:
1 2022 NeurIPS Workshop on Symmetry and Geometry in Neural Representations

# Learning unfolded networks with a cyclic group structure

Emmanouil Theodosis<EMAIL_ADDRESS>
School of Engineering and Applied Sciences
Harvard University
Cambridge  MA 02138  Demba Ba<EMAIL_ADDRESS>
School of Engineering and Applied Sciences
Harvard University
Cambridge  MA 02138

###### Abstract

Deep neural networks lack straightforward ways to incorporate domain knowledge
and are notoriously considered black boxes. Prior works attempted to inject
domain knowledge into architectures _implicitly_ through data augmentation.
Building on recent advances on equivariant neural networks, we propose
networks that _explicitly_ encode domain knowledge, specifically equivariance
with respect to rotations. By using unfolded architectures, a rich framework
that originated from sparse coding and has theoretical guarantees, we present
interpretable networks with sparse activations. The equivariant unfolded
networks compete favorably with baselines, with only a fraction of their
parameters, as showcased on (rotated) MNIST and CIFAR-10.

###### keywords:

Equivariance, model-based learning, cyclic groups, unfolded networks

††editors: Sophia Sanborn, Christian Shewmake, Simone Azeglio, Arianna Di
Bernardo, Nina Miolane

##  1 Introduction

While advances in deep neural networks have yielded groundbreaking results in
various fields such as computer vision (Redmon and Farhadi, 2017; Pavlakos et
al., 2017; Mildenhall et al., 2020), natural language processing (Devlin et
al., 2019; Brown et al., 2020), and their intersection (Radford et al., 2021),
interpreting their structure and explaining their performance is not
straightforward. At the same time, applying deep learning techniques to novel
fields comes with challenges, as it is not clear how to integrate domain
knowledge into existing architectures. In this work, we propose a novel
architecture to address both of these shortcomings at the same time, leading
to an interpretable architecture that respects expert knowledge.

Figure 1: Filters learned at the final layer of $R_{60}$-CNN, training on
CIFAR-10.

Convolutional Neural Networks (CNNs) are _equivariant_ in their
representations with respect to translation; however, there are other
operators that is natural for image models to be equivariant to, such as
rotations. While data augmentation techniques have been used to model
equivariances they require large amounts of data and increase the
computational demands for training, frequently by an order of magnitude. At
the same time, if we know the desired equivariances for a specific
application, investing computational resources to re-learn these equivariances
is wasteful. This was also acknowledged by Dieleman et al. (2016) and Cohen
and Welling (2016) who concurently introduced CNN frameworks that incorporate
rotated filters in order to create equivariant representations with respect to
rotations; however both works were limited to elementary rotations and the
reflections of $\mathcal{D}_{4}$. Follow up works by different authors
extended the ideas to vector fields (Marcos et al., 2017), applied rotations
directly on the sphere to avoid interpolation artifacts (Esteves et al.,
2020), and incorporated harmonic functions to model equivariance with respect
to arbitrary rotations (Worrall et al., 2017).

There have been several attempts to tackle interpretability, ranging from
prototype learning approaches (Chen et al., 2019; Arik and Pfister, 2020) that
learn _prototypical_ parts for each class in a classification task, to post-
hoc methods (Ribeiro et al., 2016) that reverse engineer predictions from
arbitrary classifiers. In this work, we focus on _model-based_ networks
(Shlezinger et al., 2020): in these approaches, interpretability is directly
encoded into the model by constructing a neural network to mimic the steps of
an optimization algorithm. First introduced by Gregor and LeCun (2010),
_unrolled_ neural networks have inspired a vast array of works, ranging from
theoretical contributions (Nguyen et al., 2019; Arora et al., 2015) to state-
of-the-art results (Tolooshams et al., 2020).

In this work, we propose an unrolled architecture, inspired from algorithms
for sparse coding, whose layer weights employ a _cyclic group structure_ to
achieve rotational equivariance. This allows us to significantly reduce the
number of trainable filters compared to baseline architectures. Concretely,
our contributions can be summarized as follows:

  1. 1.

We propose an _unrolled_ architecture, modeled after sparse coding, that is by
construction interpretable and equivariant with respect to rotations,

  2. 2.

we showcase its efficacy in learning filters that are governed by a cyclic
group structure with a fraction of the learnable parameters, and

  3. 3.

we evaluate the proposed architecture on MNIST, rotated MNIST, and CIFAR-10,
standard benchmarks for rotationally equivariant architectures and demonstrate
its performance.

We make the code for our experiments and architectures available online on
GitHub at https://github.com/manosth/cyclical_groups.

##  2 Background

Equivariance. In lay terms, an operator is equivariant with respect to some
actions if it behaves in a predictable manner under them. Formally, we say
that an operator $f$ is equivariant with respect to a family of actions
$\mathcal{T}$ if, for any $T\in\mathcal{T}$ it holds that

$f(T(x))=T^{\prime}(f(x)),$ (1)

for some _other_ transform $T^{\prime}$. Note that, in general, $T$ and
$T^{\prime}$ belong in different families as $T$ acts on the input space
($\operatorname{Dom}(f)$) and $T^{\prime}$ on the encoding space
($\operatorname{Im}(f)$). Constant functions are trivially equivariant, and a
special case, _invariance_ , arises when $T^{\prime}$ is the identity map.
Note that convolution is _not_ equivariant to rotation (Cohen and Welling,
2016; Dieleman et al., 2016); instead, the two are related by

$R(\bm{x})*\bm{h}=R(\bm{x}*(R^{-1}(\bm{h})),$

where $\bm{x}$ denotes an input image, $R$ is a rotation, and $\bm{h}$ is the
convolving filter.

Cyclic groups. We call a finite group $G$ a cyclic group if there exists a
generating element $g$ such that

$G=\\{e,g,g^{2},\ldots,g^{k-1}\\},$ (2)

where $e$ denotes the identity element. We denote the family of cyclic groups
as $\mathcal{G}$; several groups belong to this family, with most notable
being $\mathcal{D}_{4}$, the symmetry group of the square. Cyclic groups are
of interest for our model since all elements can be identified by the
generator $g$. This will enable us, in Section 3 to significantly reduce the
trainable parameters of our networks, while retaining (and even improving)
performance and interpretability.

Unrolled sparse autoencoders. In their most general form, unrolled networks
temporally unroll the steps of optimization algorithms, mapping algorithm
iterations to network layers. In that way, the output of the neural network
can be interpreted as the output of the optimization algorithm, with
theoretical guarantees under certain assumptions. The _Iterative Soft
Thresholding Algorithn_ (ISTA), an algorithm for sparse coding, has inspired
several architectures (Simon and Elad, 2019; Sulam et al., 2020; Tolooshams et
al., 2021), due to the desirability of sparse representations. Within that
framework, the representation at layer $l+1$ is given by

$\bm{z}^{(l+1)}=\mathcal{S}_{\lambda}\left(\bm{z}^{(l)}+\alpha\bm{W}_{l}^{T}(\bm{x}-\bm{W}_{l}\bm{z}^{(l)})\right),$
(3)

where $\bm{x}$ is the _original_ input, $\bm{z}^{(l)}$ is the representation
at the previous layer, $\bm{W}_{l}$ are the weights of layer $l$, $\alpha$ is
a constant such that
$\frac{1}{\alpha}\geq\sigma_{\max}(\bm{W}_{l}^{T}\bm{W}_{l})$, and
$\mathcal{S}_{\lambda}$ is the _soft thresholding_ operator defined as

$\mathcal{S}_{\lambda}(u)=\operatorname{sign}(u)\cdot\operatorname{ReLU}(\lvert
u\rvert-\lambda).$ (4)

If the weights of all the $L$ layers are equal, i.e. $W_{1}=\ldots=W_{L}$, we
call the network _tied_. As a final remark, Equation 3 can be rewriten as

$\bm{z}^{(l+1)}=\mathcal{S}_{\lambda}\left((I-\alpha\bm{W}_{l}^{T}\bm{W}_{l})\bm{z}^{(l)}+\alpha\bm{W}_{l}^{T}\bm{x}\right)=\mathcal{S}_{\lambda}\left(\bm{W}_{z}\bm{z}^{(l)}+\bm{W}_{x}\bm{x}\right),$
(5)

where we let $\bm{W}_{z}=(I-\alpha\bm{W}_{l}^{T}\bm{W}_{l})$ and
$\bm{W}_{x}=\alpha\bm{W}_{l}^{T}$, which can be interpreted as a residual
network (He et al., 2016), with a residual connection to the input.

##  3 Equivariant autoencoders

We will combine the ideas of cyclic groups and unrolled autoencoders from
Section 2 to create an _equivariant_ unrolled architecture, where the bulk of
the weights in each layer are cyclic rotations of some “basis” weights . Let
$R_{\theta}$ denote a two-dimensional rotation by $\theta$ degrees. If
$360\mod\theta=0$ and we let $k=360\div\theta$, then the group defined as

$G=\\{e,R_{\theta},\ldots,R_{\theta}^{k-1}\\},$ (6)

is a cyclic group generated by the generator $g=R_{\theta}$. This construction
allows us to extend this framework, in future work, to _arbitrary_ linear
operators and the possibility to _learn_ the generator $g$, leading to data-
driven approaches for the cyclic group structure. Regardless, the weights of
layer $l$ satisfy

$\bm{W}_{l}=\begin{bmatrix}\bm{w}_{l}&R_{\theta}(\bm{w}_{l})&\ldots&R_{\theta}^{k-1}(\bm{w}_{l}),\\\
\end{bmatrix}$ (7)

where $k$ is the number of rotations we are considering, per layer. Note that
Equation 7 shows a single basis weight per layer; this can be readily extended
to having $m$ basis weights $\bm{w}_{l_{1}},\ldots,\bm{w}_{l_{m}}$ along with
the $k$ rotations for each of them, as is done in Section 4. The networks are
constructed analogously to Sulam et al. (2020); Tolooshams et al. (2021) but
with _one_ (or $m$) learnable filter(s) per layer. During training, the
remaining filters are constructed and errors are backpropagated through _all_
filters. The experiments of Section 4 use untied networks for improved
performance.

##  4 Experiments

Method | MNIST | rot-MNIST | CIFAR-10 | Number of parameters
---|---|---|---|---
Baseline | 99.21 | 85.48 | 71.87 | 56.29K
$R_{90}$-CNN | 99.12 | 86.62 | 72.20 | 21.73K
$R_{60}$-CNN | 98.77 | 80.07 | 73.14 | 17.89K
Table 1: Performances of the baseline model, $R_{90}$-CNN, and $R_{60}$-CNN on
different datasets along with their total number of trainable parameters
(CIFAR-10).

We present experiments on MNIST, rotated MNIST, and CIFAR-10, standard
benchmarks for equivariant architectures, to evaluate the architecture
introduced in Section 3. We used batch normalization (Ioffe and Szegedy, 2015)
in all of our architectures, following best practices. The normalization was
applied at the output of every layer, except the last. FISTA (Beck and
Teboulle, 2009), a faster version of ISTA, is used for faster convergence of
the sparse coding. All of our networks use $L=4$ layers, $\lambda$ (the
parameter of $\mathcal{S}_{\lambda}$) is set to $0.5$, and the stepsize of
FISTA in (3) is set to $\alpha=0.01$. A summary of our main results is given
in Table 1.

We test three models: an unrolled sparse network that is considered as a
baseline; $R_{90}$-CNN, an equivariant unfolded network with the elementary
rotations; and $R_{60}$-CNN, with $60$ rotations. Every architecture has $60$
filters per layer; in the case of the baseline, all $60$ are learnable,
$R_{90}$-CNN has $15$ learnable filters per layer the rest are constructed as
elementary rotations, and $R_{60}$-CNN has only $10$ learnable filters per
layer with the rest being their $60$ rotations. The learned filters are
$7\times 7$ in the case of MNIST and $8\times 8$ in the case of CIFAR-10. A
visualization of $R_{60}$-CNN’s learned filters when trained on CIFAR-10 is
show in Figure 1.

MNIST. We first evaluate all three models on MNIST, a relatively easy dataset,
and find that all of them performed similarly. However, we note that
$R_{90}$-CNN has only $\frac{1}{4}$ the trainable filters of the baseline, as
the rest are generated as rotations of the basis weights; $R_{60}$-CNNH has
only $\frac{1}{6}$. When evaluating the architectures on the rotated MNIST, a
harder dataset, we observe that the $R_{90}$-CNN, with a _fraction_ of the
parameters of the baseline model leads to the best performance. This showcases
that the encoded equivariance in the representation is actually beneficial for
the classification of the inputs.

Evaluating the architectures on the rotated MNIST produces similar results and
$R_{90}$-CNN outperforms the baseline. However, $R_{90}$-CNN displays reduced
performance on this dataset; we speculate that this is because of
interpolation artifacts, which was one of the motivating reasons for Esteves
et al. (2020) to consider rotations on the sphere.

To further demonstrate the benefit of the equivariant unfolded networks, we
trained _dense_ variants (where the filters are now $28\times 28$ images) of
the three models on MNIST, and

Method | MNIST | rot-MNIST
---|---|---
Baseline | 97.75 | 36.89
$R_{90}$-NN | 98.04 | 37.02
$R_{60}$-NN | 97.73 | 37.4
Table 2: Models trained on MNIST but evaluated on rot-MNIST.

evaluate their performance on the rotated dataset. The results are presented
in Table 2. This experiment showcases the generalization capabilities of the
equivariant unrolled networks. While we see similar performance across all
models on the trained dataset, we see that _both_ the equivariant models are
able to generalize better than the baseline. Dense architectures were chosen
for this experiment to highlight the distribution shift when evaluating on
rot-MNIST.

CIFAR-10. When training on CIFAR-10, a comparetively harder dataset, we found
that _both_ the equivariant models outperform the baseline, even if
marginally, with only a fraction of the learnable filters per layer. Moreover,
upon further examination, the filters of $R_{90}$-CNN and $R_{60}$-CNN exhibit
a topographic structure by construction as filters are generated as rotations
of one another. That topographic structure is not present in the filters of
the baseline model which doesn’t incorporate domain knowledge regarding
rotations. The learned filters of the different models, when trained on
CIFAR-10, are presented in Appendix A.

##  5 Conclusions and future work

In this work we introduced equivariant unrolled networks, where the filters of
each layer are constructed as discrete rotations of one another. By exploiting
this cyclical group structure, we are able to facilitate training and maintain
performance while drastically reducing the number of trainable parameters of
the model. We experimentally validated the equivariant unrolled networks
against a baseline network and showed comparable, and even favorable, results,
with only a fraction of the learnable parameters on multiple datasets that are
used as benchmarks for equivariant architectures. Finally, we identify
_learning_ the generator $g$ from data, as hinted in Section 2, an exciting
avenue for future work.

## Acknowledgements

Demba Ba was supported by the National Science Foundation under Grant number
#S5206, PO 560825.

## References

  * Arik and Pfister (2020) Sercan Arik and Tomas Pfister.  Protoattend: Attention-based prototypical learning.  _Journal of Machine Learning Research_ , 21:1–35, 2020\.
  * Arora et al. (2015) Sanjeev Arora, Rong Ge, Tengyu Ma, and Ankur Moitra.  Simple, efficient, and neural algorithms for sparse coding.  In _Conference on Learning Theory_ , 2015.
  * Beck and Teboulle (2009) Amir Beck and Marc Teboulle.  A fast iterative shrinkage-thresholding algorithm for linear inverse problems.  _SIAM Journal on Imaging Sciences_ , 2(1):183–202, 2009.
  * Brown et al. (2020) Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel Ziegler, Jeffrey Wu, Clemens Winter, Chris Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei.  Language models are few-shot learners.  In _Advances in Neural Information Processing Systems_ , 2020.
  * Chen et al. (2019) Chaofan Chen, Oscar Li, Chaofan Tao, Alina Jade Barnett, Jonathan Su, and Cynthia Rudin.  This looks like that: Deep learning for interpretable image recognition.  In _Neural Information Processing Systems_ , 2019.
  * Cohen and Welling (2016) Taco Cohen and Max Welling.  Group equivariant convolutional networks.  In _International Conference on Machine Learning_ , 2016.
  * Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.  BERT: pre-training of deep bidirectional transformers for language understanding.  In _Conference of the North American Chapter of the Association for Computational Linguistics_ , 2019.
  * Dieleman et al. (2016) Sander Dieleman, Jeffrey De Fauw, and Koray Kavukcuoglu.  Exploiting cyclic symmetry in convolutional neural networks.  In _International Conference on Machine Learning_ , 2016.
  * Esteves et al. (2020) Carlos Esteves, Ameesh Makadia, and Kostas Daniilidis.  Spin-weighted spherical cnns.  In _Neural Information Processing Systems_ , 2020.
  * Gregor and LeCun (2010) Karol Gregor and Yann LeCun.  Learning fast approximations of sparse coding.  In _International Conference on Machine Learning_ , 2010.
  * He et al. (2016) Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  In _IEEE Conference on Computer Vision and Pattern Recognition_ , 2016\.
  * Ioffe and Szegedy (2015) Sergey Ioffe and Christian Szegedy.  Batch normalization: Accelerating deep network training by reducing internal covariate shift.  In _International Conference on Machine Learning_ , 2015.
  * Marcos et al. (2017) Diego Marcos, Michele Volpi, Nikos Komodakis, and Devis Tuia.  Rotation equivariant vector field networks.  In _International Conference on Computer Vision_ , 2017.
  * Mildenhall et al. (2020) Ben Mildenhall, Pratul Srinivasan, Matthew Tancik, Jonathan Barron, Ravi Ramamoorthi, and Ren Ng.  NeRF: representing scenes as neural radiance fields for view synthesis.  In _European Conference on Computer Vision_ , 2020.
  * Nguyen et al. (2019) Thanh Nguyen, Raymond Wong, and Chinmay Hegde.  On the dynamics of gradient descent for autoencoders.  In _International Conference on Artificial Intelligence and Statistics_ , 2019.
  * Pavlakos et al. (2017) Georgios Pavlakos, Xiaowei Zhou, Konstantinos Derpanis, and Kostas Daniilidis.  Coarse-to-fine volumetric prediction for single-image 3D human pose.  In _IEEE Conference on Computer Vision and Pattern Recognition_ , 2017\.
  * Radford et al. (2021) Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, Gretchen Krueger, and Ilya Sutskever.  Learning transferable visual models from natural language supervision.  OpenAI, 2021.
  * Redmon and Farhadi (2017) Joseph Redmon and Ali Farhadi.  YOLO9000: better, faster, stronger.  In _IEEE Conference on Computer Vision and Pattern Recognition_ , 2017\.
  * Ribeiro et al. (2016) Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin.  ”why should i trust you?” explaining the predictions of any classifier.  In _International Conference on Knowledge Discovery and Data Mining_ , 2016.
  * Shlezinger et al. (2020) Nir Shlezinger, Jay Whang, Yonina Eldar, and Alexandros Dimakis.  Model-based deep learning.  In _arXiv_ , 2020.
  * Simon and Elad (2019) Dror Simon and Michael Elad.  Rethinking the CSC model for natural images.  In _Neural Information Processing Systems_ , 2019.
  * Sulam et al. (2020) Jeremias Sulam, Aviad Aberdam, Amir Beck, and Michael Elad.  On multi-layer basis pursiot, efficient algorithms and convolutional neural netwroks.  _IEEE Transactions on Pattern Analysis and Machine Learning_ , 42(8):1968–1980, 2020.
  * Tolooshams et al. (2020) Bahareh Tolooshams, Andrew Song, Simona Temereanca, and Demba Ba.  Convolutional dictionary learning based auto-encoders for natural exponential-family distributions.  In _International Conference on Machine Learning_ , 2020.
  * Tolooshams et al. (2021) Bahareh Tolooshams, Sourav Dey, and Demba Ba.  Deep residual autoencoders for expectation maximization-inspired dictionary learning.  _IEEE Transactions on Neural Networks and Learning Systems_ , 32(6):2415–2429, 2021.
  * Worrall et al. (2017) Daniel Worrall, Stephan Garbin, Daniyar Turmukhambetov, and Gabriel Brostow.  Harmonic networks: Deep translation and rotation equivariance.  In _IEEE Conference on Computer Vision and Pattern Recognition_ , 2017\.

##  Appendix A Learned filters on CIFAR-10

We present filters learned on CIFAR-10 (without whitening) by the three
architectures. We choose to present filters from the first layer, as those
resemble edge detectors the most and thus are more straightforward to reason
about.

Figure 2: Filters learned using the baseline architecture. Figure 3: Filters
learned using $R_{90}$-CNN. Figure 4: Filters learned using $R_{60}$-CNN.

While all models seem to be learning similarly-looking filters, the
equivariant models do not need to “waste” computation on learning different
orientations. Indeed, if we observe Figure 2, the first filter from the top
and the third from the bottom of the first column seem to be rotated versions
of one another. In stark contrast, the third column of Figure 3 seems to
learning the elementary rotations of that same filter, without investing
resources on learning that information from the data. That is also observed in
the first column of Figure 4, which has even more rotated versions of that
same filter.


# Remarks on Generalized Calabi-Gray Manifolds

Teng Fei

###### Abstract

Generalized Calabi-Gray manifolds are non-Kähler complex manifolds with very
explicit geometry yet not being homogeneous. In this note, we demonstrate that
how generalized Calabi-Gray manifolds can be used to answer some questions in
non-Kähler geometry.

##  1 Introduction

Generalized Calabi-Gray manifolds are introduced in [Fei16] as new examples of
complex non-Kähler manifolds with very accessible geometry. Their
constructions depend on a nonconstant holomorphic map
$\varphi:Y\to\mathbb{C}\mathbb{P}^{1}$ and a hyperkähler manifold $M$. When
$\varphi$ is the Gauss map of an immersed minimal surface $Y$ in a flat
3-torus and that $M$ is of real dimension 4, we recover the classical
constructions due to Calabi [Cal58] and Gray [Gra69]. Generalized Calabi-Gray
manifolds have found their applications in solving the Hull-Strominger system
[Fei17, FHP21b, Fei20], the Anomaly flow [FHP21a], and building new models in
theoretical physics [CPS19] etc.

Noticeably, there has been an uprising interest in complex non-Kähler geometry
in recent years. Many new ideas and conjectures have been proposed, and many
exciting conferences are held. This particular piece of work is inspired by
author's discussion with his colleagues at one of the _Geometry & TACoS_
workshops.111For more information, see
https://events.dimai.unifi.it/tacos/sessions/geometry-and-physics-of-non-
kahler-calabi-yau/. It turns out that generalized Calabi-Gray manifolds (with
slight modification in some cases) can be used to answer many newly proposed
questions in non-Kähler geometry.

This paper is organized as follows. In Section 2, we first recall the
construction of the generalized Calabi-Gray manifolds and their relation with
twistor spaces. Then we derive some existence and non-existence results about
special metrics on generalized Calabi-Gray manifolds. In Section 3 and 4, we
demonstrate that generalized Calabi-Gray manifolds (with slight modification
if necessary) provide counterexamples to two questions in complex non-Kähler
geometry. Along the way, we also prove the existence of balanced metrics on
suspension of balanced manifolds with invariant volume forms and present a
brief list of interesting questions.

The author would like to thank all the participants of the _Geometry & TACoS_
workshop for their inspiring discussions. The author would also like to thank
the Simons Foundation for the generous funding, as this work is partially
supported by the Simons Collaboration Grant 853806.

##  2 The Generalized Calabi-Gray Manifolds

Let us recall the construction of generalized Calabi-Gray manifolds from
[Fei16], with some extensions to a broader setting.

Let $M$ by a compact hyperkähler manifold, namely that $M$ is equipped with a
Riemannian metric $g$ and three integrable complex structures $I$, $J$, and
$K$ satisfying the quaternion relation $IJK=-1$, such that all of $I$, $J$,
and $K$ together with $g$ define Kähler metrics on $M$. In fact, one gets an
$S^{2}=\mathbb{C}\mathbb{P}^{1}$ worth of Kähler structures on $M$ as follows.
Let $\omega_{I}$, $\omega_{J}$, and $\omega_{K}$ be the Kähler forms
associated to $I$, $J$, and $K$ respectively. For any point $(a,b,c)\in
S^{2}=\\{(a,b,c)\in\mathbb{R}^{3}:a^{2}+b^{2}+c^{2}=1\\}$, the complex
structure $aI+bJ+cK$ defines a Kähler metric on $M$, whose Kähler form is
simply $a\omega_{I}+b\omega_{J}+c\omega_{K}$.

It is very natural to consider the twistor space $Z$ of a hyperkähler manifold
$M$ (see for example [HKLR87]), which encodes the whole
$\mathbb{C}\mathbb{P}^{1}$-family of Kähler structures on $M$. As a smooth
manifold, the twistor space is simply the product
$Z=M\times\mathbb{C}\mathbb{P}^{1}$. However, $Z$ is equipped with a
tautological yet twisted integrable complex structure $\mathfrak{I}$, which
can be described as follows. For any point $z=(m,(a,b,c))\in
M\times\mathbb{C}\mathbb{P}^{1}=Z$, we have

$\mathfrak{I}_{z}=(aI+bJ+cK)_{m}\oplus j_{(a,b,c)},$

where $j$ is the canonical complex structure on $\mathbb{C}\mathbb{P}^{1}$. It
is well-known that one can recover all the information about $M$ from the
complex geometry of $Z$. The twistor space $Z$ has the following properties.

######  Proposition 2.1.

(See for example [HKLR87])

  1. (a).

The natural projection
$\pi:Z=M\times\mathbb{C}\mathbb{P}^{1}\to\mathbb{C}\mathbb{P}^{1}$ is
holomorphic.

  2. (b).

For any point $m\in M$, $\\{m\\}\times\mathbb{C}\mathbb{P}^{1}\subset
M\times\mathbb{C}\mathbb{P}^{1}=Z$ is a complex submanifold of
$(Z,\mathfrak{I})$.

  3. (c).

The bundle $\bigwedge^{2}F^{*}_{\pi}\otimes\pi^{*}\mathcal{O}(2)$ over $Z$ has
a global holomorphic section which defines a holomorphic symplectic form over
each fiber of $\pi$. Here $\mathcal{O}(1)$ is the hyperplane bundle over
$\mathbb{C}\mathbb{P}^{1}$ and $F^{*}_{\pi}$ is the relative cotangent bundle
of the holomorphic fibration $\pi:Z\to\mathbb{C}\mathbb{P}^{1}$.

Let $2m$ be the complex dimension of the hyperkähler manifold $M$, then we
know that $Z$ is a complex manifold of complex dimension $2m+1$. Regarding
Hermitian metrics on twistor spaces, we have the following well-known results.

######  Proposition 2.2.

  1. (a).

$Z$ admits balanced metrics, which are Hermitian metrics such that their
$2m$-th powers are closed.

  2. (b).

$Z$ does not admit any Kähler metric. In fact more generally, $Z$ does not
admit any astheno-Kähler metric.

###### Proof.

  1. (a).

The natural Hermitian metric $\omega$ (the product metric) on $Z$ takes the
form $\omega=\omega_{FS}+a\omega_{I}+b\omega_{J}+c\omega_{K}$, where
$\omega_{FS}$ is the Fubini-Study metric on $\mathbb{C}\mathbb{P}^{1}$. It is
straightforward to check that $\mathrm{d}(\omega^{2m})=0$.

  2. (b).

Recall that a Hermitian metric $\omega$ on a complex $n$-fold is called
astheno-Kähler if $i\partial\bar{\partial}(\omega^{n-2})=0$. Clearly Kähler
metrics are special cases of astheno-Kähler metrics. Suppose that $Z$ admits
an astheno-Kähler metric $\omega_{1}$ satisfying
$i\partial\bar{\partial}(\omega_{1}^{2m-1})=0$.Then by integration by part, we
get

$0=\int_{Z}i\partial\bar{\partial}(\omega_{1}^{2m-1})\wedge\omega=\int_{Z}\omega_{1}^{2m-1}\wedge
i\partial\bar{\partial}\omega.$ (1)

On the other hand, using the computation given in [Fei16, Theorem 5.4] or
[DLDM17, Theorem 3.2], we have

$i\partial\bar{\partial}\omega=\omega_{FS}\wedge\omega,$

hence $\omega_{1}^{2m-1}\wedge i\partial\bar{\partial}\omega$ is a positive
top form, contradiction!

∎

Now let $\varphi:Y\to\mathbb{C}\mathbb{P}^{1}$ be a nonconstant holomorphic
map from a compact complex manifold $Y$ to $\mathbb{C}\mathbb{P}^{1}$, the
base of the twistor family $\pi:Z\to\mathbb{C}\mathbb{P}^{1}$. Using
$\varphi$, one can pullback the twistor fibration $\pi$ to a holomorphic
fibration $p:X\to Y$, fitting into the following commuting square:

$\textstyle{X=\varphi^{*}Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\mathbb{C}\mathbb{P}^{1}}$

In this note, we shall call such an $X$ a _greatly generalized Calabi-Gray
manifold_ 222Greatly generalized Calabi-Gray manifolds are called _higher
analogs of twistor spaces_ in [LZ17].. Just like the twistor space $Z$, the
space $X$, as a smooth manifold, is simply the product $X=M\times Y$ with a
twisted complex structure. It is named after Calabi and Gray because when $Y$
is a minimal surface in a flat $T^{3}$ and $\varphi$ its Gauss map, and that
$M$ is a hyperkähler 4-manifold like $T^{4}$ or a K3 surface, the
corresponding $X$ recovers the classical construction of Calabi [Cal58] and
Gray [Gra69]. Greatly generalized Calabi-Gray manifold in this broad sense
also include the examples studied by LeBrun [LeB99] where $\varphi$ is a
polynomial map from $\mathbb{C}\mathbb{P}^{1}$ to itself.

Let $X$ be a greatly generalized Calabi-Gray manifold. Very much like
Proposition 2.1, we have

######  Proposition 2.3.

  1. (a).

The natural projection $p:X\to Y$ is holomorphic.

  2. (b).

For any point $m\in M$, $\\{m\\}\times Y\subset M\times Y=X$ is a complex
submanifold of $X$.

  3. (c).

The canonical bundle $K_{X}$ of $X$ can be computed as

$K_{X}\cong p^{*}(K_{Y}\otimes\varphi^{*}\mathcal{O}(-2m)).$ (2)

As a generalization of Proposition 2.2, the following statements hold.

######  Proposition 2.4.

(cf. [LZ17])

  1. (a).

If $Y$ admits a balanced metric, so does $X$.

  2. (b).

$X$ cannot admit an astheno-Kähler and a plurisubharmonic metric at the same
time. In particular, $X$ is always non-Kähler.

###### Proof.

  1. (a).

Let $\omega_{Y}$ be any balanced metric on $Y$. It is straightforward to check
that the product metric $\omega_{Y}+a\omega_{I}+b\omega_{J}+c\omega_{K}$ on
$X=M\times Y$ is a balanced metric.

  2. (b).

Recall that a Hermitian metric $\omega$ is called _plurisubharmonic_ if
$i\partial\bar{\partial}\omega\geq 0$ as a (2,2)-form.

First, we show that $X$ admits a plurisubharmonic metric if and only if $Y$
does so. If $X$ admits a plurisubharmonic metric, so do all complex
submanifolds of $X$. In particular, by Proposition 2.3(b), $Y$ also admits a
plurisubharmonic metric. On the other hand, let $\omega_{2}$ be a
plurisubharmonic metric on $Y$, then the metric
$\omega=\omega_{2}+a\omega_{I}+b\omega_{J}+c\omega_{K}$ is a plurisubharmonic
metric on $X$.

Next, let us assume that there is also an astheno-Kähler metric $\omega_{1}$
on $X$. Exactly the same calculation as in (1) yields the contradiction. Here
we need that

$i\partial\bar{\partial}\omega=i\partial\bar{\partial}\omega_{2}+\varphi^{*}\omega_{FS}\wedge(a\omega_{I}+b\omega_{J}+c\omega_{K})\geq\varphi^{*}\omega_{FS}\wedge(a\omega_{I}+b\omega_{J}+c\omega_{K}),$

and that $\varphi$ is not a constant so
$\omega_{1}^{2m-1}\wedge\varphi^{*}\omega_{FS}\wedge(a\omega_{I}+b\omega_{J}+c\omega_{K})$
is strictly positive on an open subset of $X$.

∎

######  Remark 2.5.

Complex manifolds admitting both an astheno-Kähler metric and a
plurisubharmonic metric form a strictly larger category than Kähler manifolds.
In fact, there exists a compact complex non-Kähler manifold of complex
dimension $n$ with a Hermitian metric $\omega$ satisfying
$i\partial\bar{\partial}(\omega^{k})=0$ for $k=1,2,\dots,n-1$. Such metrics
are studied in [FT11, She22].

######  Definition 2.6.

A _generalized Calabi-Gray manifold_ $X$ is a greatly generalized Calabi-Gray
manifold with the extra condition that

$K_{Y}\cong\varphi^{*}\mathcal{O}(2m).$ (3)

In view of (2), this condition holds if and only if the canonical bundle of
$X$ is holomorphically trivial, in which case $X$ is a non-Kähler Calabi-Yau
manifold.

##  3 The First Question

Let $X$ be a compact Kähler manifold. It is well-known that if the first Chern
class of $X$ vanishes in the real de Rham cohomology group, then its canonical
bundle $K_{X}$ must be torsion, meaning that a finite power of $K_{X}$ is
holomorphically trivial. It is natural to ask if a similar statement holds
when $X$ is non-Kähler, where we may work with a stronger assumption that the
first Chern class of $X$ vanishes in the Bott-Chern cohomology instead. It is
noted in [Tos15] that Magnússon [Mag17] gives an example of a compact non-
Kähler manifold with vanishing first Chern class in the Bott-Chern cohomology
whose canonical bundle has infinite order.

As completely dropping the Kähler condition leads to Magnússon's
counterexample [Mag17], one may ask what happens if we replace the Kähler
condition by a slightly weaker one. In this spirit, Alexandra Otiman asks
during the _Geometry & TACoS_ workshop that if there exists a compact complex
manifold with balanced metrics such that its first Chern class vanishes in the
Bott-Chern cohomology while its canonical bundle is not torsion.

It turns out such an example exists by considering greatly generalized Calabi-
Gray manifolds. To be more specific, we show that

######  Proposition 3.1.

For a generic compact Riemann surface of genus $g\geq 5$, there exists a
nonconstant holomorphic map $\varphi:Y\to\mathbb{C}\mathbb{P}^{1}$, such that
the associated greatly generalized Calabi-Gray manifold $X$ with $M$ being a
hyperkähler 4-manifold satisfies all the conditions in Otiman's question,
namely:

  1. (a).

$X$ admits a balanced metrics.

  2. (b).

$c_{1}(X)=0\in H^{1,1}_{BC}(X;\mathbb{R})$.

  3. (c).

$K_{X}$ has infinite order.

Because $Y$ is a Riemann surface, we know that Proposition 3.1(a) holds
automatically due to Proposition 2.4(a). For Proposition 3.1(b), we show that
it is essentially a topological condition:

######  Lemma 3.2.

Let $\varphi:Y\to\mathbb{C}\mathbb{P}^{1}$ be a holomorphic map from a compact
Riemann surface of genus $g$ to $\mathbb{C}\mathbb{P}^{1}$ and let $M$ be a
hyperkähler 4-manifold. Proposition 3.1(b) holds if and only if the
topological degree of $\varphi$ is $g-1$.

###### Proof.

As the Bott-Chern class of $c_{1}(X)$ is represented by the curvature of
$-K_{X}$, from (2) we know that

$c_{1}(X)\in p^{*}H^{1,1}_{BC}(Y;\mathbb{R}).$

Because $Y$ is a compact Riemann surface, we also know that
$H^{1,1}_{BC}(Y;\mathbb{R})\cong H^{2}(Y;\mathbb{Z})\otimes\mathbb{R}$,
therefore $c_{1}(X)=0\in H^{1,1}_{BC}(X;\mathbb{R})$ if and only if
$K_{Y}\cong\varphi^{*}\mathcal{O}(2m)$ topologically over $Y$. Note that $m=1$
since $M$ is a hyperkähler 4-manifold, we see immediately that
$K_{Y}\cong\varphi^{*}\mathcal{O}(2)$ topologically if and only if $\varphi$
is of degree $g-1$. ∎

Consequently, to prove Proposition 3.1, we only need to show:

######  Proposition 3.3.

For a generic compact Riemann surface of genus $g\geq 5$, there exists a
holomorphic map $\varphi:Y\to\mathbb{C}\mathbb{P}^{1}$ of degree $g-1$ such
that $K_{Y}\otimes\varphi^{*}\mathcal{O}(-2)$ is of infinite order in
$Pic^{0}(Y)$, the Picard variety of $Y$.

###### Proof.

The proof is based on the following facts from algebraic geometry.

  1. (a).

[ACGH85, p.206] For any compact Riemann surface $Y$ of genus $g\geq 5$, every
component of $W_{g-1}^{1}(Y)$ has dimension at least $g-4\geq 1$. Here
$W_{d}^{r}(Y)$ is the variety that parameterizes all holomorphic line bundles
of degree $d$ with $h^{0}\geq r+1$ on $Y$.

  2. (b).

[ACGH85, pp.372-373] For a generic compact Riemann surface $Y$ of genus $g\geq
5$, a generic line bundle in $W_{g-1}^{1}(Y)$ is base-point-free.

Consider a generic compact Riemann surface $Y$ of genus $g\geq 5$, we take a
generic line bundle $L\in W_{g-1}^{1}(Y)$. Since the linear series associated
to $L$ is base-point-free, the sections of $L$ give rise to a holomorphic map
$\varphi:Y\to\mathbb{C}\mathbb{P}^{1}$ such that
$\varphi^{*}\mathcal{O}(1)\cong L$. In particular, $\varphi$ is of degree
$g-1$ and $K_{Y}\otimes L^{-2}\in Pic^{0}(Y)$. As torsion elements in
$Pic^{0}(Y)$ are countable and the dimension of $W_{g-1}^{1}(Y)$ is at least
1, we see that a generic choice of $L$ would make $K_{Y}\otimes L^{-2}$ of
infinite order in $Pic^{0}(Y)$. ∎

A remaining question is that does Magnússon's example [Mag17] carry a balanced
metric? Magnússon's example falls in the broader construction called
``suspension'' that we shall describe. Let $M$ be a complex manifold and $f$ a
holomorphic automorphism of $M$. Let $\tau$ be a complex number with positive
imaginary part. We can define a $\mathbb{Z}^{2}$-action on $M\times\mathbb{C}$
by

$(1,0)\cdot(x,z)=(f(x),z+1),\quad(0,1)\cdot(x,z)=(x,z+\tau).$

It is clear that this $\mathbb{Z}^{2}$-action is free, so the quotient
manifold, denoted by $M_{f}$, is smooth and equipped with a projection map
over $C=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$. Moreover, the
$\mathbb{Z}^{2}$-action preserves the natural complex structure on
$M\times\mathbb{C}$ so the complex structure descends to $X$ and the
projection map $\pi:M_{f}\to C$ is a holomorphic fibration. Such a suspension
construction has been recently studied in Qin-Wang [QW18] and Fino-
Grantcharov-Verbitsky [FGV22].

Regarding the existence of balanced metrics on $M_{f}$, we have the following:

######  Proposition 3.4.

Suppose $M$ admits a balanced metric whose volume form is invariant under $f$,
then $M_{f}$ admits balanced metrics.

In the case of Magnússon's example, $M$ is a Kähler manifold with
holomorphically trivial canonical bundle, therefore any Ricci-flat Kähler
metric on $M$ satisfies the condition in Proposition 3.4. As a corollary, we
conclude that Magnússon's example provides an alternative manifold satisfying
all conditions asked by Otiman.

It is clear that as a smooth manifold, we have $M_{f}\cong
M^{\prime}_{f}\times S^{1}$, where $M^{\prime}_{f}$ is the quotient of
$M\times\mathbb{R}$ by the $\mathbb{Z}$-action

$1\cdot(x,z)=(f(x),z+1)$

coming together with a projection $\pi^{\prime}:M^{\prime}_{f}\to S^{1}$. It
follows that $\pi:M_{f}\to C$ decomposes as $\pi=(\pi^{\prime},\mathrm{id})$
under the obvious identifications $M_{f}\cong M^{\prime}_{f}\times S^{1}$ and
$C\cong S^{1}\times S^{1}$. Let $m$ be the complex dimension of $M$, we have
the following useful lemma.

######  Lemma 3.5.

Under the assumption of Proposition 3.4, for any small open interval $I\subset
S^{1}$, there exists a closed non-negative $(m,m)$-form on $M_{f}$ such that
it is strictly positive away from $\pi^{\prime-1}(I)\times S^{1}\subset
M^{\prime}_{f}\times S^{1}=M_{f}$.

###### Proof.

Without loss of generality, we may think of $S^{1}$ as the closed interval
$[0,1]$ with two end points glued together and that $0\in I$. In this way, we
may think of $M^{\prime}_{f}$ as the quotient of $M\times[0,1]$ by the
relation $(x,0)\sim(f(x),1)$. Let $(s,t)$ be the standard coordinates on
$S^{1}\times S^{1}=C$, and let $\rho(s)$ be a smooth nonnegative function on
$S^{1}$ vanishing in $I$.

Let $\omega$ be a balanced metric on $M$ such that $\omega^{m}$ is fixed by
$f$. Consider the following nonnegative $(m,m)$-form

$\Omega=\omega^{m}+\rho(s)\omega^{m-1}\mathrm{d}s\wedge\mathrm{d}t$

on $M\times[0,1]\times S^{1}$. Since $\omega^{m}$ is invariant under $f$ and
$\rho(0)=\rho(1)=0$, the above form descends to a non-negative $(m,m)$-form on
$M_{f}$, with the desired property in the lemma. We need $\omega$ to be a
balanced metric on $M$ to check that $\mathrm{d}\Omega=0$. ∎

To prove Proposition 3.4, we simply take two non-intersecting open intervals
$I_{1}$ and $I_{2}$ in $S^{1}$ and apply the above Lemma to construct
$\Omega_{1}$ and $\Omega_{2}$ respectively. Then
$\Omega=\Omega_{1}+\Omega_{2}$ is a closed positive $(m,m)$ form on $M_{f}$,
which is equivalent to a balanced metric. This construction gives a
constructive realization of a special case in [Mic82, Theorem 5.5] without
invoking verification of the ``topologically essential'' condition.

##  4 The Second Question and More

In his talk ``Non-Kähler manifolds with vanishing first Bott-Chern
class''333For the video, see https://www.youtube.com/watch?v=7911FB5alFg. at
_Geometry & TACoS_, Tosatti asks the following question:

Question. Let $X$ be a compact complex manifold with vanishing first Chern
class in the Bott-Chern cohomology. Is it true that holomorphic tensors on $X$
must be parallel with respect to Chern connection of a Chern-Ricci flat
metric?

The motivation of this question is that its Kähler version holds by applying
the Bochner technique. By considering generalized Calabi-Gray manifolds, we
see the answer to this question is negative. In particular, we have:

######  Proposition 4.1.

There exist generalized Calabi-Gray manifolds with Chern-Ricci flat balanced
metrics such that its nonzero holomorphic 1-forms (which exist in abundance)
are not parallel under any connection.

###### Proof.

In fact, we can consider the simplest kind of generalized Calabi-Gray
manifolds introduced in [Fei16, FHP21b]. These are generalized Calabi-Gray
manifolds associated to $\varphi:Y\to\mathbb{C}\mathbb{P}^{1}$, where $Y$ is a
Riemann surface of genus $g$, and that the hyperkähler factor $M$ is either a
K3 surface or a flat 4-torus. The canonical bundle $K_{X}$ of $X$ is
holomorphically trivial as $\varphi$ satisfies condition (3) that

$K_{Y}\cong\varphi^{*}\mathcal{O}(2).$

It is shown in [Fei16] that such a construction exists for any $g\geq 3$ and
that the natural Hermitian metric on $X$ is balanced and Chern-Ricci-flat.
Moreover, it is also shown that all holomorphic 1-forms on $X$ are pullbacks
of holomorphic 1-forms on $Y$, which forms a complex vector space of dimension
$g$. Since these are abelian differentials on a Riemann surface of high genus,
they all have zeroes and cannot be parallel under any connection. ∎

Let me end this paper by proposing and formulating some questions about
(greatly) generalized Calabi-Gray manifolds of interest.

  1. (a).

How to compute all the Hodge numbers of a generalized Calabi-Gray manifold?
Some of them are discussed in [Fei16]. This question is highly related to
computation of Hodge numbers of a twistor space, which is discussed in [ES93].

  2. (b).

This is a question asked my Misha Verbitsky. Consider the greatly generalized
Calabi-Gray manifold $X$ associated to
$\varphi:T^{2}\to\mathbb{C}\mathbb{P}^{1}$ with $M=T^{4}$. In this case, $X$
is topologically a 6-torus with a twisted complex structure such that its
first Chern class is nonpositive but not zero. Is it true that any complex
structure on $T^{6}$ with zero first Chern class must be Kähler?

## References

  * [ACGH85] E. Arbarello, M. Cornalba, P.A. Griffiths, and J.D. Harris.  Geometry of Algebraic Curves Vol. I, volume 267 of Grundlehren der Mathematischen Wissenschaften.  Springer, 1985.
  * [Cal58] E. Calabi.  Construction and properties of some 6-dimensional almost complex manifolds.  Transactions of the American Mathematical Society, 87(2):407–438, 1958.
  * [CPS19] Z. Chen, T. Pantev, and E. Sharpe.  Landau-Ginzburg models for certain fiber products with curves.  Journal of Geometry and Physics, 137:204–211, 2019.
  * [DLDM17] G. Deschamps, N. Le Du, and C. Mourougane.  Hessian of the natural Hermitian form on twistor spaces.  Bulletin de la société mathématique de France, 145(1):1–27, 2017.
  * [ES93] M.G. Eastwood and M.A. Singer.  The Fröhlicher spectral sequence on a twistor space.  Journal of Differential Geometry, 38(3):653–669, 1993.
  * [Fei16] T. Fei.  A construction of non-Kähler Calabi-Yau manifolds and new solutions to the Strominger system.  Advances in Mathematics, 302:529–550, 2016.
  * [Fei17] T. Fei.  Some torsional local models of heterotic strings.  Communications in Analysis and Geometry, 25(5):941–968, 2017.
  * [Fei20] T. Fei.  Generalized Calabi-Gray geometry and heterotic superstrings.  In Proceedings of the International Consortium of Chinese Mathematicians, 2017: First Annual Meeting, pages 261–282. International Press, 2020.
  * [FGV22] A.M. Fino, G.V. Grantcharov, and M. Verbitsky.  Special Hermitian structures obtained from hyperbolic suspensions.  arXiv: 2208.12168, 2022.
  * [FHP21a] T. Fei, Z.-J. Huang, and S.F. Picard.  The Anomaly flow over riemann surfaces.  International Mathematics Research Notices, 2021(3):2134–2165, 2021\.
  * [FHP21b] T. Fei, Z.-J. Huang, and S.F. Picard.  A construction of infinitely many solutions to the Strominger system.  Journal of Differential Geometry, 117(1):23–39, 2021.
  * [FT11] A.M. Fino and A. Tomassini.  On astheno-Kähler metrics.  Journal of the London Mathematical Society, 83(2):290–308, 2011\.
  * [Gra69] A. Gray.  Vector cross products on manifolds.  Transactions of the American Mathematical Society, 141:465–504, 1969.
  * [HKLR87] N. J. Hitchin, A. Karlhede, U. Lindström, and M. Roček.  Hyperkähler metrics and supersymmetry.  Communications in Mathematical Physics, 108(4):535–589, 1987.
  * [LeB99] C. LeBrun.  Topology versus Chern numbers for complex 3-folds.  Pacific Journal of Mathematics, 191(1):123–131, 1999.
  * [LZ17] H. Lin and T. Zheng.  Higher dimensional generalizations of twistor spaces.  Journal of Geometry and Physics, 114:492–505, 2017.
  * [Mag17] G.Th. Magnússon.  Automorphisms and examples of compact non-Kähler manifolds.  Mathematica Scandinavica, 121(1):49–56, 2017.
  * [Mic82] M.-L. Michelsohn.  On the existence of special metrics in complex geometry.  Acta Mathematica, 149(1):261–295, 1982.
  * [QW18] L.-Z. Qin and B.-T. Wang.  A family of compact complex and symplectic Calabi-Yau manifolds that are non-Kähler.  Geometry & Topology, 22(4):2115–2144, 2018.
  * [She22] X.S. Shen.  A Chern-Calabi flow on Hermitian manifolds.  The Journal of Geometric Analysis, 32(4):129, 2022.
  * [Tos15] V. Tosatti.  Non-Kähler Calabi-Yau manifolds.  In Analysis, Complex Geometry, and Mathematical Physics: In Honor of Duong H. Phong, volume 644 of Contemporary Mathematics, pages 261–277. AMS, 2015.