11institutetext: Institute for Pathophysiology and Allergy Research, Medical
University of Vienna, Vienna, Austria 22institutetext: Department of Computer
Science, Loughborough University, Loughborough, United Kingdom
33institutetext: Research and Development Department, TissueGnostics GmbH,
Vienna, Austria

# Pollen Grain Microscopic Image Classification Using an Ensemble of Fine-
Tuned Deep Convolutional Neural Networks ††thanks: This research has received
funding from the Austrian Research Promotion Agency (FFG), No. 872636. We
thank Nvidia corporation for their generous GPU donation.

Amirreza Mahbod 11 Gerald Schaefer 22 Rupert Ecker 33 Isabella Ellinger 11

###### Abstract

Pollen grain micrograph classification has multiple applications in medicine
and biology. Automatic pollen grain image classification can alleviate the
problems of manual categorisation such as subjectivity and time constraints.
While a number of computer-based methods have been introduced in the
literature to perform this task, classification performance needs to be
improved for these methods to be useful in practice.

In this paper, we present an ensemble approach for pollen grain microscopic
image classification into four categories: Corylus Avellana well-developed
pollen grain, Corylus Avellana anomalous pollen grain, Alnus well-developed
pollen grain, and non-pollen (debris) instances. In our approach, we develop a
classification strategy that is based on fusion of four state-of-the-art fine-
tuned convolutional neural networks, namely EfficientNetB0, EfficientNetB1,
EfficientNetB2 and SeResNeXt-50 deep models. These models are trained with
images of three fixed sizes ($224\times 224$, $240\times 240$, and $260\times
260$ pixels) and their prediction probability vectors are then fused in an
ensemble method to form a final classification vector for a given pollen grain
image.

Our proposed method is shown to yield excellent classification performance,
obtaining an accuracy of of 94.48% and a weighted F1-score of 94.54% on the
ICPR 2020 Pollen Grain Classification Challenge training dataset based on
five-fold cross-validation. Evaluated on the test set of the challenge, our
approach achieved a very competitive performance in comparison to the top
ranked approaches with an accuracy and a weighted F1-score of 96.28% and
96.30%, respectively.

###### Keywords:

Pollen grain images microscopic images deep learning image classification
transfer learning.

##  1 Introduction

Palynology is the scientific study of pollen grains which are produced by
plants for the purpose of reproduction. Nowadays, pollen grain classification
is a valuable tool for various applied sciences including systematics and
forensics [16]. In the medical context, pollens are among the most common
triggers of seasonal allergies. Allergic diseases have become a major public
health problem. The World Health Organization (WHO) estimates that worldwide
400 million people suffer from allergic rhinitis [7]. In Europe, the
prevalence of pollen allergy in the general population is about 40% [9].
Clinicians and patients may be able to anticipate the onset of pollen-related
allergy symptoms by monitoring pollen levels [2].

The performance of any pollen counter (human or machine) depends on two key
tasks: finding the pollen grains by discriminating pollen from non-pollen, and
pollen grain classification. Traditionally, pollen grain classification
involves observation and discrimination of features by a highly qualified
palynologist. While this is an accurate and effective method, it takes
considerable amounts of time and resources [36]. Therefore, automatic
recognition of pollen species by means of computer vision is of major
importance in modern palynology [18].

Automatic computer-based pollen grain image classification methods have been
proposed in several studies with the aim of reducing human interaction in the
analysis procedure [18]. Similar to other medical or natural image
classification tasks, machine learning-based approaches have shown a better
classification performance for pollen grain image classification compared to
standard image processing techniques [1, 33].

Most of the classical machine learning-based approaches for pollen grain image
classification made use of feature engineering to train a classifier such as
multi-layer perceptron or support vector machines. Well-known image features
that were used for this task include morphological features [40], texture
features [12] or hybrid features [4]. However, due to the similar
morphological appearance of pollen grains and various artefacts that may
present, deriving well-working hand-crafted features is a challenging task.

Using deep learning approaches and more specifically convolutional neural
networks (CNNs) [24] allows for a more suitable solution for distinguishing
different types of pollen grains compared to classical machine learning
approaches. These deep models can be trained end-to-end without the use of
hand-crafted features and can be applied directly on raw or pre-processed
images [15, 33, 1, 10].

Since the number of labelled pollen grain images in the publicly available
datasets is too small to train a CNN from scratch, transfer learning can be
employed. Here, pre-trained CNNs can be used as optimised deep feature
extractors or they can be fine-tuned to solve the classification task.
Standard pre-trained CNNs such as AlexNet [22] or VGGNet [35] have been used
for these purposes [1, 33, 30, 34]. However, the performance of such
approaches still needs to be improved to be useful for practical applications.

In this paper, we present an ensemble method for multi-class pollen grain
microscopic image classification. Based on our earlier work on various medical
image classification tasks and recent advances in transfer learning [28, 27,
25, 42], we develop a classification method which exploits four state-of-the-
art pre-trained CNNs fine-tuned with pollen grain images at three different
image resolutions. We then fuse the prediction vectors from different models
in an ensemble strategy to form the final classification vector. Evaluated on
the training set and the test set of the ICPR 2020 Pollen Grain Classification
Challenge111https://iplab.dmi.unict.it/pollenclassificationchallenge, our
proposed method provides excellent classification performance, achieving
weighted F1-scores of 94.54% and 96.30% on the challenge training and test
set, respectively.

##  2 Materials and Methods

###  2.1 Datasets

While a number of earlier datasets exist for pollen grain image
classification, such as the POLEN23E dataset [33] or the dataset from Ranzato
et al. [32], which contain 805 and 1,429 pollen grain images, respectively, in
this paper, we use the ICPR 2020 Pollen Grain Classification Challenge
dataset222https://iplab.dmi.unict.it/pollenclassificationchallenge/train.zip,
which is one the biggest publicly available datasets for this task [1]. This
dataset contains 11,279 microscopic training image of size $84\times 84$
pixels in RGB format. The images belong to one of four classes, namely Corylus
Avellana well-developed pollen grain (normal pollen), Corylus Avellana
anomalous pollen grains (anomalous pollen), Alnus well-developed pollen grains
(Alnus), and non-pollen (debris), example of which are shown in Fig. 1. There
are 1566 normal pollen images, 773 anomalous pollen images, 8216 Alnus images,
and 724 debris images. In addition to the raw images, segmentation masks of
the pollen grains are also provided by the challenge organisers, but we did
not make use of these. A test set of 1,991 images (without segmentation masks)
has also been released but labels for these images are kept private by the
challenge organisers.

Figure 1: Example images from the ICPR 2020 Pollen Grain Classification
Challenge.

###  2.2 Pre-processing

We apply two pre-processing steps. First, we subtract the mean intensity RGB
values of the ImageNet dataset [11] from all training and test images. Then we
resize all images to a fixed size of $260\times 260$ pixels.

###  2.3 Pre-trained CNNs

There are a number of well-established pre-trained CNNs that can be employed
for fine-tuning. Recently developed deep neural networks that are widely used
for various transfer learning applications include but are not limited to
residual networks and their derivatives such as ResNeXt or wide ResNet [17,
41], DenseNets with various depths such as Densenet-121 or DenseNet-169 [20],
squeeze and excitation networks that can be also combined with other models
such as SeResNeXt or SeResNet [19], GoogLeNet and its derivatives such as
Inceptions [37, 38] and EfficientNet [39] models (the state-of-the-art models
for ImageNet classification task). As mentioned above, some earlier pre-
trained CNNs such as AlexNet or VGGNet have already been used for pollen grain
image classification. However, their classification performance can be
improved by employing more advanced architectures which for other tasks have
shown superiority over AlexNet or VGGNet.

In our approach, we exploit four recently developed yet already well-
established pre-trained CNNs from the EfficientNet [39] and SeResNeXt model
[19] families as they have shown excellent performance in, for example,
classification of skin lesions or ophthalmological image classification [27,
42, 29].

The backbone model of the SeResNeXt network is ResNet. ResNet models consist
of special building blocks, called residual blocks, that alleviate the
vanishing gradient problem by connecting the input and output of each residual
block. Thus, the network depth can be increased to yield a better
classification performance with deeper models. ResNet-50, ResNet-101, and
ResNet-152 are the three most used variations, while ResNeXt is a modified
architecture whose residual blocks are wider compared to ResNet through
multiple parallel pathways similar to the inception module in the GoogLeNet
family. SeResNeXt incorporates the squeeze and excitation blocks from [19]
into the model, while SeResNeXt-50 and SeResNeXt-101 are the best-known models
of the family.

In general, by solving the vanishing gradient issue by some techniques such as
skip or dense connections, the performance of a CNN can be improved by
increasing the network depth, width, or input image resolution. In the design
of many former CNNs such as ResNet or wide ResNet, only one aspect of the
model is increased which leads to a better classification performance (e.g.
DenseNet-169 has a better classification score in comparison to DenseNet-121
as it is deeper). In contrast, in the EfficientNet architecture, all three
aspects of the models are increased systematically and by a constant factor.
This leads to generation of eight different architectures, EfficientNetB0 to
EfficientNetB7, that have various depths, widths and pre-defined input image
sizes (from $224\times 224$ to $600\times 600$ pixels) while being
computationally less expensive compared to other well-known CNNs.

We use three variations of EfficientNet, namely EfficientNetB0,
EfficientNetB1, and EfficientNetB2, as well as the SeResNeXt-50 model for
fine-tuning. We choose the shallower versions of the EfficientNet family
networks to prevent over-fitting to the limited training data available. All
four utilised networks are initially trained on natural images with image
sizes between $224\times 224$ to $260\times 260$ pixels

###  2.4 Fine-tuning

We apply a similar approach for fine-tuning the networks as in our work on
skin lesion classification [27, 28, 26]. The general scheme for fine-tuning
the pre-trained networks is depicted in Fig. 2.

Figure 2: Graphic representation of the fine-tuning approach. The backbone
model is either EfficientNetB0, EfficientNetB1, EfficientNetB2 or
SeResNeXt-50. Figure 3: Graphic representation of of the proposed ensemble
strategy.

To fine-tune the pre-trained networks, first fully connected (FC) layers of
the original networks are removed. Then, we apply two blocks of batch
normalisation, drop out layers (with a drop factor of 0.3) and FC layers (with
64 and 4 nodes in the first block and second block, respectively). We train
the models with three different images sizes, namely $224\times 224$,
$240\times 240$, and $260\times 260$ pixels. For the two smaller sizes (i.e.
$224\times 224$ and $240\times 240$ pixel images), random cropping within the
$260\times 260$ pixel images is performed during training and testing. We use
a global average pooling layer just before the first batch normalisation layer
to prevent dimensionality mismatch for different image sizes.

The weights of the newly added FC layers are initialised with the Xavier
method [14] and the weight factors are kept 10 times larger compared to all
other learnable weights in the models. We fine-tune the network with the Adam
optimiser [21] with an initial learning rate of 0.001. We halve the learning
rate after every seven epochs and train the networks in total for 40 epochs.
To deal with the imbalanced dataset set, we use a weighted focal loss function
[23] with weights of 7.20, 14.59, 1.37, and 15.57 for the normal pollen,
anomalous pollen, Alnus pollen, and debris, respectively.

As in [27], we use various augmentation techniques in the training and testing
phases to prevent over-fitting. For training augmentation, we use random
cropping (for the models trained with $224\times 224$ and $240\times 240$
pixels images), adaptive histogram equalisation (with a probability of 0.1),
random cut-out (with a probability of 0.1), random brightness and contrast
shifts (with a probability of 0.4), random horizontal and vertical flipping
(with a probability of 0.5), and random rotations (0, 90, 180 and 270 degrees,
with a probability of 0.5).

In the inference phase, we use 25-folds test time augmentation with the same
augmentation methods as described for training. We monitor the prediction
probability vectors for each augmented image. If the maximum value in a
prediction vector is below 0.5, we disregard that prediction vector and take
the average over the rest of the prediction vectors for a specific test image.

###  2.5 Fusion

To improve the classification performance, we employ a straightforward
ensemble strategy as shown in Fig. 3.

In our fusion scheme, first, the results from the five folds for each image
size and for each network are fused. Then, the results from three different
image sizes for a specific model architecture are combined and finally the
classification probability vectors of the four utilised model architectures
(i.e. EfficientNetB0, EfficientNetB0, EfficientNetB0, and SeResNeXt-50) are
fused to form the final classification predictions. From the final prediction
for each image, we chose the element with maximum probability to determine the
image class. Fusion of multiple models is performed by taking the average over
their classification prediction vectors.

###  2.6 Evaluation

As evaluation measures, we use accuracy, balanced accuracy [3], which is
defined as the average recall over all classes, and the weighted F1 score [5],
which is calculated as the F1 score for each class weighted by its number of
true samples. The latter is also the main evaluation measure that is used in
the ICPR challenge.

###  2.7 Implementation

We use Keras333https://keras.io/ and Tensorflow444https://www.tensorflow.org/
deep learning frameworks for algorithm development. All experiments are
conducted using a single workstation with an Intel Core i7-8700 3.20 GHz CPU,
32 GB of RAM and a Titian V Nvidia GPU card with 12 GB of installed memory.

##  3 Results

Since the ground truth of the test data is kept private by the challenge
organisers, in developing our model and for showing the effectiveness of the
utilised ensemble approach, we use five-fold cross-validation (5CV) based on
the ICPR challenge training data. That is, the training data is split into
five folds and four of these are used for training while the fifth is utilised
for testing. This is repeated five times so that each test fold is used once
and the obtained results are averaged.

Table 1 shows the 5CV results for each network architecture and each input
image size.

Table 1: 5CV results [%] based on different input crop sizes and different
network architectures.

network image size accuracy balanced accuracy weighted F1-score EfficientNetB0
$224\times 224$ 92.48 92.20 92.66 EfficientNetB0 $240\times 240$ 91.96 92.22
92.18 EfficientNetB0 $260\times 260$ 93.07 92.18 93.20 EfficientNetB1
$224\times 224$ 93.01 92.11 93.16 EfficientNetB1 $240\times 240$ 94.06 92.68
94.13 EfficientNetB1 $260\times 260$ 93.20 92.60 93.34 EfficientNetB2
$224\times 224$ 92.74 92.03 92.91 EfficientNetB2 $240\times 240$ 92.68 92.38
92.84 EfficientNetB2 $260\times 260$ 93.18 92.17 93.31 SeResNeXt-50 $224\times
224$ 92.00 91.84 92.20 SeResNeXt-50 $240\times 240$ 92.40 91.81 92.57
SeResNeXt-50 $260\times 260$ 91.50 91.49 91.74

The results in Table 2 give the performance of the classification models
obtained by fusing networks trained at the three different image sizes. The
final row in Table 2 shows the final fusion performance, derived from fusion
of 12 (3 image sizes $\times$ 4 architectures) sub-models.

Table 2: 5CV results [%] based on network fusion with different input image sizes and fusion of all architectures. network | image size | accuracy | balanced accuracy | weighted F1-score
---|---|---|---|---
EfficientNetB0 | all sizes | 93.35 | 92.75 | 93.47
EfficientNetB1 | all sizes | 94.26 | 92.83 | 94.34
EfficientNetB2 | all sizes | 93.61 | 92.52 | 93.72
SeResNeXt-50 | all sizes | 93.45 | 92.38 | 93.56
all networks | all sizes | 94.48 | 93.22 | 94.54

We also compare the obtained results with other state-of-the-art
classification models for pollen grain images. In particular, we compare with
the results reported in [1] where both classical feature extraction-based
methods and deep learning models were evaluated. The investigated techniques
include histogram of oriented gradients (HOG) [8] and local binary pattern
(LBP) [31] features, and multi-layer perceptron (MLP) [13] and support vector
machine (SVM) [6] classifiers, while the explored deep learning models include
AlexNet [22] and VGG [35].

The results are listed in Table 3. It should be noted that the results in [1]
are obtained based on a fixed split into 85% training data and 15% test data
and that this split is not publicly available. In contrast, we perform 5CV to
evaluate our method, and thus use slightly less training data while ensuring
that all available data is used once for testing.

Table 3: Comparison to other state-of-the-art methods from [1] based on the ICPR 2020 Pollen Grain Classification Challenge training data. approach type | features/model | accuracy | weighted F1-score
---|---|---|---
classical machine learning | HOG features + RBF SVM | 86.58 | 85.66
classical machine learning | HOG features + MLP | 84.93 | 84.31
classical machine learning | LBP features + MLP | 80.02 | 77.64
deep learning | AlexNet + augmentation | 89.63 | 88.97
deep learning | small VGG + augmentation | 89.73 | 89.14
deep learning | our approach | 94.48 | 94.54

Examples of correctly or incorrectly classified images in the training data
(by the full fusion approach) are shown in Fig. 4 and Fig. 5, respectively.

|  |
---|---|---
normal pollen | anomalous pollen | Alnus
|  |
Alnus | debris | debris
Figure 4: Examples of correctly classified images. |
---|---
normal pollen (debris) | anomalous pollen (Alnus)
|
Alnus (normal pollen) | debris (Alnus)
Figure 5: Examples of incorrectly classified images. The ground truth class is
mentioned first and the predicted class in (brackets).

As the final experiment, we extend our fusion approach to combine the output
of all 5CV results in the ensemble strategy to investigate the performance of
the model for the 1,991 test images of the ICPR challenge. Hence, instead of
combining 12 sub-models, we fuse the results of all 60 sub-models (5CV
$\times$ 3 image sizes $\times$ 4 architectures). The results for this
(evaluated by the challenge organisers) and comparison to the top three
performers of the ICPR challenge for the test data are shown in Table 4.

Table 4: Results on ICPR Challenge test data of our extended fusion approach and the top three performers of the challenge.  User name | accuracy | weighted F1-score
---|---|---
Zhangbaochang | 97.53 | 97.51
Fang Chao | 97.34 | 97.30
Penghui Gui | 97.29 | 97.26
our approach | 96.28 | 96.30

##  4 Discussion

The results in Table 1 show that all networks deliver very good classification
performance, while EfficientNetB1 is slightly superior compared to the rest
and SeResNeXt-50 is slightly worse.

Looking at Table 2, we can see that fusing the results from different image
sizes improves the classification performance for all four models. The
performance is further slightly improved when the results from different model
architectures are also fused as shown in the bottom row of Table 2.

The comparison between our proposed method and other state-of-the-art
techniques in Table 3 demonstrates a clear superiority of our approach over
the other classification models. While, as mentioned, the training/test
configuration we employ is somewhat different from that of the other methods,
it represents arguably a harder test, yet as the results in the table clearly
indicate we achieve significantly better results both in terms of accuracy and
weighted F1 score. In addition, the techniques reported in [1] utilise the
segmentation masks (which are only available for the training dataset of the
challenge) to obtain segmented images. In contrast, our approach uses only the
raw images as input and is segmentationless.

Finally, the results in Table 4 show the performance of our proposed method on
the actual test data of the ICPR challenge. By comparing the 5CV results (last
row in Table 3) and the test results (last row of Table 4), performance
improvement in both accuracy and weighted F1 score can be observed. Compared
to the top performers of the ICPR challenge our approach delivers very
competitive performance with slightly inferior results (by 1.25% and 1.21% in
terms of accuracy and weighted F1 score, respectively). Further details of the
test set results can be found on the challenge
website555https://iplab.dmi.unict.it/pollenclassificationchallenge/results.
Note that the details of the other approaches are not disclosed at the time of
writing this manuscript.

##  5 Conclusions

In this paper, we have proposed an effective approach to classify pollen grain
microscopic images to four pollen grain types. We exploit several fine-tuned
CNNs at various image sizes and combined them in a simple yet effective fusion
framework. Our approach does not require segmentation and we have demonstrated
excellent classification results on the challenge’s training and test data
sets. The performance can be potentially further improved by incorporating
other network architectures or other image resolutions.

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# BV operators of the Gerstenhaber algebras of holomorphic polyvector fields
on toric varieties

Yang Deng  School of Mathematics and Statistics, Wuhan University, Wuhan,
430072, China<EMAIL_ADDRESS>and  Wei Hong  School of Mathematics and
Statistics, Wuhan University, Wuhan, 430072, China  Hubei Key Laboratory of
Computational Science, Wuhan University, Wuhan, 430072, China
<EMAIL_ADDRESS>
###### Abstract.

The vector space of holomorphic polyvector fields on any complex manifold has
a natural Gerstenhaber algebra structure. In this paper, we study BV operators
of the Gerstenhaber algebras of holomorphic polyvector fields on smooth
compact toric varieties. We give a necessary and sufficient condition for the
existence of BV operators of the Gerstenhaber algebra of holomorphic
polyvector fields on any smooth compact toric variety.

###### Key words and phrases:

toric variety, holomorphic polyvector fields, Gerstenhaber algebra, BV
operator

##  1\. introduction

Polyvector fields appeared in the recent work of many mathematicians.
Barannikov and Kontsevich [B-K98] show that polyvector fields play important
roles in deformation theory and mirror symmetry. Polyvector fields also
appeared in the work of Bandiera, Stiénon and Xu [B-S-Xu19]. In differential
geomtry, holomorphic polyvector fields appeared in the work of Hitchin
[Hitchin11]. It is well known that the vector space of polyvector fields on a
smooth manifold $M$ has a Gerstenhaber algebra structure by the wedge product
and the Schouten bracket of polyvector fields. In the same way, the vector
space of holomorphic polyvector fields on a complex manifold $X$ becomes a
Gerstenhaber algebra. And the Gerstenhaber algebra structure of polyvector
fields on the algebraic torus is studied in [M-R19].

This paper is a subsequent work of [Hong20]. For any smooth compact toric
variety $X$, the vector space of holomorphic polyvector fields on $X$ has been
given an explicit description in [Hong20]. In this paper, we study Batalin-
Vilkovisky operators for the Gerstenhaber algebras of holomorphic polyvector
fields on smooth compact toric varieties. We give a necessary and sufficient
condition for the existence of BV operators. And we also give the explicit
forms of the BV operators.

Acknowledgements. Deng would like to thank Prof. Yuping Tu and Prof. Fengwen
An for their help during his study in Wuhan University. Hong’s research is
partially supported by NSFC grant 12071241 and NSFC grant 12071358.

##  2\. Preliminary

###  2.1. Gerstenhaber algebra and Batalin-Vilkovisky algebra

Let us recall some classical knowledge of Gerstenhaber algebra and Batalin-
Vilkovisky algebra (BV algebra). One may consult [Roger09] and [Xu99].

Let $\mathcal{A}=\bigoplus_{k\in\mathbb{Z}}\mathcal{A}^{k}$ be a
$\mathbb{Z}$-graded vector space over a field $\mathbf{k}$. For any element
$a\in\mathcal{A}^{k}$, we denote by $|a|=k$ the degree of $a$. Let
$\mathcal{A}[1]=\bigoplus_{k\in\mathbb{Z}}\mathcal{A}[1]^{k}$, where
$\mathcal{A}[1]^{k}=\mathcal{A}^{k+1}$.

######  Definition 2.1.

A graded vector spcae $\mathcal{A}=\bigoplus_{k\in\mathbb{Z}}\mathcal{A}^{k}$
is a Gerstenhaber algebra if one has:

  1. (1)

An associative, graded commutative multiplication

$\mathcal{A}\times\mathcal{A}\rightarrow\mathcal{A},$

ı.e., for every $a,b,c\in\mathcal{A}$, we have

     * •

$a\cdot(b\cdot c)=(a\cdot b)\cdot c$,

     * •

$a\cdot b=(-1)^{|a||b|}b\cdot a$.

  2. (2)

A graded Lie algebra bracket
$[,]:\mathcal{A}[1]\times\mathcal{A}[1]\rightarrow\mathcal{A}[1]$, ı.e., we
have $[\mathcal{A}^{k},\mathcal{A}^{l}]\subseteq\mathcal{A}^{k+l-1}$, and for
every $a,b,c\in\mathcal{A}$,

     * •

$[a,b]=-(-1)^{(|a|-1)(|b|-1)}[b,a]$ (graded anti-symmetry),

     * •

$(-1)^{(|a|-1)(|c|-1)}[[a,b],c]+(-1)^{(|b|-1)(|a|-1)}[[b,c],a]+(-1)^{(|c|-1)(|b|-1)}[[c,a],b]=0$
(graded Jacobi identity).

  3. (3)

The multiplication and bracket satisfying the Leibniz relation, ı.e., for
every $a,b,c\in\mathcal{A}$,

$[a,b\cdot c]=[a,b]\cdot c+(-1)^{(|a|-1)|b|}b\cdot[a,c].$

We have the following examples of Gerstenhaber algebras.

######  Example 2.2.

Let $M$ be a smooth real manifold. Let

$\mathcal{A}^{k}=\begin{cases}\Gamma(M,\wedge^{k}TM),\quad k>0;\\\
C^{\infty}(M),\quad k=0;\\\ 0,\quad k<0.\end{cases}$

Then $\mathcal{A}=\bigoplus_{k\in\mathbb{Z}}\mathcal{A}^{k}$ is a
$\mathbb{Z}$-graded vector space over $\mathbb{R}$. Let the multiplication on
$\mathcal{A}$ be the wedge product on $\Gamma(M,\wedge^{k}TM)$. The bracket
$[,]$ is given by the Schouten bracket of polyvector fields. Then
$(\mathcal{A},\wedge,[,])$ is a Gerstenhaber algebra.

######  Example 2.3.

Let $X$ be a complex manifold. Let
$\mathcal{A}^{k}=H^{0}(X,\wedge^{k}\mathcal{T}_{X})$ be the vector space of
holomorphic $k$-vector fields on $X$ for $k>0$. Let
$\mathcal{A}^{0}=H^{0}(X,\mathcal{O}_{X})$ be the vector space of holomorphic
functions on $X$. And let $\mathcal{A}^{k}=0$ for $k<0$. Then
$\mathcal{A}=\bigoplus_{k\in\mathbb{Z}}\mathcal{A}^{k}$ is a
$\mathbb{Z}$-graded vector space over $\mathbb{C}$. Let the multiplication on
$\mathcal{A}$ be the wedge product of polyvector fields. The bracket $[,]$ is
given by the Schouten bracket of polyvector fields. Then
$(\mathcal{A},\wedge,[,])$ is a Gerstenhaber algebra.

Next we give two equivalent definitions of Batalin-Vilkovisky algebras.

######  Definition 2.4.

A graded vector space $\mathcal{A}=\bigoplus_{k\in\mathbb{Z}}\mathcal{A}^{k}$
over $\mathbf{k}$ is a Batalin-Vilkovisky algebra (BV algebra) if

  1. (1)

$\mathcal{A}$ is an associative, graded commutative algebra, i.e., there is an
associative, graded commutative multiplication on $\mathcal{A}$.

  2. (2)

There is a differential $\mathbf{k}$-linear operator
$D:\mathcal{A}\rightarrow\mathcal{A}$ of order $2$ and degree $-1$. Here $D$
is a differential operator of order $2$ in the sense that, for all
$a,b,c\in\mathcal{A}$,

(2.1) $\displaystyle
D(abc)=D(ab)c+(-1)^{|a|}aD(bc)+(-1)^{|b|(|a|+1)}bD(ac)-D(a)bc-(-1)^{|a|}aD(b)c-(-1)^{|a|+|b|}abD(c)$

  3. (3)

$D^{2}=0$.

A BV algebra $(\mathcal{A},\cdot,D)$ is always a Gerstenhaber algebra with the
same multiplication, and the bracket is given by

$[a,b]=(-1)^{|a|}(D(ab)-D(a)b-(-1)^{|a|}aD(b))$

for all $a,b\in\mathcal{A}$.

The following is an equivalent definition of BV algebra.

######  Definition 2.5.

Given a Gerstenhaber algebra $(\mathcal{A},\cdot,[,])$, if there is exist a
degree $-1$ $\mathbf{k}$-linear operator $D:\mathcal{A}\rightarrow\mathcal{A}$
such that

  1. (1)

(2.2) $[a,b]=(-1)^{|a|}(D(ab)-D(a)b-(-1)^{|a|}aD(b))$

for any $a,b\in\mathcal{A}$,

  2. (2)

$D^{2}=0$,

then the Gerstenhaber algebra $(\mathcal{A},\cdot,[,])$ is said to be exact,
and $D$ is called a generating operator.

That a Gerstenhaber algebra $(\mathcal{A},\cdot,[,])$ is exact is equivalent
to that $(\mathcal{A},\cdot,D)$ is a BV algebra. And the generating operator
$D$ is also called a BV operator.

###  2.2. Holomorphic polyvector fields on toric varieties

Recall that a toric variety [Cox] is an irreducible variety $X$ such that

  1. (1)

$T=(\mathbb{C}^{*})^{n}$ is a Zariski open set of $X$,

  2. (2)

the action of $T=(\mathbb{C}^{*})^{n}$ on itself extends to an action of
$T=(\mathbb{C}^{*})^{n}$ on $X$.

One may consult [Oda] and [Fulton] for more details of toric varieties.

Let $N=\operatorname{Hom}(\mathbb{C}^{*},T)\cong\mathbb{Z}^{n}$ and
$M=\operatorname{Hom}(T,\mathbb{C}^{*})$. Then
$M\cong\operatorname{Hom}_{\mathbb{Z}}(N,\mathbb{Z})$ and
$N\cong\operatorname{Hom}_{\mathbb{Z}}(M,\mathbb{Z})$. Each element $m$ in $M$
gives rise to a character $\chi^{m}\in Hom(T,\mathbb{C}^{*})$, which can be
also considered as a rational function on $X$. Let
$N_{\mathbb{R}}=N\otimes_{\mathbb{Z}}\mathbb{R}\cong\mathbb{R}^{n}$, and
$M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}\cong\mathbb{R}^{n}$ be the
dual space of $N_{\mathbb{R}}$.

Let $X=X_{\Delta}$ be a smooth compact toric variety associated with a fan
$\Delta$ in $N_{\mathbb{R}}\cong\mathbb{R}^{n}$. The $T$-action on $X$ induces
a $T$-action on $H^{0}(X,\wedge^{k}\mathcal{T}_{X})$, the vector space of
holomorphic $k$-vector fields on $X$. Denote by $V_{I}^{k}$ the weight space
corresponding to the character $I\in M$. We have

(2.3) $H^{0}(X,\wedge^{k}\mathcal{T}_{X})=\bigoplus_{I\in M}V_{I}^{k}.$

Since $H^{0}(X,\wedge^{k}\mathcal{T}_{X})$ is a finite dimensional vector
space, there are only finite elements $I\in M$ such that $V_{I}^{k}\neq 0$.

Some technical notations are necessary for us in this paper.

  1. (1)

Let $N_{\mathbb{C}}=N\otimes_{\mathbb{Z}}\mathbb{C}$. Since $T\cong
N\otimes_{\mathbb{Z}}\mathbb{C}^{*}$, we have $\text{Lie}(T)\cong
N_{\mathbb{C}}$, where $\text{Lie}(T)$ denotes the Lie algebra of $T$. We
define a map $\rho:N_{\mathbb{C}}\rightarrow\mathfrak{X}(X)$ by

(2.4)
$\rho:N_{\mathbb{C}}=N\otimes_{\mathbb{Z}}\mathbb{C}\cong\text{Lie}(T)\rightarrow\mathfrak{X}(X),$

where $\text{Lie}(T)\rightarrow\mathfrak{X}(X)$ is defined by the
infinitesimal action of the Lie algebra $\text{Lie}(T)$ on $X$. And the map
$\rho:N_{\mathbb{C}}\rightarrow\mathfrak{X}(X)$ is determined by

$\rho(x)(\chi^{m})=\langle x,m\rangle\chi^{m}$

for any $x\in N_{\mathbb{C}}$ and $m\in M$. Since the action map $T\times
X\rightarrow X$ is holomorphic, the images of $\rho$ are holomorphic vector
fields on $X$. By the abuse of notation, the induced maps

(2.5) $\wedge^{k}N_{\mathbb{C}}\rightarrow
H^{0}(X,\wedge^{k}\mathcal{T}_{X}):\quad x_{1}\wedge\ldots\wedge
x_{k}\rightarrow\rho(x_{1})\wedge\ldots\wedge\rho(x_{k}),\quad\forall
x_{1},\ldots,x_{k}\in N_{\mathbb{C}},$

are also denoted by $\rho$ for $2\leq k\leq n$. Let $W=\rho(N_{\mathbb{C}})$,
$W^{k}=\wedge^{k}W$ and $W^{0}=\mathbb{C}$. Then $W^{k}$ is a subspace of
$H^{0}(X,\wedge^{k}\mathcal{T}_{X})$. And the map
$\wedge^{k}N_{\mathbb{C}}\xrightarrow{\rho}W^{k}$ is an isomorphism.

  2. (2)

Let $\Delta(k)$ $(0\leq k\leq n)$ be the set of all $k$-dimensional cones in
$\Delta$. Suppose that $\alpha_{t}~{}(1\leq t\leq r)$ are all one dimensional
cones in $\Delta(1)$. Let $e(\alpha_{t})\in N$ $(1\leq t\leq r)$ be the
corresponding primitive elements, i.e., the unique generator of
$\alpha_{t}\cap N$. Let $P_{\Delta}$ be a polytope in $M_{\mathbb{R}}$ defined
by

$P_{\Delta}=\bigcap_{\alpha_{t}\in\Delta(1)}\\{I\in M_{\mathbb{R}}\mid\langle
I,e(\alpha_{t})\rangle\geq-1\\}.$

Since $X_{\Delta}$ is compact, $P_{\Delta}$ is a compact polytope in
$M_{\mathbb{R}}$. Let

$S_{\Delta}=\\{I\mid I\in M\cap P_{\Delta}\\}.$

Then $S_{\Delta}$ is a non-empty finite set and $0\in S_{\Delta}$. We denote
by $P_{\Delta}(i)$ the set of all $i$-dimensional faces of the polytope
$P_{\Delta}$. Let

$S(\Delta,i)=\bigcup_{F_{j}\in P_{\Delta}(n-i)}\\{I\in int(F_{j})\cap M\\}$

for $0\leq i\leq n$, where $int(F_{j})$ denotes the relative interior of the
face $F_{j}$. Let

$S_{k}(\Delta)=\bigcup_{0\leq i\leq k}S(\Delta,i)$

for all $0\leq k\leq n$.

  3. (3)

For any $I\in S_{\Delta}$, there exists a unique face $F_{I}(\Delta)$ of
$P_{\Delta}$ such that $I\in int(F_{I}(\Delta))\cap M$. Let

$E_{I}(\Delta)=\\{e(\alpha_{t})\mid\alpha_{t}\in\Delta(1)~{}\text{and}~{}\langle
I,e(\alpha_{t})\rangle=-1\\}.$

Suppose that $E_{I}(\Delta)=\\{e(\alpha_{s_{1}}),\ldots e(\alpha_{s_{l}})\\}$,
where $1\leq s_{1}<\ldots<s_{l}\leq r$. Let $F^{\perp}_{I}(\Delta)\subseteq
N_{\mathbb{R}}$ be the normal space of $F_{I}(\Delta)$, defined by

$F^{\perp}_{I}(\Delta)=\sum_{1\leq t\leq l}\mathbb{R}\cdot e(\alpha_{s_{t}}).$

And we set

$F^{\perp}_{I}(\Delta)=0\quad\text{if}\quad E_{I}(\Delta)=\varnothing.$

  4. (4)

Suppose that $I\in S(\Delta,i)$. Then we have $\dim F_{I}(\Delta)=n-i$ and
$\dim F^{\perp}_{I}(\Delta)=i$. Considering
$F_{I}^{\perp}(\Delta)\otimes\mathbb{C}$ as a subspace of $N_{\mathbb{C}}$, we
have $\wedge^{i}(F^{\perp}_{I}(\Delta)\otimes\mathbb{C})\cong\mathbb{C}$.

Let $\hat{F}_{I}(\Delta)$ be the affine space in $M_{\mathbb{C}}$ with the
lowest dimension such that $F_{I}(\Delta)\subseteq\hat{F}_{I}(\Delta)$, where
$F_{I}(\Delta)\subset M_{\mathbb{R}}$ is considered as a subset of
$M_{\mathbb{C}}$ in the natural way. Then we have

(2.6)
$\hat{F}_{I}(\Delta)=\\{I+\alpha\mid\alpha\in(F^{\perp}_{I}(\Delta)\otimes\mathbb{C})^{\perp}\\},$

where $(F^{\perp}_{I}(\Delta)\otimes\mathbb{C})^{\perp}\subseteq
M_{\mathbb{C}}$ is the annihilator of
$F_{I}^{\perp}(\Delta)\otimes\mathbb{C}\subseteq N_{\mathbb{C}}$. Especially,
if $I\in M$ is a vertex of $P_{\Delta}$, then we have
$\hat{F}_{I}(\Delta)=\\{I\\}$.

  5. (5)

Let

$N_{I}^{k}(\Delta)=\begin{cases}\wedge^{i}(F_{I}^{\perp}(\Delta)\otimes\mathbb{C})\wedge(\wedge^{k-i}N_{\mathbb{C}})\quad&\text{for}~{}0<i\leq
k,\\\ \wedge^{k}N_{\mathbb{C}}&\text{for}~{}i=0,\\\
0\quad&\text{for}~{}i>k.\end{cases}$

Then $N_{I}^{k}(\Delta)$ is a subspace of $\wedge^{k}N_{\mathbb{C}}$. We have
to notice that

$N_{I}^{k}(\Delta)\begin{cases}\neq 0\quad\text{for}~{}~{}I\in
S_{k}(\Delta)\\\ =0\quad\text{for}~{}~{}I\notin S_{k}(\Delta)\\\ \end{cases}$

Moreover, we have

(2.7) $N_{I}^{n}(\Delta)=\wedge^{n}N_{\mathbb{C}}$

for all $I\in S_{\Delta}$.

  6. (6)

Let

$W_{I}^{k}(\Delta)=\rho(N_{I}^{k}(\Delta))$

be a subspace of $H^{0}(X,\wedge^{k}\mathcal{T}_{X})$. We define

$V_{I}^{k}(\Delta)=\chi^{I}\cdot W_{I}^{k}(\Delta)=\\{\chi^{I}\cdot w\mid w\in
W_{I}^{k}(\Delta)\\}.$

The elements in $V_{I}^{k}(\Delta)$ are considered as meromorphic $k$-vector
fields on $X=X_{\Delta}$. In [Hong20], it is proved that for any $I\in
S_{k}(\Delta)$, the $k$-vector fields in $V_{I}^{k}(\Delta)$ have moveable
singularities. And we use them to represent the corresponding holomorphic
$k$-vector fields.

The next theorem gives a description of the holomorphic polyvector fields on
toric varieties.

######  Theorem 2.6.

[Hong20] Let $X=X_{\Delta}$ be a smooth compact toric variety associated with
a fan $\Delta$ in $N_{\mathbb{R}}$. Then we have the following decomposition

(2.8) $H^{0}(X,\wedge^{k}\mathcal{T}_{X})=\bigoplus_{I\in
S_{k}(\Delta)}V_{I}^{k}(\Delta)$

for all $0\leq k\leq n$, where $V_{I}^{k}(\Delta)=V_{-I}^{k}$ for all $I\in
S_{k}(\Delta)$.

###  2.3. The Gerstenhaber algebra $(\mathcal{A}_{T},\wedge,[,])$

In the case of $X=T=(\mathbb{C}^{*})^{n}$, the Gerstenhaber algebra structure
of the polyvector fields (in algebraic version) has been studied in [M-R19].
Let us recall the results.

Any $k$-polyvector field (in algebraic version) on $X=T$ can be written as the
finite sum

$\sum_{I\in M}c_{I}\cdot\chi^{I}\cdot\rho(A_{I}),$

where $c_{I}\in\mathbb{C}$, $A_{I}\in\wedge^{k}N_{\mathbb{C}}$.

Let

$\mathbb{C}[M]=\\{\sum_{I\in M}c_{I}\cdot\chi^{I}\mid
c_{I}\in\mathbb{C},~{}~{}~{}\text{and there are only finite $c_{I}\neq 0$ in
the sum}\\},$

which can be considered as the group algebra of the abelian group $M$.

Let

$\mathcal{A}_{T}^{k}=\mathbb{C}[M]\otimes_{\mathbb{C}}\wedge^{k}N_{\mathbb{C}},$

where $0\leq k\leq n$. Then the vector space of $k$-polyvector field (in
algebraic version) on $X=T$ is isomorphic to $\mathcal{A}_{T}^{k}$ by the map

$\sum_{I\in M}c_{I}\cdot\chi^{I}\cdot\rho(A_{I})\longleftrightarrow\sum_{I\in
M}c_{I}\cdot\chi^{I}\otimes A_{I}$

Let

$\mathcal{A}_{T}=\bigoplus_{k\in\mathbb{Z}}\mathcal{A}^{k}_{T},$

where $\mathcal{A}_{T}^{k}=0$ for $k<0$ or $k>n$. The Gerstenhaber algebra
structure of the polyvector fields on $X=T$ can be described in the following
way.

######  Proposition 2.7.

[M-R19]

  1. (1)

The vector space $(\mathcal{A}_{T},\cdot,[,])$ is a Gerstenhaber algebra, with
the multiplication and the bracket given by the following way.

     * •

The multiplication on $\mathcal{A}_{T}$ is given by

$(\chi^{I}\otimes A_{I})\cdot(\chi^{J}\otimes
A_{J})=\chi^{I+J}\otimes(A_{I}\wedge A_{J}),$

where $I,J\in M$,
$A_{I}\in\wedge^{k}N_{\mathbb{C}},A_{J}\in\wedge^{l}N_{\mathbb{C}}$.

     * •

The bracket $[,]$ on $\mathcal{A}_{T}$ is given by

$[\chi^{I}\otimes A_{I},\chi^{J}\otimes
A_{J}]=(-1)^{k+1}\chi^{I+J}\otimes((\imath_{J}A_{I})\wedge
A_{J}+(-1)^{k}A_{I}\wedge(\imath_{I}A_{J})),$

where $I,J\in M$,
$A_{I}\in\wedge^{k}N_{\mathbb{C}},A_{J}\in\wedge^{l}N_{\mathbb{C}}$, and
$\imath_{J}(A_{I})$ is the contraction of $J\in M\subset M_{\mathbb{C}}$ with
$A_{I}\in\wedge^{k}N_{\mathbb{C}}$.

  2. (2)

Let $D:\mathcal{A}_{T}\rightarrow\mathcal{A}_{T}$ be a $\mathbb{C}$-linear
operator given by

$D(\chi^{I}\otimes A_{I})=-\chi^{I}\otimes(\imath_{I}A_{I}),$

where $I\in M$ and $A_{I}\in\wedge^{k}N_{\mathbb{C}}$. Then $D$ is a BV
operator for the Gerstenhaber algebra $(\mathcal{A}_{T},\cdot,[,])$.

######  Remark 2.8.

In the second part of Proposition 2.7, the original statement in [M-R19] is
that $-D$ is a BV operator for the Gerstenhaber algebra
$(\mathcal{A}_{T},\cdot,-[,])$, where the bracket on $\mathcal{A}_{T}$ is
choosing to be $-[,]$ (negative the bracket $[,]$), the operator $D$, the
bracket $[,]$ and the multiplication $\cdot$ is shown in the first part of
Proposition 2.7.

##  3\. BV operators of the Gerstenhaber algebras of holomorphic polyvector
fields on toric varieties

###  3.1. The Gerstenhaber algebra $(\mathcal{A}_{\Delta},\wedge,[,])$

Let $X=X_{\Delta}$ be a smooth compact toric variety. Let
$\mathcal{A}^{k}_{\Delta}=H^{0}(X,\wedge^{k}\mathcal{T}_{X})$ be the vector
space of holomorphic $k$-vector fields on $X$ for $k>0$. Let
$\mathcal{A}^{0}_{\Delta}=H^{0}(X,\mathcal{O}_{X})=\mathbb{C}$ and let
$\mathcal{A}^{k}_{\Delta}=0$ for $k<0$ and $k>n$. Let

$\mathcal{A}_{\Delta}=\bigoplus_{k\in\mathbb{Z}}\mathcal{A}^{k}_{\Delta}.$

As shown in Example 2.3, $(\mathcal{A}_{\Delta},\wedge,[,])$ is a Gerstenhaber
algebra.

By Theorem 2.6, we have

(3.1) $\mathcal{A}^{k}_{\Delta}=\bigoplus_{I\in
S_{k}(\Delta)}V_{I}^{k}(\Delta).$

The elements in $V_{I}^{k}(\Delta)$ can be written as
$\chi^{I}\cdot\rho(A_{I})$, where $I\in S_{k}(\Delta)$ and $A_{I}\in
N_{I}^{k}(\Delta)$.

######  Proposition 3.1.

  1. (1)

The Gerstenhaber algebra structure on $(\mathcal{A}_{\Delta},\wedge,[,])$ can
be described by the following way. For any $\chi^{I}\cdot\rho(A_{I})\in
V_{I}^{k}(\Delta)$ and $\chi^{J}\cdot\rho(A_{J})\in V_{J}^{l}(\Delta)$, we
have

$\displaystyle(\chi^{I}\cdot\rho(A_{I}))\wedge(\chi^{J}\cdot\rho(A_{J}))=\chi^{I+J}\cdot\rho(A_{I}\wedge
A_{J}),$
$\displaystyle[\chi^{I}\cdot\rho(A_{I}),\chi^{J}\cdot\rho(A_{J})]=(-1)^{k+1}\chi^{I+J}\cdot\rho((\imath_{J}A_{I})\wedge
A_{J}+(-1)^{k}A_{I}\wedge(\imath_{I}A_{J})),$

where $I\in S_{k}(\Delta),A_{I}\in N_{I}^{k}(\Delta)$, and $J\in
S_{l}(\Delta),A_{J}\in N_{J}^{l}(\Delta)$. As a consequence, we have
$V_{I}^{k}(\Delta)\wedge V_{J}^{l}(\Delta)\subseteq V_{I+J}^{k+l}(\Delta)$ and
$[V_{I}^{k}(\Delta),V_{J}^{l}(\Delta)]\subseteq V_{I+J}^{k+l-1}(\Delta)$.

  2. (2)

Let $\gamma:\mathcal{A}_{\Delta}\rightarrow\mathcal{A}_{T}$ be a
$\mathbb{C}$-linear map defined by

$\gamma(\chi^{I}\cdot\rho(A_{I}))=\chi^{I}\otimes A_{I},$

where $\chi^{I}\cdot\rho(A_{I})\in V_{I}^{k}(\Delta)$, $I\in S_{k}(\Delta)$
and $A_{I}\in N_{I}^{k}(\Delta)$. Then
$\gamma:\mathcal{A}_{\Delta}\rightarrow\mathcal{A}_{T}$ is an injective map.
And
$\gamma:(\mathcal{A}_{\Delta},\wedge,[,])\rightarrow(\mathcal{A}_{T},\wedge,[,])$
is a morphism of Gerstenhaber algebras. Thus
$(\mathcal{A}_{\Delta},\wedge,[,])$ can be considered as a sub-Gerstenhaber
algebra of $(\mathcal{A}_{T},\wedge,[,])$.

Proposition 3.1 can be proved in a similar way to Proposition 2.7, which can
be find in [M-R19]. Here we skip the details.

###  3.2. BV operators of the Gerstenhaber algebra
$(\mathcal{A}_{\Delta},\wedge,[,])$

Let $X=X_{\Delta}$ be a smooth compact toric variety. A natural question is
when the Gerstenhaber algebra $(\mathcal{A}_{\Delta},\wedge,[,])$ will be
exact? In other words, is there any degree $-1$ $\mathbb{C}$-linear operator
$D:\mathcal{A}_{\Delta}\rightarrow\mathcal{A}_{\Delta}$ such that

  1. (1)

(3.2) $[a,b]=(-1)^{|a|}(D(a\wedge b)-D(a)\wedge b-(-1)^{|a|}a\wedge D(b))$

for any $a,b\in\mathcal{A}_{\Delta}$,

  2. (2)

(3.3) $D^{2}=0.$

Since $\mathcal{A}_{\Delta}^{k}=0$ for $k<0$ and $k>n$, we only need to
consider the linear maps
$D:\mathcal{A}_{\Delta}^{k}\rightarrow\mathcal{A}_{\Delta}^{k-1}$ in the case
of $1\leq k\leq n$.

######  Lemma 3.2.

Let $D$ be a BV operator on the Gerstenhaber algebra
$(\mathcal{A}_{\Delta},\wedge,[,])$. Then there exists an element $\delta\in
M_{\mathbb{C}}$ such that, for all $\chi^{I}\cdot\rho(A)\in V_{I}^{k}(\Delta)$
($1\leq k\leq n$), we have

(3.4) $D(\chi^{I}\cdot\rho(A))=\chi^{I}\cdot\rho(\imath_{\delta-I}A),$

where $I\in S_{k}(\Delta)$ and $A\in N_{I}^{k}(\Delta)$.

###### Proof.

  * •

Let us consider the linear map
$D:\mathcal{A}_{\Delta}^{1}\rightarrow\mathcal{A}_{\Delta}^{0}=\mathbb{C}$.
Let $\delta\in M_{\mathbb{C}}$ be defined by

(3.5) $\langle\delta,x\rangle=D(\rho(x)),$

for all $x\in N_{\mathbb{C}}$, where $\rho(x)\in
W\subseteq\mathcal{A}_{\Delta}^{1}$.

  * •

First, we prove the lemma in the case of $k=n$. In Equation (3.2), let
$b=\chi^{I}\cdot\rho(A)$, where $I\in S_{n}(\Delta)=S_{\Delta}$ and $A\in
N_{I}^{n}(\Delta)=\wedge^{n}N_{\mathbb{C}}$; and let $a=\rho(x)$, where $x\in
N_{\mathbb{C}}$. By proposition 3.1, we have

$\displaystyle[a,b]=[\rho(x),\chi^{I}\cdot\rho(A)]=\langle
I,x\rangle\chi^{I}\cdot\rho(A),$

where $\langle I,x\rangle$ is the pairing of $I\in S_{\Delta}\subseteq
M_{\mathbb{R}}\subseteq M_{\mathbb{C}}$ and $x\in N_{\mathbb{C}}$. On the
right hand of Equation (3.2), we have $a\wedge b=0$ since that the complex
dimension of $X_{\Delta}$ is $n$, and we have
$D(a)=D(\rho(x))=\langle\delta,x\rangle$ by Equation (3.5). By Equation (3.2),
we have

$\displaystyle\langle
I,x\rangle\chi^{I}\cdot\rho(A)=\langle\delta,x\rangle)\chi^{I}\cdot\rho(A)-\rho(x)\wedge
D(\chi^{I}\cdot\rho(A)),$

which implies

(3.6) $\langle\delta-I,x\rangle\chi^{I}\cdot\rho(A)=\rho(x)\wedge
D(\chi^{I}\cdot\rho(A)).$

On the other hand, we have $x\wedge A=0$, which implies

$\displaystyle\imath_{\delta-I}(x\wedge A)=\langle\delta-I,x\rangle
A-x\wedge(\imath_{\delta-I}A)=0.$

Therefore we have

$\langle\delta-I,x\rangle A=x\wedge(\imath_{\delta-I}A).$

As a consequence, we get

(3.7)
$\displaystyle\langle\delta-I,x\rangle\chi^{I}\cdot\rho(A)=\chi^{I}\cdot\rho(\langle\delta-I,x\rangle
A)=\chi^{I}\cdot\rho(x\wedge(\imath_{\delta-I}A))=\chi^{I}\cdot\rho(x)\wedge\rho(\imath_{\delta-I}A).$

By Equation (3.6) and Equation (3.7), we get

(3.8)
$\rho(x)\wedge(D(\chi^{I}\cdot\rho(A))-\chi^{I}\cdot\rho(\imath_{\delta-I}A))=0.$

By Proposition 3.1 and Equation (3.8), we have

(3.9)
$x\wedge\gamma(D(\chi^{I}\cdot\rho(A))-\chi^{I}\cdot\rho(\imath_{\delta-I}A))=0$

for all $x\in N_{\mathbb{C}}$. Hence

$\gamma(D(\chi^{I}\cdot\rho(A))-\chi^{I}\cdot\rho(\imath_{\delta-I}A))=0.$

As a consequence, we get

$D(\chi^{I}\cdot\rho(A))=\chi^{I}\cdot\rho(\imath_{\delta-I}A).$

Therefore, we proved the lemma for $k=n$.

  * •

Next we prove the lemma by recursion. Suppose the lemma holds in the case of
$k=l$ $(l\geq 2)$, we will prove it in the case of $k=l-1$.

In Equation (3.2), let $b=\chi^{I}\cdot\rho(A)$, where $I\in S_{l-1}(\Delta)$
and $A\in N_{I}^{l-1}(\Delta)$; and let $a=\rho(x)$, where $x\in
N_{\mathbb{C}}$. By Proposition 3.1, we have

$\displaystyle[a,b]=[\rho(x),\chi^{I}\cdot\rho(A)]=\langle
I,x\rangle\chi^{I}\cdot\rho(A).$

On the right hand of Equation (3.2), we have

$\displaystyle D(a\wedge
b)=D(\rho(x)\wedge(\chi^{I}\cdot\rho(A)))=D(\chi^{I}\cdot\rho(x\wedge
A))=\chi^{I}\cdot\rho(\imath_{\delta-I}(x\wedge A)),$ $\displaystyle
D(a)\wedge
b=D(\rho(x))\chi^{I}\cdot\rho(A)=\langle\delta,x\rangle\chi^{I}\cdot\rho(A),$
$\displaystyle a\wedge D(b)=\rho(x)\wedge D(\chi^{I}\cdot\rho(A)).$

By Equation (3.2), we have

$\langle
I,x\rangle\chi^{I}\cdot\rho(A)=-\chi^{I}\cdot\rho(\imath_{\delta-I}(x\wedge
A))+\langle\delta,x\rangle\chi^{I}\cdot\rho(A)-\rho(x)\wedge
D(\chi^{I}\cdot\rho(A)).$

As a consequence, we have

$\displaystyle\rho(x)\wedge
D(\chi^{I}\cdot\rho(A))=\chi^{I}\cdot(\langle\delta-I,x\rangle\rho(A)-\rho(\imath_{\delta-I}(x\wedge
A)))$
$\displaystyle=\chi^{I}\cdot\rho(x\wedge\imath_{\delta-I}A)=\chi^{I}\cdot\rho(x)\wedge\rho(\imath_{\delta-I}A)=\rho(x)\wedge(\chi^{I}\cdot\rho(\imath_{\delta-I}A)).$

Therefore, we have

$\rho(x)\wedge(D(\chi^{I}\cdot\rho(A))-\chi^{I}\cdot\rho(\imath_{\delta-I}A))=0$

for all $x\in N_{\mathbb{C}}$. By similar reason as we have shown in the case
of $k=n$, we get

$D(\chi^{I}\cdot\rho(A))=\chi^{I}\cdot\rho(\imath_{\delta-I}A).$

Thus if the lemma holds in the case of $k=l$, then it also holds in the case
of $k=l-1$.

∎

To prove Lemma 3.4 and Lemma 3.5, we need a basic fact in linear algebra, here
we write it as a lemma.

######  Lemma 3.3 (A basic fact of linear algebra).

Let $V$ be a $n$-dimensional vector space over $\mathbf{k}$, and $F$ be a
$i$-dimensional subspace of $V$. Let $A$ be a nonzero element in
$\wedge^{i}F\cong\mathbf{k}$. Then for any $\alpha\in V^{*}$, we have that
$\imath_{\alpha}A=0$ if and only if $\alpha\in F^{\perp}$, where
$F^{\perp}\subseteq V^{*}$ is the annihilator of $F$.

######  Lemma 3.4.

Let $\delta$ be an element in $M_{\mathbb{C}}$. Suppose that for all
$\chi^{I}\cdot\rho(A)\in V_{I}^{k}(\Delta)$ $(1\leq k\leq n)$, we have
$\chi^{I}\cdot\rho(\imath_{\delta-I}A)\in V_{I}^{k-1}(\Delta)$, where $I\in
S_{k}(\Delta)$ and $A\in N_{I}^{k}(\Delta)$.

  1. (1)

We have

(3.10) $\delta\in\bigcap_{I\in S_{\Delta}}\hat{F}_{I}(\Delta),$

  2. (2)

If $I\in M$ is a vertex of $P_{\Delta}$, then we have $\delta=I$. Thus the
polytope $P_{\Delta}\subset M_{\mathbb{R}}$ has at most one vertex belonging
to $M$.

###### Proof.

  1. (1)

We will prove that for any $I\in S_{\Delta}$, we have
$\delta\in\hat{F}_{I}(\Delta)$.

Suppose that $I\in S(\Delta,i)$. Then we have $\dim F_{I}(\Delta)=n-i$ and
$\dim F_{I}^{\perp}(\Delta)=i$. According to the assumption, for any
$\chi^{I}\cdot\rho(A)\in V_{I}^{k}(\Delta)$, we have
$\chi^{I}\cdot\rho(\imath_{\delta-I}A)\in V_{I}^{k-1}(\Delta)$. Hence for any
$A\in N_{I}^{k}(\Delta)$, we have $\imath_{\delta-I}A\in N_{I}^{k-1}(\Delta)$.
In the case of $k=i$, we have $N_{I}^{i-1}(\Delta)=0$. Let $A$ be a nonzero
element in
$N_{I}^{i}(\Delta)=\wedge^{i}(F_{I}^{\perp}(\Delta)\otimes\mathbb{C})\cong\mathbb{C}$.
Then we have

(3.11) $\imath_{\delta-I}A=0.$

By Lemma 3.3 and Equation (3.11), we have

(3.12) $\delta-I\in(F_{I}^{\perp}(\Delta)\otimes\mathbb{C})^{\perp}.$

As a consequence, we get $\delta\in\hat{F}_{I}(\Delta)$ for all $I\in
S(\Delta,i)$.

Since $S_{\Delta}=\bigcup_{0\leq i\leq n}S(\Delta,i)$, we get

$\delta\in\bigcap_{I\in S_{\Delta}}\hat{F}_{I}(\Delta).$

  2. (2)

Suppose that $I\in M$ is a vertex of the polytope $P_{\Delta}$. Then we have
$\delta\in S_{\Delta}$ and $\hat{F}_{I}(\Delta)=\\{I\\}$. Since

$\delta\in\bigcap_{I\in S_{\Delta}}\hat{F}_{I}(\Delta),$

we have $\delta=I$. Thus the polytope $P_{\Delta}\subset M_{\mathbb{C}}$ has
at most one vertex belonging to $M$.

∎

######  Lemma 3.5.

Let $\delta\in\bigcap_{I\in S_{\Delta}}\hat{F}_{I}(\Delta)$. Then for all
$\chi^{I}\cdot\rho(A)\in V_{I}^{k}(\Delta)$ $(1\leq k\leq n)$, we have
$\chi^{I}\cdot\rho(\imath_{\delta-I}A)\in V_{I}^{k-1}(\Delta)$, where $I\in
S_{k}(\Delta)$ and $A\in N_{I}^{k}(\Delta)$.

###### Proof.

To prove the lemma, it is enough for us to prove that $\imath_{\delta-I}A\in
N_{I}^{k-1}(\Delta)$ for all $I\in S_{k}(\Delta)$ and $A\in
N_{I}^{k}(\Delta)$. Suppose that $I\in S(\Delta,i)$. Then we have $k\leq i$
since $I\in S_{k}(\Delta)$.

  * •

We first prove the lemma in the case of $k=i$. In the case of $k=i$, we have

$\displaystyle
N_{I}^{i}(\Delta)=\wedge^{i}(F_{I}^{\perp}(\Delta)\otimes\mathbb{C})\cong\mathbb{C}\quad\text{and}\quad
N_{I}^{i-1}(\Delta)=0.$

Suppose that $A_{1}$ is a nonzero element in
$N_{I}^{i}(\Delta)=\wedge^{i}(F_{I}^{\perp}(\Delta)\otimes\mathbb{C})\cong\mathbb{C}$.
Since $\delta\in\hat{F}_{I}(\Delta)$, we have
$\delta-I\in(F_{I}^{\perp}(\Delta)\otimes\mathbb{C})^{\perp}$. By Lemma 3.3,
we get

$\imath_{\delta-I}A_{1}=0.$

Since $N_{I}^{i}(\Delta)=\mathbb{C}A_{1}$, we have proved the lemma in the
case of $k=i$.

  * •

In the case of $k>i$, we have

$N_{I}^{k}(\Delta)=N_{I}^{i}(\Delta)\wedge(\wedge^{k-i}N_{\mathbb{C}}).$

Any element $A\in N_{I}^{k}(\Delta)$ can be written as

$A=A_{1}\wedge A_{2},$

where $A_{2}\in\wedge^{k-i}N_{\mathbb{C}}$. We have

(3.13) $\imath_{\delta-I}A=(\imath_{\delta-I}A_{1})\wedge
A_{2}+(-1)^{i}A_{1}\wedge(\imath_{\delta-I}A_{2})=(-1)^{i}A_{1}\wedge(\imath_{\delta-I}A_{2}).$

Since $A_{1}\in N_{I}^{i}(\Delta)$ and
$\imath_{\delta-I}A_{2}\in\wedge^{k-i-1}N_{\mathbb{C}}$, we have

$\imath_{\delta-I}A\in N_{I}^{k-1}(\Delta).$

∎

######  Lemma 3.6.

Let $\delta$ be an element in $M_{\mathbb{C}}$. Let $D$ be a linear operator
defined on the Gerstenhaber algebra $(\mathcal{A}_{\Delta},\wedge,[,])$
satisfying

$D(\chi^{I}\cdot\rho(A))=\chi^{I}\cdot\rho(\imath_{\delta-I}A)$

for all $\chi^{I}\cdot\rho(A)\in V_{I}^{k}(\Delta)$ $(1\leq k\leq n)$, where
$I\in S_{k}(\Delta)$ and $A\in N_{I}^{k}(\Delta)$. Then $D$ satisfies Equation
(3.2) and (3.3).

###### Proof.

  * •

Since that $\mathcal{A}_{\Delta}^{k}=0$ for $k<0$ and $k>n$, we only need to
verify that
$D^{2}:\mathcal{A}_{\Delta}^{k}\rightarrow\mathcal{A}_{\Delta}^{k-2}$ equal to
zero for $2\leq k\leq n$.

As $D$ is a linear map from the vector space
$\mathcal{A}_{\Delta}^{k}=\bigoplus_{I\in S_{k}(\Delta)}V_{I}^{k}(\Delta)$ to
the vector space $\mathcal{A}_{\Delta}^{k-1}=\bigoplus_{I\in
S_{k-1}(\Delta)}V_{I}^{k-1}(\Delta)$, we get that

$\displaystyle
D(\chi^{I}\cdot\rho(A))=\chi^{I}\cdot\rho(\imath_{\delta-I}A)\in
V_{I}^{k-1}(\Delta)$

for all $\chi^{I}\cdot\rho(A)\in V_{I}^{k}(\Delta)$ $(1\leq k\leq n)$. For any
$\chi^{I}\cdot\rho(A)\in V_{I}^{k}(\Delta)$, suppose that $I\in S(\Delta,i)$,
then we have $i\leq k$ since $I\in S_{k}(\Delta)$.

In the case of $k<i+2$, since $D^{2}(\chi^{I}\cdot\rho(A))\in
V_{I}^{k-2}(\Delta)$ and $V_{I}^{k-2}(\Delta)=0$, we get that
$D^{2}(\chi^{I}\cdot\rho(A))=0$. And in the case of $k\geq i+2$, we have

$\displaystyle
D^{2}(\chi^{I}\cdot\rho(A))=D(\chi^{I}\cdot\rho(\imath_{\delta-I}A))=\chi^{I}\cdot\rho(\imath_{\delta-I}(\imath_{\delta-I}A))=0.$

Since $\mathcal{A}_{\Delta}^{k}=\bigoplus_{I\in
S_{k}(\Delta)}V_{I}^{k}(\Delta)$, we get $D^{2}=0$.

  * •

If $a\in\mathcal{A}^{0}_{\Delta}=\mathbb{C}$ or
$b\in\mathcal{A}^{0}_{\Delta}=\mathbb{C}$, Equation (3.3) holds obviously.

Next we will prove that Equation (3.3) holds for any $a\in V_{I}^{k}(\Delta)$
and $b\in V_{J}^{l}(\Delta)$, where $1\leq k,l\leq n$, $I\in S_{k}(\Delta)$
and $J\in S_{l}(\Delta)$.

Suppose that $a=\chi^{I}\cdot\rho(A_{I})$ and $b=\chi^{J}\cdot\rho(A_{J})$,
where $A_{I}\in N_{I}^{k}(\Delta)$ and $A_{J}\in N_{J}^{l}(\Delta)$. By
Proposition 3.1, we have

(3.14)
$\displaystyle[a,b]=[\chi^{I}\cdot\rho(A_{I}),\chi^{J}\cdot\rho(A_{J})]=(-1)^{k+1}\chi^{I+J}\cdot\rho((\imath_{J}A_{I})\wedge
A_{J}+(-1)^{k}A_{I}\wedge(\imath_{I}A_{J})).$

On the other hand, we have

(3.15) $D(a\wedge
b)=D((\chi^{I}\cdot\rho(A_{I}))\wedge(\chi^{J}\cdot\rho(A_{J})))=D(\chi^{I+J}\cdot\rho(A_{I}\wedge
A_{J}))=\chi^{I+J}\cdot\rho(\imath_{\delta-I-J(}A_{I}\wedge A_{J})).$

We notice that Equation (3.15) holds even in the case of $k+l>n$ since that
$a\wedge b=0$ and $A_{I}\wedge A_{j}=0$ if $k+l>n$. Moreover, we have

(3.16) $D(a)\wedge
b=D(\chi^{I}\cdot\rho(A_{I}))\wedge(\chi^{J}\cdot\rho(A_{J}))=(\chi^{I}\cdot\rho(\imath_{\delta-I}A_{I}))\wedge(\chi^{J}\cdot\rho(A_{J}))=\chi^{I+J}\cdot\rho((\imath_{\delta-I}A_{I})\wedge
A_{J})$

and

(3.17) $a\wedge D(b)=\chi^{I}\cdot\rho(A_{I})\wedge
D(\chi^{J}\cdot\rho(A_{J}))=(\chi^{I}\cdot\rho(A_{I}))\wedge(\chi^{J}\cdot\rho(\imath_{\delta-J}A_{J}))=\chi^{I+J}\cdot\rho(A_{I}\wedge(\imath_{\delta-J}A_{J})).$

By Equation (3.15), Equation (3.16) and Equation (3.17), we get that

$\displaystyle D(a\wedge b)-D(a)\wedge b-(-1)^{k}a\wedge D(b)$
$\displaystyle=\chi^{I+J}\cdot(\rho(\imath_{\delta-I-J(}A_{I}\wedge
A_{J}))-\rho((\imath_{\delta-I}A_{I})\wedge
A_{J})-(-1)^{k}\rho(A_{I}\wedge(\imath_{\delta-J}A_{J})))$
$\displaystyle=\chi^{I+J}\cdot\rho(\imath_{\delta-I-J}(A_{I}\wedge
A_{J})-(\imath_{\delta-I}A_{I})\wedge
A_{J}-(-1)^{k}A_{I}\wedge(\imath_{\delta-J}A_{J})).$

Since that

$\displaystyle\imath_{\delta}(A_{I}\wedge A_{J})-\imath_{\delta}A_{I}\wedge
A_{J}-(-1)^{k}A_{I}\wedge\imath_{\delta}A_{J}=0,$
$\displaystyle\imath_{-I}(A_{I}\wedge A_{J})-(\imath_{-I}A_{I})\wedge
A_{J}=(-1)^{k}A_{I}\wedge(\imath_{-I}A_{J}),$
$\displaystyle\imath_{-J}(A_{I}\wedge
A_{J})-(-1)^{k}A_{I}\wedge(\imath_{-J}A_{J})=\imath_{-J}A_{I}\wedge A_{J},$

we get

(3.18) $D(a\wedge b)-D(a)\wedge b-(-1)^{k}a\wedge
D(b)=\chi^{I+J}\cdot\rho((-1)^{k}A_{I}\wedge(\imath_{-I}A_{J})+\imath_{-J}A_{I}\wedge
A_{J}).$

By Equation (3.15) and Equation (3.18), we get

$[a,b]=(-1)^{k}(D(a\wedge b)-D(a)\wedge b-(-1)^{k}a\wedge D(b)).$

∎

######  Theorem 3.7.

Let $X_{\Delta}$ be a smooth compact toric variety. Let
$(\mathcal{A}_{\Delta},\wedge,[,])$ be the Gerstenhaber algebra associated to
$X_{\Delta}$.

  1. (1)

The necessary and sufficient condition for the existence of the BV operator is
that

(3.19) $\bigcap_{I\in S_{\Delta}}\hat{F}_{I}(\Delta)\neq\varnothing.$

  2. (2)

If Equation (3.19) holds, choosing any element $\delta\in\bigcap_{I\in
S_{\Delta}}\hat{F}_{I}(\Delta)$, we can define a BV-operator
$D:\mathcal{A}_{\Delta}\rightarrow\mathcal{A}_{\Delta}$ by

(3.20) $D(\chi^{I}\cdot\rho(A))=\chi^{I}\cdot\rho(\imath_{\delta-I}A),$

where $\chi^{I}\cdot\rho(A)\in V_{I}^{k}(\Delta)$ $(1\leq k\leq n)$, $I\in
S_{k}(\Delta)$ and $A\in N_{I}^{k}(\Delta)$.

  3. (3)

All the BV operators of the Gerstenhaber algebra
$(\mathcal{A}_{\Delta},\wedge,[,])$ are in the form as shown in Equation
(3.20)

###### Proof.

  1. (1)

     * •

The necessary condition:
Suppose that $D:A_{\Delta}\rightarrow A_{\Delta}$ is a BV-operator of
Gerstenhaber algebra $(\mathcal{A}_{\Delta},\wedge,[,])$. By Lemma 3.2, there
exist an element $\delta\in M_{\mathbb{C}}$ such that, for all
$\chi^{I}\cdot\rho(A)\in V_{I}^{k}(\Delta)$ ($1\leq k\leq n$), we have

$D(\chi^{I}\cdot\rho(A))=\chi^{I}\cdot\rho(\imath_{\delta-I}A),$

where $I\in S_{k}(\Delta)$ and $A\in N_{I}^{k}(\Delta)$. By Theorem 2.6, to
make $D:A_{\Delta}\rightarrow A_{\Delta}$ to be a well defined operator, we
have

$D(\chi^{I}\cdot\rho(A))=\chi^{I}\cdot\rho(\imath_{\delta-I}A)\in
V_{I}^{k-1}(\Delta)$

for all $I\in S_{k}(\Delta)$ and $A\in N_{I}^{k}(\Delta)$. And by Lemma 3.5,
we get

$\delta\in\bigcap_{I\in S_{\Delta}}\hat{F}_{I}(\Delta).$

Hence we have

$\bigcap_{I\in S_{\Delta}}\hat{F}_{I}(\Delta)\neq\varnothing.$

     * •

The sufficient condition:
If Equation (3.19) holds, choosing any element $\delta\in\bigcap_{I\in
S_{\Delta}}\hat{F}_{I}(\Delta)$, we can define a $\mathbb{C}$-linear operator
$D:\mathcal{A}_{\Delta}\rightarrow\mathcal{A}_{\Delta}$ by

$D(\chi^{I}\cdot\rho(A))=\chi^{I}\cdot\rho(\imath_{\delta-I}A),$

where $\chi^{I}\cdot\rho(A)\in V_{I}^{k}(\Delta)$ $(1\leq k\leq n)$, $I\in
S_{k}(\Delta)$ and $A\in N_{I}^{k}(\Delta)$. By Lemma 3.5,
$D:A_{\Delta}\rightarrow A_{\Delta}$ is a well defined operator since
$D(V_{I}^{k}(\Delta))\subseteq V_{I}^{k-1}(\Delta)$. By Lemma 3.6, $D$ is a
BV-operator of the Gerstenhaber algebra $(\mathcal{A}_{\Delta},\wedge,[,])$.

  2. (2)

It is already proved above in the sufficient condition part.

  3. (3)

It follows directly from Lemma 3.2.

∎

## References