# Self Meta Pseudo Labels: Meta Pseudo Labels Without The Teacher

1st Kei-Sing Ng  HKU Business School
The University of Hong Kong
Hong Kong, China
<EMAIL_ADDRESS>2nd Qingchen Wang  HKU Business School
The University of Hong Kong
Hong Kong, China
<EMAIL_ADDRESS>
###### Abstract

We present Self Meta Pseudo Labels, a novel semi-supervised learning method
similar to Meta Pseudo Labels [1] but without the teacher model. We introduce
a novel way to use a single model for both generating pseudo labels and
classification, allowing us to store only one model in memory instead of two.
Our method attains similar performance to the Meta Pseudo Labels method while
drastically reducing memory usage.

###### Index Terms:

semi-supervised learning, deep learning, neural network, machine learning

##  I Introduction

Semi-supervised learning methods are essential techniques for many real-world
machine learning problems as many companies have few and limited labeled data
for model training. A recent research—Meta Pseudo Labels [1]improves the
pseudo labeling method and has achieved new state-of-the-art performance on
image classification problems [1, 2]. The method updates the teacher model
based on the performance of the student model to generate better pseudo labels
for training [1]. Meta Pseudo Labels works well with only 1,000 labeled
examples and 60,300 unlabeled examples on the Street View House Numbers (SVHN)
dataset [1, 3], showing that semi-supervised learning can achieve better
results than supervised learning.

Despite the strong performance of Meta Pseudo Labels, it requires storing two
models in memory—a teacher model and a student model—during the training
process. Some models can be prohibitively large with today’s memory size, such
as Efficient Nets [4] and GPT-3 model [5], making it difficult to store two
models in memory. Other semi-supervised learning approaches [6]can also
achieve state-of-the-art performance using knowledge distillation [7], but
they also require storing a large and over-parameterized teacher model or
additional neural network layers in VRAM during training [8]. It is essential
to find a way to reduce VRAM usage during training while still achieve strong
performance.

In this paper, we introduce a novel semi-supervised learning method as a
variant of Meta Pseudo Labels. We design a mechanism for a model to learn from
self-generated pseudo labels and improve quality of learning during the
process. This model learns from both pseudo-labeled data and real labeled
data.

##  II Related Works

###  II-A Consistency Regularization

There are many other semi-supervised learning approaches. Consistency
regularization methods assume a model gives similar predictions for an
unlabeled data sample and its perturbed version [2]. Given a neural network
output of a data point, the objective of consistency regularization is to
minimize the distance between it and the neural network output of its
perturbed version. Some common distance measures include mean squared error
and Kullback-Leiber divergence.

One example of consistency regularization is the Π-model by Samuli and Timo
[9]. They use the stochastic nature of some neural network techniques to
generate slightly different network outputs. For example, the dropout
technique removes some neuron activation outputs randomly, making the final
network output a stochastic variable. For every epoch of training, predictions
are generated twice for each data example. The model is trained to minimize
the difference between two outputs and the supervised loss on labeled data.

Another example is Unsupervised Data Augmentation (UDA) [10]. It improves the
performance of a model by minimizing the consistency loss between predictions
on original unlabeled examples and their noised versions. To generate the
noised examples, UDA uses advanced data augmentations from supervised
learning. It is shown to achieve an error rate of 4.20% on the IMDb text
dataset with only 20 labeled samples.

###  II-B Generative Models

Standard generative model (M1): Generative Models generate new examples from
the input data distribution and learn the feature representations during
training. This method transfers the learned feature representations to semi-
supervised tasks and estimates the joint distribution of input data and
labels. Because generative models do not require labels for input data during
the training process, they can utilize a large amount of unlabeled data to
learn transferable features for semi-supervised tasks.

Extended generative model (M2): During the training process of latent features
in the M1 model, the labels of input data are not utilized. The training
process of M2 model is similar to the one of M1 model but with an additional
supervised loss if the labels are available. The labels are treated as latent
variables if they are not available.

Stacked generative model (M1+M2): The stacked generative model method combines
the M1 and M2 approaches. The method first trains the M1 model to learn the
latent features. It then trains the M2 model using the learned latent features
from the M1 model as a new representation of the input data.

Variational Autoencoder for Semi-supervised Learning: Variational Autoencoder
(VAE) [11] is a famous autoencoder model architecture. An autoencoder is a
generative neural network model, with the objective of reconstructing the
input data. It consists of an encoder and a decoder. The encoder projects the
input data to a latent space, and the decoder reconstructs the input data from
the latent vector. A VAE has an additional objective function to enforce that
the latent vectors follow a unit Gaussian distribution. For a classification
problem, an M2 VAE has an extra neural network classifier.

Generative Adversarial Networks for Semi-supervised Learning: A Generative
Adversarial Network (GAN) [12] has a generator model and a discriminator
model. The generator model generates fake images, and is trained to generate
images that are indistinguishable from the real images. A discriminator model
takes both real and fake images as inputs, and its objective is to distinguish
the fake data inputs from real ones. During the training process, the
generator learns to generate better images, and the discriminator learns
better representations of the input data. Some GANs such as BiGANs can
generate highly realistic images [13].

##  III Self Meta Pseudo Labels (SMPL)

We present a novel method for semi-supervised learning. We mainly have two
contributions:

  * •

We present a variant of Meta Pseudo Labels. Meta Pseudo Labels is currently
the state-of-the-art pseudo labeling method. Our variant reduces VRAM usage by
19% during training.

  * •

We introduce a novel way to train a model for semi-supervised learning, using
a two-step gradient update. The second update is based on an evaluation of the
performance of the model in the first update.

To begin with, we first present the background of our method.

###  III-A Background: Pseudo Labeling

Pseudo labeling is a semi-supervised approach for deep neural networks.
Conventional supervised deep neural network training methods update model
parameters using a back-propagation algorithm with labeled data. We cannot
utilize unlabeled data in vanilla supervised learning because the labels are
not available. In a pseudo labeling approach, we have a teacher model and a
student model. The teacher model generates pseudo labels for unlabeled data.
The method then trains the student model with both labeled and unlabeled data
simultaneously [14]. With pseudo labels, the method utilizes more unlabeled
data during the training process to improve the final prediction result. One
problem with conventional pseudo labeling is that the pre-trained teacher
model is fixed during the training process. The student model may overfit on
incorrect pseudo labels and results in confirmation bias [15].

To address the problem, the teacher model of Meta Pseudo Labels is not fixed
during the training process[1]. The teacher model receives feedback on the
student model’s performance on the labeled examples, and is updated
accordingly. The model learns to generate better pseudo labels, thus allowing
the student model to converge better. This key modification makes Meta Pseudo
Labels a new state-of-the-art with top-1 accuracy of 90.2% on the ImageNet
dataset [16].

Figure 1: The difference between Pseudo Labels, Meta Pseudo Labels, and Self
Meta Pseudo Labels. Left: Pseudo Labels, where the pre-trained teacher model
is fixed to generate pseudo labels. Middle: Meta Pseudo Labels, where the
teacher model is trained and updated during training. The student model is
trained with the pseudo labels generated by the teacher model. Right: Self
Meta Pseudo Labels, where the teacher model and the student model are the same
models. The model is updated based on its performance on labeled examples of
the previous update.

###  III-B Self Meta Pseudo Labels

One drawback of Meta Pseudo Labels is using more VRAM than vanilla pseudo
labeling. This method stores the teacher model in VRAM along with the student
model during the training process. Our question is, do we need an external
agent to generate the pseudo labels and evaluate the model? Human beings have
the capability to act according to their hypotheses, observe, and adjust their
actions. We want to simulate the process and update a model. We named it Self
Meta Pseudo Labels.

Unlike conventional pseudo labeling, we do not have an extra teacher model. To
generate the pseudo labels, we pass a mini-batch of unlabeled data to the
student model and get probabilistic predictions. We filter predictions of low
confidence, and then convert the probabilistic predictions to hard pseudo
labels. Every epoch contains two gradient updates. The first update is
standard back-propagation using the hard pseudo labels [17], and the second
update depends on the performance of the first update:

  * •

The first SGD:
$\theta^{{}^{\prime}}_{M}=\theta_{M}-\eta_{1}\cdot\nabla{L}_{1}(\theta_{M})$

  * •

The second SGD:
$\theta^{{}^{\prime\prime}}_{M}=\theta^{{}^{\prime}}_{M}-\eta_{2}\cdot\nabla{L}_{2}(\theta_{M}-\eta_{1}\cdot\nabla{L}_{1}(\theta_{M}))$

where $M$ is the neural network in Self Meta Pseudo Labels and $\theta_{M}$ is
its parameters.

We let ($x_{l}$, $y_{l}$) be a batch of labeled examples and their
corresponding labels. We use $x_{u}$ to denote a batch of unlabeled examples.
$x_{ua}$ is the augmented version of $x_{u}$. We use $p$ to denote the softmax
prediction from the model and $H(p)$ to denote the hard labels.

Input: Labeled data $x_{l}$, $y_{l}$ and unlabeled data $x_{u}$.

1 Initialize $\theta_{M}$.

2 for  _$k=0$ to $N-1$_ do

3 Sample a batch of unlabeled examples ($x_{u}$,$x_{ua}$) and a batch of
labeled examples ($x_{l}$, $y_{l}$)

4 Compute the hard pseudo labels $H(p_{u}$).

5 Compute the loss $\mathcal{L}_{1}(\theta_{M})$ and gradient with the pseudo
labels.

6 Update the model:
$\theta^{{}^{\prime}}_{M}=\theta_{M}-\eta_{1}\cdot\nabla{L}_{1}(\theta_{M})$

7

8 Compute the new loss $\mathcal{L}_{2}(\theta^{{}^{\prime}}_{M})$ and
gradient with the pseudo labels and ($x_{l}$, $y_{l}$).

9 Update the model:
$\theta^{{}^{\prime\prime}}_{M}=\theta^{{}^{\prime}}_{M}-\eta_{2}\cdot\nabla{L}_{2}(\theta^{{}^{\prime}}_{M})$

10

11 end for

12return _$\theta_{M}$_

Algorithm 1 The Self Meta Pseudo Labels method

The first objective function $\mathcal{L}_{1}(\theta_{M})$ is a cross-entropy
loss function $CE$ with labeling smoothing:

$\mathcal{L}_{1}(\theta_{M})=CE(H(p_{u}),p_{ua})$ (1)

With label smoothing, we train the model to predict $(1-\alpha)$ instead of 1
for the correct class and $\alpha/(n-1)$ for the other classes where $\alpha$
is a small positive number and $n$ is the total number of classes.

The second objective function consists of two parts:

$\mathcal{L}_{2}(\theta^{{}^{\prime}}_{M})=\mathcal{L}_{UDA}+\lambda\mathcal{L}_{MPL}$
(2)

where $\mathcal{L}_{UDA}$ is the unsupervised data augmentation loss and
$\mathcal{L}_{MPL}$ is the semi-supervised loss. $\lambda$ is a constant to
control the ratio between the two terms. For the unsupervised data
augmentation loss:

$\mathcal{L}_{UDA}=CE(y)+\beta_{k}E[{-p_{u}log(p_{ua})])}$ (3)

$\beta_{k}=\beta_{0}*Min(1,(k+1)/a)$ (4)

$\beta_{k}$ is a warm-up variable to control the magnitude. It gradually
increases until the total number of steps reaches a constant $a$.
$\mathcal{L}_{UDA}$ is masked and only predictions with high confidence are
used. $CE(y)$ is the cross-entropy loss for the labeled examples. For the
semi-supervised loss:

$\mathcal{L}_{MPL}=\Delta CE*CE(H(p_{u}),p_{u})$ (5)

We use $\mathcal{L}_{MPL}$ to evaluate the performance of the model after the
first gradient update. $\mathcal{L}_{MPL}$ equals to the dot product $\Delta
CE$ times the cross-entropy loss of $p_{u}$ and the hard pseudo labels. The
dot product $\Delta CE$ is the difference in the cross-entropy loss of labeled
examples before and after the first gradient update. In practice we subtract
the moving average of $\Delta CE$ from $\Delta CE$ when we compute
$\mathcal{L}_{MPL}$ to reduce the variance.

###  III-C Augmentation Strategies

We use different data augmentation policies such as Unsupervised Data
Augmentation, AutoAugment [18] and RandAugment [19] in our method to enhance
the performance. We use UDA as an extended objective when training in the
teacher role.

We combine data augmentation policies from AutoAugment and RandAugment in our
method. AutoAugment is a method that automatically searches for combinations
of data augmentation policies to improve the accuracy of a classification
model. In every mini-batch, sub-policies are randomly chosen for each data
example. It composes of many sub-policies such as translation, rotation, and
shearing. It shows an improvement in accuracy on the CIFAR-10, CIFAR-100 [20],
SVHN, and ImageNet datasets. RandAugment is another data argumentation method
that finds data argumentation policies with a reduced search space [19]. It
removes the need for a separate search phase, and can be applied to various
models and dataset sizes. We include a list of all the data augmentation
policies used in our experiments in Appendix A.

##  IV Experiments

###  IV-A Toy Experiment

To better understand the method, we conduct a toy experiment on a small-scale
dataset. We then compare the result between Self Meta Pseudo Labels and
conventional supervised learning.

####  IV-A1 Dataset

We use the moon dataset from Scikit-learn [21]. It is a simple toy dataset
having 2d data points with two interleaving half circles on a 2d plane. We
randomly generate 2,000 examples into two classes. We keep 6 examples as label
examples randomly and use the rest as unlabeled data. The task of the
experiment is to classify the examples correctly.

####  IV-A2 Training details

We remove all data augmentations and regularization losses. We use a simple
neural network model with two fully connected hidden layers. Each layer has 8
units. We use a ReLU activation function [22], and train the model with an
initial learning rate of 0.1 for 1,000 steps. We use the last checkpoint as
the final checkpoint and evaluate it on the whole dataset. For the supervised
learning experiment, We keep the hyperparameters unchanged but we train the
model with the labeled examples only.

To better explain our approach, we illustrate the gradient descent process
with an example. We project the cost function to a 3d space in Figure 2. The
red arrow represents a gradient descent of vanilla supervised learning. It
moves towards the global minimum. In SMPL, we have two gradient updates in
every training epoch, represented with two blue arrows. The first update moves
away from the global minimum while the second update corrects the direction.
The final result is closer to the global minimum compared to vanilla
supervised learning.

Figure 2:  Left: A conventional gradient descent (red arrow). Middle: A two-
step gradient descent of Self Meta Pseudo Labels (blue arrows). Right: We
illustrate the two gradient descents in one figure.

####  IV-A3 Results

We achieve 81.15% and 76.55% accuracies with Self Meta Pseudo Labels and the
supervised learning respectively. We achieve 4.6% more in accuracy with Self
Meta Pseudo Labels than supervised learning using the same model
infrastructure. Self Meta Pseudo Labels trains a better model by utilizing the
information of unlabeled data during updates. We visualize the result in
Figure 3.

Figure 3:  We conduct a toy experiment on the moon dataset. The result of
supervised learning is on the left. The result of Self Meta Pseudo Labels is
on the right. We color the two categories of examples into orange and dark
blue. The labeled examples are marked in white. The model separates data
points into red and blue.

###  IV-B CIFAR-10-4K, CIFAR100-10K, and SVHN-1K Experiments

####  IV-B1 Datasets

We run experiments on three standard datasets: CIFAR-10-4K, CIFAR-100-10K and
SVHN-1K. The CIFAR-10 and CIFAR-100 datasets contain 32x32 colour images in 10
classes and 100 classes respectively. The SVHN dataset contains images of
digits from real-world house numbers photos. All three datasets contain less
than 100k image samples which is comparable to many real-world machine
learning problems. Many companies contain a limited amount of training samples
and few labeled data. They require an efficient semi-supervised learning
method.

In our experiments, we keep a portion of labeled data and use the rest as
unlabeled data. We keep 4,000 labeled images and use 46,000 unlabeled images
in the CIFAR-10-4K dataset. For the CIFAR-100-10K dataset, we keep 10,000
labeled images, and use 40,000 images as unlabeled data for a total of 100
classes. For the SVHN-1K dataset, we keep 1,000 labeled images, and use about
603,000 images as unlabeled data for a total of 10 classes.

####  IV-B2 Baselines

Since we present a variant of Meta Pseudo Labels, we directly compare the
performance between Meta Pseudo Labels and our training method. We re-
implement Meta Pseudo Labels, and use a WideResNet-28-2 [23] neural network
model in our training. We compare the final results on the CIFAR-10-4K and
SVHN-1K datasets. We use a same set of hyperparameter settings and
augmentation methods for the two methods. The main difference between the two
methods is we use two models in Meta Pseudo labels and one model in Self Meta
Pseudo labels. We also train a WideResNet-28-2 neural network model on the
CIFAR-100-10K dataset with Self Meta Pseudo Labels.

For a fair comparison, we only compare Self Meta Pseudo Labels against methods
that use the same model architectures. It is known that larger architectures
can possibly improve any deep learning method’s performance. Our method can
also be used along with many other deep learning optimization techniques such
as a different optimizer, neural architecture search, etc.

####  IV-B3 Training details

Specifically, we have two stochastic gradient descent steps in every training
epoch. In step one, we first draw a batch of labeled data ($x_{l}$, $y_{l}$)
and a batch of unlabeled data $x_{u}$ stochastically. For every batch of
unlabeled data $x_{u}$, we generate a batch of augmented version $x_{ua}$. We
then generate hard pseudo labels using the model predictions of $x_{l}$,
$x_{u}$ and $x_{ua}$. We compute the gradient and the first objective function
$\mathcal{L}_{1}(\theta_{M})$. We also calculate the cross-entropy loss for
($x_{l}$, $y_{l}$) and save it for the step two computation. We update the
model’s parameters $\theta_{M}$ using a conventional stochastic gradient
descent method. We use $\theta^{{}^{\prime}}_{M}$ to denote the updated
parameters.

In step two, We generate new predictions $p{}^{\prime}$ on $x_{l}$ and $x_{u}$
using the updated model. We then update the model based on the semi-supervised
loss function $\mathcal{L}_{MPL}$ and the unsupervised data augmentation loss
$\mathcal{L}_{UDA}$. We clip the gradient norm at 0.8.

After the training phase, we finetune the best checkpoint on labeled data to
improve the accuracy. In our finetuning process, we retrain the model with the
labeled data, using stochastic gradient descent with a fixed learning rate of
5e-6 to update the model. We retrain the model for 8,000 epochs with a batch
size of 512. Following the technique of Meta Pseudo Labels, we return the
model at the final checkpoint because the number of labeled samples is limited
for all three datasets.

TABLE I: The Hyper-parameters for Self Meta Pseudo Labels Hyper-parameter | CIFAR-10 | CIFAR-100 | SVHN
---|---|---|---
$\beta_{0}$ | 8 | 8 | 8
$a$ | 5,000 | 5,000 | 5,000
$\alpha$ | 0.15 | 0.15 | 0.15
Initial learning rate | 0.05 | 0.05 | 0.0025
Batch size | 128 | 128 | 128
Dropout rate on last layer | 0.5 | 0.5 | 0.5

####  IV-B4 Results

We were unable to successfully re-run the Meta Pseudo Labels experiments with
the official released code and instructions [24], see details in Appendix B.
We replicate our version of Meta Pseudo Labels using Pytorch on the
CIFAR-10-4K dataset and achieve an accuracy of 95.87% compared to 96.11% in
the original paper, taking 0.41s for one training epoch and 6,861MiB VRAM on
average. We achieve an accuracy of 95.91% using Self Meta Pseudo Labels on the
CIFAR-10-4K dataset with the same set of hyperparameters and setup. We spend
0.44s on one training epoch and use 5,537MiB VRAM on average, achieving a
19.3% reduction in VRAM usage. We reduce one generation of pseudo labels for
every training epoch compared to Meta Pseudo Labels. The training time is
longer because, in the current version of our back-propagation implementation,
it takes longer to keep the computation graph in memory after the first
stochastic gradient descent step. On the SVHN-1K dataset, we achieve 94.55%
and 95.69% accuracy with the Meta Pseudo Labels method and our method
respectively, and also obtain a 19.1% reduction in VRAM usage. We achieve an
accuracy of 78.32% on the CIFAR-100-4K dataset.

TABLE II: A comparison of testing accuracy on datasets Method | CIFAR-10 | CIFAR-100 | SVHN
---|---|---|---
MixMatch [25] | 93.76% | 74.12% | 96.73%
FixMatch(CTA) [26] | 95.69% | 76.82% | 97.64%
SimPLE [27] | 94.95% | 78.11% | 97.54%
Meta Pseudo Labels | 96.11% | N/A | 98.01%
Meta Pseudo Labels | 95.87% | N/A | 94.55%
(our re-implemetation) |  |  |
Self Meta Pseudo Labels | 95.91% | 78.32% | 95.69%
TABLE III: A comparison of the VRAM usage on datasets Method | CIFAR-10 | CIFAR-100 | SVHN
---|---|---|---
Meta Pseudo Labels | 6,861MiB | N/A | 6,862MiB
(our re-implemetation) |  |  |
Self Meta Pseudo Labels | 5,537MiB | 19,697MiB | 5,549MiB

##  V Conclusion

In this paper we present a novel semi-supervised learning method as a variant
of Meta Pseudo Labels that combines the teacher model and student model into
one single model. We present a novel way to train a model with a two-step
gradient update. The large reduction in VRAM usage compared with Meta Pseudo
Labels is critical in solving many practice problems because many models are
too large to fit in a single commercial GPU. Specifically, we can utilize a
larger model with the reduction in VRAM usage under the same VRAM capacity. We
believe the method can be applied to many vector inputs other than images.

##  Appendix A Data Augmentation Policies

Table IV is a list of data augmentation policies we use in the experiments. We
refer readers to [18] for the detailed descriptions of these data augmentation
policies.

TABLE IV: Data augmentation policies that we used in our method AutoContrast
---
Brightness
Color
Contrast
Cutout
Equalize
Invert
Sharpness
Posterize
Solarize
Rotate
ShearX
ShearY
TranslateX
TranslateY

##  Appendix B Re-running The Meta Pseudo Labels Experiments With The Official
Code

We use the code and instructions from the repository
https://github.com/google-research/google-
research/tree/master/meta_pseudo_labels [24].

We create a variable for the project’s ID in Google cloud shell.

    export PROJECT_ID=project-id

We configure the project and log into the instance.

    gcloud config set project $PROJECT_ID
    gcloud compute tpus execution-groups create \
    --name=instance-1 \
    --zone=us-central1-a \
    --disk-size=300 \
    --machine-type=n1-standard-16 \
    --tf-version=2.6.0 \
    --accelerator-type=v3-8

    gcloud compute ssh instance-1 \
    --zone=us-central1-a

We go to the target folder and run the following command from the README.md.

    python -m main.py \
      --task_mode="train" \
      --dataset_name="cifar10_4000_mpl" \
      --output_dir="path/to/the/output/dir" \
      --model_type="wrn-28-2" \
      --log_every=100 \
      --master="path/to/the/tpu/worker" \
      --image_size=32 \
      --num_classes=10 \
      --optim_type="momentum" \
      --lr_decay_type="cosine" \
      --save_every=1000 \
      --use_bfloat16 \
      --use_tpu \
      --nouse_augment \
      --reset_output_dir \
      --eval_batch_size=64 \
      --alsologtostderr \
      --running_local_dev \
      --train_batch_size=128 \
      --uda_data=7 \
      --weight_decay=5e-4 \
      --num_train_steps=300000 \
      --augment_magnitude=16 \
      --batch_norm_batch_size=256 \
      --dense_dropout_rate=0.2 \
      --ema_decay=0.995 \
      --label_smoothing=0.15 \
      --mpl_student_lr_wait_steps=3000 \
      --uda_steps=5000 \
      --uda_temp=0.7 \
      --uda_threshold=0.6 \
      --uda_weight=8

The program stops because of the missing class ’autocontrast’.

    NameError: name ’autocontrast’ is not defined

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Hodge theory on nearly Kähler manifolds

Misha Verbitsky111Misha Verbitsky is partially supported by RFBR grant
10-01-93113-NCNIL-a, RFBR grant 09-01-00242-a, and Science Foundation of the
SU-HSE award No. 10-09-0015.

<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

Let $(M,I,\omega,\Omega)$ be a nearly Kähler 6-manifold, that is, an
$SU(3)$-manifold with (3,0)-form $\Omega$ and Hermitian form $\omega$ which
satisfies $d\omega=3\lambda\operatorname{Re}\Omega$,
$d\operatorname{Im}\Omega=-2\lambda\omega^{2}$, for a non-zero real constant
$\lambda$. We develop an analogue of the Kähler relations on $M$, proving
several useful identities for various intrinsic Laplacians on $M$. When $M$ is
compact, these identities give powerful results about cohomology of $M$. We
show that harmonic forms on $M$ admit a Hodge decomposition, and prove that
$H^{p,q}(M)=0$ unless $p=q$ or $(p=1,q=2)$ or $(p=2,q=1)$.

###### Contents

  1. 1 Introduction
    1. 1.1 Nearly Kähler 6-manifolds
    2. 1.2 Nearly Kähler manifolds: history and classification
    3. 1.3 Examples of nearly Kähler manifolds
    4. 1.4 Nearly Kähler manifolds in geometry and physics
    5. 1.5 Local structure of nearly Kähler 6-manifolds
    6. 1.6 Hodge decomposition of the de Rham differential and intrinsic Laplacians
  2. 2 Algebraic differential operators on the de Rham algebra
    1. 2.1 Algebraic differential operators: basic properties
    2. 2.2 Algebraic differential operators on $\Lambda^{*}(M)$
    3. 2.3 An algebraic differential operator and its adjoint
  3. 3 Kähler identities on nearly Kähler manifolds
    1. 3.1 The operators $\partial$, $\overline{\partial}$ on almost complex manifolds
    2. 3.2 The Nijenhuis operator squared
  4. 4 The de Rham Laplacian via $\Delta_{\partial}$, $\Delta_{\overline{\partial}}$, $\Delta_{\partial-\overline{\partial}}$
    1. 4.1 An expression for $\Delta_{d}$
    2. 4.2 $N=\lambda[L_{\Omega},\Lambda_{\omega}]$
    3. 4.3 Commutator relations for $N$, $\overline{N}$, $\partial$, $\overline{\partial}$
    4. 4.4 The Hodge decomposition of the Laplacian
  5. 5 Kähler-type identities for $N$, $\overline{N}$
  6. 6 Harmonic forms on nearly Kähler manifolds
    1. 6.1 Harmonic forms and the $\partial$, $\overline{\partial}$-Laplacians
    2. 6.2 The Hodge decomposition on cohomology
  7. 7 Appendix: Hodge theory on orbifolds

##  1 Introduction

###  1.1 Nearly Kähler 6-manifolds

Nearly Kähler manifolds (also known as $K$-spaces or almost Tachibana spaces)
were defined and studied by Alfred Gray ([Gr1], [Gr2], [Gr3], [Gr4]) in the
general context of intrinsic torsion of $U(n)$-structures and weak holonomies.
An almost complex Hermitian manifold $(M,I)$ is called nearly Kähler if
$\nabla_{X}(I)X=0$, for any vector field $X$ ($\nabla$ denotes the Levi-Civita
connection). In other words, the tensor $\nabla\omega$ must be totally skew-
symmetric, for $\omega$ the Hermitian form on $M$. If $\nabla_{X}(\omega)\neq
0$ for any non-zero vector field $X$, $M$ is called strictly nearly Kähler.

Using deep results of Kirichenko and Cleyton-Swann ([Ki], [CS]) P.-A. Nagy in
[N2] has shown that that any strictly nearly Kähler manifold is locally a
product of locally homogeneous manifolds, strictly nearly Kähler 6-manifolds,
and twistor spaces of quaternionic Kähler manifolds of positive Ricci
curvature, equipped with the Eels-Salamon metric ([ES]).

These days the term “nearly Kähler” usually denotes strictly nearly Kähler
6-manifolds. We shall follow this usage, often omitting “strictly” and
“6-dimensional”. In more recent literature (such as [MNS2]), these objects are
called Gray manifolds.

For a history of this notion, a number of equivalent definitions and a
bibliography of current work in this field, we refer the reader to [MNS1] and
[V3].

It is convenient to define nearly Kähler 6-manifolds in terms of differential
forms, as follows.

Proposition 1.1: Let $(M,I,\omega)$ be a Hermitian almost complex 6-manifold.
Then the following conditions are equivalent.

(i)

The tensor $\nabla_{X}(I)Y$ is skew-symmetric with respect to $X$, $Y$, and
non-zero.

(ii)

The form $\nabla\omega\in\Lambda^{1}(M)\otimes\Lambda^{2}(M)$ is non-zero and
totally skew-symmetric. Notice that in this case, by Cartan’s formula, we have
$d\omega=\nabla\omega$.

(iii)

There is $(3,0)$-form $\Omega$ with $|\Omega|=1$, and

$\displaystyle d\omega=$ $\displaystyle 3\lambda\operatorname{Re}\Omega,$
(1.1) $\displaystyle d\operatorname{Im}\Omega=$
$\displaystyle-2\lambda\omega^{2}$

where $\lambda$ is a non-zero real constant.

Proof: [Gr2]; see also [BFGK], or [V3, Theorem 4.2].

Definition 1.2: An $SU(3)$-manifold $(M,\omega,\Omega,I)$ is called nearly
Kähler if (1.1) holds.

The examples of compact nearly Kähler manifolds are scarce; one may hope that
the nearly Kähler orbifolds would occur more often. The results of this paper
are stated for the manifolds, but they are valid for all nearly Kähler
orbifolds, with the same proofs.

###  1.2 Nearly Kähler manifolds: history and classification

Nearly Kähler manifolds (also known as $K$-spaces or almost Tachibana spaces)
were defined and studied by Alfred Gray ([Gr1], [Gr2], [Gr3], [Gr4]) in a
general context of intrinsic torsion of $U(n)$-structures and weak holonomies.
An almost complex Hermitian manifold $(M,I)$ is called nearly Kähler if
$\nabla_{X}(I)X=0$, for any vector fields $X$ ($\nabla$ denotes the Levi-
Civita connection). In other words, the tensor $\nabla\omega$ must be totally
skew-symmetric, where $\omega$ denotes the Hermitian form on $M$. If
$\nabla_{X}(\omega)\neq 0$ for any non-zero vector field $X$, $M$ is called
strictly nearly Kähler.

As V. F. Kirichenko proved, nearly Kähler manifolds admit a connection with
totally antisymmetric, parallel torsion ([Ki]). This observation was used to
prove a splitting theorem for nearly Kähler manifolds: any nearly Kähler
manifold is locally a Riemannian product of a Kähler manifold and a strictly
nearly Kähler one ([Gr4], [N1]).

A powerful classification theorem for Riemannian manifolds admitting an
orthogonal connection with irreducible connection and parallel torsion was
obtained by R. Cleyton and A. Swann in [CS]. Cleyton and Swann proved that any
such manifold is either locally homogeneous, has vanishing torsion, or has
weak holonomy $G_{2}$ (in dimension 7) or $SU(3)$ (in dimension 6).

Using Kirichenko theorem, this result can be used to obtain a classification
of nearly Kähler manifolds. P.-A. Nagy ([N2]) has shown that that any strictly
nearly Kähler manifold is locally a product of locally homogeneous manifolds,
strictly nearly Kähler 6-manifolds, and twistor spaces of quaternionic Kähler
manifolds of positive Ricci curvature, equipped with the Eels-Salamon metric.

These days the term “nearly Kähler” usually denotes strictly nearly Kähler
6-manifolds. In sequel we shall follow this usage, often omitting “strictly”
and “6-dimensional”.

In dimension 6, a manifold is (strictly) nearly Kähler if and only if it
admits a Killing spinor ([Gru]). Therefore, such a manifold is Einstein, with
positive Einstein constant.

As one can easily show (see e.g. [BFGK] or [V3]), strictly nearly Kähler
6-manifolds can be defined as 6-manifolds with structure group $SU(3)$ and
fundamental forms $\omega\in\Lambda^{1,1}_{\mathbb{R}}(M)$,
$\Omega\in\Lambda^{3,0}(M)$, satisfying
$d\omega=3\lambda\operatorname{Re}\Omega$,
$d\operatorname{Im}\Omega=-2\lambda\omega^{2}$. An excellent introduction to
nearly Kähler geometry is found in [MNS1].

The most puzzling aspect of nearly Kähler geometry is a complete lack of non-
homogeneous examples. With the exception of 4 homogeneous cases described
below (Subsection 1.3), no other compact examples of strictly nearly Kähler
6-manifolds are known to exist.

###  1.3 Examples of nearly Kähler manifolds

Just as the conical singularities of parallel $G_{2}$-manifolds correspond to
nearly Kähler manifolds, the conical singularities of $Spin(7)$-manifolds
correspond to the so-called “nearly parallel” $G_{2}$-manifolds (see [I]). A
$G_{2}$-manifold $(M,\omega)$ is called nearly parallel if $d\omega=c*\omega$,
where $c$ is some constant. The analogy between nearly Kähler 6-manifolds and
nearly parallel $G_{2}$-manifolds is almost perfect. These manifolds admit a
connection with totally antisymmetric torsion and have weak holonomy $SU(3)$
and $G_{2}$ respectively. N. Hitchin realized nearly Kähler 6-manifolds and
nearly parallel $G_{2}$-manifolds as extrema of a certain functional, called
Hitchin functional by physicists (see [Hi]).

However, examples of nearly parallel $G_{2}$-manifolds are found in profusion
(every 3-Sasakian manifold is nearly parallel $G_{2}$), and compact nearly
Kähler manifolds are rare.

Only 4 compact examples are known (see the list below); all of them
homogeneous. In [Bu] it was shown that any homogeneous nearly Kähler
6-manifold belongs to this list.

  1. 1.

The 6-dimensional sphere $S^{6}$. The almost complex structure on $S^{6}$ is
reconstructed from the octonion action, and the metric is standard.

  2. 2.

$S^{3}\times S^{3}$, with the complex structure mapping $\xi_{i}$ to
$\xi^{\prime}_{i}$, $\xi_{i}^{\prime}$ to $-\xi_{i}$, where $\xi_{i}$,
$\xi^{\prime}_{i}$, $i=1,2,3$ is a basis of left invariant 1-forms on the
first and the second component.

  3. 3.

Given a self-dual Einstein Riemannian 4-manifold $M$ with positive Einstein
constant, one defines its twistor space $\operatorname{Tw}(M)$ as a total
space of a bundle of unit spheres in $\Lambda^{2}_{-}(M)$ of anti-self-dual
2-forms. Then $\operatorname{Tw}(M)$ has a natural Kähler-Einstein structure
$(I_{+},g)$, obtained by interpreting unit vectors in $\Lambda^{2}_{-}(M)$ as
complex structure operators on $TM$. Changing the sign of $I_{+}$ on $TM$, we
obtain an almost complex structure $I_{-}$ which is also compatible with the
metric $g$ ([ES]). A straightforward computation insures that
$(\operatorname{Tw}(M),I_{-},g)$ is nearly Kähler ([M]).

As N.Hitchin proved, there are only two compact self-dual Einstein
4-manifolds: $S^{4}$ and ${\mathbb{C}}P^{2}$. The corresponding twistor spaces
are ${\mathbb{C}}P^{3}$ and the flag space $F(1,2)$. The almost complex
structure operator $I_{-}$ induces a nearly Kähler structure on these two
symmetric spaces.

###  1.4 Nearly Kähler manifolds in geometry and physics

In [V3] it was shown that, unless a nearly Kähler manifold $M$ is locally
isometric to a 6-sphere, the almost complex structure on $M$ is uniquely
determined by the metric. In [F] this result was proved for $S^{6}$ as well.
Also in [V3] it was shown that the metric on $M$ is uniquely determined by the
almost complex structure.

Denote by $C(M)$ the Riemannian cone of $(M,g)$. By definition, the Riemannian
cone is a product ${\mathbb{R}}^{>0}\times M$, equipped with a metric
$t^{2}g+dt^{2}$, where $t$ is a unit parameter of ${\mathbb{R}}^{>0}$

The definition of nearly Kähler manifolds can be reformulated in terms of
Riemannian geometry, as follows.

Recall that a spinor $\psi$ is called a Killing spinor if
$\nabla_{X}\psi=\lambda X\cdot\psi$ for all vector fields $X\in TM$ and a
fixed, non-zero real constant $\lambda$. Any manifold which admits a Killing
spinor is Einstein ([BFGK]). The following theorem was proven by C. Bär.

Proposition 1.3: Let $(M,g)$ be a Riemannian 6-manifold. Then $M$ admits a
strictly nearly Kähler almost complex structure if and only if any of the
following equivalent conditions holds.

(i)

$M$ admits a real Killing spinor.

(ii)

The Riemannian cone $C(M)$ has holonomy $G_{2}$.

Proof: [B].

For an in-depth study of Killing spinors, with applications to 6-dimensional
geometry, see [BFGK]. From (i) it is apparent that a nearly Kähler manifold is
Einstein; indeed, only Einstein manifolds can admit Killing spinors.

Nearly Kähler manifolds appear as the end result of several important
classification-type problems - in classification of manifolds admitting a
Killing spinor, in classification of conical singularities of
$G_{2}$-manifolds, in classification of manifolds admitting a connection with
totally antisymmetric and parallel torsion ([CS]) and so on. These manifolds
are even more important in physics, being solutions of type II B string theory
([FI]). In that sense, nearly Kähler manifolds are just as important as the
usual Calabi-Yau threefolds.

The conical singularities of $G_{2}$-manifolds and the resulting nearly Kähler
geometries also known to have applications in physics, giving supergravity
solutions which are a product an anti-de Sitter space with an Einstein space
(see [AFHS]). More recently, the conical singularities of $G_{2}$-manifolds
arising from nearly Kähler geometry were used to obtain string models with
chiral matter fields ([AtW], [AcW]).

###  1.5 Local structure of nearly Kähler 6-manifolds

Let $(M,I,\omega,\Omega)$ be a nearly Kähler manifold. Since
$d\omega=3\lambda\operatorname{Re}\Omega$, the complex structure on $M$ is
non-integrable; indeed, a differential of the $(1,1)$-form $\omega$ lies in
$\Lambda^{3,0}(M)\oplus\Lambda^{0,3}(M)$, and this is impossible if $(M,I)$ is
integrable.

To fix the notation, we recall some well-known results and calculations
relating the de Rham differential and the Nijenhuis tensor.

An obstruction to integrability of an almost complex structure is given by the
Nijenhuis tensor,

$N^{*}:\;T^{1,0}(M)\otimes T^{1,0}(M){\>\longrightarrow\>}T^{0,1}(M),$

mapping a pair of $(1,0)$-vector fields to the $(0,1)$-part of their
commutator. For our purposes, it is more convenient to deal with its dual,
which we denote by the same letter:

$N:\;\Lambda^{0,1}(M){\>\longrightarrow\>}\Lambda^{2,0}(M).$ (1.2)

From Cartan’s formula it is apparent that $N$ is equal to the $(2,-1)$-part of
the de Rham differential. On the other hand, $N$ can be expressed through
$\nabla I$, in a usual way:

$N^{*}(X,Y)=(\nabla_{X}I)Y-(\nabla_{Y}I)X$

where $X,Y$ are $(1,0)$-vector fields. On a nearly Kähler manifold,
$\nabla(I)$ can be expressed through the 3-form $d\omega=\nabla\omega$. This
gives the following relations ([KN]):

$N^{*}(X,Y)=d\omega(X,Y,\cdot)^{\sharp},$ (1.3)

where $d\omega(X,Y,\cdot)^{\sharp}$ is a vector field dual to the 1-form
$d\omega(X,Y,\cdot)$. Since $d\omega=3\lambda\operatorname{Re}\Omega$, the
relation (1.3) allows one to express $N$ through $\Omega$ and $\omega$.

Let $\xi_{1},\xi_{2},\xi_{3}\in\Lambda^{1,0}(M)$ be an orthonormal coframe,
such that $\Omega=\xi_{1}\wedge\xi_{2}\wedge\xi_{3}$. Then (1.3) gives

$N(\overline{\xi}_{1})=\lambda\xi_{2}\wedge\xi_{3},\ \
N(\overline{\xi}_{2})=-\lambda\xi_{1}\wedge\xi_{3},\ \
N(\overline{\xi}_{3})=\lambda\xi_{1}\wedge\xi_{2},$ (1.4)

This calculation is well known; it is explained in more detail in [V3].

###  1.6 Hodge decomposition of the de Rham differential and intrinsic
Laplacians

The results of this paper can be summarized as follows. Let

$d=d^{2,-1}+d^{1,0}+d^{0,1}+d^{-1,2},$

be the Hodge decomposition of de Rham differential (Subsection 3.1). We use
the following notation: $d^{2,-1}=:N$, $d^{-1,2}=:\overline{N}$,
$d^{1,0}=:\partial$, $d^{0,1}=:\overline{\partial}$.

The usual Kähler identities have a form “a commutator of some Hodge component
of de Rham differential with the Hodge operator $\Lambda$ is proportional to a
Hermitian adjoint of some other Hodge component of de Rham differential”. We
prove that a similar set of identities is valid on nearly Kähler manifolds
(3.1, 5). These identities are used to study various intrinsic Laplacians on
$M$. We show that the difference

$\Delta_{\partial}-\Delta_{\overline{\partial}}=R$ (1.5)

is a scalar operator, acting on $(p,q)$-forms as $\lambda^{2}(p-q)(3-p-q)$
(see 3.2). For the de Rham Laplacian $\Delta_{d}=dd^{*}+d^{*}d$, the following
formula holds:

$\Delta_{d}=\Delta_{\partial-\overline{\partial}}+\Delta_{N}+\Delta_{\overline{N}}$
(1.6)

(see (5.6)). This formula is used to study the harmonic forms on $M$ when $M$
is compact. We show that $\eta$ is harmonic if and only if all Hodge
components of $d$ and $d^{*}$ vanish on $\eta$ (6.2). This implies that the
harmonic forms on $M$ admit a Hodge decomposition:

${\cal H}^{*}(M)=\bigoplus{\cal H}^{p,q}(M).$

Using (1.5), we obtain that ${\cal H}^{p,q}(M)=0$ unless $p=q$ or ($q=2,p=1$)
or ($q=1,p=2$). We also prove that all harmonic forms $\eta\in{\cal
H}^{p,q}(M)$, for ($q=2,p=1$) or ($q=1,p=2$) or $p=q=2$ are coprimitive, that
is, satisfy $\eta\wedge\omega=0$, where $\omega$ is the Hermitian form (6.2).

##  2 Algebraic differential operators on the
de Rham algebra

The following section is purely algebraic. We reproduce some elementary
results about algebraic differential operators on graded commutative algebras.
There results are later on used to study the de Rham superalgebra.

###  2.1 Algebraic differential operators: basic properties

Let $A^{*}:=\bigoplus^{i}A_{i}$ be a graded commutative ring with unit. For
$a\in A_{i}$, we denote by $L_{a}$ the operator of multiplication by $a$:
$L_{a}(\eta)=a\eta$. A supercommutator of two graded endomorphisms
$x,y\in\operatorname{End}(A^{*})$ is denoted by

$\\{x,y\\}:=xy-(-1)^{\tilde{x}\tilde{y}}yx,$

where $\tilde{x}$ denotes the parity of $x$.

Speaking of elements of graded spaces further on in this section, we shall
always mean pure elements, that is, elements of pure even or pure odd degree.
The parity $\tilde{x}$ is always defined as 1 on odd elements, and 0 on even
elements.

Vectors of pure even degree are called even, and vectors of pure odd degree
are called odd.

A supercommutator of two even endomorphisms, or an odd and an even
endomorphism is equal to their commutator. We shall sometimes use the usual
bracket notation $[\cdot,\cdot]$ in this case.

Definition 2.1:  The space $D^{i}(A^{*})\subset\operatorname{End}(A^{*})$ of
algebraic differential operators of algebraic order $i$ is a graded subspace
of $\operatorname{End}(A^{*})$, which is defined inductively as follows.

(i)

$D^{0}(A)$ is a space of $A^{*}$-linear endomorphisms of $A^{*}$, that is,
$D^{0}(A^{*})\cong A^{*}$.

(ii)

$D^{n+1}(A^{*})$ is defined as a graded subspace of
$\operatorname{End}(A^{*})$ consisting of all endomorphisms
$\rho\in\operatorname{End}(A^{*})$ (even or odd) which satisfy
$\\{L_{a},\rho\\}\in D^{n}(A^{*})$, for all $a\in A$.

This notion was defined by A. Grothendieck. Using induction, it is easy to
check that $D^{*}(A^{*})=\bigcup D^{i}(A^{*})$ is a filtered algebra:

$D^{i}(A^{*})\cdot D^{j}(A^{*})\subset D^{i+j}(A^{*}),$ (2.1)

and also

$\\{D^{i}(A^{*}),D^{j}(A^{*})\\}\subset D^{i+j-1}(A^{*}).$ (2.2)

Definition 2.2: Let $\delta:\;A^{*}{\>\longrightarrow\>}A^{*}$ be an even or
odd endomorphism. We say that $\delta$ is a derivation if

$\delta(ab)=\delta(a)b+(-1)^{\tilde{a}\tilde{\delta}}a\delta(b),$

for any $a,b\in A^{*}$.

Clearly, all derivations of $A^{*}$ are first order algebraic differential
operators and vanish on the unit of $A^{*}$. The converse is also true: if
$D\subset D^{1}(A^{*})$ is a first order differential operator, $D(1)=0$, then
$D$ is a derivation, as the following claim implies.

Claim 2.3:  Let $D\in D^{1}(A^{*})$ be a first order differential operator.
Then

$D-L_{D(1)}$

is a derivation of $A$.

Proof: It suffices to prove 2.1 assuming that $D(1)$=0. Let $a,b\in A$ be even
or odd elements. Since $\\{D,L_{a}\\}$ is $A^{*}$-linear, we have

$\displaystyle D(ab)-(-1)^{\tilde{a}\tilde{D}}aD(b)=$
$\displaystyle\\{D,L_{a}\\}(b)=\\{D,L_{a}\\}(1)b$ $\displaystyle=$
$\displaystyle D(a)b+(-1)^{\tilde{a}\tilde{D}}aD(1)=D(a)b.$

Remark 2.4:  From 2.1, it is clear that a first order differential operator on
$A$ is determined by the values taken on 1 and any set of multiplicative
generators of $A$.

The following claim is also clear.

Claim 2.5:  Let $D\in\operatorname{End}(A^{*})$ be an endomorphism of $A^{*}$,
and $V$ a set of generators of $A^{*}$. Assume that for any $\nu\in V$, we
have $\\{L_{\nu},D\\}\in D^{i}(A^{*})$. Then $D$ is an $(i+1)$-st order
algebraic differential operator on $A$.

###  2.2 Algebraic differential operators on $\Lambda^{*}(M)$

Let $M$ be a smooth manifold, and $\Lambda^{*}(M)$ its de Rham algebra. It is
easily seen that the differential operators (in the usual sense) and the
algebraic differential operators on $\Lambda^{*}(M)$ coincide. However, the
“algebraic order” of differential operators in the sense of Grothendieck’s
definition and in the sense of the usual definition are different. For
instance, a contraction with a vector field is $C^{\infty}(M)$-linear, hence
it has order 0 in the usual sense. However, the contraction with a vector
field has algebraic order one in the sense of 2.1. Further on, we always use
the term “order” in the sense of “algebraic order”, and not in the
conventional sense.

From now till the end of this Appendix, the manifold $M$ is always assumed to
be Riemannian.

Claim 2.6:  Let $M$ be a Riemannian manifold, and $\eta\in\Lambda^{1}(M)$ a
1-form. Denote by $\Lambda_{\eta}$ the metric adjoint to $L_{\eta}$,
$\Lambda_{\eta}=-*L_{\eta}*$. Then $\Lambda_{\eta}$ is a first order
differential operator.

Proof: Clearly, $\Lambda_{\eta}$ is a contraction with a vector field
$\eta^{\sharp}$ dual to $\eta$. Then 2.2 is clear, because a contraction with
a vector field is clearly a derivation.

This claim is a special case of the following proposition, which is proved
independently.

Proposition 2.7:  Let $M$ be a Riemannian manifold, and
$\eta\in\Lambda^{n}(M)$ an n-form. Denote by $\Lambda_{\eta}$ the metric
adjoint to $L_{a}$, $\Lambda_{\eta}=(-1)^{\tilde{\eta}}*L_{\eta}*$. Then
$\Lambda_{\eta}$ is a differential operator of algebraic order $n$.

Proof: We use the induction on $n$. For $n=0$ everything is clear. As 2.1
implies, to prove that $\Lambda_{\eta}\in D^{n}(\Lambda^{*}(M))$, we need to
show that

$\\{\Lambda_{\eta},L_{a}\\}\in D^{n-1}(\Lambda^{*}(M)),$ (2.3)

for any $a\in\Lambda^{0}(M),\Lambda^{1}(M)$. For $a\in\Lambda^{0}(M)$, (2.3)
is clear, because $\Lambda_{\eta}$ is $C^{\infty}(M)$-linear, hence
$\\{\Lambda_{\eta},L_{a}\\}=0$. For $a\in\Lambda^{1}(M)$, it is easy to check
that

$\\{\Lambda_{\eta},L_{a}\\}=\Lambda_{\eta\hskip
2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip 2.0pta^{\sharp}}$

where $a^{\sharp}$ is the dual vector field, and $\lrcorner$ a contraction.
The induction statement immediately brings (2.3).

###  2.3 An algebraic differential operator and its adjoint

The main result of this section is the following proposition.

Proposition 2.8:  Let $(M,g)$ be a Riemannian manifold, and

$D:\;\Lambda^{*}(M){\>\longrightarrow\>}\Lambda^{*+1}(M)$

a first order algebraic differential operator. Denote by $D^{*}$ its metric
adjoint, $D^{*}=-*D*$. Then $D^{*}$ is a second order algebraic differential
operator.

Proof: Step 1: As follows from 2.1, it suffices to check that

$\\{\\{D^{*},L_{a}\\},L_{b}\\}\ \ \text{is $\Lambda^{*}(M)$-linear},$ (2.4)

for all $a,b\in\Lambda^{0}(M),\Lambda^{1}(M)$.

Step 2:

Lemma 2.9:  Let $\Lambda^{*}(M)\stackrel{{\scriptstyle
D_{1}}}{{{\>\longrightarrow\>}}}\Lambda^{*-1}(M)$ be a first order algebraic
differential operator decreasing the degree by 1. Then $D_{1}=0$.

Proof: Follows from 2.1.

Step 3: Clearly,

$\\{\\{D^{*},L_{a}\\},L_{b}\\}^{*}=D_{1}^{*},$

where $D_{1}:=\\{\\{D,\Lambda_{a}\\},\Lambda_{b}\\}$. From 2.2 and (2.2), we
find that $D_{1}$ is an algebraic differential operator of algebraic order 1
(being a commutator of several first order operators). When
$a,b\in\Lambda^{1}(M)$, $D_{1}$ decreases the degree of a form by 1. By 2.1,
$D_{1}$ is a derivation. Clearly, a derivation which vanishes on
$\Lambda^{0}(M)$ is $C^{\infty}(M)$-linear. This shows that $D_{1}$ is
$C^{\infty}(M)$-linear.

By 2.3, the commutator of $D_{1}$ with $\Lambda_{c}$ vanishes, for all
$c\in\Lambda^{1}(M)$:

$\\{D_{1},\Lambda_{c}\\}=0$ (2.5)

The operator $D_{1}^{*}=\\{\\{D^{*},L_{a}\\},L_{b}\\}$ is
$C^{\infty}(M)$-linear (being adjoint to $D_{1}$), and commutes with all
$L_{a}$, as follows from (2.5). Therefore, $D_{1}^{*}$ is
$\Lambda^{*}(M)$-linear. This proves (2.4) for $a,b\in\Lambda^{1}(M)$.

Step 4: Clearly, $L_{a}=\Lambda_{a}$ when $a\in\Lambda^{0}(M)$. Then

$\\{\\{D^{*},L_{a}\\},L_{b}\\}=\\{\\{D^{*},\Lambda_{a}\\},\Lambda_{b}\\}=\\{\\{D,L_{a}\\},L_{b}\\}^{*}=0,$
(2.6)

because $D$ is a first order algebraic differential operator. We proved (2.4)
for $a,b\in\Lambda^{0}(M)$.

Step 5: Since the algebra $\Lambda^{*}(M)$ is graded commutative,
$\\{L_{a},L_{b}\\}=0$ for all $a,b\in\Lambda^{*}(M)$. Using the graded Jacobi
identity, we find that

$\\{\\{D^{*},L_{a}\\},L_{b}\\}=(-1)^{\tilde{a}\tilde{b}}\\{\\{D^{*},L_{b}\\},L_{a}\\},$
(2.7)

for all $a,b$. In Steps 3 and 4 we proved (2.4) for $a,b\in\Lambda^{1}(M)$,
$a,b\in\Lambda^{0}(M)$. By (2.7), to prove 2.3 it remains to show that
$\\{\\{D^{*},L_{a}\\},L_{b}\\}$ is $\Lambda^{*}(M)$-linear for
$a\in\Lambda^{0}(M),b\in\Lambda^{1}(M)$.

Step 6: In this case,

$\\{D^{*},L_{a}\\}=\\{D^{*},\Lambda_{a}\\}=\\{D,L_{a}\\}^{*}=L_{D(a)}^{*}=\Lambda_{D(a)}.$

Then

$\\{\\{D^{*},L_{a}\\},L_{b}\\}=\\{\Lambda_{D(a)},L_{b}\\}=g(D(a),b),$

because $\Lambda_{D(a)}$ is a contraction with the dual vector field
$D(a)^{\sharp}$. We proved that $\\{\\{D^{*},L_{a}\\},L_{b}\\}$ is a scalar
function, hence it is $\Lambda^{*}(M)$-linear. 2.3 is proved.

##  3 Kähler identities on nearly Kähler manifolds

###  3.1 The operators $\partial$, $\overline{\partial}$ on almost complex
manifolds

Let $(M,I)$ be an almost complex manifold, and
$d:\;\Lambda^{i}(M){\>\longrightarrow\>}\Lambda^{i+1}(M)$ the de Rham
differential. The Hodge decomposition gives

$d=\bigoplus_{i+j=1}d^{i,j},\ \
d^{i,j}:\;\Lambda^{p,q}(M){\>\longrightarrow\>}\Lambda^{p+i,q+j}(M)$ (3.1)

Using the Leibniz identity, we find that the differential and all its Hodge
components are determined by the values taken on all vectors in the spaces
$\Lambda^{0}(M)$, $\Lambda^{1}(M)$, generating the de Rham algebra. On
$\Lambda^{0}(M)$, only $d^{1,0}$, $d^{0,1}$ can be non-zero, and on
$\Lambda^{1}(M)$ only $d^{2,-1},d^{1,0},d^{0,1},d^{-1,2}$ can be non-zero.
Therefore, only 4 components of (3.1) can be possibly non-zero:

$d=d^{2,-1}+d^{1,0}+d^{0,1}+d^{-1,2},$

Since $N:=d^{2,-1}$, $\overline{N}:=d^{-1,2}$ vanish on $\Lambda^{0}(M)$,
these components are $C^{\infty}(M)$-linear. In fact,

$N:\;\Lambda^{0,1}(M){\>\longrightarrow\>}\Lambda^{2,0}(M)$

is the Nijenhuis tensor (1.2) of $(M,I)$, extended to an operator on
$\Lambda^{*}(M)$ by the Leibniz rule. We denote $d^{1,0}$ as
$\partial:\;\Lambda^{p,q}(M){\>\longrightarrow\>}\Lambda^{p+1,q}(M)$, and
$d^{0,1}$ as
$\overline{\partial}:\;\Lambda^{p,q}(M){\>\longrightarrow\>}\Lambda^{p,q+1}(M)$.
Decomposing $d^{2}=0$, we find

$\displaystyle N^{2}+\\{N,\partial\\}+$
$\displaystyle(\\{\overline{\partial},N\\}+\partial^{2})+(\\{N,\overline{N}\\}+\\{\partial,\overline{\partial}\\})$
(3.2) $\displaystyle+$
$\displaystyle(\\{\partial,\overline{N}\\}+\overline{\partial}^{2})+\\{\overline{N},\overline{\partial}\\}+\overline{N}^{2}$
$\displaystyle=$ $\displaystyle d^{2}=0$

where $\\{\cdot,\cdot\\}$ denotes the supercommutator. The terms in brackets
in (3.2) are different Hodge components of $d^{2}$, and since $d^{2}=0$, they
all vanish:

$\displaystyle N^{2}=$
$\displaystyle\\{N,\partial\\}=\\{\overline{\partial},N\\}+\partial^{2}=$
(3.3) $\displaystyle=$
$\displaystyle\\{N,\overline{N}\\}+\\{\partial,\overline{\partial}\\}=\\{\partial,\overline{N}\\}+\overline{\partial}^{2}$
$\displaystyle=$
$\displaystyle\\{\overline{N},\overline{\partial}\\}=\overline{N}^{2}=0$

However, the operators $\partial^{2}$ and $\overline{\partial}^{2}$ can be
non-zero.

The following almost complex version of the Kähler idenities is quite useful
further on in our study.

Theorem 3.1:  Let $(M,I)$ be an almost complex Hermitian manifold,
$\omega\in\Lambda^{1,1}(M)$ a Hermitian form, and
$\Lambda_{\omega}:\;\Lambda^{i}(M){\>\longrightarrow\>}\Lambda^{i-2}(M)$ a
Hermitian adjoint to $L_{\omega}(\eta)=\omega\wedge\eta$. Consider the
operators
$\partial,\overline{\partial}:\;\Lambda^{i}(M){\>\longrightarrow\>}\Lambda^{i+1}(M)$
constructed above, and let
$\partial^{*},\overline{\partial}^{*}:\;\Lambda^{i}(M){\>\longrightarrow\>}\Lambda^{i-1}(M)$
be the corresponding Hermitian adjoint operators. Assume that
$d\omega\in\Lambda^{3,0}(M)\oplus\Lambda^{0,3}(M)$, that is,
$\partial\omega=\overline{\partial}\omega=0$. Then

$[\Lambda_{\omega},\partial]=\sqrt{-1}\>\overline{\partial}^{*},\ \
[\Lambda_{\omega},\overline{\partial}]=-\sqrt{-1}\>\partial^{*}$ (3.4)

and

$[L_{\omega},\partial^{*}]=\sqrt{-1}\>\overline{\partial},\ \
[\Lambda_{\omega},\partial]=\sqrt{-1}\>\overline{\partial}^{*}$ (3.5)

Proof: To prove 3.1, we use essentially the same argument as used in the proof
of the conventional Kähler identities in the situation when a coordinate
approach does not work; see e.g. the proof of Kähler identities in HKT-
geometry, obtained in [V1], and the proof of the Kähler identities in locally
conformally hyperkähler geometry, obtained in [V2].

The equations (3.4) and (3.5) are Hermitian adjoint, hence equivalent. The two
equations (3.5) are complex conjugate, hence they are also equivalent. To
prove 3.1 it is sufficient to prove only one of these equations, say

$[L_{\omega},\partial^{*}]=\sqrt{-1}\>\overline{\partial}.$ (3.6)

The proof of such a relation follows a general template, which is given in
Section 2. There is an algebraic notion of differential operators on a graded
commutative algebra, due to Grothendieck (2.1). In this sense, the operators
$\partial^{*}$, $\overline{\partial}^{*}$ are second order algebraic
differential operators on $\Lambda^{*}(M)$ (see 2.3), and $L_{\omega}$ is
$\Lambda^{*}(M)$-linear. Then $[L_{\omega},\partial^{*}]$ (being a commutator
of 0-th and 2-nd order algebraic differential operators on $\Lambda^{*}(M)$)
is a first order algebraic differential operator. To prove a relation between
first order algebraic differential operators on an algebra, such as (3.6), it
suffices to check it on any set of generators of this algebra (2.1). To prove
3.1 it remains to show that (3.6) holds on some set of generators, e. g.
1-forms and 0-forms.

Given a function $f\in C^{\infty}(M)$, we have

$[L_{\omega},\partial^{*}]f=-\partial^{*}(f\omega).$ (3.7)

However, $\partial^{*}=-*\partial*$ and $*(f\omega)=\overline{f}\omega^{n-1}$,
where $n=\dim_{\mathbb{C}}M$. Then

$[L_{\omega},\partial^{*}]f=*(\partial\overline{f}\wedge\omega^{n-1})=\sqrt{-1}\>\overline{\partial}f$

because for any $(1,0)$-form $\eta$ we have
$*(\eta\wedge\omega^{n-1})=\sqrt{-1}\>\overline{\eta}$, and
$\overline{\partial\overline{f}}=\overline{\partial}f$.

It is easy to check that $\partial^{*}$-closed 1-forms generate the bundle of
all 1-forms over $C^{\infty}(M)$. Indeed, on 2-forms we have
$(\partial^{*})^{2}=0$, and therefore, all $\partial^{*}$-exact 1-forms are
$\partial^{*}$-closed. A local calculation implies that
$\partial^{*}(\Lambda^{2}(M))$ generates $\Lambda^{1}(M)$ over
$C^{\infty}(M)$.

Consider a 1-form $\eta\in\Lambda^{1}(M)$. To prove 3.1 it remains to show
that $[L_{\omega},\partial^{*}](\eta)=\sqrt{-1}\>\overline{\partial}\eta$.
Since $\partial^{*}$-closed 1-forms generate $\Lambda^{*}(M)$, we may assume
that $\eta$ is $\partial^{*}$-closed. Then

$\displaystyle\ [L_{\omega},\partial^{*}](\eta)=$
$\displaystyle-\partial^{*}L_{\omega}\eta=*\partial*(\omega\wedge\eta)$ (3.8)
$\displaystyle=$ $\displaystyle*\partial(\omega^{n-2}\wedge
I(\overline{\eta}))=*(\omega^{n-2}\wedge\partial(I\overline{\eta})).$

Since $\eta$ is $\partial^{*}$-closed, we have
$\omega^{n-1}\wedge\partial(I\overline{\eta})=0$, and therefore the form
$\omega^{n-2}\wedge\partial\eta$ is coprimitive (satisfies
$(\omega^{n-2}\wedge\partial\eta)\wedge\omega=0$). Given a coprimitive
$(2n-2)$-form $\alpha=\kappa\wedge\omega^{n-2}$, the form $*\alpha$ can be
written down explicitly in terms of $\kappa$: $*\alpha=-I(\overline{\kappa})$.
Then

$*(\omega^{n-2}\wedge\partial I(\overline{\eta}))=-\overline{I\partial
I\overline{\eta}}=\sqrt{-1}\>\overline{\partial}\eta.$

Comparing this with (3.8), we find that

$[L_{\omega},\partial^{*}](\eta)=\sqrt{-1}\>\overline{\partial}\eta$

We proved 3.1.

###  3.2 The Nijenhuis operator squared

Later on, we shall need the following useful identity.

Proposition 3.2:  Let $(M,I,\omega,\Omega)$ be a nearly Kähler 6-manifold,
$d\omega=\lambda\operatorname{Re}\Omega$, and
$C:=N+\overline{N}=d^{2,-1}+d^{-1,2}$ the $(2,-1)\oplus(-1,2)$-part of the de
Rham differential. Then the following $C^{\infty}(M)$-linear maps

$\Lambda^{p,q}(M){\>\longrightarrow\>}\Lambda^{p+1,q+1}(M)$

are equal:

(i)

$C^{2}$

(ii)

$-\\{\partial,\overline{\partial}\\}$.

(iii)

the scalar operator $\sqrt{-1}\>\lambda^{2}(p-q)L_{\omega}$, mapping
$\eta\in\Lambda^{p,q}(M)$ to

$\sqrt{-1}\>\lambda^{2}(p-q)\eta\wedge\omega.$

Proof: The equivalence $C^{2}=-\\{\partial,\overline{\partial}\\}$ is clear,
because the $(1,1)$-part of $d^{2}$ is equal to
$C^{2}+\\{\partial,\overline{\partial}\\}$, and $d^{2}=0$ (see (3.3)). To
prove

$C^{2}=\sqrt{-1}\>\lambda^{2}(p-q)L_{\omega},$ (3.9)

we notice that both sides of (3.9) are differentiations ($C^{2}$ being a
supercommutator of an odd differentiation with itself). Therefore it suffices
to check (3.9) only on the generators of $\Lambda^{*}(M)$, e.g. on
$\Lambda^{0}(M)$ and $\Lambda^{1}(M)$. On $\Lambda^{0}(M)$, both $C$ and
$(p-q)$ vanish, hence (3.9) is clear. Let us check (3.9) on $\Lambda^{1,0}(M)$
(a proof of (3.9) on $\Lambda^{0,1}(M)$ is obtained in the same fashion).
Choose an orthonormal frame $\xi_{1},\xi_{2},\xi_{3}\in\Lambda^{1,0}(M)$, in
such a way that

$\omega=-\sqrt{-1}\>(\xi_{1}\wedge\overline{\xi}_{1}+\xi_{2}\wedge\overline{\xi}_{2}+\xi_{3}\wedge\overline{\xi}_{3}),\
\ \ \Omega=\xi_{1}\wedge\xi_{2}\wedge\xi_{3}$

Let $\eta$ be a $(0,1)$-form, say, $\eta=\overline{\xi}_{1}$ (this assumption
is not restrictive, because both sides of (3.9) are manifestly
$C^{\infty}(M)$-linear). Then $N(\eta)=\lambda\xi_{2}\wedge\xi_{3}$, as (1.4)
implies. Similarly, the Leibniz rule and (1.4) give

$\overline{N}N(\eta)=\lambda^{2}(\overline{\xi}_{1}\wedge\overline{\xi}_{3}\wedge\xi_{3}+\xi_{1}\wedge\overline{\xi}_{1}\wedge\overline{\xi}_{2})=\sqrt{-1}\>\lambda^{2}\eta\wedge\omega.$
(3.10)

On the other hand,

$C^{2}(\eta)=(N+\overline{N})^{2}\eta=\overline{N}N(\eta),$ (3.11)

because $\overline{N}\eta$ vanishes, being a $(-1,3)$-form, and
$N^{2}=\overline{N}^{2}=0$ as (3.2) implies. Combining (3.10) and (3.11), we
obtain (3.9). 3.2 is proved.

Corollary 3.3:  Let $(M,I,\omega,\Omega)$ be a nearly Kähler 6-manifold,
$d\omega=\lambda\operatorname{Re}\Omega$, and $\partial,\overline{\partial}$
the $(1,0)$\- and $(0,1)$-parts of de Rham differential. Consider the
corresponding Laplacians:

$\Delta_{\partial}:=\partial\partial^{*}+\partial^{*}\partial,\ \ \
\Delta_{\overline{\partial}}:=\overline{\partial}\overline{\partial}^{*}+\overline{\partial}^{*}\overline{\partial}.$

Then $\Delta_{\partial}-\Delta_{\overline{\partial}}=R$, where $R$ is a scalar
operator acting on $(p,q)$-forms as a multiplication by
$\lambda^{2}(3-p-q)(p-q)$.

Proof: As 3.2 implies,
$\\{\partial,\overline{\partial}\\}=-\sqrt{-1}\>(p-q)\lambda^{2}L_{\omega}$.
It is well known that $H:=[L_{\omega},\Lambda_{\omega}]$ acts on $(p,q)$-forms
as a multiplication by $(3-p-q)$ (see e.g. [GH]). Then

$\\{\Lambda_{\omega},\\{\partial,\overline{\partial}\\}\\}=\sqrt{-1}\>R.$
(3.12)

Applying the graded Jacobi identity and 3.1 to (3.12), we obtain

$\sqrt{-1}\>R=\\{\Lambda_{\omega},\\{\partial,\overline{\partial}\\}\\}=\\{\\{\Lambda_{\omega},\partial\\},\overline{\partial}\\}+\\{\partial,\\{\Lambda_{\omega},\overline{\partial}\\}\\}=\sqrt{-1}\>\Delta_{\partial}-\sqrt{-1}\>\Delta_{\overline{\partial}}.$
(3.13)

This proves 3.2.

##  4 The de Rham Laplacian via $\Delta_{\partial}$,
$\Delta_{\overline{\partial}}$, $\Delta_{\partial-\overline{\partial}}$

###  4.1 An expression for $\Delta_{d}$

Let $M$ be a nearly Kähler 6-manifold,
$d=N+\partial+\overline{\partial}+\overline{N}$ the Hodge decomposition of the
de Rham differential, and $\Delta_{\partial}$, $\Delta_{\overline{\partial}}$
the Laplacians defined above,
$\Delta_{\partial}:=\\{\partial,\partial^{*}\\}$,
$\Delta_{\overline{\partial}}:=\\{\overline{\partial},\overline{\partial}^{*}\\}$.
Denote the usual (Hodge-de Rham) Laplacian by $\Delta_{d}=\\{d,d^{*}\\}$. In
addition to 3.2, the following relation between the Laplacians can be
obtained.

Theorem 4.1:  Let $M$ be a nearly Kähler 6-manifold, and $\Delta_{\partial}$,
$\Delta_{\overline{\partial}}$, $\Delta_{d}$ the Laplacians considered above.
Then

$\Delta_{d}=\Delta_{\partial}+\Delta_{\overline{\partial}}+\Delta_{N+\overline{N}}-\\{\partial,\overline{\partial}^{*}\\}-\\{\overline{\partial},\partial^{*}\\}.$
(4.1)

where $\Delta_{N+\overline{N}}$ is defined as a supercommutator of the
$C^{\infty}$-linear operator $C:=N+\overline{N}$ and its Hermitian adjoint:

$\Delta_{N+\overline{N}}:=CC^{*}+C^{*}C.$

The proof of 4.1 takes the rest of this Section.

###  4.2 $N=\lambda[L_{\Omega},\Lambda_{\omega}]$

The following linear-algebraic relation is used further on in the proof of 4.1

Claim 4.2:  Let $(M,I,\omega,\Omega)$ be a nearly Kähler 6-manifold which
satisfies $d\omega=\lambda\operatorname{Re}\Omega$, $N$ the $(2,-1)$-part of
the de Rham differential, and $\Lambda_{\omega}$ the Hodge operator defined
above. Then

$\lambda[L_{\Omega},\Lambda_{\omega}]=N,$ (4.2)

where $L_{\Omega}(\eta):=\Omega\wedge\eta$.

Proof: As in the proof of 3.1, we consider $L_{\Omega},\Lambda_{\omega}$ as
algebraic differential operators on the graded commutative algebra
$\Lambda^{*}(M)$ (see 2.1). Then $L_{\Omega}$ is a 0-th order operator, and
$\Lambda_{\omega}$ a second order operator, as follows from 2.2. Therefore,
the commutator $[L_{\Omega},\Lambda_{\omega}]$ is a first order operator. By
2.1 it suffices to check (4.2) on 0- and 1-forms. This can be done by an
explicit calculation, using (1.4).

###  4.3 Commutator relations for $N$, $\overline{N}$, $\partial$,
$\overline{\partial}$

Proposition 4.3:  Under the assumptions of 4.1, the following anticommutators
vanish:

$\\{N^{*},\overline{\partial}\\}=\\{\overline{N}^{*},\partial\\}=\\{N,\overline{\partial}^{*}\\}=\\{\overline{N},\partial^{*}\\}=0.$
(4.3)

Moreover, we have

$\displaystyle\\{\overline{\partial}^{*},\partial\\}$
$\displaystyle=-\\{N,\partial^{*}\\}=-\\{\overline{N}^{*},\overline{\partial}\\},$
(4.4) $\displaystyle\\{\partial^{*},\overline{\partial}\\}$
$\displaystyle=-\\{\overline{N},\overline{\partial}^{*}\\}=-\\{N^{*},\partial^{*}\\},$

Proof: Clearly, all relations of (4.3) can be obtained by applying the complex
conjugation and taking the Hermitian adjoint of the following relation:

$\\{N^{*},\overline{\partial}\\}=0.$ (4.5)

Decomposing $d^{2}=0$ onto Hodge components, we obtain $\\{N,\partial\\}=0$
(see (3.3)). By 4.2, this is equivalent to

$\\{\\{L_{\Omega},\Lambda_{\omega}\\},\partial\\}=0.$ (4.6)

Clearly, $\partial\Omega=0$, hence

$\\{L_{\Omega},\partial\\}=0.$ (4.7)

Applying the graded Jacobi identity to (4.6) and using (4.7), we obtain

$0=\\{L_{\Omega},\\{\Lambda_{\omega},\partial\\}\\}=\sqrt{-1}\>\\{L_{\Omega},\overline{\partial}^{*}\\}.$
(4.8)

Acting on (4.8) by $\\{\Lambda_{\omega},\cdot\\}$ and using the graded Jacobi
identity, we obtain

$0=\\{\Lambda_{\omega},\\{L_{\Omega},\overline{\partial}^{*}\\}\\}=-\lambda\\{N,\overline{\partial}^{*}\\}.$

This proves (4.5) and (4.3).

It remains to prove (4.4). Taking a Hodge component of $d^{2}=0$, we obtain

$\frac{1}{2}\\{\partial,\partial\\}+\\{N,\overline{\partial}\\}=0$ (4.9)

(see (3.3)). Using the same argument as gives (4.7), we find

$\displaystyle\\{N,\overline{\partial}\\}=$
$\displaystyle-\lambda^{-1}\\{\\{\Lambda_{\omega},L_{\Omega}\\},\overline{\partial}\\}$
$\displaystyle=$
$\displaystyle\lambda^{-1}\\{\\{L_{\Omega},\\{\Lambda_{\omega},\overline{\partial}\\}\\}-\lambda^{-1}\\{\Lambda_{\omega},\\{L_{\Omega},\overline{\partial}\\}\\}$
$\displaystyle=$
$\displaystyle-\sqrt{-1}\>\lambda^{-1}\\{L_{\Omega},\partial^{*}\\}.$

Together with (4.9), this brings

$\frac{1}{2}\\{\partial,\partial\\}=\sqrt{-1}\>\lambda^{-1}\\{L_{\Omega},\partial^{*}\\}.$
(4.10)

Acting on (4.10) with $\\{\Lambda_{\omega},\cdot\\}$, we obtain

$\sqrt{-1}\>\\{\partial,\overline{\partial}^{*}\\}=\sqrt{-1}\>\lambda^{-1}\\{\\{\Lambda_{\omega},L_{\Omega}\\},\partial^{*}\\}=-\sqrt{-1}\>\\{N,\partial^{*}\\}.$

We obtained the first equation of (4.4):

$\\{\partial,\overline{\partial}^{*}\\}=-\\{N,\partial^{*}\\}.$ (4.11)

We proved 4.3.

###  4.4 The Hodge decomposition of the Laplacian

Now we can finish the proof of 4.1. Decomposing $d$, $d^{*}$ onto Hodge
components, we obtain

$\displaystyle\Delta_{d}=$
$\displaystyle\bigg{(}\\{N^{*},\overline{\partial}\\}+\\{N,\overline{\partial}^{*}\\}+\\{\overline{N}^{*},\partial\\}+\\{\overline{N},\partial^{*}\\}\bigg{)}$
(4.12) $\displaystyle+$
$\displaystyle\bigg{(}\\{\partial^{*},\overline{\partial}\\}+\\{\overline{\partial}^{*},\partial\\}+\\{N,\partial^{*}\\}+\\{\overline{N},\overline{\partial}^{*}\\}+\\{N^{*},\partial\\}+\\{\overline{N}^{*},\overline{\partial}\\}\bigg{)}$
$\displaystyle+$
$\displaystyle\Delta_{\partial}+\Delta_{\overline{\partial}}+\Delta_{N+\overline{N}}.$

The first term in brackets vanishes by (4.3), and the second term is equal
$-\\{\partial,\overline{\partial}^{*}\\}-\\{\overline{\partial},\partial^{*}\\}$
as (4.4) implies. This gives

$\Delta_{d}=\Delta_{\partial}+\Delta_{\overline{\partial}}+\Delta_{N+\overline{N}}-\\{\partial,\overline{\partial}^{*}\\}-\\{\overline{\partial},\partial^{*}\\}.$
(4.13)

We proved 4.1.

The relation (4.13) can be rewritten as the following relation between
Laplacians.

Corollary 4.4:  Under the assumptions of 4.1, denote by
$\Delta_{\partial-\overline{\partial}}$ the Laplacian
$\\{\partial-\overline{\partial},\partial^{*}-\overline{\partial}^{*}\\}$.
Then

$\Delta_{d}=\Delta_{\partial-\overline{\partial}}+\Delta_{N+\overline{N}}.$
(4.14)

Proof: As (4.13) implies, to prove (4.14) we need to show that

$\Delta_{\partial-\overline{\partial}}=\Delta_{\partial}+\Delta_{\overline{\partial}}-\\{\partial,\overline{\partial}^{*}\\}-\\{\overline{\partial},\partial^{*}\\}$

This is clear.

##  5 Kähler-type identities for $N$, $\overline{N}$

Further on, we shall need the following analogue of Kähler relations (3.1),
but for the $C^{\infty}(M)$-linear “outer” parts of the de Rham differential,
$N$ and $\overline{N}$.

Proposition 5.1:  Let $(M,I,\omega,\Omega)$ be a nearly Kähler 6-manifold,
$N$, $\overline{N}$ the $(2,-1)$\- and $(-1,2)$-parts of the de Rham
differential, $N^{*}$, $\overline{N}^{*}$ their Hermitian adjoints operators,
and $\Lambda_{\omega}$ the Hermitian adjoint to
$L_{\omega}(\eta):=\omega\wedge\eta$. Then

$\displaystyle\ \ [\Lambda_{\omega},N^{*}]=2\sqrt{-1}\>\overline{N},$
$\displaystyle[\Lambda_{\omega},\overline{N}^{*}]=-2\sqrt{-1}\>N.$ (5.1)
$\displaystyle\ \ [L_{\omega},N]=2\sqrt{-1}\>\overline{N}^{*},$
$\displaystyle[L_{\omega},\overline{N}]=-2\sqrt{-1}\>N^{*}.$

Proof: The equalities of (5.1) are obtained one from another by taking complex
conjugation and Hermitian adjoint, hence they are equivalent. Therefore, it
suffices to prove

$[L_{\omega},N^{*}]=2\sqrt{-1}\>\overline{N}.$ (5.2)

The proof of this formula follows the same lines as the proof of 3.1. Again,
both sides of (5.2) are first order algebraic differential operators on the
algebra $\Lambda^{*}(M)$, in the sense of Grothendieck (2.1). Therefore, it
suffices to check (5.2) only on 0-forms and 1-forms (2.1). On 0-forms, both
sides of (5.2) clearly vanish. To finish the proof of (5.2), it remains to
check

$N^{*}L_{\omega}(\eta)=2\sqrt{-1}\>\overline{N}(\eta),$ (5.3)

where $\eta$ is a 1-form. Since both sides of (5.3) vanish on $(1,0)$-forms,
we may also assume that $\eta\in\Lambda^{0,1}(M)$.

Let $\xi_{1},\xi_{2},\xi_{2}$ be an orthonormal frame in $\Lambda^{1,0}(M)$,
satisfying

$\omega=-\sqrt{-1}\>(\xi_{1}\wedge\overline{\xi}_{1}+\xi_{2}\wedge\overline{\xi}_{2}+\xi_{3}\wedge\overline{\xi}_{3}),\
\ \ \Omega=\xi_{1}\wedge\xi_{2}\wedge\xi_{3}.$

Since both sides of (5.2) are $C^{\infty}(M)$-linear, we need to prove (5.2)
only for $\eta=\overline{\xi}_{1},\overline{\xi}_{2},\overline{\xi}_{2}$.
Assume for example that $\eta=\overline{\xi}_{1}$. Then

$L_{\omega}\eta=-\sqrt{-1}\>\overline{\xi}_{1}\wedge(\xi_{2}\wedge\overline{\xi}_{2}+\xi_{3}\wedge\overline{\xi}_{3}).$

Using (1.4), we obtain

$N^{*}L_{\omega}\eta=-\sqrt{-1}\>\xi_{3}\wedge\xi_{2}+\sqrt{-1}\>\xi_{2}\wedge\xi_{3}=2\sqrt{-1}\>\xi_{2}\wedge\xi_{3}=2\sqrt{-1}\>\>\overline{N}(\eta).$

This proves (5.2). 5 is proved.

5 is used in this paper only once, to obtain the following corollary.

Corollary 5.2:  Let $M$ be a nearly Kähler 6-manifold, $N$, $\overline{N}$ the
$(2,-1)$\- and $(-1,2)$-parts of the de Rham differential, $N^{*}$,
$\overline{N}^{*}$ their Hermitian adjoint operators, and $\Delta_{N}$,
$\Delta_{\overline{N}}$, $\Delta_{N+\overline{N}}$ the corresponding
Laplacians. Then

$\Delta_{N+\overline{N}}=\Delta_{N}+\Delta_{\overline{N}}$

Proof: Clearly, we have

$\Delta_{N+\overline{N}}=\Delta_{N}+\Delta_{\overline{N}}+\\{N,\overline{N}^{*}\\}+\\{\overline{N},N^{*}\\}.$

Therefore, to prove 5, it suffices to show that

$\\{N,\overline{N}^{*}\\}=0,\ \ \\{\overline{N},N^{*}\\}=0.$

One of these equations is obtained from another by complex conjugation;
therefore, they are equivalent. Let us prove, for instance,
$\\{N,\overline{N}^{*}\\}=0$. As follows from 5,

$\\{N,\overline{N}^{*}\\}=-\frac{\sqrt{-1}\>}{2}\\{N,\\{\Lambda_{\omega},N\\}\\}.$
(5.4)

However, $\\{N,N\\}=0$ as follows from (3.3). Using the graded Jacobi
identity, we obtain

$0=\\{\Lambda_{\omega},\\{N,N\\}\\}=\\{\\{\Lambda_{\omega},N\\},N\\}+\\{N,\\{\Lambda_{\omega},N\\}\\}=2\\{N,\\{\Lambda_{\omega},N\\}\\}.$

Therefore, (5.4) implies $\\{N,\overline{N}^{*}\\}=0$. This proves 5.

Remark 5.3: From 5, 4.4 and 4.1, we infer that

$\Delta_{d}=\Delta_{\partial}+\Delta_{\overline{\partial}}+\Delta_{N}+\Delta_{\overline{N}}-\\{\partial^{*},\overline{\partial}\\}-\\{\partial,\overline{\partial}^{*}\\}$
(5.5)

and

$\Delta_{d}=\Delta_{\partial-\overline{\partial}}+\Delta_{N}+\Delta_{\overline{N}}$
(5.6)

##  6 Harmonic forms on nearly Kähler manifolds

###  6.1 Harmonic forms and the $\partial$, $\overline{\partial}$-Laplacians

For harmonic forms on a compact nearly Kähler manifold, the relation (4.1) of
4.1 can be strengthened significantly.

Theorem 6.1:  Let $M$ be a compact nearly Kähler 6-manifold, and $\eta$ a
differential form. Then $\eta$ is harmonic if and only if

$\eta\in\ker\Delta_{\partial}\cap\ker\Delta_{\overline{\partial}}\cap\ker\Delta_{N+\overline{N}}.$
(6.1)

Proof: The “if” part is clear; indeed, if (6.1) is true, then
$\partial\eta=\overline{\partial}\eta=\partial^{*}\eta=\overline{\partial}^{*}\eta=0$,
and $\Delta_{d}\eta=0$ by 4.1.

As 4.4 implies,

$\Delta_{d}\eta=0\Leftrightarrow\bigg{(}\Delta_{\partial-\overline{\partial}}\eta=0,\
\text{and}\ \ \Delta_{N+\overline{N}}\eta=0\bigg{)}.$ (6.2)

Therefore, for any $\Delta_{d}$-harmonic form $\eta$, we have
$\Delta_{\partial-\overline{\partial}}\eta=0$, that is,
$(\partial-\overline{\partial})\eta=0$ and
$(\partial^{*}-\overline{\partial}^{*})\eta=0$. Moreover, since
$d\eta=(N+\overline{N})\eta=0$,
$\partial+\overline{\partial}=d-N-\overline{N}$ vanishes on $\eta$ as well.
Substracting from $(\partial+\overline{\partial})\eta=0$ the relation
$(\partial-\overline{\partial})\eta=0$, we obtain that
$\overline{\partial}\eta=0$. In a similar way one proves the whole set of
equations

$\partial\eta=\overline{\partial}\eta=\partial^{*}\eta=\overline{\partial}^{*}\eta=0$

This gives (6.1). 6.1 is proved.

###  6.2 The Hodge decomposition on cohomology

The main result of this paper is an immediate corollary of 6.1.

Theorem 6.2:  Let $M$ be a compact nearly Kähler 6-manifold, and ${\cal
H}^{i}(M)$ the space of harmonic $i$-forms on $M$. Then ${\cal H}^{i}(M)$ is a
direct sum of harmonic forms of pure Hodge type:

${\cal H}^{i}(M)=\bigoplus\limits_{i=p+q}{\cal H}^{p,q}(M).$ (6.3)

Moreover, ${\cal H}^{p,q}(M)=0$ unless $p=q$ or $(p=2,q=1)$ or $(q=1,p=2)$.

Proof: From 6.1, we find that a form $\eta$ is harmonic if and only if
$\partial\eta=\overline{\partial}\eta=\partial^{*}\eta=\overline{\partial}^{*}\eta=0$
and $\Delta_{N+\overline{N}}\eta=0$. From 5, we find that the latter equation
is equivalent to $N\eta=\overline{N}\eta=N^{*}\eta=\overline{N}^{*}\eta=0$. We
find that a form $\eta$ is harmonic if and only if all the Hodge components of
$d$, $d^{*}$ vanish on $\eta$:

$\partial\eta=\overline{\partial}\eta=\partial^{*}\eta=\overline{\partial}^{*}\eta=N\eta=\overline{N}\eta=N^{*}\eta=\overline{N}^{*}\eta=0.$
(6.4)

Therefore, all Hodge components of $\eta$ also satisfy (6.4). This implies
that these components are also harmonic. We proved (6.3).

To prove that ${\cal H}^{p,q}(M)$ vanishes unless $p=q$ or $(p=2,q=1)$ or
$(q=1,p=2)$, we use 3.2. Let $\eta$ be a non-zero harmonic $(p,q)$-form. Then
the scalar operator $R=\Delta_{\partial}-\Delta_{\overline{\partial}}$
vanishes on $\eta$, $R=\lambda^{2}(3-p-q)(p-q)$. Therefore, either $p=q$ or
$p+q=3$. We obtain immediately that $p=q$ or $(p=2,q=1)$ or $(q=1,p=2)$ or
$(p=3,q=0)$ or $(q=0,p=3)$. The last two cases are impossible: on
$\Lambda^{3,0}(M)$, $\Lambda^{0,3}(M)$, the operator $N+\overline{N}$ is
clearly injective (see (1.4)), hence it cannot vanish; however, by (6.4) we
have $N+\overline{N}(\eta)=0$. This proves 6.2.

Remark 6.3: The middle cohomology of a compact nearly Kähler 6-manifold is
remarkably similar to the middle cohomology of a Kähler manifold. In
particular, the intermediate Jacobian $T:=H^{2,1}(M)/H^{3}(M,{\mathbb{Z}})$ is
well defined in this case as well. As in the Kähler case, $T$ is a compact
complex torus, and we have a pseudoholomorphic map $S{\>\longrightarrow\>}T$
from the space of pseudoholomorphic rational curves on $M$ to the intermediate
Jacobian.

Remark 6.4:  All harmonic forms $\eta\in{\cal H}^{p,q}(M)$, $(p=2,q=1)$ or
$(q=1,p=2)$ are primitive and coprimitive, that is, satisfy
$L_{\omega}(\eta)=\Lambda_{\omega}(\eta)=0$. Indeed, $\\{N,\overline{N}\\}$
vanishes on $\eta$ as follows from (6.4). However, 3.2 implies that
$\\{N,\overline{N}\\}(\eta)=(p-q)\omega\wedge\eta$, hence
$\omega\wedge\eta=0$.

Similarly, all harmonic forms $\eta\in{\cal H}^{p,p}(M)$ are primitive for
$p=1$ and comprimitive for $p=2$. This is implied directly by (6.4) and the
local expression for $N$ given in (1.4). For instance, for a $(1,1)$-form
$\eta$, we have $N(\eta)=\Lambda_{\omega}(\eta)\cdot\Omega$, hence $\eta$ is
primitive if $N(\eta)=0$.

##  7 Appendix: Hodge theory on orbifolds

In the appendinx, we explain how the results of this paper can be applied to
compact nearly Kähler orbifolds.

Recall that an orbifold is a topological space equipped with an atlas of local
charts, which are isomorphic to ${\mathbb{R}}^{n}/G$, where $G$ is a finite
group acting faithfully and smoothly, and with all the gluing maps smooth and
compatible with the group action.

The differential forms on an orbifold are defined in local charts as
$G$-invariant differential forms on ${\mathbb{R}}^{n}$, which are compatible
with the gluing maps.

The de Rham algebra and de Rham cohomology are defined literally in the same
way as for manifolds, and they are equal to the singular cohomology. This was
first observed by I. Satake, who defined the orbifolds in 1950-ies and called
them “V-manifolds” ([S1], [S2]). Since Satake, all the usual constructions of
smooth topology, such as the Chern-Weil theory of characteristic classes,
Atiyah-Singer index formula, signature theorem and Riemann-Roch-Grothendieck
theorem, were generalized for the orbifolds (see e.g. [Ka1], [Ka2], [Ka3]).

The Hodge theory identifies harmonic forms with the de Rham cohomology, using
the closedness of the image of the de Rham differential on a compact manifold.
The basic machinery here works for the orbifolds as well as in the smooth case
([Ka3]).

The results of the present paper, such as 6.2, are obtained by application of
local formulas for Laplacians to the global statements about cohomology, using
the identification of cohomology and harmonic forms. These arguments are
transferred to the orbifold case word by word.

The vanishing theorems, such as 6.1, are obtained by showing that a difference
of certain second order operators is positive, which implies that a kernel of
one of these operators lies inside a kernel of another. Here the vanishing
arguments are also translated to the orbifold case, without any difficulty.

Acknowledgements: I am grateful to Robert Bryant and Paul-Andi Nagy for
enlightening correspondence. P.-A. Nagy also suggested to add 6.2.

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Misha Verbitsky
Institute of Theoretical and Experimental Physics
B. Cheremushkinskaya, 25, Moscow, 117259, Russia, and
Higher School of Economics
Faculty of Mathematics, 7 Vavilova Str. Moscow, Russia,
<EMAIL_ADDRESS>