# HYCEDIS: HYbrid Confidence Engine for Deep Document Intelligence System

Bao-Sinh Nguyen<EMAIL_ADDRESS>,  Quang-Bach Tran<EMAIL_ADDRESS>Cinnamon AI IncVietnam ,  Tuan-Anh Nguyen Dang<EMAIL_ADDRESS>,  Duc
Nguyen<EMAIL_ADDRESS>Cinnamon AI IncVietnam and  Hung Le
<EMAIL_ADDRESS>Deakin UniversityAustralia

(2021)

###### Abstract.

Measuring the confidence of AI models is critical for safely deploying AI in
real-world industrial systems. One important application of confidence
measurement is information extraction from scanned documents. However, there
exists no solution to provide reliable confidence score for current state-of-
the-art deep-learning-based information extractors. In this paper, we propose
a complete and novel architecture to measure confidence of current deep
learning models in document information extraction task. Our architecture
consists of a Multi-modal Conformal Predictor and a Variational Cluster-
oriented Anomaly Detector, trained to faithfully estimate its confidence on
its outputs without the need of host models modification. We evaluate our
architecture on real-wold datasets, not only outperforming competing
confidence estimators by a huge margin but also demonstrating generalization
ability to out-of-distribution data.

uncertainty, neural networks, supervised learning, information extraction

††copyright: acmcopyright††journalyear: 2021††doi:
10.1145/1122445.1122456††ccs: Information systems Document
structure††copyright: none††conference: The Second Document Intelligence
Workshop at KDD; 2021; Virtual Event

##  1\. Introduction

Recent advances in machine learning enables creations of automatic information
extractors that can read the input document in image format, locate and
understand relevant text lines before organizing the information into
computer-readable format for further analysis (yang2017learning, ; he2018end,
). Despite these successes, in critical domains such as healthcare and
banking, humans still have to involve to scrutinize AI outputs as there is no
room for AI errors in making important decisions that can affect human life.
Confidence score estimation is one critical step towards implementing
practical industrial systems wherein AI automates most of the operations yet
human will intervene if necessary (zheng2017hybrid, ).

Unfortunately, to the best of our knowledge, there exists no holistic solution
to reliably estimate the confidence score for the task of document information
extraction. Current confidence score approaches are either generic methods
verified only for simple image classification tasks (gal2016dropout, ) or
applied only for part of the information extraction process
(mor2018confidence, ).

In this paper, we introduce a novel neural architecture that can judge the
result of extracted structured information from documents provided by the
information extracting neural networks (hereafter referred to as the IE
Networks). Our architecture is hybrid, consisting of two models, which are a
Multi-modal Conformal Predictor (MCP) and an Variational Cluster-oriented
Anomaly Detector (VCAD). The former aims to combine the neural signals from 3
main stages of information extraction processes including text-box
localization, OCR, and key-value recognition to predict the confidence level
for each extracted key-value output. The later computes anomaly scores for the
raw input document image, providing the MCP with additional features to
produce better confidence estimation. The VCAD works on global, low-level
features and plays a critical role in lifting the burden of detecting outliers
off the MCP, which focuses more on local, high-level features.

We demonstrate the capacity of our proposed architecture on real-world invoice
datasets: SROIE (huang2019icdar2019, ), CORD (park2019cord, )) and 2 in-house
datasets. The experimental results demonstrate that our method outperforms
various confidence estimator baselines (including Droupout (gal2016dropout, ),
temperature scaling (guo2017calibration, )). In short, we summarize our
contribution as follows:

  * •

We propose a Multi-modal Conformal Predictor (MCP) using a Feature Fusion
module over 3 Feature Encoders to fuse signals extracted from IE Networks and
compute the confidence score of the IE Networks’ outputs.

  * •

We provide a Variational Cluster-oriented Anomaly Detector (VCAD) to equip the
MCP with an ability to handle out-of-distribution data.

  * •

We unify the proposed MCP and VCAD in a single hybrid confidence engine,
dubbed as HYCEDIS, that for the first time, can well estimate the confidence
of document intelligent system.

  * •

We conduct intensive experiments on 4 datasets with detailed ablation studies
to show the effectiveness and generalization of our hybrid architecture on
real-world problems.

##  2\. Background

A typical Document Intelligence System consists of multiple smaller steps:
text detection, text recognition and information extraction (IE). Given a
document image, the usual first step is to detect text lines, using
segmentation (baek2019character, ; he2017single, ; long2018textsnake, ) or
object detection method (liao2017textboxes, ; liao2018rotation, ; liu2017deep,
). The detected text line images can each go through an OCR model to
transcribe into text (graves2006connectionist, ). After all text contents are
transcribed, the relevant text entities can be extracted, using entity
recognition (sequence tagging) method (yao2019graph, ; liu2019graph, ;
xu2020layoutlm, ), segmentation-based method (yang2017learning, ; dang2019end,
), or graph-based method (Qian2018, ; Liu2019, ; Vedova2019, ) which
formulates the document layout as a graph of text-lines/words.

In this paper, we adopt a common IE Network that consisted of 3 main modules:
text detection (Layout Analysis), text recognition (CRNN) and graph-based
information extraction model (Graph KV). The text detection model shares the
same architecture with (baek2019character, ) which utilizes segmentation masks
to detect text-lines in the document image. The text recognition (CRNN) uses
popular CNN+Bi-LSTM+CTC-loss architecture to transcribe each text-line images
into text. Finally, the GCN model (Liu2019, ) performs the node classification
tasks from the input document graph constructed from the text-lines’ location
and text to extract relevant information. Here, for our problems, we classify
each node into different key types that represent the categories of the text-
line.

##  3\. Methodology

Figure 1. HYCEDIS architecture. (a) The Multi-modal Conformal Predictor (MCP).
(b) The Variational Cluster-oriented Anomaly Detector (VCAD). (c) Confidence
estimator (CE). MCP’s output vector, plus the VCAD’s abnormal score, is fed to
fully-connected layers to produce the final output of HYCEDIS, indicating
whether the extracted field true or false.

###  3.1. Multi-modal Conformal Predictor (MCP)

Given extracted intermediate features of IE Networks, our Multi-modal
Conformal Predictor aims to estimate the confidence score through predicting
whether the final output is true or false. The MCP architecture (see Figure
1(a)) contains two main components which are Feature Encoding and Feature
Fusion. The Feature Encoding extracts features from different layers of
trained IE Networks while the Feature Fusion combine them for predicting the
final output.

##### Feature Encoding

Motivated by designs of late-fusion multi-view sequential learning approaches
(chung2017lip, ), three components of the Feature Encoding layers are
independent processing streams including visual, lingual and structural
feature encoders. In particular, the visual feature encoder is a many-to-one
LSTM $f_{VF}(\cdot)$ that captures the visual information embedded in the CRNN
of the IE Network. It takes the CRNN’s logits (whose shape is $T\times
F_{in\\_vis}$ where $T$ is the number of timesteps) as input and outputs a
vector of size $F_{out\\_vis}$, which represents the knowledge of the IE
Networks on its OCR model’s neural activations given the input image. In
particular, for the $i$-th extracted text-line image $I_{i}$, we compute it
as: $E^{vis}_{i}=f_{VF}(CRNN(I_{i}))$.

The lingual feature encoder, which is also implemented as a many-to-one LSTM
$f_{LF}(\cdot)$, processes the predicted OCR texts of the IE Networks. Each
OCR-ed character in the text is represented as an one-hot vector with the size
that equals to the size of the corpus. For the $i$-th extracted OCR-ed text,
the LSTM takes a sequence of these one-hot vectors (denoted by $text_{i}$,
whose shape is $T\times F_{in\\_OCR}{}$) and produces an output vector of size
$F_{out\\_OCR}$, representing the knowledge of the IE Networks on the
linguistic meaning and the syntactic pattern of its OCR-ed outputs. We compute
it as: $E^{OCR}_{i}=f_{LF}(text_{i})$.

The structural feature encoding $f_{SF}(\cdot)$ is a feed-forward neural
network that accesses the information from the final layer of the IE Networks
(node classification) – the Graph KV module. Here, the logits before softmax
layer of the $i$-th node in the graph (corresponding to the $i$-th text box
extracted from the document), denoted by ${logit}^{(KV)}_{i}$, is the input of
the structural feature encoding, and the corresponding output is node
embedding vector $E^{node}_{i}$ representing the knowledge of the IE networks
on its final decision (node classification):
$E^{node}_{i}=f_{SF}({logit}^{(KV)}_{i})$.

##### Feature Fusion

The Feature Fusion network $f_{Fusion}$ takes the three outputs from the
Feature Encoding module and produces the ultimate feature vector. We use
simple concatenation and Bi-linear pooling (yu2017multi, ) as two options for
Feature Fusion. Bi-linear pooling use outer-product to combine inputs of
different modalities. For simple concatenation, we just concatenate three
vectors. For Bi-linear Pooling, we first pool the pair of $E_{i}^{vis}$ and
$E_{i}^{OCR}$, and then pool the resulting vector with $E_{i}^{node}$ to get
the pooled output $F_{i}$:

(1) $F_{i}=f_{Fusion}(E_{i}^{vis},E_{i}^{OCR},E_{i}^{node})$

###  3.2. Variational Cluster-oriented Anomaly Detector (VCAD)

The anomaly detector aims to detect which input image is normal or abnormal,
thus bolsters the MCP by a measurement of the normality that the input has.
Specifically, the input to the anomaly detector is a compressed representation
of the document image, and the output is a score in the range $[0,1]$
indicating the level of anomaly of the input. This score serves as an
additional input to the confidence estimator.

##### Representing image data with cluster-oriented embeddings

In this section, we describe the representation learning of document images.
Firstly, the training dataset was classified into some categories based on the
appearance and the layout structure of the document image. Then we train a
CNN-based image encoder to map each document image into a lower-dimensional
vector representation. Here, the CNN architecture is MobileNet
(sandler2018mobilenetv2, ). We adopt the triplet loss (huang2012bounding, ) to
learn the compressed representation, wherein the embeddings of images from the
same category tend to form a cluster in the embedding space.

##### Anomaly detector training

After constructing embeddings for training images, we build a Variational Auto
Encoder (VAE) (kingma2013auto, ; rezende2014stochastic, ) as our anomaly
detector (Figure 1(b)). The VAE outlier detector is first trained on a set of
normal (inlier) data to reconstruct the input it receives, with the standard
VAE loss function which is the sum of KL term and reconstruction loss:

(2)
$\mathcal{L_{\mathrm{VAE}}}(x;\theta,\phi)=-KL(q_{\phi}(z|x)||p_{\theta}(z))+\frac{1}{L}\sum_{l=1}^{L}\log{p_{\theta}(x|z^{(l)})}$

where $x$, $z$ and $L$ denote the VAE’s input, latent variable, and number of
samples, respectively. $q_{\phi}$ represents the encoder and $p_{\theta}$ the
decoder of VAE.

If the input data cannot be reconstructed well, the reconstruction error
(implemented as L1 loss between VAE’s input and output) is high and the data
can be flagged as an outlier. We apply the min-max normalization
(akcay2018ganomaly, ) to the reconstruction losses in order to get the
corresponding abnormal scores in the range of $[0,1]$.

###  3.3. Hybrid confidence estimation

After getting the scalar output from our VCAD, we simply concatenate this
scalar with the output of the Feature Fusion module in the MCP. The resulting
vector is fed to a confidence estimator (CE), which is implemented as a
2-layer feed-forward neural network. We freeze the VCAD and train the CE and
MCP end-to-end on the training data as the set of IE’s predictions in the
training dataset.

In particular, let $x_{i}$ denote the input document image, the function
$IE(\cdot)$ denote our pipeline of IE networks. The output of IE system is
$\hat{v}_{i}=IE(x_{i})$. More specifically,
$\hat{v}_{i}=\\{\hat{v}_{ik}\\}_{k=1\mathrel{\mathop{\ordinarycolon}}K_{i}}$
is the set of $K_{i}$ predictions where each $\hat{v}_{ik}$ contains location
information along with extracted text corresponding to a particular key (e.g:
{’location’: [123,234,184,246], ’text’: ’27/03/2018’, ’key’: ’date’}). We also
have the ground truth
$v_{i}=\\{v_{ij}\\}_{j=1\mathrel{\mathop{\ordinarycolon}}J_{i}}$ is the set of
$J_{i}$ elements presented in the $i\mathrm{-th}$ document.

Let $F_{ik}$ denote the input of our CE corresponding to prediction
$\hat{v}_{ik}$, the CE is represented by the function $f_{CE}(\cdot)$ yielding
the softmax output $p_{ik}=f_{CE}(F_{ik})$. The label for confidence
estimation task is

(3) $y_{ik}=\mathbb{1}\\{\exists
j\in\\{1\mathrel{\mathop{\ordinarycolon}}J_{i}\\}\mid
v_{ij}\overset{\mathrm{match}}{=}\hat{v}_{ik}\\}$

The IE’s output is considered to match the ground truth element if both the
text contents and the keys match and the locations’ IoU is greater than a
threshold ( $0.3$ in this paper). $y_{ik}$ is $1$ if IE’s prediction matches a
ground truth element (be correct) and vice versa. Then the loss function is
the standard binary cross-entropy loss with label $y_{ik}$ and probability
$p_{ik}$.

##  4\. Experiments

###  4.1. Datasets and evaluation metrics

####  4.1.1. Datasets

We collect 4 Invoice-like datasets and divide them into 2 tasks, corresponding
to English and Japanese language used in the data. For each task, we use the
bigger dataset as the main one, and the smaller as the out-of-distribution
(OOD) dataset with respect to the main dataset.

We first use pre-trained IE Networks (see Sec. 2) to generate the intermediate
features for the MCP as mentioned in Sec. 3.1. The outputs of the IE Networks
and the ground-truth IE outputs are used to produce labels for the confidence
estimation task (Sec. 3.3).

We only train the confidence models on the training dataset and benchmark them
on the testing and corresponding OOD datasets. The evaluation on OOD data is a
challenging benchmark since the OOD dataset is totally different from the main
one in terms of layout, background and writing styles. Moreover, since the OOD
datasets can have different type of keys from those in the main one, we only
test the models on fields that share common keys with the main dataset.

  1. (a)

Public datasets (English)

SROIE - Main dataset. SROIE (huang2019icdar2019, ) is a dataset of scanned
receipts. There are 4 keys: address, company, date, total. The training set
has 626 files corresponding to 3859 IE’s output key-value fields. We further
hold 10% of the training as the validation set. The statistics for the test
set are 341 files and 1,640 fields, respectively.

CORD - OOD dataset. CORD (park2019cord, ) contains receipts collected from
Indonesian shops and restaurants. Compared to SROIE, CORD document images are
captured in the wild, thus the data is noisy and low in quality. CORD field
shares only one key with SROIE, which is total. We use the CORD-dev set which
contains 100 files correspoding to 103 IE’s output fields.

  2. (b)

In-house datasets (Japanese)

In-house 1 - Main dataset. In-house 1 is a dataset containing Japanese invoice
documents collected from several vendors. There are 25 keys. Example keys are
issued_date, total_amount, tax, item_name, item_amount. The training set has
835 files corresponding to 24,697 IE’s output fields, and the test set has 338
files and 10,898 fields.

In-house 2 - OOD dataset. In-house 2 consists of 68 invoice documents from
another Japanese company. The document pattern is quite different to the In-
house 1 dataset. The two in-house dataset share 4 key types in common,
resulting in 3,887 IE’s output fields.

####  4.1.2. Evaluation metrics

We use the popular Area Under the Receiver Operating Characteristic Curve
(AUC) (mor2018confidence, ; ayhan2018test, ; mandelbaum2017distance, ;
hein2019relu, ) and Expected Calibration Error (ECE) (naeini2015obtaining, )
metrics for measuring the performance of confidence predictors.

###  4.2. Experimental baselines

Softmax Threshold. Our IE pipeline consists of multiple sequential models, so
we adapted (hendrycks2016baseline, ) by combining both softmax probabilities
from OCR and KV models using multiplication (i.e:
${p_{final}}={p_{OCR}}*{p_{KV}}$). We then specify a threshold score and
considered examples with higher-than-threshold softmax probability as
correctly predicted one, and vice versa. The threshold score is tuned on the
training dataset.

Temperature Scaling. Temperature scaling (guo2017calibration, ) is a technique
that post-processes the neural networks to make them calibrated in term of
confidence. Temperature scaling divides the logits (inputs to the softmax
function) by a learned scalar parameter $T$ (temperature). We learn this
parameter on a validation set, where $T$ is chosen to minimize negative log-
likelihood.

Softmax Classifier. Instead of only utilizing the softmax probability of the
predicted class as Softmax Threshold, Softmax Classifier is more advanced by
making use of the whole softmax vector. Particularly, we build a simple
classifier using a feed-forward neural network. The input for the network is
the concatenation of the OCR model’s softmax vector and the KV model’s one.

Monte Carlo Dropout. MC Dropout (gal2016dropout, ) belongs to the class of
Bayesian/variational approaches. By keeping the dropout enabled at test time,
we can obtain the variance of the neural network’s outputs, and this variance
indicates the level of uncertainty. We apply MC Dropout on our KV model, which
is the final model in the pipeline.

###  4.3. Benchmarking results

####  4.3.1. Ablation study

We ablate the effect of VCAD and MCP on the whole hybrid system. Table 1
reports the results on SROIE dataset. Without VCAD, the proposed model
achieves best AUC score of 86.90% using bi-linear pooling fusion strategy.
Simpler concatenation method underperforms by about 3% demonstrating the
importance of using outer-product to retain bit-level relationships among 3
modalities. When the VCAD is integrated, it consistently improves the
performance of all fusion methods. Hence, the full hybrid HYCEDIS architecture
can reach 88.12% AUC. Similar behaviors can be found with measurement using
ECE metric.

Table 1. Ablation study on SROIE dataset Methods | ECE | AUC
---|---|---
MCP (concatenation) | 0.1525 | 83.75
MCP (bilinear pooling) | 0.1175 | 86.90
MCP (concatenation) + VCAD | 0.1385 | 84.37
MCP (bilinear pooling) + VCAD | 0.1002 | 88.12
Table 2. Performance comparison of baselines and proposed methods on SROIE and CORD datasets Methods | SROIE | CORD
---|---|---
ECE | AUC | ECE | AUC
Softmax threshold | 0.1525 | 83.75 | 0.1731 | 66.91
Softmax classifier | 0.1400 | 85.50 | 0.3289 | 54.91
MC Dropout | 0.1175 | 86.90 | 0.5446 | 43.52
Temperature scaling | 0.1385 | 84.37 | 0.3787 | 74.58
MCP | 0.1124 | 86.40 | 0.1432 | 75.12
HYCEDIS | 0.1002 | 88.12 | 0.1259 | 77.45
Table 3. Performance comparison of baselines and proposed methods on In-house datasets Methods | In-house 1 | In-house 2
---|---|---
ECE | AUC | ECE | AUC
Softmax threshold | 0.1285 | 68.79 | 0.5885 | 53.38
Softmax classifier | 0.2810 | 71.43 | 0.3945 | 51.22
MC Dropout | 0.3733 | 66.14 | 0.3621 | 48.20
Temperature scaling | 0.1728 | 64.00 | 0.5879 | 58.18
MCP | 0.0782 | 86.32 | 0.3348 | 60.12
HYCEDIS | 0.0712 | 90.12 | 0.3019 | 61.90

####  4.3.2. Public English datasets result

Table 2 shows the performance of all models on public datasets. On both SROIE
and its OOD CORD dataset, our full HYCEDIS is consistently the best performer
regarding both ECE and AUC scores. Our MCP is the runner-up under ECE metric.
The improvements of MCP in AUC and ECE suggests that the signals from
intermediate features extracted from text-line images, OCR-ed text and graph
structure help improve the accuracy of the softmax-based methods which only
rely on some softmax layers of the IE Networks. In addition, when combined
with VCAD, the AUC score is further increased and the ECE also downgrades.
That manifests the contribution of our VCAD model. We can see a significant
performance drop from baselines such as MC-Dropout when being tested on OOD
CORD data. Our methods alleviate this issue, maintaining a moderate
generalization to strange data.

####  4.3.3. In-house Japanese datasets result

We also benchmark the models on two in-house datasets. In Table 3, our model
continues to show the superior performance compared with other baselines. Our
MCP model improves about 14.89% and 2% AUC score and reduces 0.0503 and 0.0273
ECE score in In-house 1 and In-house 2 datasets, respectively. When adding
VCAD, the performance is improved around 3.82% on In-house 1 dataset and 2.78%
on In-house dataset, which again validates our hypothesis on using anomaly
detector to enhance conformal predictor.

##  5\. Conclusion

We have introduced a holistic confidence score architecture that aims to
verify the result of IE Networks in document understanding tasks. Our
architecture takes advantages of a Multi-modal Conformal Predictor and a
Variational Cluster-oriented Anomaly Detector to predict whether the IE
Networks’ output correct or not using features of different granularity. Our
hybrid approach surpasses prior confidence estimation methods by a huge margin
in benchmarks with invoice datasets. Remarkably, it demonstrates a capability
of generalization to out-of-distribution datasets.

## References

  * (1) Samet Akcay, Amir Atapour-Abarghouei, and Toby P Breckon.  Ganomaly: Semi-supervised anomaly detection via adversarial training.  In Asian conference on computer vision, pages 622–637. Springer, 2018.
  * (2) Murat Seckin Ayhan and Philipp Berens.  Test-time data augmentation for estimation of heteroscedastic aleatoric uncertainty in deep neural networks.  2018\.
  * (3) Youngmin Baek, Bado Lee, Dongyoon Han, Sangdoo Yun, and Hwalsuk Lee.  Character region awareness for text detection.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9365–9374, 2019.
  * (4) Joon Son Chung, Andrew Senior, Oriol Vinyals, and Andrew Zisserman.  Lip reading sentences in the wild.  In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 3444–3453. IEEE, 2017.
  * (5) Tuan Anh Nguyen Dang and Dat Nguyen Thanh.  End-to-end information extraction by character-level embedding and multi-stage attentional u-net.  In BMVC, page 96, 2019.
  * (6) Yarin Gal and Zoubin Ghahramani.  Dropout as a bayesian approximation: Representing model uncertainty in deep learning.  In international conference on machine learning, pages 1050–1059, 2016.
  * (7) Alex Graves, Santiago Fernández, Faustino Gomez, and Jürgen Schmidhuber.  Connectionist temporal classification: labelling unsegmented sequence data with recurrent neural networks.  In Proceedings of the 23rd international conference on Machine learning, pages 369–376, 2006.
  * (8) Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger.  On calibration of modern neural networks.  arXiv preprint arXiv:1706.04599, 2017.
  * (9) Pan He, Weilin Huang, Tong He, Qile Zhu, Yu Qiao, and Xiaolin Li.  Single shot text detector with regional attention.  In Proceedings of the IEEE international conference on computer vision, pages 3047–3055, 2017.
  * (10) Tong He, Zhi Tian, Weilin Huang, Chunhua Shen, Yu Qiao, and Changming Sun.  An end-to-end textspotter with explicit alignment and attention.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 5020–5029, 2018.
  * (11) Matthias Hein, Maksym Andriushchenko, and Julian Bitterwolf.  Why relu networks yield high-confidence predictions far away from the training data and how to mitigate the problem.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 41–50, 2019.
  * (12) Dan Hendrycks and Kevin Gimpel.  A baseline for detecting misclassified and out-of-distribution examples in neural networks.  arXiv preprint arXiv:1610.02136, 2016.
  * (13) Gary B Huang, Andrew Kae, Carl Doersch, and Erik Learned-Miller.  Bounding the probability of error for high precision optical character recognition.  The Journal of Machine Learning Research, 13(1):363–387, 2012.
  * (14) Zheng Huang, Kai Chen, Jianhua He, Xiang Bai, Dimosthenis Karatzas, Shijian Lu, and CV Jawahar.  Icdar2019 competition on scanned receipt ocr and information extraction.  In 2019 International Conference on Document Analysis and Recognition (ICDAR), pages 1516–1520. IEEE, 2019.
  * (15) Diederik P Kingma and Max Welling.  Auto-encoding variational bayes.  arXiv preprint arXiv:1312.6114, 2013.
  * (16) Minghui Liao, Baoguang Shi, Xiang Bai, Xinggang Wang, and Wenyu Liu.  Textboxes: A fast text detector with a single deep neural network.  In Proceedings of the AAAI conference on artificial intelligence, volume 31, 2017.
  * (17) Minghui Liao, Zhen Zhu, Baoguang Shi, Gui-song Xia, and Xiang Bai.  Rotation-sensitive regression for oriented scene text detection.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 5909–5918, 2018.
  * (18) Xiaojing Liu, Feiyu Gao, Qiong Zhang, and Huasha Zhao.  Graph convolution for multimodal information extraction from visually rich documents.  arXiv preprint arXiv:1903.11279, 2019.
  * (19) Xiaojing Liu, Feiyu Gao, Qiong Zhang, and Huasha Zhao.  Graph Convolution for Multimodal Information Extraction from Visually Rich Documents.  mar 2019.
  * (20) Yuliang Liu and Lianwen Jin.  Deep matching prior network: Toward tighter multi-oriented text detection.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1962–1969, 2017.
  * (21) Shangbang Long, Jiaqiang Ruan, Wenjie Zhang, Xin He, Wenhao Wu, and Cong Yao.  Textsnake: A flexible representation for detecting text of arbitrary shapes.  In Proceedings of the European conference on computer vision (ECCV), pages 20–36, 2018.
  * (22) Amit Mandelbaum and Daphna Weinshall.  Distance-based confidence score for neural network classifiers.  arXiv preprint arXiv:1709.09844, 2017.
  * (23) Noam Mor and Lior Wolf.  Confidence prediction for lexicon-free ocr.  In 2018 IEEE Winter Conference on Applications of Computer Vision (WACV), pages 218–225. IEEE, 2018.
  * (24) Mahdi Pakdaman Naeini, Gregory Cooper, and Milos Hauskrecht.  Obtaining well calibrated probabilities using bayesian binning.  In Proceedings of the AAAI Conference on Artificial Intelligence, volume 29, 2015.
  * (25) Seunghyun Park, Seung Shin, Bado Lee, Junyeop Lee, Jaeheung Surh, Minjoon Seo, and Hwalsuk Lee.  Cord: A consolidated receipt dataset for post-ocr parsing.  2019\.
  * (26) Yujie Qian, Enrico Santus, Zhijing Jin, Jiang Guo, and Regina Barzilay.  GraphIE: A Graph-Based Framework for Information Extraction.  2018\.
  * (27) Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra.  Stochastic backpropagation and approximate inference in deep generative models.  arXiv preprint arXiv:1401.4082, 2014.
  * (28) Mark Sandler, Andrew Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh Chen.  Mobilenetv2: Inverted residuals and linear bottlenecks.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4510–4520, 2018.
  * (29) Luca Della Vedova, Hong Yang, and Garrick Orchard.  An Invoice Reading System Using a Graph Convolutional Network.  2:434–449, 2019.
  * (30) Yiheng Xu, Minghao Li, Lei Cui, Shaohan Huang, Furu Wei, and Ming Zhou.  Layoutlm: Pre-training of text and layout for document image understanding.  In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pages 1192–1200, 2020.
  * (31) Xiao Yang, Ersin Yumer, Paul Asente, Mike Kraley, Daniel Kifer, and C Lee Giles.  Learning to extract semantic structure from documents using multimodal fully convolutional neural networks.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 5315–5324, 2017.
  * (32) Liang Yao, Chengsheng Mao, and Yuan Luo.  Graph convolutional networks for text classification.  In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 7370–7377, 2019.
  * (33) Zhou Yu, Jun Yu, Jianping Fan, and Dacheng Tao.  Multi-modal factorized bilinear pooling with co-attention learning for visual question answering.  In Proceedings of the IEEE international conference on computer vision, pages 1821–1830, 2017.
  * (34) Nan-ning Zheng, Zi-yi Liu, Peng-ju Ren, Yong-qiang Ma, Shi-tao Chen, Si-yu Yu, Jian-ru Xue, Ba-dong Chen, and Fei-yue Wang.  Hybrid-augmented intelligence: collaboration and cognition.  Frontiers of Information Technology & Electronic Engineering, 18(2):153–179, 2017.


# 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.

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 $Z$ such that

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

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:

  1. (i)

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

  2. (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:

  1. (i)

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

  2. (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:

  1. (i)

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

  2. (ii)

$f$ vanishes on $Z$,

  3. (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:

  1. (a)

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

  2. (b)

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

  3. (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

${(U,U\cap Z)}$${(W,W\cap Z)}$${(V,V\cap
Z).}$$\scriptstyle{F}$$\scriptstyle{G}$

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

${U\setminus Z}$${\kappa(W\setminus Z)}$${V\setminus
Z,}$$\scriptstyle{f}$$\scriptstyle{g}$

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:

  1. (i)

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

  2. (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:

  1. (i)

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

  2. (ii)

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

  3. (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:

  1. (a)

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

  2. (b)

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

  3. (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.

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

## References

  * [1] T. Barron and M. Francis.  On automorphisms of complex $b^{k}$-manifolds, 2023 (arXiv preprint).
  * [2] M. Braverman, Y. Loizides, and Y. Song.  Geometric quantization of $b$-symplectic manifolds.  J. Symplectic Geom., 19(1):1–36, 2021.
  * [3] F. Forstnerič.  Stein manifolds and holomorphic mappings, volume 56 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics].  Springer, Cham, second edition, 2017.  The homotopy principle in complex analysis.
  * [4] V. Guillemin, E. Miranda, and A. R. Pires.  Symplectic and Poisson geometry on $b$-manifolds.  Adv. Math., 264:864–896, 2014.
  * [5] Y. Lin, Y. Loizides, R. Sjamaar, and Y. Song.  Symplectic reduction and a darboux-moser-weinstein theorem for lie algebroids, 2022.
  * [6] R. B. Melrose.  The Atiyah-Patodi-Singer index theorem, volume 4 of Research Notes in Mathematics.  A K Peters, Ltd., Wellesley, MA, 1993.
  * [7] G. A. Mendoza.  Complex $b$-manifolds.  In Geometric and spectral analysis, volume 630 of Contemp. Math., pages 139–171. Amer. Math. Soc., Providence, RI, 2014.
  * [8] A. Newlander and L. Nirenberg.  Complex analytic coordinates in almost complex manifolds.  Ann. of Math. (2), 65:391–404, 1957.