^{*}\to{{}^{b}TM^{*}}$ of the anchor map $\rho:{{}^{b}TM}\to TM$ is likewise an isomorphism when restricted over $M\setminus Z$ . In particular, $\rho^{*}$ is injective on a dense open set, and so (what is equivalent) induces an injective mapping $C^{\infty}(M;TM^{*})\to C^{\infty}(M;{{}^{b}TM^{*}})$ . Put in simpler terms, a 1-form $\omega\in C^{\infty}(M;TM^{*})$ is fully determined by its pairing with $b$ -vector fields. The $\mathbf{b}$ -exterior derivative of a smooth function $f$ on $M$ is defined as:

\displaystyle{{}^{b}d}f\coloneq\rho^{*}(df).

(2.2)

Actually, one may quite legitimately regard $df$ and ${{}^{b}d}f$ as identical, with the latter notation merely hinting that one only intends to pair the form with $b$ -vector fields.

A general philosophy of $b$ -calculus is that many classical geometries admit $b$ -analogues in which the role of tangent bundle is played by the $b$ -tangent bundle. A notable example is $b$ -symplectic geometry ([4], [5]). Complex $b$ -geometry was introduced by Mendoza [7] who furthermore made a systematic study of the $b$ -Dolbeault complex. We remark that the basic idea of a complex $b$ -structure is mentioned in passing on pp. 218 of [6].

Definition 2.6.

A complex $\mathbf{b}$ -structure on an (even-dimensional) $b$ -manifold $M$ is a complex subbundle ${{}^{b}T^{0,1}}M$ of the complexified $b$ -tangent bundle satisfying:

(i)

$\mathbb{C}{{}^{b}TM}={{}^{b}T^{1,0}}M\oplus{{}^{b}T^{0,1}}M$ , where ${{}^{b}T^{1,0}}M\coloneq\overline{{{}^{b}T^{0,1}}M}$ ,
(ii)

${{}^{b}T^{0,1}}M$ is involutive.

A complex $\mathbf{b}$ -manifold is a $b$ -manifold equipped with a complex $b$ -structure. An isomorphism $\theta:M_{1}\to M_{2}$ of complex $b$ -manifolds is an isomorphism of their underlying $b$ -manifolds satisfying ${{}^{b}\theta_{*}}({{}^{b}T^{0,1}}M_{1})={{}^{b}T^{0,1}}M_{2}$ . Here, by abuse of notation, ${{}^{b}\theta_{*}}$ denotes the complexification of the isomorphism ${{}^{b}\theta_{*}}:{{}^{b}TM_{1}}\to{{}^{b}TM_{2}}$ defined above (2.1).

Because ${{}^{b}TM}$ is naturally isomorphic to $TM$ away from $Z$ , a $b$ -complex structure on $M$ in particular gives a complex structure in the usual sense on $M\setminus Z$ . In particular, one may consider ordinary complex manifolds as special cases of complex $b$ -manifolds with $Z=\varnothing$ .

Actually, for any complex $b$ -manifold $M$ , the restricted complex structure on $M\setminus Z$ determines the $b$ -complex structure, as the following proposition shows. Of course, not all complex structures on $M\setminus Z$ are the restriction of a (unique) $b$ -complex structure on $M$ , just those that degenerate in a particular way at $Z$ .

Proposition 2.7.

Let $M_{i}$ be a complex $b$ -manifold with singular locus $Z_{i}$ for $i=1,2$ . Suppose that $\theta:M_{1}\to M_{2}$ is a diffeomorphism such that:

(i)

$\theta(Z_{1})=Z_{2}$ (i.e. $\theta$ is an isomorphism of $b$ -manifolds),
(ii)

$\theta$ restricts to an isomorphism of (ordinary) complex manifolds $M_{1}\setminus Z_{1}\to M_{2}\setminus Z_{2}$ .

Then, $\theta$ is an isomorphism of complex $b$ -manifolds.

Proof.

From (i), we have an induced isomorphism on the complexified $b$ -tangent bundles ${{}^{b}\theta_{*}}:\mathbb{C}{{}^{b}T}M_{1}\to\mathbb{C}{{}^{b}TM_{2}}$ . From (ii), and applying the natural isomorphisms of $b$ -tangent bundle and ordinary tangent bundle away from the singular loci, the subbundles ${{}^{b}\theta_{*}}({{}^{b}T^{0,1}}M_{1})$ and ${{}^{b}T^{0,1}}M_{2}$ of $\mathbb{C}{{}^{b}TM_{2}}$ agree away from $Z_{2}$ . The conclusion follows by a continuity argument. ∎

Taking duals, the splitting $\mathbb{C}{{}^{b}TM}={{}^{b}T^{1,0}}M\oplus{{}^{b}T^{0,1}}M$ given by a complex $b$ -structure induces an associated splitting of the complexified $b$ -cotangent bundle $\mathbb{C}{{}^{b}TM^{*}}$ and of the complexified $b$ -exterior derivative ${{}^{b}d}:C^{\infty}(M,\mathbb{C})\to C^{\infty}(M;\mathbb{C}{{}^{b}TM^{*}})$ , as tabulated below:

	$\displaystyle\mathbb{C}{{}^{b}TM^{}}={{}^{b}T^{1,0}}M^{}\oplus{{}^{b}T^{0,1}}M^{*}$
	$\displaystyle{{}^{b}d}={{}^{b}\partial}+{{}^{b}{\overline{\partial}}}$
	$\displaystyle{{}^{b}\partial}:C^{\infty}(M,\mathbb{C})\to C^{\infty}(M;{{}^{b}T^{1,0}}M^{*})$
	$\displaystyle{{}^{b}{\overline{\partial}}}:C^{\infty}(M,\mathbb{C})\to C^{\infty}(M;{{}^{b}T^{0,1}}M^{*})$

In effect, ${{}^{b}\partial}f$ , respectively ${{}^{b}{\overline{\partial}}}f$ , is $df$ restricted to the $b$ - $(1,0)$ -vector fields, respectively the $b$ - $(0,1)$ -vector fields (that is to say, sections of ${{}^{b}T^{1,0}}M$ , respectively sections of ${{}^{b}T^{0,1}}M$ ).

Definition 2.8.

A $b$ -holomorphic function on a complex $b$ -manifold $M$ (or an open subset thereof) is a smooth function $f:M\to\mathbb{C}$ satisfying ${{}^{b}{\overline{\partial}}}f=0$ . We write ${{}^{b}\mathcal{O}}(M)$ for the collection of $b$ -holomorphic functions on $M$ .

In more concrete terms, $f$ is $b$ -holomorphic if $Xf=0$ for every $b$ - $(0,1)$ -vector field $X$ or (what is sufficient), all $X$ in a given frame for ${{}^{b}T^{0,1}}M$ . Because ${{}^{b}T^{0,1}}M$ is involutive, ${{}^{b}\mathcal{O}}(M)$ is a ring with respect to pointwise-multiplication.

Example 2.9.

Consider the $b$ -manifold $M=\mathbb{R}^{2}=\mathbb{C}$ with singular locus $Z=\{0\}\times\mathbb{R}$ . Then, ${{}^{b}\mathfrak{X}}(M)=\langle x\partial_{x},\partial_{y}\rangle$ , the free $C^{\infty}(\mathbb{R}^{2})$ -module generated by $x\partial_{x}$ and $\partial_{y}$ . An example of a complex $b$ -structure for $M$ has ${{}^{b}T^{0,1}}M$ spanned by

\displaystyle{{}^{b}\partial}_{\overline{z}}\coloneq\tfrac{1}{2}(x\partial_{x}+i\partial_{y}).

and, correspondingly, ${{}^{b}T^{1,0}}M$ spanned by

\displaystyle{{}^{b}\partial}_{z}\coloneq\tfrac{1}{2}(x\partial_{x}-i\partial_{y}).

A function $f$ is $b$ -holomorphic precisely when ${{}^{b}\partial}_{\overline{z}}f=0$ . An important globally-defined $b$ -holomorphic function on $M$ was mentioned in the introduction

\displaystyle g:\mathbb{R}^{2}\to\mathbb{C}

\displaystyle g(x,y)=xe^{iy}.

More generally, if $h$ is a usual holomorphic function defined near $0\in\mathbb{C}$ , then $f=h\circ g$ is a $b$ -holomorphic function defined near $0\in M$ . There also exist $b$ -holomorphic functions defined near $0$ not of the latter form and, indeed, not real-analytic near $0$ . For example, on the domain $-\frac{\pi}{4}<y<\frac{\pi}{4}$ , one has a $b$ -holomorphic $f$ defined by $f(x,y)=\exp(\frac{-1}{g(x,y)})$ for $x>0$ and $f(x,y)=0$ for $x\leq 0$ . These examples are also noted in [1], Example 8.1.

Remark 2.10.

Observe that the $b$ -holomorphic function $g(x,y)=xe^{iy}$ in the above example may be conceptualized as $g(x,y)=\phi_{iy}(x)$ , where $\phi_{t}(x)=e^{t}x$ is the flow of $x\partial_{x}$ . This somewhat cryptic remark will be expanded on in Section 7.

Example 2.11.

Consider the $b$ -manifold $M=\mathbb{R}^{2n+2}=\mathbb{C}^{n+1}$ with $Z=\{0\}\times\mathbb{R}^{2n+1}$ as its singular locus. Introduce coordinates $(x_{0},y_{0},\ldots,x_{n},y_{n})=(z_{0},\ldots,z_{n})$ . Thus,

\displaystyle{{}^{b}\mathfrak{X}}(M)=\langle x_{0}\partial_{x_{0}},\partial_{y_{0}},\partial_{x_{1}},\partial_{y_{1}},\ldots,\partial_{x_{n}},\partial_{y_{n}}\rangle.

An example of a complex $b$ -structure ${{}^{b}T^{0,1}}M$ for $M$ is the one spanned by:

	$\displaystyle{{}^{b}\partial}_{\overline{z}_{0}}$	$\displaystyle\coloneq\tfrac{1}{2}(x_{0}\partial_{x_{0}}+i\partial_{y_{0}})$
	$\displaystyle\partial_{\overline{z}_{j}}$	$\displaystyle\coloneq\tfrac{1}{2}(\partial_{x_{j}}+i\partial_{y_{j}})$	$\displaystyle j=1,\ldots,n.$

Correspondingly, a frame for ${{}^{b}T^{1,0}}M$ is:

	$\displaystyle{{}^{b}\partial}_{z_{0}}$	$\displaystyle\coloneq\tfrac{1}{2}(x_{0}\partial_{x_{0}}-i\partial_{y_{0}})$
	$\displaystyle\partial_{z_{j}}$	$\displaystyle\coloneq\tfrac{1}{2}(\partial_{x_{j}}-i\partial_{y_{j}})$	$\displaystyle j=1,\ldots,n.$

The function $g(x_{0},y_{0},\ldots,x_{n},y_{n})=x_{0}e^{iy_{0}}$ from Example 2.9 is still $b$ -holomorphic, as are the complex coordinate functions $z_{j}=x_{j}+iy_{j}$ , $j=1,\ldots,n$ . Additional $b$ -holomorphic functions may be obtained by applying an $(n+1)$ -variable holomorphic function (in the usual sense) to $g,z_{1},\ldots,z_{n}$ .

Example 2.12.

Consider the complex $b$ -manifold $N=\mathbb{R}^{2}$ with singular locus $Z\coloneq\{0\}\times\mathbb{R}$ and with ${{}^{b}T^{0,1}}N$ spanned by

L\coloneq\tfrac{1}{2}\Big{(}x\partial_{x}+i(-xy\partial_{x}+(1+y^{2})\partial_{y})\Big{)}.

This complex $b$ -manifold $N$ is isomorphic to a neighbourhood of the origin of the complex $b$ -manifold $M=\mathbb{R}^{2}$ with ${{}^{b}T^{0,1}}M$ spanned by ${{}^{b}\partial}_{\overline{z}}\coloneq\frac{1}{2}(x\partial_{x}+i\partial_{y})$ from Example 2.9. Indeed, the diffeomorphism $G:\mathbb{R}\times(-\frac{\pi}{2},\frac{\pi}{2})\to\mathbb{R}^{2}$ defined by $G(x,y)=(x\cos y,\tan y)$ satisfies $G_{*}({{}^{b}\partial}_{\overline{z}})=L$ . Thus, $b$ -holomorphic functions on $N$ may be obtained by pushforward through $G$ . For example, the pushforward of $g(x,y)=xe^{iy}$ by $G$ is $p(x,y)=x+ixy$ which indeed satisfies $Lp=0$ .

For a general complex $b$ -manifold $M$ , it is not immediately clear whether, locally near points on the singular locus $Z$ , any nonconstant $b$ -holomorphic functions must exist at all. This article will confirm that they do (Sections 3 and 5).

Note that a $b$ -holomorphic function on a complex $b$ -manifold $M$ restricts to a holomorphic function in the usual sense on the complex manifold $M\setminus Z$ . Indeed, in a similar spirit to Proposition 2.7, if $f$ is smooth and restricts to a holomorphic function on $M\setminus Z$ , then it is $b$ -holomorphic on $M$ by a continuity argument.

Recall that, in the case of ordinary complex manifolds (defined using integrable almost-complex structures), the existence of local holomorphic functions is closely-tied to the existence of charts. The proposition below, which will be used in Sections 4 and 6, illustrates this principle. We include its (standard) proof for the sake of completeness.

Proposition 2.13.

Let $M$ be a complex manifold. Let $\Omega\subseteq M$ and $\Omega^{\prime}\subseteq\mathbb{C}^{n}$ be open. Suppose $f_{1},\ldots,f_{n}:\Omega\to\mathbb{C}$ are holomorphic functions such that $\theta:\Omega\to\Omega^{\prime}$ , $\theta(p)=(f_{1}(p),\ldots,f_{n}(p))$ defines a diffeomorphism. Then, $\theta_{*}$ maps $T^{0,1}M$ onto $T^{0,1}\mathbb{C}^{n}$ over $\Omega$ where, by definition, the latter is spanned by $\partial_{\overline{z}_{j}}\coloneq\frac{1}{2}(\partial_{x_{j}}+i\partial_{y_{j}})$ , $j=1,\ldots,n$ .

Proof.

Let $L\in C^{\infty}(\Omega;T^{0,1}M)$ be a $(0,1)$ -vector field defined on $\Omega$ . Using $Lf_{j}=0$ for $j=1,\ldots,n$ and $\theta_{*}(f_{j})=z_{j}$ (the $j$ th coordinate function) for $j=1,\ldots,n$ , we obtain

\theta_{*}(L)z_{j}=0.

If we write $\theta_{*}(L)=\sum_{j=1}^{n}\alpha_{j}\partial_{x}+\beta_{j}\partial_{y}$ , where $\alpha_{j},\beta_{j}$ are smooth, complex-valued functions on $\mathbb{C}$ , then the above gives $\alpha_{j}+i\beta_{j}=0$ for $j=1,\ldots,n$ . Thus, we obtain

\theta_{*}(L)=\sum_{j=1}-2i\beta_{j}\partial_{\overline{z}_{j}},

so that $\theta_{*}(L)$ is a $(0,1)$ vector field on $\Omega^{\prime}$ . ∎

In the case of complex $b$ -manifolds, $b$ -holomorphic functions cannot directly play the role of coordinate functions of charts. To see this, note that any solution $f:\mathbb{R}^{2}\to\mathbb{C}$ of ${{}^{b}\partial}_{\overline{z}}f=0$ (where ${{}^{b}\partial}_{\overline{z}}\coloneq\frac{1}{2}(x\partial_{x}+i\partial y)$ ) is necessarily constant along $Z=\{0\}\times\mathbb{R}$ .

3 Existence of $b$ -holomorphic functions: dimension $2$

In this section, we show that, near any point on the singular locus of a two-dimensional complex $b$ -manifold, there is a nontrivial $b$ -holomorphic function. Here, “nontrivial” means the derivative normal to the singular locus is nonzero. From Theorem 1.1, this amounts to the following (as usual ${{}^{b}\partial}_{z}\coloneq\frac{1}{2}(x\partial_{x}-i\partial_{y})$ , ${{}^{b}\partial}_{\overline{z}}\coloneq\frac{1}{2}(x\partial_{x}+i\partial_{y})$ ).

Theorem 3.1.

Consider a complex $b$ -manifold $M=\mathbb{R}^{2}$ with singular locus $Z=\{0\}\times\mathbb{R}$ and with ${{}^{b}T^{0,1}}M$ spanned by

\displaystyle L\coloneq{{}^{b}\partial}_{\overline{z}}+\gamma\cdot{{}^{b}\partial}_{z}

where $\gamma:\mathbb{R}^{2}\to\mathbb{C}$ is a smooth function vanishing to infinite order on $Z$ . Then, there is a neighbourhood $U\subseteq M$ of $(0,0)$ and a $b$ -holomorphic function $f:U\to\mathbb{C}$ that vanishes on $Z$ and satisfies $\partial_{x}f(0,0)=1$ .

The proof of Theorem 3.1 will rely on the following three elementary lemmas whose proofs we omit.

Firstly, we record several vector field pushforward formulae for the singular coordinate change $g(x,y)=xe^{iy}$ . Note $g$ is simply the polar coordinate transformation (with negative radii permitted). It is a local diffeomorphism away from $Z=\{0\}\times\mathbb{R}$ . We treat a more general type of coordinate change in Section 7, providing additional context for the methods of this section and rendering them somewhat less ad hoc.

Lemma 3.2.

The smooth surjection $g:\mathbb{R}^{2}\to\mathbb{C}$ defined by $g(x,y)=xe^{iy}$ satisfies:

	$\displaystyle g_{*}(x\partial_{x})$	$\displaystyle=x\partial_{x}+y\partial_{y}$
	$\displaystyle g_{*}(\partial_{y})$	$\displaystyle=-y\partial_{x}+x\partial_{y}$
	$\displaystyle g_{*}({{}^{b}\partial}_{\overline{z}})$	$\displaystyle=\overline{z}\partial_{\overline{z}}$
	$\displaystyle g_{*}({{}^{b}\partial}_{z})$	$\displaystyle=z\partial_{z}.$

Here, ${{}^{b}\partial}_{\overline{z}}\coloneq\frac{1}{2}(x\partial_{x}+i\partial_{y})$ , ${{}^{b}\partial}_{z}\coloneq\frac{1}{2}(x\partial_{x}-i\partial_{y})$ , $\partial_{\overline{z}}\coloneq\frac{1}{2}(\partial_{x}+i\partial_{y})$ , $\partial_{z}\coloneq\frac{1}{2}(\partial_{x}-i\partial_{y})$ . ∎

Secondly, we need the following fact concerning the expression in polar coordinates of plane functions vanishing to infinite order at the origin.

Lemma 3.3.

Define $g:\mathbb{R}^{2}\to\mathbb{C}$ by $g(x,y)=xe^{iy}$ and $\sigma:\mathbb{R}^{2}\to\mathbb{R}^{2}$ by $\sigma(x,y)=(-x,y+\pi)$ . Let $\gamma:\mathbb{R}^{2}\to\mathbb{C}$ be a $\sigma$ -invariant (i.e. $\gamma\circ\sigma=\gamma$ ) smooth function that vanishes to infinite order on $Z\coloneq\{0\}\times\mathbb{R}$ . Then, the pushforward $g_{*}(\gamma)$ is a well-defined smooth function on $\mathbb{C}$ that vanishes to infinite order at $0$ . ∎

Thirdly, we need the following divisibility property of plane functions vanishing to infinite order at the origin.

Lemma 3.4.

Let $\gamma$ be a smooth, complex-valued function on $\mathbb{C}$ vanishing to infinite order at $0$ . Then, $(1/z)\gamma$ and $(1/\overline{z})\gamma$ extend to smooth, complex-valued functions on $\mathbb{C}$ vanishing to infinite order at $0$ .∎

We now prove the main result of this section.

Proof of Theorem 3.1.

Since we are only looking for a local solution, there is no harm in assuming $\gamma$ satisfies the periodicity assumption $\gamma\circ\sigma=\gamma$ of Lemma 3.3. Indeed, we may take $\gamma$ to be compactly-supported in $(-\frac{\pi}{2},\frac{\pi}{2})\times\mathbb{R}$ , which is a fundamental domain for $\sigma$ , and extend it periodically. This assumption makes the pushforward of $L$ by $g(x,y)=xe^{iy}$ well-defined, as the following computation shows.

	$\displaystyle g_{*}(L)$	$\displaystyle=g_{*}({{}^{b}\partial}_{\overline{z}}+\gamma\cdot{{}^{b}\partial}_{z})$
		$\displaystyle=\overline{z}\partial_{\overline{z}}+g_{*}(\gamma)z\partial_{z}$	(Lemmas 3.2 and 3.3).

Because $g$ is a local diffeomorphism $\mathbb{R}^{2}\setminus Z\to\mathbb{C}\setminus\{0\}$ , we have that $g_{*}(L)$ defines a complex structure on $\mathbb{C}\setminus\{0\}$ . Furthermore, using Lemma 3.4, we can write $g_{*}(\gamma)=\overline{z}\gamma^{\prime}$ where $\gamma^{\prime}$ is another smooth, complex-valued function on $\mathbb{C}$ vanishing to infinite order at $0$ . Thus,

g_{*}(L)=\overline{z}(\partial_{\overline{z}}+\gamma^{\prime}z\partial_{z}).

But now, note that $\partial_{\overline{z}}+\gamma^{\prime}z\partial_{z}$ defines an (ordinary) complex structure on all of $\mathbb{C}$ (because the $\gamma^{\prime}z\partial_{z}$ term vanishes at $0$ ). So, by the ordinary Newlander-Nirenberg theorem for dimension two, there exists a $C^{\infty}$ , complex-valued function $h$ defined near $0\in\mathbb{C}$ such that

(\partial_{\overline{z}}+\gamma^{\prime}z\partial_{z})h=0

and also satisfying $h(0)=0$ and $\partial_{x}h(0)=1$ . Putting $f=h\circ g$ completes the proof. ∎

4 Proof of main result for dimension $2$

This section is devoted to the proof of Theorem 1.2 in the two-dimensional case ( $n=0$ , in the notation of Theorem 1.2). Thanks to Mendoza’s Theorem 1.1, we may begin on a complex $b$ -manifold $M=\mathbb{R}^{2}$ with singular locus $Z=\mathbb{R}\times\{0\}$ and ${{}^{b}T^{0,1}}M$ spanned by

\displaystyle L\coloneq{{}^{b}\partial}_{\overline{z}}+\gamma\cdot{{}^{b}\partial}_{z}

where $\gamma:\mathbb{R}^{2}\to\mathbb{C}$ is a smooth function vanishing to infinite order on $Z$ . Our task is to find new coordinates near $(0,0)$ in which ${{}^{b}T^{0,1}}M$ is spanned by ${{}^{b}\partial}_{\overline{z}}$ .

From Theorem 3.1, there is an open set $U\subseteq\mathbb{R}^{2}$ containing $(0,0)$ and a smooth function $f:U\to\mathbb{C}$ such that:

(i)

$Lf=0$ ( $b$ -holomorphicity),
(ii)

$f$ vanishes on $Z$ ,
(iii)

$f_{x}(0,0)=1$ (denoting partial derivatives by subscripts for brevity).

We split $f$ into its real and imaginary parts

f=u+iv=(u,v).

(recall $\mathbb{C}=\mathbb{R}^{2}$ in this article). The key will be the local coordinate change $F$ defined, roughly speaking, by $(u,\frac{v}{u})$ . In more precise terms, from (ii), we may write

\displaystyle u(x,y)=x\cdot a(x,y)

\displaystyle v(x,y)=x\cdot b(x,y)

where $a$ and $b$ are smooth, real-valued functions on $U$ . From (iii), we have

\displaystyle a(0,0)=u_{x}(0,0)=1

\displaystyle b(0,0)=v_{x}(0,0)=0.

Shrinking $U$ , we may assume $a$ is nowhere-vanishing on $U$ and thus define

\displaystyle F:U\to\mathbb{R}^{2}

\displaystyle F(x,y)=\left(u(x,y),\frac{b(x,y)}{a(x,y)}\right).

Claim 4.1.

After possibly further shrinking $U$ around $(0,0)$ , one has:

(a)

$F$ is a diffeomorphism from $U$ onto an open set $F(U)\subseteq\mathbb{R}^{2}$ ,
(b)

$F(0,0)=(0,0)$ and $F(U\cap Z)=F(U)\cap Z$ ,
(c)

$f=\kappa\circ F$ , where $\kappa:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is given by $\kappa(x,y)=(x,xy)$ .

Proof.

Statement (c) is a direct consequence of the definition of $F$ . For statement (b), since $u$ vanishes on $Z$ and $u_{x}(0,0)=1$ , we may, after shrinking $U$ , assume $u^{-1}(0)\cap U\subseteq Z$ . For statement (a), it suffices to show the Jacobian of $F$ at $(0,0)$ is invertible. Then, by the inverse function theorem, after possibly shrinking $U$ again, $F$ gives a diffeomorphism $U\to F(U)$ . We claim that the Jacobian of $F$ at $(0,0)$ has the form $\begin{bmatrix}1&0\\ *&1\end{bmatrix}$ . Obviously the top row is correct, so we need only confirm that $\partial_{y}\Big{(}\frac{b}{a}\Big{)}(0,0)=1$ . Away from $Z$ , we have

\displaystyle\partial_{y}\Big{(}\frac{b}{a}\Big{)}=\partial_{y}\Big{(}\frac{v}{u}\Big{)}=\frac{v_{y}u-vu_{y}}{u^{2}}.

Collecting real and imaginary parts in $Lf=0$ we have

\displaystyle xu_{x}-v_{y}\sim 0

\displaystyle xv_{x}+u_{y}\sim 0,

where $\sim$ denotes agreement of Taylor series at $(0,0)$ . Thus,

\displaystyle v_{y}u-vu_{y}\sim(xu_{x})u+v(xv_{x})=x^{2}(u_{x}a+v_{x}b)

from which it follows that

\displaystyle\partial_{y}\Big{(}\frac{b}{a}\Big{)}\sim\frac{u_{x}a+v_{x}b}{a^{2}}.

In particular, $\partial_{y}\Big{(}\frac{b}{a}\Big{)}(0,0)=\frac{(1)(1)+(0)(0)}{(1)^{2}}=1$ , as was claimed. ∎

It will be relevant to see what the above constructions amount to in the model case when $\gamma=0$ and $L$ is simply ${{}^{b}\partial}_{\overline{z}}=\frac{1}{2}(x\partial_{x}+i\partial_{y})$ . Then, in place of $f$ , we may use our function

g(x,y)=xe^{iy}=(x\cos y,x\sin y).

With this replacement, the diffeomorphism $F$ above becomes the diffeomorphism $G$ appearing in Example 2.12

\displaystyle G:\mathbb{R}\times(-\tfrac{\pi}{2},\tfrac{\pi}{2})\to\mathbb{R}^{2}

\displaystyle G(x,y)=(x\cos y,\tan y).

Define $W\coloneq F(U)$ and $V\coloneq G^{-1}(W)$ , so that we have $b$ -manifold isomorphisms

Applying the induced maps ${{}^{b}F_{*}},{{}^{b}G_{*}}$ on $b$ -tangent bundles (see (2.1) in Section 2), we arrive at two complex $b$ -structures on $(W,W\cap Z)$ which can be compared.

Claim 4.2.

The complex $b$ -structures on $W$ spanned ${{}^{b}F_{*}}(L)$ and ${{}^{b}G_{*}}({{}^{b}\partial}_{\overline{z}})$ are equal.

Proof.

Referring to Proposition 2.7, it suffices to check this away from the singular loci. Thus, we need only check that the usual pushforwards $F_{*}(L)$ and $G_{*}({{}^{b}\partial}_{\overline{z}})$ coincide up to a smooth rescaling over $W\setminus Z$ . Note $\kappa(x,y)=(x,xy)$ restricts to a diffeomorphism of $\mathbb{R}^{2}\setminus Z$ and recall $f=\kappa\circ F$ on $U$ , $g=\kappa\circ G$ on $V$ . Thus, we have diffeomorphisms

and it suffices to check that $(f|_{U\setminus Z})_{*}(L)$ and $(g|_{V\setminus Z})_{*}({{}^{b}\partial}_{\overline{z}})$ agree up to a smooth rescaling. Indeed, since $f$ and $g$ are holomorphic in the usual sense away from $Z$ , Proposition 2.13 shows that $(f|_{U\setminus Z})_{*}(L)$ and $(g|_{V\setminus Z})_{*}({{}^{b}\partial}_{\overline{z}})$ are rescalings of $\partial_{\overline{z}}=\frac{1}{2}(\partial_{x}+i\partial_{y})$ (actually, from Lemma 3.2, we even know $g_{*}({{}^{b}\partial}_{\overline{z}})=\overline{z}\partial_{\overline{z}}$ ). ∎

The preceding claim completes this section’s proof of Theorem 1.2 in the two-dimensional case. The desired local coordinates are provided by $G^{-1}\circ F:U\to V$ .

5 Existence of $b$ -holomorphic functions: general case

In this section, we extend the results of Section 3 to the general case. That is, we show that nontrivial $b$ -holomorphic functions exist near points on the singular locus of any complex $b$ -manifold. By Theorem 1.1, this amounts to the following.

Theorem 5.1.

Consider a complex $b$ -manifold $M=\mathbb{C}^{n+1}=\mathbb{R}^{2n+2}$ with singular locus $Z=\{0\}\times\mathbb{R}^{2n+1}$ equipped with coordinates $(x_{0},y_{0},\ldots,x_{n},y_{n})=(z_{0},\ldots,z_{n})$ with complex $b$ -structure ${{}^{b}T^{0,1}}M$ spanned by

	$\displaystyle L_{0}$	$\displaystyle\coloneq{{}^{b}\partial}_{\overline{z}}+\Gamma_{0}$
	$\displaystyle L_{j}$	$\displaystyle\coloneq\partial_{\overline{z}_{j}}+\Gamma_{j}$	$\displaystyle j=1,\ldots,n$

where $\Gamma_{0},\ldots,\Gamma_{n}$ are linear combinations of ${{}^{b}\partial}_{z_{0}},\partial_{z_{1}},\ldots,\partial_{z_{n}}$ with coefficients in smooth, complex-valued functions on $M$ vanishing to infinite order on $Z$ . Then, there exists an open set $U\subseteq M$ containing $0$ and $b$ -holomorphic functions $f_{0},f_{1},\ldots,f_{n}:U\to\mathbb{C}$ such that:

(i)

$f_{0}$ vanishes on $Z$ and $f_{j}(0)=0$ for $j=1,\ldots,n$ ,
(ii)

$\partial_{x_{j}}f_{k}(0)=\delta_{j,k}$ for $j,k\in\{0,1,\ldots,n\}$ , using Kronecker delta notation.

The proof is largely the same as Theorem 3.1, so we will be somewhat brief. First, we state straightforward higher-dimensional generalizations of Lemmas 3.3 and 3.4. Again, we omit the proofs of these statements.

Lemma 5.2.

Define

	$\displaystyle g$	$\displaystyle:\mathbb{R}^{2}\to\mathbb{C}$	$\displaystyle g(x,y)$	$\displaystyle=xe^{iy}$	$\displaystyle\widetilde{g}$	$\displaystyle:\mathbb{R}^{2n+2}\to\mathbb{C}^{n+1}$	$\displaystyle\widetilde{g}=g\times\mathrm{id}$
	$\displaystyle\sigma$	$\displaystyle:\mathbb{R}^{2}\to\mathbb{R}^{2}$	$\displaystyle\sigma(x,y)$	$\displaystyle=(-x,y+\pi)$	$\displaystyle\widetilde{\sigma}$	$\displaystyle:\mathbb{R}^{2n+2}\to\mathbb{R}^{2n+2}$	$\displaystyle\widetilde{\sigma}=\sigma\times\mathrm{id}$

Let $\gamma:\mathbb{R}^{2n+2}\to\mathbb{C}$ be a $\widetilde{\sigma}$ -invariant smooth function that vanishes to infinite order on $Z\coloneq\{0\}\times\mathbb{R}^{2n+1}$ . Then, the pushforward $\widetilde{g}_{*}(\gamma)$ is a well-defined smooth function on $\mathbb{C}^{n+1}$ that vanishes to infinite order on $Z^{\prime}\coloneq\{0\}\times\mathbb{C}^{n}$ . ∎

Corollary 5.3.

Suppose $\gamma_{0},\ldots,\gamma_{n}:\mathbb{R}^{2n+2}\to\mathbb{C}$ are $\widetilde{\sigma}$ -invariant smooth functions vanishing infinite order on $Z=\{0\}\times\mathbb{R}^{2n+1}$ . Define $\Gamma=\gamma_{0}\cdot{{}^{b}\partial}_{\overline{z}_{0}}+\sum_{j=1}^{n}\gamma_{j}\cdot\partial_{\overline{z}_{j}}$ (note $\Gamma$ is $\widetilde{\sigma}$ -invariant in the sense that $\widetilde{\sigma}_{*}(\Gamma)=\Gamma$ ). Then, the pushforward $\widetilde{g}_{*}(\Gamma)$ is a well-defined smooth vector field on $\mathbb{C}^{n+1}$ vanishing to infinite order on $Z^{\prime}=\{0\}\times\mathbb{C}^{n}$ .

Proof.

From Lemma 3.2 and Lemma 5.2, we have $\widetilde{g}_{*}(\Gamma)=\widetilde{g}_{*}(\gamma_{0})\overline{z}_{0}\partial_{\overline{z}_{0}}+\sum_{j=1}^{n}\widetilde{g}_{*}(\gamma_{j})\partial_{\overline{z}_{j}}$ . ∎

Lemma 5.4.

Let $\mathbb{C}^{n+1}$ have coordinates $(z_{0},\ldots,z_{n})$ . Suppose $\gamma$ is a smooth, complex-valued function on $\mathbb{C}^{n+1}$ vanishing to infinite order on $Z^{\prime}\coloneq\{0\}\times\mathbb{C}^{n}$ . Then, $(1/z_{0})\gamma$ and $(1/\overline{z}_{0})\gamma$ extend to smooth, complex-valued functions on $\mathbb{C}^{n+1}$ vanishing to infinite order on $Z^{\prime}$ . ∎

Having made the above preparations, we now proceed to the proof of this section’s main result.

Proof of Theorem 5.1.

As in the proof of Lemma 3.1, since we only desire local $b$ -holomorphic functions, there is no harm assuming the deformations terms $\Gamma_{j}$ to be $\widetilde{\sigma}$ -invariant. With this assumption, Lemma 3.2 and Corollary 5.3 together imply that the vector fields $L_{j}$ and $\Gamma_{j}$ have well-defined pushforwards by $\widetilde{g}$ . Indeed,

	$\displaystyle\widetilde{g}_{*}(L_{0})$	$\displaystyle=\overline{z}_{0}\partial_{\overline{z}_{0}}+\widetilde{g}_{*}(\Gamma_{0})$
	$\displaystyle\widetilde{g}_{*}(L_{j})$	$\displaystyle=\partial_{\overline{z}_{j}}+\widetilde{g}_{*}(\Gamma_{j})$	$\displaystyle j=1,\ldots,n,$

where $\widetilde{g}_{*}(\Gamma_{j})$ is a smooth vector field on $\mathbb{C}^{n+1}$ vanishing to infinite order on $Z^{\prime}\coloneq\{0\}\times\mathbb{C}^{n}$ for $j=0,\ldots,n$ .

Because $\widetilde{g}$ is a local diffeomorphism $\mathbb{R}^{2n+2}\setminus Z\to\mathbb{C}^{n+1}\setminus Z^{\prime}$ and bracket commutes with pushforward, $\widetilde{g}_{*}(L_{j})$ , $j=0,\ldots,n$ at least define a complex structure on $\mathbb{C}^{n+1}\setminus Z^{\prime}$ . Going further, we can use Lemma 5.4 to write $\widetilde{g}_{*}(\Gamma_{0})=\overline{z}_{0}\Gamma_{0}^{\prime}$ , where $\Gamma_{0}^{\prime}$ is another smooth vector field on $\mathbb{C}^{n+1}$ vanishing to infinite order on $Z^{\prime}$ . We claim that

	$\displaystyle\tfrac{1}{\overline{z}_{0}}\widetilde{g}_{*}(L_{0})$	$\displaystyle=\partial_{\overline{z}_{0}}+\Gamma_{0}^{\prime}$			(5.1)
	$\displaystyle\widetilde{g}_{*}(L_{j})$	$\displaystyle=\partial_{\overline{z}_{j}}+\widetilde{g}_{*}(\Gamma_{j})$	$\displaystyle j=1,\ldots,n$		(5.2)

define an ordinary complex structure on the whole of $\mathbb{C}^{n+1}$ (extending the aforementioned complex structure on $\mathbb{C}^{n+1}\setminus Z^{\prime}$ ). Indeed, the deformation terms vanish on $Z^{\prime}$ , so the vector fields (5.1) together with their complex conjugates, form a global frame for the complexified tangent bundle. By a simple computation using the derivation property of the Lie bracket, involutivity holds away from $Z^{\prime}$ , hence everywhere by a continuity argument.

Now, by an application of the classical Newlander-Nirenberg theorem, there is an open neighbourhood $U\subseteq\mathbb{C}^{n+1}$ of $0$ and smooth functions $h_{0},\ldots,h_{n}:U\to\mathbb{C}$ that are holomorphic for the complex structure defined by (5.1) and furthermore satisfy $h_{j}(0)=0$ , $\partial_{x_{j}}h_{k}(0)=\delta_{j,k}$ for $j,k\in\{0,1,\ldots,n\}$ . In fact, again using the vanishing of the deformation terms along $Z^{\prime}$ , the complex structure on $\mathbb{C}^{n+1}$ defined by (5.1) is such that $Z^{\prime}=\{0\}\times\mathbb{C}^{n}$ sits as a complex submanifold. Indeed, the inherited complex structure on $Z^{\prime}$ is its standard one spanned by $\partial_{\overline{z}_{1}},\ldots,\partial_{\overline{z}_{n}}$ . Using the implicit function theorem for several complex variables, complex submanifolds of complex codimension-1 can locally be written as preimages of regular values of holomorphic functions. Thus, possibly shrinking $U$ , we may even choose $h_{0}$ to vanish on $Z^{\prime}$ . Putting $f_{j}=h_{j}\circ\widetilde{g}$ for $j=0,1,\ldots,n$ completes the proof of the result. ∎

6 The $b$ -Newlander-Nirenberg theorem: general case

This section is devoted to the proof of our main result Theorem 1.2. In light of Mendoza’s Theorem 1.1, we may begin on a complex $b$ -manifold $M=\mathbb{C}^{n+1}=\mathbb{R}^{2n+2}$ with singular locus $Z=\{0\}\times\mathbb{R}^{2n+1}$ equipped with coordinates $(x_{0},y_{0},\ldots,x_{n},y_{n})=(z_{0},\ldots,z_{n})$ with complex $b$ -structure ${{}^{b}T^{0,1}}M$ spanned by

	$\displaystyle L_{0}$	$\displaystyle\coloneq{{}^{b}\partial}_{\overline{z}}+\Gamma_{0}$
	$\displaystyle L_{j}$	$\displaystyle\coloneq\partial_{\overline{z}_{j}}+\Gamma_{j}$	$\displaystyle j=1,\ldots,n$

where $\Gamma_{0},\ldots,\Gamma_{n}$ are smooth, complex vector fields on $M$ vanishing to infinite order on $Z$ .

By Theorem 5.1, there is an open set $U\subseteq\mathbb{R}^{2n+2}$ containing $0$ and smooth functions $f_{0},\ldots,f_{n}:U\to\mathbb{C}$ satisfying:

(i)

$L_{j}f_{k}=0$ for $j,k\in\{0,1,\ldots,n\}$ ( $b$ -holomorphicity),
(ii)

$f_{0}$ vanishes on $Z$ and $f_{j}(0)=0$ for $j=1,\ldots,n$ ,
(iii)

$\partial_{x_{j}}f_{k}(0)=\delta_{j,k}$ for $j,k\in\{0,1,\ldots,n\}$ .

As in Section 4, we decompose $f_{0}$ into real and imaginary parts $f_{0}=u+iv$ . Since $u$ and $v$ vanish on $Z$ , we may write $u=x_{0}\cdot a$ and $v=x_{0}\cdot b$ , where $a$ and $b$ are real-valued smooth functions. As in Section 4, by shrinking $U$ , we may enforce that $a$ is nowhere vanishing on $U$ and that:

(a)

$F=(a,\tfrac{b}{a},f_{1},\ldots,f_{n})$ is a diffeomorphism from $U$ onto an open set $F(U)\subseteq\mathbb{R}^{2n+2}$ ,
(b)

$F(0)=0$ and $F(U\cap Z)=F(U)\cap Z$ ,
(c)

$f=\widetilde{\kappa}\circ F$ where $f=(f_{0},f_{1},\ldots,f_{n})$ and $\widetilde{\kappa}=\kappa\times\mathrm{id}:\mathbb{R}^{2n+2}\to\mathbb{R}^{2n+2}$ where $\kappa:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is $\kappa(x,y)=(x,xy)$ .

As in Section 4, we argue that the pushforward by $F$ of the complex $b$ -structure on $U$ spanned by $L_{0},\ldots,L_{n}$ is independent of the deformation terms $\Gamma_{0},\ldots,\Gamma_{n}$ . Again, by Proposition 2.7, if suffices check this away from the singular locus $Z$ . Using the fact that $\widetilde{\kappa}$ restricts to a self-diffeomorphism of $\mathbb{R}^{2n+2}\setminus Z$ , we have that $f$ defines a diffeomorphism of $U\setminus Z$ onto $\kappa(F(U\setminus Z))$ . Because the components of $f$ are holomorphic for the induced complex structure on $U\setminus Z$ , Proposition 2.13 implies that $f|_{U\setminus Z}$ pushes forward the subbundle spanned by $L_{0},\ldots,L_{n}$ to the subbundle spanned by $\partial_{\overline{z}_{0}},\ldots,\partial_{\overline{z}_{n}}$ . Thus, the pushforward by $F|_{U\setminus Z}$ of the subbundle spanned by $L_{0},\ldots,L_{n}$ is completely determined to be the pullback of the standard complex structure on $\kappa(F(U\setminus Z))$ by $\kappa$ . Running the same argument in the case where the deformation terms are all zero, we obtain the desired coordinate change.

7 Remarks on singular pushforwards

The polar coordinate change $g:\mathbb{R}^{2}\to\mathbb{C}$ , $g(x,y)=xe^{iy}$ played a crucial role in this article by enabling us to relate the nonelliptic operator ${{}^{b}\partial}_{z}=\frac{1}{2}(x\partial_{x}-i\partial_{y})$ to the standard complex partial derivative operator $\partial_{z}=\frac{1}{2}(\partial_{x}-i\partial_{y})$ by way of the elementary, but perhaps somewhat mysterious, pushforward formula $g_{*}({{}^{b}\partial}_{z})=z\partial_{z}$ . In this section, we shed additional light on this pushforward formula by placing it in a more general context. The material in this section will also play a role in planned future work on the $b$ -Newlander-Nirenberg theorem for complex $b^{k}$ -manifolds [1].

Suppose that, more generally, we wish to relate the vector field

\displaystyle\tfrac{1}{2}(x^{k}\partial_{x}-i\partial_{y})

to $\partial_{z}$ , for $k$ a positive integer. More generally still, suppose we wish to relate

\displaystyle\tfrac{1}{2}(f(x)\partial_{x}-i\partial_{y})

to $\partial_{z}$ , where $f(z)$ is an entire function on $\mathbb{C}$ whose restriction $f(x)$ to $\mathbb{R}$ is real-valued.

We first recall a few more-or-less notational points concerning holomorphic vector fields and imaginary-time flows. One may refer to [3], pp. 39 for additional details. Let $f$ be a holomorphic function on $\mathbb{C}$ so that $V\coloneq f(z)\partial_{z}$ is a holomorphic vector field on $\mathbb{C}$ . We may decompose $V$ as

V=\tfrac{1}{2}(X-iJX),

where the real vector fields $X=\mathrm{Re}(f)\partial_{x}+\mathrm{Im}(f)\partial_{y}$ and $JX=-\mathrm{Im}(f)\partial_{x}+\mathrm{Re}(f)\partial_{y}$ commute with one another. We may then define the complex-time flow of $V$ by

\displaystyle\phi^{V}_{w}(z)\coloneq\phi^{X}_{s}\circ\phi^{JX}_{t}(z)

\displaystyle\text{where }w=s+it,

with the usual caveat that the flow need only be defined for $w$ sufficiently close to $0$ .

This flow is jointly holomorphic in $w$ and $z$ . One practical consequence of this joint holomorphicity is that, if $f(z)$ is real on the $x$ -axis, so that $X$ coincides with $f(x)\partial_{x}$ on the $x$ -axis, then the complex-time flow of the holomorphic vector field $f(z)\partial_{z}$ may be obtained by analytic continuation in both variables of the real-time flow of the one-dimensional vector field $f(x)\partial_{x}$ .

Example 7.1.

The real-time flow of the one-dimensional vector field $X_{1}\coloneq x\partial_{x}$ is given by $\phi^{X_{1}}_{t}(x)=e^{t}x$ . Accordingly, the complex-time flow of the holomorphic vector field $V_{1}\coloneq z\partial_{z}$ is given by $\phi^{V_{1}}_{w}(z)=e^{w}z$ .

Example 7.2.

The real-time flow of the one-dimensional vector field $X_{2}\coloneq x^{2}\partial_{x}$ is given by $\phi^{X_{2}}_{t}(x)=\frac{x}{1-tx}$ . Accordingly, the complex-time flow of the holomorphic vector field $V_{2}\coloneq z^{2}\partial_{z}$ is given by $\phi^{V_{2}}_{w}(z)=\frac{z}{1-wz}$ .

Example 7.3.

For any integer $k\geq 2$ , the real-time flow of the one-dimensional vector field $X_{k}\coloneq x^{k}\partial_{x}$ is given by $\phi^{X_{k}}_{t}(x)=\frac{x}{\sqrt[k-1]{1-(k-1)tx^{k-1}}}$ . Accordingly, the complex-time flow of the holomorphic vector field $V_{k}\coloneq z^{k}\partial_{z}$ is given by $\phi^{V_{k}}_{w}(z)=\frac{z}{\sqrt[k-1]{1-(k-1)wz^{k-1}}}$ .

Bearing in mind the above notations, we now give the main result of this section.

Proposition 7.4.

Let $f$ be a holomorphic function on $\mathbb{C}$ that is real-valued on $\mathbb{R}$ . Let $U\subseteq\mathbb{R}^{2}$ be a sufficiently small open neighbourhood of $\mathbb{R}\times\{0\}$ that $h:U\to\mathbb{C}$ , $h(x,y)=\phi^{f(z)\partial_{z}}_{iy}(x)$ is defined. Then, the vector field $\frac{1}{2}(f(x)\partial_{x}+i\partial_{y})$ on $U$ is $h$ -related to the holomorphic vector field $f(z)\partial_{z}$ on $\mathbb{C}$ .

Proof.

Decompose $V=f(z)\partial_{z}$ in the form $V=\frac{1}{2}(X-iJX)$ , as above. Thus, by definition, $h(x,y)=\phi^{JX}_{y}(x)$ . For fixed $(x,y)\in U$ , taking $t$ sufficiently small, we have

h(x,y+t)=\phi^{JX}_{y+t}(x)=\phi^{JX}_{t}(h(x,y+t)).

Thus, the vector field $\partial_{y}$ is $h$ -related to $JX$ . Similarly,

\phi^{X}_{t}h(x,y)=\phi^{X}_{t}\phi^{JX}_{y}(x)=\phi^{JX}_{y}\phi^{X}_{t}(x)=\phi^{JX}_{y}\phi^{f(x)\partial_{x}}_{t}(x)=h(\phi^{f(x)\partial_{x}}_{t}(x),y)=h(\phi^{f(x)\partial_{x}}_{t}(x,y))

Thus, the vector field $f(x)\partial_{x}$ is $h$ -related to $X$ . The result follows by linearity. ∎

Corollary 7.5.

Let $g_{2}:\mathbb{R}^{2}\to\mathbb{C}$ , $g_{2}(x,y)=\frac{x}{1-ixy}$ . Then, $\frac{1}{2}(x^{2}\partial_{x}+i\partial_{y})$ is $g_{2}$ -related to $z^{2}\partial_{z}$ .

Corollary 7.6.

Fix an integer $k\geq 2$ . Let $g_{k}:\mathbb{R}^{2}\to\mathbb{C}$ , $g_{k}(x,y)=\frac{x}{\sqrt[k-1]{1-i(k-1)x^{k-1}y}}$ . Then $\tfrac{1}{2}(x^{k}\partial_{x}-i\partial_{y})$ is $g_{k}$ -related to $z^{k}\partial_{z}$ .

We remark that $g_{k}(x,y)=\frac{x}{\sqrt[k-1]{1-i(k-1)x^{k-1}y}}$ collapses $Z=\{0\}\times\mathbb{R}$ to $\{0\}$ and defines a diffeomorphism $\mathbb{R}^{2}\setminus Z\to\{re^{i\theta}\in\mathbb{C}:r\neq 0,-\frac{\pi}{2(k-1)}<\theta<\frac{\pi}{2(k-1)}\}$ . Note that the line $\mathrm{Re}(z)=1$ is completely contained in the principal branch of $\sqrt[k-1]{z}$ . For purposes of clarification, sample plots of the real vector field $JX$ in the decomposition $z^{k}\partial_{z}=\frac{1}{2}(X-iJX)$ are given below. Note all integral curves passing through the $x$ -axis are complete.

Refer to caption — Figure 1: The vector field $JX=-\mathrm{Im}(z^{k})\partial_{x}+i\mathrm{Re}(z^{k})\partial_{y}$ for $k=2,3,4$ .

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[8] A. Newlander and L. Nirenberg. Complex analytic coordinates in almost complex manifolds. Ann. of Math. (2), 65:391–404, 1957.

The Newlander-Nirenberg theorem for complex bb-manifolds

Abstract

1 Introduction

Theorem 1.1 ([7], Proposition 5.1).

Theorem 1.2.

2 Background

Definition 2.1.

Definition 2.2.

Example 2.3.

Definition 2.4.

Definition 2.5.

Definition 2.6.

Proposition 2.7.

Proof.

Definition 2.8.

Example 2.9.

Remark 2.10.

Example 2.11.

Example 2.12.

Proposition 2.13.

Proof.

3 Existence of bb-holomorphic functions: dimension 22

Theorem 3.1.

Lemma 3.2.

Lemma 3.3.

Lemma 3.4.

Proof of Theorem 3.1.

4 Proof of main result for dimension 22

Claim 4.1.

Proof.

Claim 4.2.

Proof.

5 Existence of bb-holomorphic functions: general case

Theorem 5.1.

Lemma 5.2.

Corollary 5.3.

Proof.

Lemma 5.4.

Proof of Theorem 5.1.

6 The bb-Newlander-Nirenberg theorem: general case

7 Remarks on singular pushforwards

Example 7.1.

Example 7.2.

Example 7.3.

Proposition 7.4.

Proof.

Corollary 7.5.

Corollary 7.6.

References

The Newlander-Nirenberg theorem for
complex $b$ -manifolds

3 Existence of $b$ -holomorphic functions: dimension $2$

4 Proof of main result for dimension $2$

5 Existence of $b$ -holomorphic functions: general case

6 The $b$ -Newlander-Nirenberg theorem: general case