# If you’ve got it, flaunt it:
Making the most of fine-grained sentiment annotations

Jeremy Barnes, Lilja Øvrelid, and Erik Velldal
University of Oslo
Department of Informatics
<EMAIL_ADDRESS>
###### Abstract

Fine-grained sentiment analysis attempts to extract sentiment holders, targets
and polar expressions and resolve the relationship between them, but progress
has been hampered by the difficulty of annotation. Targeted sentiment
analysis, on the other hand, is a more narrow task, focusing on extracting
sentiment targets and classifying their polarity.

In this paper, we explore whether incorporating holder and expression
information can improve target extraction and classification and perform
experiments on eight English datasets. We conclude that jointly predicting
target and polarity BIO labels improves target extraction, and that augmenting
the input text with gold expressions generally improves targeted polarity
classification. This highlights the potential importance of annotating
expressions for fine-grained sentiment datasets. At the same time, our results
show that performance of current models for predicting polar expressions is
poor, hampering the benefit of this information in practice.

##  1 Introduction

Sentiment analysis comes in many flavors, arguably the most complete of which
is what is often called fine-grained sentiment analysis Wiebe et al. (2005);
Liu (2015). This approach models the sentiment task as minimally extracting
all opinion holders, targets, and expressions in a text and resolving the
relationships between them. This complex task is further complicated by
interactions between these elements, strong domain effects, and the subjective
nature of sentiment. Take the annotated sentence in Figure 1 as an example.
Knowing that the target “UMUC” is modified by the expression “5 stars” and not
“don’t believe” is important to correctly classifying the polarity.
Additionally, the fact that this is a belief held by “some others” as apposed
to the author of the sentence can help us determine the overall polarity
expressed in the sentence.

[theme = simple] [column sep=1em] Have & seen & some others & giving & UMUC &
{5 stars} & \- & {don’t believe} & them.
[edge start x offset=-20pt]65target 63holder 89target

Figure 1: An opinion annotation from the Darmstadt Review Corpus.

Compared to document- or sentence-level sentiment analysis, where distant
labelling schemes can be used to obtain annotated data, fine-grained
annotation of sentiment does not occur naturally, which means that current
machine learning models are often hampered by the small size of datasets.
Furthermore, fine-grained annotation is demanding, leads to relatively small
datasets, and has low inter-annotator agreement Wiebe et al. (2005); Wang et
al. (2017). This begs the question: is it worth it to annotate full fine-
grained sentiment?

Targeted sentiment Mitchell et al. (2013); Zhang et al. (2015) is a reduction
of the fine-grained sentiment task which concentrates on extracting sentiment
targets and classifying their polarity, effectively ignoring sentiment holders
and expressions. The benefit of this setup is that it is faster to annotate
and simpler to model. But would targeted sentiment models benefit from knowing
the sentiment holders and expressions?

In this work, we attempt to determine whether holder and expression
information is useful for extracting and then classifying sentiment targets.
Specifically, we ask the following research questions:

  1. RQ1: Given the time and difficulty required to annotate opinion holders, expressions, and polarity, is this information useful to extract sentiment targets?

    1. (a)

Does augmenting the input text with holders and expressions improve target
extraction?

    2. (b)

Do target extraction models benefit from predicting holders and expressions?

    3. (c)

Do target extraction models benefit from predicting the polarity of targets
and/or expressions?

  2. RQ2: Can holder and expression information improve polarity classification on extracted targets?

    1. (a)

Does augmenting the input text with holders and expressions improve polarity
classification?

    2. (b)

Do potential benefits of augmenting the input depend on how we model the
target, i.e., using the [CLS] embeddings, mean pooling the target embeddings,
etc.?

    3. (c)

Can sentiment lexicons provide enough information on expressions to give
improvements?

We conduct a series of experiments on eight English sentiment datasets (three
with full fine-grained sentiment and five targeted) with state-of-the-art
models based on fine-tuned BERT models. We show that (1) it is possible to
improve target extraction by also trying to predict the polarity, and that (2)
classification models benefit from having access to information about
sentiment expressions. We also (3) release the
code111https://github.com/ltgoslo/finegrained_modelling to reproduce the
experiments, as well as the scripts to download, preprocess, and collect the
datasets into a compatible JSON format, with the hope that this allows future
research on the same data.

##  2 Related work

Fine-grained approaches to sentiment analysis attempt to discover opinions
from text, where each opinion is a tuple of (opinion holder, opinion target,
opinion expression, polarity, intensity). Annotation of datasets for this
granularity requires creating in-depth annotation guidelines, training
annotators, and generally leads to lower inter-annotator scores than other
sentiment tasks, e.g., document- or sentence-level classification, as deciding
on the spans for multiple elements and their relationships is undeniably
harder than choosing a single label for a full text. Targeted sentiment, on
the other hand, generally concentrates only on target extraction and polarity
classification. This has the benefit of allowing non-experts and crowd-
sourcing to perform annotation, making it easier to collect larger datasets
for machine learning. This simplified annotation can be crowd-sourced, leading
to larger datasets for machine learning.

###  2.1 Datasets

The Multi-purpose Question Answering dataset (MPQA) Wiebe et al. (2005) is the
first dataset that annotated opinion holders, targets, expressions and their
relationships. The news wire data leads to complex opinions and a generally
difficult task for sentiment models. Normally, the full opinion extraction
task is modelled as extraction of the individual elements (holders, targets,
and expressions) and the subsequent resolution of the relationship between
them.

The Darmstadt Review Corpora Toprak et al. (2010) contain annotated opinions
for consumer reviews of universities and services. The authors annotate
holders, targets, expressions, polarity, modifiers, and intensity. They
achieve between 0.5 and 0.8 agreement using the $agr$ method Wiebe et al.
(2005), with higher disagreement on what they call “polar targets” – targets
that have a polarity but no annotated sentiment expression – holders, and
expressions.

The Open Domain Targeted dataset Mitchell et al. (2013) makes use of crowd
sourcing to annotate NEs from scraped tweets in English and Spanish Etter et
al. (2013) with their polarities. The authors use majority voting to assign
the final labels for the NEs, discarding tweets without sentiment consensus on
all NEs.

The 2014 SemEval shared task Pontiki et al. (2014) on aspect-based sentiment
analysis include labeled data from restaurant and laptop reviews for two
subtasks: 1) target extraction, which they call “aspect term extraction” and
2) classification of polarity with respect to targets (“aspect term
polarity”).

As most targeted datasets only contain a single target, or multiple targets
with the same polarity, sentence-level classifiers are strong baselines. In
order to mitigate this, Jiang et al. (2019) create a Challenge dataset which
has both multiple targets and multiple polarities in each sentence. Similarly,
Wang et al. (2017) also point out that most targeted sentiment methods perform
poorly with multiple targets and propose TDParse, a corpus of UK election
tweets with multiple targets per tweet.

###  2.2 Modelling

Katiyar and Cardie (2016) explore jointly extracting holders, targets, and
expressions with LSTMs. They find that adding sentence-level and relation-
level dependencies (is-from or is-about) improve extraction, but find that the
LSTM models lag behind CRFs with rich features.

|  | domain |  | sentences | holders | targets | expressions | polarity
---|---|---|---|---|---|---|---|---
|  |  |  | # | avg. | # | avg. | max | # | avg. | max | # | avg. | max | $+$ | neu | $-$
Fine-grained Sentiment | MPQA | newswire | train | 4500 | 25 | 1306 | 2.6 | 27 | 1382 | 6.1 | 56 | 1656 | 2.4 | 14 | 675 | 271 | 658
|  | dev | 1622 | 23 | 377 | 2.6 | 16 | 449 | 5.3 | 41 | 552 | 2.1 | 8 | 241 | 105 | 202
|  | test | 1681 | 24 | 371 | 2.8 | 32 | 405 | 6.4 | 42 | 479 | 2.0 | 8 | 166 | 89 | 199
DS. Services | service | train | 5913 | 16 | 18 | 1.2 | 2 | 2504 | 1.2 | 7 | 1273 | 1.2 | 10 | 1623 | 46 | 838
| reviews | dev | 744 | 18 | 1 | 1.7 | 3 | 288 | 1.2 | 4 | 144 | 1.4 | 5 | 103 | 1 | 104
|  | test | 748 | 17 | 2 | 1 | 1 | 328 | 1.2 | 5 | 168 | 1.4 | 6 | 241 | 7 | 80
DS. Uni | university | train | 2253 | 20 | 65 | 1.2 | 2 | 1252 | 1.2 | 5 | 837 | 1.9 | 9 | 495 | 149 | 610
| reviews | dev | 232 | 9 | 17 | 1.1 | 3 | 151 | 1.2 | 3 | 106 | 1.7 | 6 | 40 | 19 | 92
|  | test | 318 | 20 | 12 | 1.3 | 4 | 198 | 1.2 | 6 | 139 | 2.0 | 5 | 77 | 18 | 103
Targeted Sentiment | TDParse | political | train | 2889 | 6.9 | - | - | - | 9088 | 1.2 | 7 | - | - | - | 1238 | 3931 | 3919
| tweets | dev | 321 | 6.6 | - | - | - | 1040 | 1.2 | 5 | - | - | - | 128 | 454 | 458
|  | test | 867 | 6.9 | - | - | - | 2746 | 1.2 | 6 | - | - | - | 378 | 1162 | 1206
SemEval R. | restaurant | train | 2740 | 13 | - | - | - | 3293 | 1.4 | 19 | - | - | - | 1902 | 574 | 734
| reviews | dev | 304 | 11.3 | - | - | - | 350 | 1.4 | 5 | - | - | - | 226 | 54 | 63
|  | test | 800 | 9.6 | - | - | - | 1128 | 1.4 | 8 | - | - | - | 724 | 195 | 195
SemEval L. | laptop | train | 2744 | 22.5 | - | - | - | 2049 | 1.5 | 6 | - | - | - | 870 | 402 | 747
| reviews | dev | 304 | 21.1 | - | - | - | 244 | 1.6 | 5 | - | - | - | 99 | 44 | 96
|  | test | 800 | 18.6 | - | - | - | 633 | 1.6 | 7 | - | - | - | 327 | 162 | 128
Open | tweets | train | 1903 | 12.8 | - | - | - | 2594 | 1.6 | 8 | - | - | - | 578 | 1801 | 215
|  | dev | 211 | 12.3 | - | - | - | 291 | 1.6 | 6 | - | - | - | 46 | 220 | 25
|  | test | 234 | 11.6 | - | - | - | 337 | 1.6 | 7 | - | - | - | 74 | 232 | 31
Challenge | restaurant | train | 4297 | 8.8 | - | - | - | 11186 | 1.3 | 9 | - | - | - | 3380 | 5042 | 2764
| reviews | dev | 500 | 8.9 | - | - | - | 1332 | 1.3 | 8 | - | - | - | 403 | 604 | 325
|  | test | 500 | 8.9 | - | - | - | 1336 | 1.3 | 8 | - | - | - | 400 | 607 | 329

Table 1: Stastistics of the datasets, including number of sentences, as well
as average, and max lengths (in tokens) for holder, target, and expression
annotations. Additionally, we include the distribution of polarity –
restricted to positive, neutral, and negative – in each dataset.

Regarding modelling the interaction between elements, there are several
previous attempts to jointly learn to extract and classify targets, using
factor graphs Klinger and Cimiano (2013), multi-task learning He et al. (2019)
or sequence tagging with collapsed tagsets representing both tasks Li et al.
(2019). In general, the benefits are small and have suggested that there is
only a weak relationship between target extraction and polarity classification
Hu et al. (2019).

##  3 Data

One of the difficulties of working with fine-grained sentiment analysis is
that there are only a few datasets (even in English) and they come in
incompatible, competing data formats, e.g., BRAT or various flavors of XML.
With the goal of creating a simple unified format to work on fine-grained
sentiment tasks, we take the eight datasets mentioned in Section 2 – MPQA
Wiebe et al. (2005), Darmstadt Services and Universities Toprak et al. (2010),
TDParse Wang et al. (2017), SemEval Restaurant and Laptop Pontiki et al.
(2014), Open Domain Targeted Sentiment Mitchell et al. (2013), and the
Challenge dataset from Jiang et al. (2019) – and convert them to a standard
_JSON_ format. The datasets are sentence and word tokenized using NLTK Loper
and Bird (2002), except for MPQA, DS. Service and DS. Uni, which already
contain sentence and token spans. All polarity annotations are mapped to
positive, negative, neutral, and conflict222We discard conflict during
evaluation because there are not enough examples to properly learn this class
in most datasets. As such, each sentence contains a sentence id, the tokenized
text, and a possibly empty set of opinions which contain a holder, target,
expression, polarity, and intensity. We allow for empty holders and
expressions in order generalize to the targeted corpora. Finally, we use 10
percent of the training data as development and another 10 percent for test
for the corpora that do not contain a suggested train/dev/test split. For
training and testing models, however, we convert the datasets to CoNLL format.

Table 1 presents an overview of the different datasets and highlights
important differences between them. The fully fine-grained sentiment datasets
(MPQA, DS. Services, and DS. Uni) tend to be larger but have fewer targets
annotated, due to a larger number of sentences with no targets. However, the
MPQA dataset contains much longer targets than the other datasets – an average
of 6, but a maximum of 56 tokens. It also contains more opinion holders and
expressions and these also tend to be longer, all of which marks MPQA as an
outlier among the datasets. The distribution of polarity is also highly
dependent on the dataset, with DS. Services being the most skewed and SemEval
Laptop the least skewed. Finally, the challenge dataset is by far the largest
with over 11,000 training targets. Additionally, Table 6 in Appendix A shows
the percentage of unique targets per dataset, as well as the percentage of
targets shared between the training set and the dev and test sets. Again MPQA
has the largest number of unique targets and the least overlap.333We do not,
however, consider partial overlap which may exaggerate the true uniqueness of
targets.

##  4 Experimental Setup

Figure 2: Our BERT-based _target extraction_ and _classification models_ ,
with the three strategies for extraction ((1) predict only targets, (2)
predict holders, targets and expressions, and (3) predict the polarity of the
targets and expressions as well) and five strategies for sentiment
classification (passing to the softmax layer the contextualized embedding from
(1) the [CLS] embedding, (2) the first token in the target (3) averaging all
embeddings in the target phrase, (4) taking the max of the target embeddings,
(5) concatenating the max, mean, and min).

We split the task of targeted sentiment analysis into the extraction of
sentiment targets and subsequent polarity classification of extracted targets,
given their context. Figure 2 shows the two tasks and the eight models used in
the experiments. As a base model, we take the target extraction and
classification models from Xu et al. (2019), which achieve state-of-the-art
performance on the SemEval task. The approach first fine-tunes BERT Devlin et
al. (2019) on domain-specific unlabeled data as a domain-adaptation step. We
use the datasets themselves to perform this step, except for the SemEval
datasets. For these, we follow Rietzler et al. (2020) and instead use larger
amounts of unlabeled data – 1,710,553 and 2,000,000 sentences for SemEval
Laptop and Restaurant respectively – taken from Amazon Laptop reviews He and
McAuley (2016) and the Yelp Dataset
Challenge.444https://www.yelp.com/dataset/challenge We further deviate from Xu
et al. (2019) by not pretraining the models on the SQUAD question answering
dataset and in-domain sentiment questions which they create, as this data is
not publicly available. Finally, a linear prediction is added after the BERT
model and the full model is updated on the sentiment task.

For target extraction, we use the contextualized BERT embeddings as input to a
softmax layer and predict the sequence of tags. We compare three prediction
strategies:

  1. 1.

Targ.: The model predicts the labels $y\in\\{\textit{B,I,O}\\}$ for the
targets only.

  2. 2.

Pred.: We additionally predict the labels for holders and expressions and
predict
$y\in\\{\textit{B-holder},\textit{I-holder},\textit{B-target},\textit{I-target},$
$\textit{B-expression},\textit{I-expression},O\\}$.

  3. 3.

+Pol.: Finally, we add the polarity (positive, negative, neutral) to the
annotation specific BIO-tag, which leads to an inventory of 19 labels for the
full fine-grained setup and 7 for the targeted setup.

For polarity classification, we take as a baseline the classification
architecture from Xu et al. (2019), which makes use of the two-sentence
training procedure for BERT, by prepending the target before the sentence
separation token, and then adding the full sentence after. We compare five
strategies for producing the input to the softmax layer for predicting the
sentiment of the target:

  1. 1.

[Cls]: this model uses the [CLS] embedding from the final BERT layer.

  2. 2.

First: uses the contextualized BERT embedding from the first token of the
target in context.

  3. 3.

Mean: instead takes the average of the BERT embeddings for the tokens in the
target.

  4. 4.

Max: uses the max of the contextualized BERT embeddings for the tokens in the
target.

  5. 5.

MaxMM: takes the max, min, and mean pooled representations and passes the
concatenation to the softmax layer, which has shown to perform well for
sentiment tasks Tang et al. (2014). However, this triples the size of the
input representation to the softmax layer.

The Targ. and [Cls] models correspond to the models used in Xu et al. (2019)
and serve as baselines. The extraction and classification models are fine-
tuned for 50 epochs using Adam with an initial learning rate of
$3\mathrm{e}{-5}$, with a linear warmup of 0.1 and all other hyperparameters
are left at default BERT settings (further details in Appendix B). The best
model on the development set is used for testing. Combined with the four input
manipulations (Table 2), this leads to eleven extraction experiments – Targ.
and Pred. on the original data which only has annotated targets are the same
and for simplicity we only show the results from Targ.– and twenty
classification experiments per dataset. In order to control for the effect of
random initialization, we run each experiment 5 times on different random
seeds and report the mean and standard deviation.

###  4.1 Training with gold annotations

original | Money Magazine rated E-Trade highly .
---|---
\+ holders |  [<H ] Money Magazine [H>] rated E-Trade highly
\+ expressions | Money Magazine [<E ] rated [E>] E-Trade [<E ] highly [E>]
\+ full |  [<H ] Money Magazine [H>] [<E ] rated [E>] E-Trade [<E ] highly [E>]

Table 2: We inform our models regarding annotations other than targets by
inserting special tags into the input text before and after annotated holders
and expressions.

Given that we are interested in knowing whether it is beneficial to include
information about additional annotations (holder, expressions, polarity), we
perform experiments where we systematically include these. We do so by adding
special tags, e.g.,, [<E ], into the input text surrounding the annotated
spans, as shown in Table 2. The models then have access to this information
both during training and at test time, albeit in an indirect way. For the
first set of experiments, we perform controlled experiments under ideal
conditions, i.e., having gold annotations during testing. This allows us to
isolate the effects of incorporating the additional annotations, without
worrying about noisy predictions

###  4.2 Training with predicted expressions

It is equally important to know whether the models are able to use noisy
predicted annotations. In order to test this, we train _expression prediction
models_ on the three full fine-grained sentiment corpora. We use the same
BERT-based model and hyperparameters from the target extraction models above
and train five models with different random seeds. Preliminary results
suggested that these models had high precision, but low recall. Therefore, we
take a simple ensemble of the five trained models, where for each token, we
keep labels predicted by at least one of the expression models in order to
increase recall.

We perform an additional set of experiments where we use sentiment lexicons
and assume any word in these lexicons is a sentiment expression. We use the Hu
and Liu lexicon Hu and Liu (2004), the SoCal and SoCal-Google lexicons Taboada
et al. (2006) and the NRC emotion lexicon Mohammad and Turney (2013), which
also contains sentiment annotations. The lexicons contain 6,789, 5,824, 2,142,
and 5,474 entries, respectively. The MPQA and Darmstadt experiments show the
effect of predicted vs. gold expressions, as well as domain transfer. The
experiments on the targeted datasets, on the other hand, will show us whether
it is possible to improve the targeted models with predicted expressions.

##  5 Results

|  | MPQA | DS. Services | DS. Unis | Challenge | SemEval R. | SemEval L. | Open | TDParse
---|---|---|---|---|---|---|---|---|---
| Xu et al. (2019) | n/a | n/a | n/a | n/a | 78.0 | 84.3 | n/a | n/a
| BiLSTM-CRF | 12.2 (1) | 85.0 (1) | 84.4 (1) | 73.4 (1) | 72.5 (1) | 74.0 (1) | 62.2 (1) | 82.6
| original | 14.1 (2) | 85.9 (1) | 84.6 (0) | 75.8 (1) | 51.9 (1) | 71.3 (1) | 62.0 (4) | 81.7 (3)
Targ. | \+ holders | 11.9 (1) | 84.3 (1) | 83.6 (1) | - | - | - | - | -
\+ exp. | 11.6 (1) | 85.0 (0) | 83.4 (0) | - | - | - | - | -
\+ full | 10.5 (2) | 84.8 (1) | 83.8 (1) | - | - | - | - | -
Pred. | \+ holders | 12.1 (2) | 86.2 (0) | 84.6 (0) | - | - | - | - | -
\+ exp. | 14.9 (1) | 84.7 (1) | 84.5 (1) | - | - | - | - | -
\+ full | 13.0 (3) | 85.5 (1) | 84.3 (1) | - | - | - | - | -
+Pol. | BiLSTM-CRF | 13.9 (1) | 85.2 (1) | 83.7 (1) | 73.6 (1) | 73.7 (1) | 74.5 (1) | 62.3 (1) | 81.8 (1)
original | 13.8 (1) | 85.4 (1) | 84.3 (1) | 76.9 (1) | 52.5 (1) | 71.6 (1) | 62.9 (1) | 83.2 (0)
\+ holders | 13.8 (2) | 85.6 (1) | 84.4 (1) | - | - | - | - | -
\+ exp. | 13.5 (2) | 85.4 (1) | 84.3 (0) | - | - | - | - | -
\+ full | 12.0 (1) | 86.0 (1) | 84.6 (0) | - | - | - | - | -

Table 3: Average token-level F1 scores for the target extraction task across
five runs, (standard deviation in parenthesis). Bold numbers indicate the best
model per dataset, while blue and pink highlighting indicates an improvement
or loss in performance compared to the original data, respectively.

In this section we describe the main results from the extraction and two
classification experiments described in Section 4.

###  5.1 Target extraction

Table 3 shows the results for the extraction experiment, where token-level
$\text{F}_{1}$ is measured only on targets. The models perform poorer than the
state-of-the-art, as we did not finetune on the SQUAD question answering
dataset and in-domain sentiment questions or perform extensive hyperparameter
tuning. The average $\text{F}_{1}$ score depends highly on the dataset – MPQA
is the most difficult dataset with 13.1 $\text{F}_{1}$ on the original data,
while the Darmstadt Universities corpus is the easiest for target extraction
with 84.6. Augmenting the input text with further annotations, but predicting
only sentiment targets (Targ. in Table 3) hurts the model performance in all
cases. Specifically, adding holder tags leads to an average drop of 1.3
percentage points (pp), expressions 1.2 and full 1.5. Attempting to
additionally predict these annotations (Pred. in Table 3) leads to mixed
results – the model leads to improvements on MPQA + exp. and Darmstadt
Services + holders, no notable difference on MPQA + full and Darmstadt
Universities + exp., and a loss on the rest.

Adding the polarity to the target BIO tags (original +Pol. in Table 3) leads
to the most consistent improvements across experiments – an average of 0.5 pp
– with the largest improvement of 1.5 pp on the TDParse dataset. This suggests
a weak-to-moderate relationship between polarity and extraction, which
contradicts previous conclusions Hu et al. (2019). Finally, further adding the
holder and expression tags (+Pol. in Table 3) tends to decrease performance.

###  5.2 Polarity classification with gold annotations

Table 4 shows the macro $\text{F}_{1}$ scores for the polarity classification
task on the gold targets. The model performs better than the best reported
results on Challenge Jiang et al. (2019), and similar to previous results on
the SemEval corpora. Regarding the choice of target representation, First is
the strongest overall, with an average of 64.7 $\text{F}_{1}$ across the
original eight datasets, followed by Max (64.6), Mean (64.4), MaxMM (64.2),
and finally [Cls] (64.1). It is, however, unclear exactly which representation
is the best, as it differs for each dataset. But we can conclude that [Cls] is
in general the weakest model, while either First or Max provide good starting
points.

Adding holder annotations to the input text delivers only small improvements
on four of the fifteen experiments, and has losses on seven. The +exp. model,
however, leads to significant improvements on 10 experiments. The outlier
seems to be Darmstadt Services, which contains a large number of “polar
targets” in the data, which do not have polar expressions. This may explain
why including this information has less effect on this dataset. Finally, +full
performs between the original input and +exp.

###  5.3 Polarity classification with predicted annotations

The expression models achieve modest $\text{F}_{1}$ scores when trained and
tested on the same dataset – between 15.0 and 47.9 –, and poor scores when
transferred to a different dataset – between 0.9 and 14.9 (further details
shown in Table 7 in Appendix A). The lexicons often provide better cross-
dataset $\text{F}_{1}$ than the expression models trained on another dataset,
as they have relatively good precision on general sentiment terms.

Figure 3 shows a heatmap of improvements (blue) and losses (red) on the eight
datasets (x-axis) when augmenting the input text with expression tags from the
expression models and lexicons (y-axis). We compare the expression augmented
results to the original results for each pooling technique and take the
average of these improvements and losses. For a full table of all results, see
Table 5 in Appendix A.

Augmenting the input text with predicted sentiment expressions leads to losses
in 41 out of averaged 56 experiments shown in Figure 3 (or in 173 out of 280
experiments in Table 5). Curiously, the experiments that use an expression
model trained on the same dataset as the classification task, e.g., MPQA
predicted expressions on the MPQA classification task, have the largest losses
– the largest of which is MPQA (-2.78 on average). This seems to indicate that
the mismatch between the train prediction, which are near perfect, and the
rather poor test predictions is more problematic than cross-dataset
predictions, which are similar on train and test.

|  | MPQA | DS. Services | DS. Unis | Challenge | SemEval R. | SemEval L. | Open | TDParse
---|---|---|---|---|---|---|---|---|---
| Previous Results | n/a | n/a | n/a | 70.3 | 80.1 | 78.3 |  |
[Cls] | original | 63.5 (2) | 57.3 (1) | 57.6 (4) | 84.3 (0) | 74.1 (2) | 72.8 (1) | 54.6 (1) | 48.8 (1)
\+ holders | 63.1 (2) | 57.1 (1) | 60.5 (0) | - | - | - | - | -
\+ exp. | 64.0 (3) | 56.4 (0) | 62.9 (4) | - | - | - | - | -
\+ full | 61.9 (2) | 56.6 (1) | 62.8 (2) | - | - | - | - | -
First | original | 64.3 (2) | 57.8 (1) | 58.7 (4) | 84.4 (1) | 75.6 (1) | 74.3 (1) | 55.6 (2) | 46.6 (1)
\+ holders | 63.4 (2) | 57.7 (2) | 60.5 (3) | - | - | - | - | -
\+ exp. | 64.8 (2) | 57.0 (1) | 63.7 (2) | - | - | - | - | -
\+ full | 64.0 (1) | 55.2 (1) | 65.7 (4) | - | - | - | - | -
Mean | original | 63.5 (2) | 57.3 (1) | 60.2 (4) | 84.4 (1) | 74.1 (2) | 72.8 (1) | 56.8 (3) | 46.1 (1)
\+ holders | 63.1 (2) | 57.8 (1) | 56.7 (5) | - | - | - | - | -
\+ exp. | 64.3 (2) | 56.2 (1) | 64.1 (3) | - | - | - | - | -
\+ full | 64.2 (2) | 56.3 (1) | 63.7 (2) | - | - | - | - | -
Max | original | 60.8 (4) | 58.2 (1) | 57.8 (3) | 81.4 (1) | 73.9 (2) | 74.5 (2) | 61.4 (5) | 49.0 (3)
\+ holders | 61.9 (4) | 57.9 (1) | 53.9 (1) | - | - | - | - | -
\+ exp. | 64.3 (2) | 57.4 (1) | 61.5 (6) | - | - | - | - | -
\+ full | 62.7 (3) | 57.9 (1) | 54.5 (2) | - | - | - | - | -
MaxMM | original | 59.3 (2) | 57.8 (1) | 55.2 (3) | 81.3 (1) | 77.2 (1) | 74.5 (1) | 60.2 (5) | 48.5 (5)
\+ holders | 61.3 (1) | 57.8 (1) | 54.7 (3) | - | - | - | - | -
\+ exp. | 64.1 (2) | 59.8 (3) | 54.0 (2) | - | - | - | - | -
\+ full | 63.9 (1) | 57.7 (1) | 54.4 (4) | - | - | - | - | -

Table 4: Average macro $\text{F}_{1}$ scores for polarity classification
across five runs (standard deviation in parenthesis) on gold targets, also
adding information about holders and expressions. Bold indicates the best
model per dataset, while blue and pink highlighting indicates an improvement
or loss in performance compared to the original (targets only) data,
respectively.

The best expression prediction model is the one trained on MPQA, improving the
performance on Darmstadt Universties, Open, and SemEval Restaurants. This is
likely due to the fact that MPQA has the largest number of annotated
expressions, and that the domain is more general, leading to expression
predictions that generalize better. The expression models trained on Darmstadt
Services leads to small benefits on two corpora and the expression model
trained on Darmstadt Universities only leads to losses

The datasets that receive the most benefit from expression annotations are
Darmstadt Universities (6/7 experiments) and the TDParse dataset (5/7). In
both cases, the lexicon-based expression models provide more consistent
benefits than the trained expression prediction models. The fact that the
dataset that benefits most is the TDParse dataset suggests that expression
information is most useful when there are multiple targets with multiple
polarities.

There is no significant correlation between the performance of the expression
prediction model and the performance on the classification task on the three
fine-grained datasets. In fact, there is a small but insignificant negative
correlation (-0.33 p$=$0.13, -0.16 p$=$0.48, -0.26 p$=$0.25 for macro
Precision, Recall, or $\text{F}_{1}$ respectively, as measured by Pearson’s
correlation between the expression performances and the $\text{F}_{1}$ of the
classification models augmented with these predicted expressions). It seems
that the possible benefits depends more on the target dataset than the actual
expression model used.

Figure 3: Heatmap of average improvements (blue) and losses (red) on the
target classification tasks (x-axis) when augmenting the input text with
predicted sentiment expressions from the expression prediction models
(y-axis).

##  6 Conclusion

In this work we have explored the benefit of augmenting targeted sentiment
models with holder and sentiment expressions. The experiments have shown that
although augmenting text with holder and expression tags (RQ1 a) or
simultaneously predicting them (RQ1 b) have no benefit for target extraction,
predicting collapsed BIO + polarity tags consistently improves target
extraction (RQ1 c). Furthermore, augmenting the input text with gold
expressions generally improves targeted polarity classification (RQ2 a),
although it is not clear which target representation strategy is best (RQ2 b).
Furthermore, we have found benefits of including lexicon-based expressions for
the more complex targeted datasets (RQ2 c).

The rather poor performance of the learned expression models and the
difference between augmenting with gold or predicted expressions reveals the
need to improve expression prediction approaches, both by creating larger
corpora annotated with sentiment expressions, as well as performing further
research on the modeling aspect. Any future work interested in modelling more
complex sentiment phenomena should therefore be aware that we may first
require more high-quality annotated data if we wish to do so with current
state-of-the-art machine learning approaches.

Furthermore, we introduce a common format for eight standard English datasets
in fine-grained sentiment analysis and release the scripts to download and
preprocess them easily. We plan to include further datasets in our script in
the future, as well as extending our work to other languages with available
fine-grained corpora.

## References

  * Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019.  BERT: Pre-training of deep bidirectional transformers for language understanding.  In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , pages 4171–4186, Minneapolis, Minnesota. Association for Computational Linguistics.
  * Etter et al. (2013) David Etter, Francis Ferraro, Ryan Cotterell, Olivia Buzek, and Benjamin Van Durme. 2013.  Nerit: Named Entity Recognition for Informal Text.  Technical Report 11, Human Language Technology Center of Excellence, Johns Hopkins University, Baltimore, Maryland.
  * He et al. (2019) Ruidan He, Wee Sun Lee, Hwee Tou Ng, and Daniel Dahlmeier. 2019.  An interactive multi-task learning network for end-to-end aspect-based sentiment analysis.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 504–515, Florence, Italy. Association for Computational Linguistics.
  * He and McAuley (2016) Ruining He and Julian McAuley. 2016.  Ups and downs: Modeling the visual evolution of fashion trends with one-class collaborative filtering.  In _Proceedings of the 25th International Conference on World Wide Web_ , WWW ’16, page 507–517, Republic and Canton of Geneva, CHE. International World Wide Web Conferences Steering Committee.
  * Hu et al. (2019) Minghao Hu, Yuxing Peng, Zhen Huang, Dongsheng Li, and Yiwei Lv. 2019.  Open-domain targeted sentiment analysis via span-based extraction and classification.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 537–546, Florence, Italy. Association for Computational Linguistics.
  * Hu and Liu (2004) Minqing Hu and Bing Liu. 2004.  Mining and summarizing customer reviews.  In _Proceedings of the 10th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_ , pages 168–177, Seattle, USA.
  * Jiang et al. (2019) Qingnan Jiang, Lei Chen, Ruifeng Xu, Xiang Ao, and Min Yang. 2019.  A challenge dataset and effective models for aspect-based sentiment analysis.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pages 6279–6284, Hong Kong, China. Association for Computational Linguistics.
  * Katiyar and Cardie (2016) Arzoo Katiyar and Claire Cardie. 2016.  Investigating LSTMs for joint extraction of opinion entities and relations.  In _Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 919–929, Berlin, Germany. Association for Computational Linguistics.
  * Klinger and Cimiano (2013) Roman Klinger and Philipp Cimiano. 2013.  Bi-directional inter-dependencies of subjective expressions and targets and their value for a joint model.  In _Proceedings of the 51st Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)_ , pages 848–854, Sofia, Bulgaria. Association for Computational Linguistics.
  * Li et al. (2019) Xin Li, Lidong Bing, Piji Li, and Wai Lam. 2019.  A unified model for opinion target extraction and target sentiment prediction.  In _Proceedings the Thirty-Third AAAI Conference on Artificial Intelligence (AAAI 2019)_ , pages 6714–6721, Honolulu, Hawaii. AAAI Press.
  * Liu (2015) Bing Liu. 2015.  _Sentiment analysis: Mining Opinions, Sentiments, and Emotions_.  Cambridge University Press, Cambridge, United Kingdom.
  * Loper and Bird (2002) Edward Loper and Steven Bird. 2002.  NLTK: The natural language toolkit.  In _Proceedings of the ACL-02 Workshop on Effective Tools and Methodologies for Teaching Natural Language Processing and Computational Linguistics - Volume 1_ , pages 63–70.
  * Mitchell et al. (2013) Margaret Mitchell, Jacqui Aguilar, Theresa Wilson, and Benjamin Van Durme. 2013\.  Open domain targeted sentiment.  In _Proceedings of the 2013 Conference on Empirical Methods in Natural Language Processing_ , pages 1643–1654, Seattle, Washington, USA. Association for Computational Linguistics.
  * Mohammad and Turney (2013) Saif M. Mohammad and Peter D. Turney. 2013.  Crowdsourcing a Word-Emotion Association Lexicon.  _Computational Intelligence_ , 29(3):436–465.
  * Pontiki et al. (2014) Maria Pontiki, Dimitris Galanis, John Pavlopoulos, Harris Papageorgiou, Ion Androutsopoulos, and Suresh Manandhar. 2014.  SemEval-2014 task 4: Aspect based sentiment analysis.  In _Proceedings of the 8th International Workshop on Semantic Evaluation (SemEval 2014)_ , pages 27–35, Dublin, Ireland. Association for Computational Linguistics.
  * Rietzler et al. (2020) Alexander Rietzler, Sebastian Stabinger, Paul Opitz, and Stefan Engl. 2020.  Adapt or get left behind: Domain adaptation through BERT language model finetuning for aspect-target sentiment classification.  In _Proceedings of The 12th Language Resources and Evaluation Conference_ , pages 4933–4941, Marseille, France. European Language Resources Association.
  * Taboada et al. (2006) Maite Taboada, Caroline Anthony, and Kimberly Voll. 2006.  Methods for Creating Semantic Orientation Dictionaries.  In _Proceedings of the Fifth International Conference on Language Resources and Evaluation (LREC’06)_ , Genoa, Italy. European Language Resources Association (ELRA).
  * Tang et al. (2014) Duyu Tang, Furu Wei, Nan Yang, Ming Zhou, Ting Liu, and Bing Qin. 2014.  Learning sentiment-specific word embedding for twitter sentiment classification.  In _Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 1555–1565, Baltimore, Maryland. Association for Computational Linguistics.
  * Toprak et al. (2010) Cigdem Toprak, Niklas Jakob, and Iryna Gurevych. 2010.  Sentence and expression level annotation of opinions in user-generated discourse.  In _Proceedings of the 48th Annual Meeting of the Association for Computational Linguistics_ , pages 575–584, Uppsala, Sweden. Association for Computational Linguistics.
  * Wang et al. (2017) Bo Wang, Maria Liakata, Arkaitz Zubiaga, and Rob Procter. 2017.  TDParse: Multi-target-specific sentiment recognition on twitter.  In _Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguistics: Volume 1, Long Papers_ , pages 483–493, Valencia, Spain. Association for Computational Linguistics.
  * Wiebe et al. (2005) Janyce Wiebe, Theresa Wilson, and Claire Cardie. 2005.  Annotating expressions of opinions and emotions in language.  _Language Resources and Evaluation (formerly Computers and the Humanities)_ , 39(2/3):164–210.
  * Xu et al. (2019) Hu Xu, Bing Liu, Lei Shu, and Philip Yu. 2019.  BERT post-training for review reading comprehension and aspect-based sentiment analysis.  In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , pages 2324–2335, Minneapolis, Minnesota. Association for Computational Linguistics.
  * Zhang et al. (2015) Meishan Zhang, Yue Zhang, and Duy-Tin Vo. 2015.  Neural networks for open domain targeted sentiment.  In _Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing_ , pages 612–621, Lisbon, Portugal. Association for Computational Linguistics.

##  Appendix A Additional tables

|  |  | MPQA | DS. Services | DS. Unis | Challenge | Open | SemEval R. | SemEval L. | TDParse
---|---|---|---|---|---|---|---|---|---|---
|  original | [Cls] | 63.5 (2) | 57.3 (1) | 57.6 (4) | 84.3 (0) | 54.6 (1) | 74.1 (2) | 72.8 (1) | 48.8 (1)
| First | 64.3 (2) | 57.8 (1) | 58.7 (4) | 84.4 (1) | 55.6 (2) | 75.6 (1) | 74.3 (1) | 46.6 (1)
| Mean | 63.5 (2) | 57.3 (1) | 60.2 (4) | 84.4 (1) | 56.8 (3) | 74.1 (2) | 72.8 (1) | 46.1 (1)
| Max | 60.8 (4) | 58.2 (1) | 57.8 (3) | 81.4 (1) | 61.4 (5) | 73.9 (2) | 74.5 (2) | 49.0 (3)
| MaxMM | 59.3 (2) | 57.8 (1) | 55.2 (3) | 81.3 (1) | 60.2 (5) | 77.2 (1) | 74.5 (1) | 48.5 (5)
Predicted Expressions |  MPQA | [Cls] | 60.3 (2) | 57.0 (1) | 61.3 (5) | 83.1 (1) | 57.5 (4) | 74.2 (2) | 72.2 (1) | 47.5 (2)
First | 61.6 (2) | 57.0 (2) | 59.8 (3) | 83.5 (1) | 55.2 (3) | 77.1 (1) | 73.9 (2) | 45.2 (1)
Mean | 60.3 (2) | 57.0 (1) | 61.3 (5) | 83.1 (1) | 57.5 (4) | 74.2 (2) | 72.2 (1) | 47.5 (2)
Max | 59.1 (2) | 58.1 (0) | 57.0 (4) | 82.3 (0) | 63.7 (1) | 75.0 (2) | 74.7 (1) | 48.5 (2)
MaxMM | 56.2 (5) | 58.1 (1) | 52.7 (2) | 81.3 (1) | 61.9 (3) | 75.6 (2) | 75.2 (1) | 45.8 (4)
DS. Services | [Cls] | 63.6 (1) | 56.3 (1) | 60.6 (1) | 82.4 (1) | 53.4 (4) | 72.1 (2) | 72.2 (1) | 49.4 (2)
First | 61.3 (2) | 54.5 (0) | 59.4 (3) | 82.6 (1) | 56.2 (9) | 76.3 (1) | 74.8 (1) | 45.4 (2)
Mean | 64.1 (2) | 56.0 (0) | 60.1 (5) | 82.4 (1) | 56.1 (3) | 73.0 (2) | 72.4 (1) | 49.8 (2)
Max | 61.4 (1) | 56.7 (1) | 55.2 (2) | 80.3 (1) | 63.0 (2) | 76.5 (1) | 74.5 (1) | 48.1 (3)
MaxMM | 58.5 (2) | 57.1 (1) | 54.8 (5) | 80.4 (1) | 59.0 (2) | 75.9 (2) | 73.2 (2) | 49.5 (4)
DS. Unis | [Cls] | 63.1 (1) | 57.0 (1) | 60.3 (3) | 82.8 (1) | 54.7 (2) | 73.6 (3) | 72.8 (1) | 48.1 (3)
First | 64.1 (2) | 56.9 (1) | 58.2 (2) | 82.6 (1) | 55.2 (3) | 70.8 (3) | 72.9 (2) | 44.4 (1)
Mean | 62.3 (1) | 57.0 (1) | 59.7 (3) | 82.8 (1) | 54.7 (2) | 73.6 (3) | 72.8 (1) | 48.1 (3)
Max | 59.6 (4) | 57.8 (1) | 53.4 (1) | 80.5 (1) | 62.0 (1) | 74.8 (2) | 71.0 (1) | 46.9 (2)
MaxMM | 59.8 (3) | 57.5 (1) | 52.3 (4) | 80.5 (0) | 59.9 (4) | 74.8 (1) | 74.2 (1) | 47.0 (4)
Lexicon Expressions |  HuLiu | [Cls] | 60.3 (2) | 56.2 (1) | 60.8 (3) | 82.8 (1) | 54.0 (3) | 73.6 (1) | 73.2 (1) | 50.6 (1)
First | 61.2 (2) | 55.0 (1) | 61.2 (1) | 82.3 (1) | 46.5 (3) | 74.8 (1) | 74.2 (1) | 43.8 (1)
Mean | 60.3 (2) | 56.2 (1) | 60.8 (3) | 82.8 (1) | 54.0 (3) | 73.6 (1) | 73.2 (1) | 50.6 (1)
Max | 59.7 (3) | 57.2 (1) | 56.4 (2) | 81.0 (1) | 61.1 (2) | 75.5 (2) | 73.7 (3) | 47.2 (3)
MaxMM | 60.8 (3) | 57.1 (1) | 55.2 (2) | 80.8 (1) | 61.3 (3) | 73.8 (3) | 73.9 (1) | 49.2 (4)
NRC | [Cls] | 64.0 (2) | 56.9 (1) | 63.0 (2) | 83.1 (1) | 54.8 (3) | 72.0 (1) | 73.0 (1) | 49.4 (1)
First | 63.7 (2) | 56.9 (1) | 61.1 (2) | 83.3 (1) | 49.1 (5) | 74.9 (3) | 74.9 (0) | 46.1 (2)
Mean | 64.0 (2) | 56.9 (1) | 63.0 (2) | 83.1 (1) | 54.8 (3) | 72.0 (1) | 73.0 (1) | 49.4 (1)
Max | 61.1 (3) | 58.0 (1) | 55.6 (2) | 80.4 (1) | 62.0 (1) | 75.3 (2) | 74.6 (2) | 49.7 (3)
MaxMM | 59.5 (3) | 57.6 (1) | 56.9 (4) | 80.8 (1) | 61.3 (2) | 75.4 (2) | 74.8 (1) | 49.8 (4)
SoCal | [Cls] | 63.2 (2) | 56.6 (1) | 60.5 (4) | 83.0 (0) | 51.5 (4) | 69.8 (1) | 71.0 (1) | 50.0 (1)
First | 61.8 (2) | 53.7 (2) | 59.9 (4) | 81.8 (0) | 51.4 (6) | 72.8 (2) | 73.0 (1) | 45.4 (2)
Mean | 63.2 (2) | 56.6 (1) | 60.5 (4) | 83.0 (0) | 51.5 (4) | 69.8 (1) | 71.0 (1) | 50.0 (1)
Max | 59.2 (2) | 57.8 (1) | 54.5 (2) | 79.3 (1) | 62.3 (2) | 71.5 (4) | 71.8 (2) | 49.5 (4)
MaxMM | 59.7 (2) | 56.5 (2) | 55.6 (1) | 79.1 (1) | 60.9 (3) | 73.4 (3) | 73.0 (1) | 51.4 (2)
SoCal-Google | [Cls] | 62.6 (3) | 56.5 (1) | 60.0 (3) | 83.0 (1) | 53.2 (3) | 71.5 (1) | 72.9 (1) | 50.2 (1)
First | 62.1 (1) | 56.2 (1) | 60.8 (5) | 82.5 (1) | 49.7 (6) | 74.9 (2) | 74.0 (1) | 46.2 (0)
Mean | 62.6 (3) | 56.5 (1) | 60.0 (3) | 83.0 (1) | 53.2 (3) | 71.5 (1) | 72.9 (1) | 50.2 (1)
Max | 60.0 (3) | 57.8 (0) | 55.5 (5) | 80.9 (1) | 61.5 (3) | 74.8 (2) | 74.5 (2) | 49.5 (4)
MaxMM | 60.6 (4) | 57.1 (1) | 54.6 (3) | 80.5 (1) | 60.5 (3) | 73.5 (4) | 72.7 (2) | 45.8 (5)

Table 5: Macro $\text{F}_{1}$ scores for polarity classification of gold targets. Bold numbers indicate the best model per dataset, while blue and pink highlighting indicates an improvement or loss in performance compared to the original data (gold targets only), respectively. | % Unique | % Overlap
---|---|---
| train | dev | test | train-dev | train-test
MPQA | 85.7 | 88.5 | 89.2 | 15 | 19
DS. Services | 36.2 | 48.6 | 47.5 | 45.0 | 35.6
DS. Uni | 35.2 | 52.9 | 45.0 | 58.5 | 47.6
TDParse | 33 | 51.8 | 41.7 | 57.4 | 47.3
SemEval R. | 36.3 | 59.8 | 49.4 | 56.4 | 33.8
SemEval L. | 45.5 | 71.7 | 64.8 | 48.9 | 33.7
Open | 85 | 92.4 | 87.1 | 23 | 24
Challenge | 23.1 | 39.0 | 39.7 | 54.1 | 52
Table 6: Analysis of targets in the datasets. % Unique describes the number of targets that are found only in that split. % Overlap describes the percentage of dev/test targets that are found in the train set. We disregard partial matches, e.g., “chinese food” and “food”. |  | MPQA | DS. Services | DS. Unis
---|---|---|---|---
trained | MPQA | 15.0 (1.7) | 1.0 (0.8) | 2.2 (1.2)
DS. Services | 0.9 (0.3) | 47.9 (7.3) | 14.9 (1.2)
DS. Unis | 1.4 (0.6) | 10.9 (1.5) | 18.5 (1.5)
lexicons | HuLiu | 4.7 | 17.9 | 16.0
NRC | 3.3 | 7.4 | 9.0
SoCal | 2.4 | 13.2 | 13.8
SoCal Google | 1.0 | 13.2 | 11.4
Table 7: Token-level macro $\text{F}_{1}$ scores for expression prediction
models (trained) and lexicon expressions (lexicons) when tested on the three
fine-grained datasets (x-axis). The trained model scores are the average and
standard deviation across five runs with different random seeds. The lexicon
models are deterministic and therefore only have a single score.

##  Appendix B Training details

GPU Infrastructure | 1 NVIDIA P100, 16 GiB RAM
---|---
CPU Infrastructure | Intel Xeon-Gold 6126 2.6 GHz
Number of search trials | 50
Domain training duration | 2580 sec
Extraction fine-tuning duration | 15381 sec
Classification fine-tuning duration | 9080 sec
Model implementation | https://github.com/blinded/for/review

Hyperparameter |  Assignment
---|---
number of epochs |  50
max. sequence length |  128
metric early stopping monitored |  validation loss
batch size |  32
sentiment dropout |  0.3
learning rate optimiser |  Bert Adam
fine-tuning learning rate |  3e-5
learning rate warmup proportion |  0.1
regularisation type |  L2
regularisation value |  0.01


# relative connections on Principal bundles and relative equivariant
structures

Mainak Poddar  Department of Mathematics, Indian Institute of Science
Education and Research (IISER), Pune, India<EMAIL_ADDRESS>and  Anoop
Singh  School of Mathematics, Tata Institute of Fundamental Research, Homi
Bhabha Road, Mumbai 400005, India  Department of Mathematical Sciences, Indian
Institute of Technology (BHU), Varanasi 221005, India
<EMAIL_ADDRESS>
###### Abstract.

We investigate relative holomorphic connections on a principal bundle over a
family of compact complex manifolds. A sufficient condition is given for the
existence of a relative holomorphic connection on a holomorphic principal
bundle over a complex analytic family. We also introduce the notion of
relative equivariant bundles and establish its relation with relative
holomorphic connections on principal bundles.

###### Key words and phrases:

Relative holomorphic connection, Principal bundle, equivariant structure,
Kähler manifold, Atiyah-Weil criterion.

###### 2010 Mathematics Subject Classification:

53C05, 14F10, 14H15

##  1\. Introduction

The notion of holomorphic connections on a holomorphic vector bundle was
introduced by Atiyah [1], which was further generalised in many contexts in
mathematics. A well-known theorem due to Atiyah [1] and Weil [10] says that a
holomorphic vector bundle $E$ over a compact Riemann surface $Y$ admits a
holomorphic connection if and only if the degree of every holomorphic direct
summand of $E$ is zero. In [2], this result was extended to holomorphic
principal $G$-bundles on $Y$, where $G$ is a connected reductive complex
algebraic group. Moreover, in [3] and [4], authors have studied the
relationship between the existence of equivariant structures and holomorphic
$G$-connections on a principal bundle over a complex manifold.

Throughout this article $\pi:X\to S$ will denote a surjective holomorphic
proper submersion between two complex manifolds $X$ and $S$ with connected
fibers.

Motivated by the above results, we have a basic question in the relative set
up described as follows.

######  Question 1.1.

Let $H$ be a connected complex Lie group. Let
$E^{H}\,\stackrel{{\scriptstyle\varpi}}{{\longrightarrow}}\,X\xrightarrow{\pi}S$
be a holomorphic principal $H$-bundle over $X/S$. Is there a good criterion
for the existence of a relative holomorphic connection on $E^{H}$?

We tackle above question in the following manner (see Section 2).

Let $H$ and $E^{H}$ be as above. Then, we construct a short exact sequence

$0\,\longrightarrow\,\text{ad}(E^{H})\,\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}\,\text{At}_{S}(E^{H})\,\stackrel{{\scriptstyle\widetilde{d\varpi}}}{{\longrightarrow}}\,T_{X/S}\,\longrightarrow\,0,$
(1.1)

of vector bundles over $X/S$, where $\mathrm{ad}(E^{H})$ is the adjoint vector
bundle for $E^{H}$ and $\text{At}_{S}(E^{H})$ is the relative Atiyah bundle
for $E^{H}$ (see (2.13)).

A relative holomorphic connection on $E^{H}$ is by definition a holomorphic
splitting of (1.1), see Lemma 2.1 for equivalent conditions.

We give a sufficient condition for the existence of a relative holomorphic
connection on $E^{H}$ (see Theorem 2.6), more precisely, we prove the
following.

Suppose that for every $s\in S$, there is a holomorphic connection on the
principal $H$-bundle
$\varpi|_{E^{H}_{s}}\,:\,E^{H}_{s}\,\longrightarrow\,X_{s}$, and

$\mathrm{H}^{1}(S,\,\pi_{*}(\Omega^{1}_{X/S}\otimes\mathrm{ad}(E^{H})))\,=\,0\,.$

Then, $E^{H}$ admits a relative holomorphic connection.

We also note in Proposition 2.2 that the existence of a holomorphic connection
on each bundle $E_{s}^{H}$, $s\in S$, is a necessary condition for the
existence of a relative holomorphic connection on $E^{H}$.

Let $G$ be a complex Lie group and let $\pi:X\longrightarrow S$ be of relative
dimension $l=m-n$, that is, $X$ is a complex analytic family of connected
complex manifolds of dimension $l$ parametrised by a complex manifold $S$ of
dimension $n$. For every point $s\in S$, we denote $\pi^{-1}(s)$ by $X_{s}$.
Consider actions of $G$ on $X$

$\tau:G\times X\to X\,,$

and on $S$

$\nu:G\times S\to S\,,$

such that $\pi:X\to S$ is $G$-equivariant.

A similar question as in Question 1.1 can be asked for the existence of
relative holomorphic $G$-connections on $E^{H}$. For that, we proceed as
follows (see Section 2).

Given the action of $G$ on $X$ and $S$ such that $\pi:X\to S$ is $G$-invariant
(i.e., the action $\nu$ is trivial), we also construct a short exact sequence
of holomorphic vector bundles over $X/S$

$0\,\longrightarrow\,\text{ad}(E^{H})\,\stackrel{{\scriptstyle\iota_{0}}}{{\longrightarrow}}\,\text{At}_{S}^{\tau}(E^{H})\,\stackrel{{\scriptstyle
q}}{{\longrightarrow}}\,X\times\mathfrak{g}\,\longrightarrow\,0,$ (1.2)

where $\mathfrak{g}$ is the Lie algebra of $G$. The vector bundle
$\text{At}_{S}^{\tau}(E^{H})$ mentioned in (1.2) is a subbundle of the vector
bundle $\text{At}_{S}(E^{H})\oplus(X\times\mathfrak{g})$. By definition, a
relative holomorphic $G$-connection on $E^{H}$ is the holomorphic splitting of
the short exact sequence (1.2). We prove a sufficient condition for the
existence of the relative holomorphic $G$-connection on $E^{H}$ (see Theorem
2.7). Again, the existence of a holomorphic $G$-connection on each $E^{H}_{s}$
is a necessary condition (see Proposition 2.5), and a part of the sufficient
condition, for the existence of a relative holomorphic $G$-connection on
$E^{H}$.

To illustrate Theorem 2.6 and Theorem 2.7, we give examples of the existence
of relative holomorphic connections and $G$-connections on $E^{H}$ where $S$
is a Stein manifold (see Example 2.8). We also give an example of a relative
$G$-connection on $E^{H}$ where $S$ is a projective space and $H$ is an
abelian complex Lie group (see Example 2.9).

Note that the notion of a $G$-connection on $E^{H}$ depends on the $G$-action
on the base $X$. This is useful as a $G$-connection can then serve as a tool
to determine if a $G$-action on $X$ can be lifted to a $G$-action on $E^{H}$.
In fact, the authors of [3] and [4] have studied $G$-equivariant structure on
a principal $H$-bundle over a connected complex manifold from a connection
theoretic perspective. Inspired by their work, our aim in this article is to
study the relative aspect of such $G$-equivariant structure on a family of
principal $H$-bundles using relative $G$-connections.

In section 3, we consider the group $\text{Aut}_{S}(E^{H})$ of relative
automorphisms of $E^{H}$ over $X/S$. Let

$G_{S}\subset G$

be the subset consisting of all $g\in G$ such that for every $s\in S$ the
pulled back principal $H$-bundle $\tau_{g}^{*}E^{H}_{\nu_{g}(s)}$ is
isomorphic to $E^{H}_{s}$.

Let $\mathcal{G}_{S}$ denote the space of all pairs of the form $(\theta,g)$
where $g\in G_{S}$, and $\theta:E^{H}\longrightarrow E^{H}$ is a holomorphic
automorphism such that for every $s\in S$, $\theta_{s}:E^{H}_{s}\to
E^{H}_{\nu_{g}(s)}$ is an isomorphism over $\tau_{g}:X_{s}\to X_{\nu_{g}(s)}$.
Under the assumption that $\pi:X\to S$ is $G$-invariant, we show that the Lie
algebra of $\mathcal{G}_{S}$ is canonically identified with the Lie algebra
$\mathrm{H}^{0}(X,\,\text{At}_{S}^{\tau}(E^{H}))$ (see Proposition 3.1). We
also show that the holomorphic principal $H$-bundle $E^{H}$ admits a
tautological relative holomorphic $\mathcal{G}_{S}$-connection. The relative
curvature of this relative holomorphic $\mathcal{G}_{S}$-connection on $E^{H}$
vanishes identically (see Proposition 3.3).

In section 4, we define the relative equivariant structure on the principal
$H$-bundle $E^{H}$ with respect to the given group $G$. We denote the action
of $G$ on $E^{H}$ by $\sigma^{E}$ (see (4.1)), and relative equivariant
structure by the pair $(E^{H},\sigma^{E})$. When $\pi:X\to S$ is
$G$-invariant, we show that for a given relative equivariant structure
$(E^{H},\sigma^{E})$ over $X/S$, $E^{H}$ admits a tautological relative
holomorphic $G$-connection and the relative curvature of this relative
holomorphic $G$-connection vanishes identically (see Proposition 4.1). Under
the assumption that $\pi:X\to S$ is $G$-invariant, and that $G$ is a
semisimple and simply connected affine algebraic group defined over
$\mathbb{C}$, we also show that (see Theorem 4.3), if
$E^{H}\,\stackrel{{\scriptstyle\varpi}}{{\longrightarrow}}\,X\xrightarrow{\pi}S$
admits a relative holomorphic $G$-connection $h$, then $E^{H}$ admits a
relative equivariant structure $\sigma^{E}:G\times E^{H}\longrightarrow
E^{H}.$

### Acknowledgments

Both the authors would like to express their gratitude to Indranil Biswas for
his collaboration with them on closely related topics and for generously
sharing his knowledge. They also thank the anonymous referee for interesting
comments and questions that helped to improve the manuscript. The research of
the first named author was supported in part by the SERB MATRICS Grant
MTR/2019/001613. The second named author thanks IISER Pune for their
hospitality while the work was carried out.

##  2\. Relative Atiyah sequence and group action

###  2.A. Relative Atiyah exact sequence of a principal $H$-bundle

Let $X$ and $S$ be two complex manifolds of dimensions $m$ and $n$
respectively. Let

$\pi:X\longrightarrow S$ (2.1)

be a holomorphic surjective submersion of relative dimension $l=m-n$ such that
the fibres are connected, that is, $X$ is a complex analytic family of
connected complex manifolds of dimension $l$ parametrised by $S$. For every
point $s\in S$, we denote $\pi^{-1}(s)$ by $X_{s}$.

Let $H$ be a connected complex Lie group. We denote by $\mathfrak{h}$ its Lie
algebra.

By a family of holomorphic principal $H$-bundles parametrised by $S$, we mean
a holomorphic principal $H$-bundle

$\varpi:E^{H}\longrightarrow X$ (2.2)

over $X$ such that for every $s\in S$ the restriction

$\varpi|_{X_{s}}:E^{H}_{s}:=E^{H}|_{X_{s}}\longrightarrow X_{s}$ (2.3)

is a holomorphic principal $H$-bundle over $X_{s}$. Note that $H$ acts on both
$X$ and $S$ trivially.

Let $d\pi\,:\,TX\,\longrightarrow\,\pi^{*}TS$ be the differential of $\pi$ in
(2.1), where $TX$ and $TS$ be the holomorphic tangent bundles of $X$ and $S$
respectively. The subbundle

$T_{X/S}\,:=\,\mathrm{Ker}(d\pi)\,\subset\,TX$

is called the relative tangent bundle for $\pi$. Thus we have a short exact
sequence of vector bundles

$0\,\longrightarrow\,T_{X/S}\,\stackrel{{\scriptstyle\imath}}{{\longrightarrow}}\,TX\,\stackrel{{\scriptstyle
d\pi}}{{\longrightarrow}}\,\pi^{*}TS\,\longrightarrow\,0$ (2.4)

over $X$.

Consider the composition

$\pi\circ\varpi:E^{H}\longrightarrow S.$ (2.5)

Let

$d(\pi\circ\varpi):TE^{H}\longrightarrow(\pi\circ\varpi)^{*}TS$ (2.6)

be the differential of $\pi\circ\varpi$ in (2.5), where $TE^{H}$ is the
holomorphic tangent bundle of $E^{H}$. Its kernel

$T_{E^{H}/S}:=\mathrm{Ker}(d(\pi\circ\varpi))$

is known as relative tangent bundle for $\pi\circ\varpi$. Moreover, the
restriction of the differential

$d\varpi:TE^{H}\longrightarrow\varpi^{*}TX$

of $\varpi$ in (2.2) to $T_{E^{H}/S}$ gives a morphism of bundles

$d\varpi^{\prime}:=(d\varpi)|_{T_{E^{H}/S}}:T_{E^{H}/S}\longrightarrow\varpi^{*}T_{X/S}$
(2.7)

over $E^{H}$. We denote its kernal by

$T_{E^{H}/X/S}:=\mathrm{Ker}((d\varpi)|_{T_{E^{H}/S}}).$

So, we get a short exact sequence of vector bundles

$0\,\longrightarrow\,T_{E^{H}/X/S}\,\stackrel{{\scriptstyle\imath}}{{\longrightarrow}}\,T_{E^{H}/S}\,\stackrel{{\scriptstyle
d\varpi^{\prime}}}{{\longrightarrow}}\,\varpi^{*}T_{X/S}\,\longrightarrow\,0$
(2.8)

over $E^{H}$.

Note that we also have relative bundle for $\varpi$ denoted as
$T_{E^{H}/X}:=\mathrm{Ker}(d\varpi)$. Moreover,

$T_{E^{H}/X}\subset T_{E^{H}/S},$ (2.9)

and

$T_{E^{H}/X/S}\cong T_{E^{H}/X}.$ (2.10)

Let

$\sigma:E^{H}\times H\longrightarrow E^{H}$ (2.11)

be the action of $H$ on $E^{H}$. Note that the action of $H$ on each fibre of
$\varpi$ is free and transitive. The differential of $\sigma$ in (2.11)
induces a homomorphism from the trivial vector bundle on $E^{H}$ with fibre
$\mathfrak{h}$

$E^{H}\times\mathfrak{h}\longrightarrow TE^{H},$

and we have an isomorphism

$T_{E^{H}/X/S}\cong E^{H}\times\mathfrak{h}$ (2.12)

of vector bundles over $E^{H}$.

The action $\sigma$ in (2.11) induces an action of $H$ on the total space of
relative tangent bundle $T_{E^{H}/S}$. The quotient

$\text{At}_{S}(E^{H}):=(T_{E^{H}/S})/H$ (2.13)

is a holomorphic vector bundle over $X/S$, which is known as relative Atiyah
bundle (see [1], [5]).

There is an adjoint action of $H$ on its Lie algebra $\mathfrak{h}$, which
will induce an action of $H$ on the vector bundle $E^{H}\times\mathfrak{h}$.
Consider the quotient

$\text{ad}(E^{H}):=E^{H}\times^{H}\mathfrak{h}=E^{H}\times\mathfrak{h}/H$
(2.14)

which is known as adjoint vector bundle associated to $E^{H}$. From the
identification (2.12), we have

$\text{ad}(E^{H})=T_{E^{H}/X/S}/H.$

Thus, after taking the quotient by $H$, the short exact sequence in (2.8)
gives a short exact sequence of holomorphic vector bundles

$0\,\longrightarrow\,\text{ad}(E^{H})\,\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}\,\text{At}_{S}(E^{H})\,\stackrel{{\scriptstyle\widetilde{d\varpi}}}{{\longrightarrow}}\,T_{X/S}\,\longrightarrow\,0$
(2.15)

over $X$, which is known as relative Atiyah exact sequence [5], where
$\widetilde{d\varpi}$ is given by $d\varpi^{\prime}$ in (2.7).

A relative holomorphic connection on $E^{H}$ is a holomorphic splitting of the
relative Atiyah exact sequence in (2.15).

Tensoring the short exact sequence in (2.15) by the relative cotangent bundle
$\Omega^{1}_{X/S}$, we get the following short exact sequence

$0\,\longrightarrow\,\Omega^{1}_{X/S}\otimes\text{ad}(E^{H})\,\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}\,\Omega^{1}_{X/S}\otimes\text{At}_{S}(E^{H})\,\stackrel{{\scriptstyle\widetilde{d\varpi}}}{{\longrightarrow}}\,\mathcal{E}\mathit{nd}_{\mathcal{O}_{X}}(T_{X/S})\,\longrightarrow\,0$
(2.16)

The above short exact sequence (2.16) of $\mathcal{O}_{X}$-modules gives a
long exact sequence of $\mathbb{C}$-vector spaces

$\cdots\to\mathrm{H}^{0}(X,\,\Omega^{1}_{X/S}\otimes\text{At}_{S}(E^{H}))\to\mathrm{H}^{0}(X,\,\mathcal{E}\mathit{nd}_{\mathcal{O}_{X}}(T_{X/S}))\xrightarrow{\delta}\mathrm{H}^{1}(X,\,\Omega^{1}_{X/S}\otimes\text{ad}(E^{H}))\to\cdots,$
(2.17)

where $\delta$ is the connecting homomorphism. Now, the extension class of the
relative Atiyah exact sequence is defined by

$\mathrm{at}_{S}(E^{H}):=\delta(\mathbf{1}_{T_{X/S}})\in\mathrm{H}^{1}(X,\,\Omega^{1}_{X/S}\otimes\text{ad}(E^{H})),$
(2.18)

which is also known as relative Atiyah class of the bundle $E^{H}$.

######  Lemma 2.1.

Let
$E^{H}\,\stackrel{{\scriptstyle\varpi}}{{\longrightarrow}}\,X\xrightarrow{\pi}S$
be a holomorphic principal $H$-bundle. Then, the followings are equivalent.

  1. (1)

$E^{H}$ admits a relative holomorphic connection.

  2. (2)

The relative Atiyah exact sequence for $E^{H}$ in (2.15) splits.

  3. (3)

The relative Atiyah class $\mathrm{at}_{S}(E^{H})$ vanishes.

######  Proposition 2.2 (family of holomorphic connections).

Suppose that $E^{H}$ admits a relative holomorphic connection. Then, we have a
family of holomorphic connections on $\\{E^{H}_{s}\\}_{s\in S}$.

###### Proof.

The proof easily follows from the following commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{ad}(E^{H})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{s}}$$\scriptstyle{\iota}$$\textstyle{\text{At}_{S}(E^{H})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{s}}$$\scriptstyle{\widetilde{d\varpi}}$$\textstyle{T_{X/S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{s}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{ad}(E^{H}_{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{s}}$$\textstyle{\text{At}(E^{H}_{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{d\varpi}_{s}}$$\textstyle{T_{X_{s}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(2.19)

where $r_{s}$ denotes the corresponding restriction map for every $s\in S$,
and the bottom exact sequence is the Atiyah exact sequence for the principal
$H$-bundle $E^{H}_{s}$ over $X_{s}$ (see [1]). The holomorphic splitting of
the top exact sequence in (2.19) will induce a holomorphic splitting of the
bottom exact sequence in (2.19).

∎

###  2.B. Relative Atiyah bundle for group action

Let $G$ be a connected complex Lie group acting holomorphically on $X$ and $S$
such that the holomorphic map $\pi:X\to S$ in (2.1) is $G$-invariant.

Let

$\tau:G\times X\to X$ (2.20)

denote the action of $G$ on $X$ from left. Let $\mathfrak{g}$ denote the Lie
algebra of $G$.

Since $\pi:X\to S$ is $G$-invariant, the differential of $\tau$ in (2.20)
induces an $\mathcal{O}_{X}$-linear morphism of vector bundles

$d^{\prime}\tau:X\times\mathfrak{g}\longrightarrow T_{X/S}$ (2.21)

over $X$, where $X\times\mathfrak{g}$ is the trivial vector bundle with fibre
$\mathfrak{g}$ over $X$ and $\mathcal{O}_{X}$ is the sheaf of holomorphic
functions on $X$. Note that the image of $d^{\prime}\tau$ need not be a
subbundle of $T_{X/S}$.

Define a holomorphic homomorphism of vector bundles

$\mu:\text{At}_{S}(E^{H})\oplus(X\times\mathfrak{g})\longrightarrow T_{X/S}$
(2.22)

over $X$ by

$\mu(u,v)=\widetilde{d\varpi}(u)-d^{\prime}\tau(v),$ (2.23)

where $\widetilde{d\varpi}$ and $d^{\prime}\tau$ are as given in equations
(2.15) and (2.21) respectively. Since $\widetilde{d\varpi}$ is surjective,
$\mu$ is surjective.

Define an $\mathcal{O}_{X}$-submodule

$\text{At}_{S}^{\tau}(E^{H}):=\mu^{-1}(0)\subset\text{At}_{S}(E^{H})\oplus(X\times\mathfrak{g}),$
(2.24)

which is in fact a subbundle, because $\widetilde{d\varpi}$ is surjective.

In view of definition of $\text{At}_{S}^{\tau}(E^{H})$ in (2.24), we have two
holomorphic homomorphisms

$\iota_{0}:\text{ad}(E^{H})\longrightarrow\text{At}_{S}^{\tau}(E^{H}),\,\,\,\,\,\,u\mapsto(\iota(u),0),$
(2.25)

where $\iota$ is defined in (2.15), and

$q:\text{At}_{S}^{\tau}(E^{H})\longrightarrow
X\times\mathfrak{g},\,\,\,\,\,\,(u,v)\mapsto v,$ (2.26)

where $u\in\text{At}_{S}(E^{H})$ and $v\in X\times\mathfrak{g}$.

Note that $q$ in (2.26) is surjective because $\widetilde{d\varpi}$ is
surjective.

Thus, we have a short exact sequence of holomorphic vector bundles over $X$

$0\,\longrightarrow\,\text{ad}(E^{H})\,\stackrel{{\scriptstyle\iota_{0}}}{{\longrightarrow}}\,\text{At}_{S}^{\tau}(E^{H})\,\stackrel{{\scriptstyle
q}}{{\longrightarrow}}\,X\times\mathfrak{g}\,\longrightarrow\,0$ (2.27)

A relative holomorphic $G$-connection on the principal $H$-bundle $E^{H}$ is a
holomorphic splitting of (2.27), that is, there exists a holomorphic
homomorphism of vector bundles

$h:X\times\mathfrak{g}\longrightarrow\text{At}_{S}^{\tau}(E^{H})$

such that

$q\circ h=\mathbf{1}_{X\times\mathfrak{g}}.$

######  Remark 2.3.

It is easy to observe that the surjectivity of the map,
$d^{\prime}\tau:X\times\mathfrak{g}\longrightarrow T_{X/S}$, is a necessary
condition for a (relative) $G$-connection to be a (relative) holomorphic
connection. But, this is not a sufficient condition. However, a (relative)
$G$-connection corresponds to a (relative) holomorphic connection if
$d^{\prime}\tau$ is an isomorphism; for instance, when the action
$\tau:G\times X\to X$ is free and the dimensions of $G$ and $X_{s}$ are equal.
When $d^{\prime}\tau$ is an isomorphism, there is an isomorphism
$\phi:\text{At}_{S}(E^{H})\to\text{At}_{S}^{\tau}(E^{H})$ defined by
$\phi(u)=(u,(d^{\prime}\tau)^{-1}\widetilde{d\varpi}(u))$. The inverse of
$\phi$ has the simple formula, $\phi^{-1}(u,v)=u$. Moreover, given a splitting
$\eta^{\tau}$ of (2.27), we have a splitting $\eta$ of the relative Atiyah
sequence (2.15) given by

$\eta(y):=\phi^{-1}(\eta^{\tau}(d^{\prime}\tau)^{-1}(y))\,.$

Let $V$ denote the trivial vector bundle $X\times\mathfrak{g}$ over $X$. Let

$\mathrm{at}_{S}(E^{H})_{\tau}\in\mathrm{H}^{1}(X,\,V^{*}\otimes\text{ad}(E^{H}))$

be the extension class of the short exact sequence (2.27) and we call it the
relative $G$-Atiyah class of the principal $H$-bundle $E^{H}$ for the action
$\tau$.

######  Lemma 2.4.

Let $G$ acts on $X$ and $S$ such that the morphism $\pi:X\to S$ is
$G$-invariant. Let
$E^{H}\,\stackrel{{\scriptstyle\varpi}}{{\longrightarrow}}\,X\xrightarrow{\pi}S$
be a holomorphic principal $H$-bundle. Then, the followings are equivalent.

  1. (1)

$E^{H}$ admits a relative holomorphic $G$-connection.

  2. (2)

The relative Atiyah exact sequence for $E^{H}$ in (2.27) splits.

  3. (3)

The relative $G$-Atiyah class $\mathrm{at}_{S}(E^{H})_{\tau}$ vanishes.

######  Proposition 2.5 (family of holomorphic $G$-connections).

Let $\pi:X\to S$ be $G$-invariant. Then, a relative holomorphic $G$-connection
on $E^{H}$ induces a family of holomorphic $G$-connections on
$\\{E^{H}_{s}\\}_{s\in S}$.

###### Proof.

The proof is an easy consequence of the following commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{ad}(E^{H})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{s}}$$\scriptstyle{\iota_{0}}$$\textstyle{\text{At}^{\tau}_{S}(E^{H})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{s}}$$\scriptstyle{q}$$\textstyle{X\times\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{s}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{ad}(E^{H}_{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\iota_{0}}_{s}}$$\textstyle{\text{At}^{\tau}(E^{H}_{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{s}}$$\textstyle{{X_{s}}\times\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(2.28)

where $r_{s}$ denotes the corresponding restriction map for every $s\in S$,
and the bottom exact sequence is the Atiyah exact sequence of $E^{H}_{s}$ for
the $G$ action on $X_{s}$ (see [4]). The holomorphic splitting of the top
exact sequence in (2.28) will induce a holomorphic splitting of the bottom
exact sequence in (2.28). ∎

###  2.C. Sufficient condition for the existence of relative holomorphic
connections

We will give sufficient condition for the existence of relative holomorphic
connections on $E^{H}$ and relative holomorphic $G$-connections on $E^{H}$. In
view of Proposition 2.2, it is clear that a relative holomorphic connection on
the principal $H$-bundle $E^{H}$ gives a family of holomorphic connections.
But the converse of Proposition 2.2 need not be true.

######  Theorem 2.6.

Let $E^{H}\,\stackrel{{\scriptstyle\varpi}}{{\longrightarrow}}\,X$ be a
holomorphic principal $H$-bundle. Suppose that for every $s\in S$, there is a
holomorphic connection on the principal $H$-bundle
$\varpi|_{E^{H}_{s}}\,:\,E^{H}_{s}\,\longrightarrow\,X_{s}$, and

$\mathrm{H}^{1}(S,\,\pi_{*}(\Omega^{1}_{X/S}\otimes\mathrm{ad}(E^{H})))\,=\,0\,.$

Then, $E^{H}$ admits a relative holomorphic connection.

###### Proof.

Consider the relative Atiyah exact sequence for the principal $H$-bundle in
(2.15). Tensoring it by $\Omega^{1}_{X/S}$ produces the exact sequence

$0\,\longrightarrow\,\Omega^{1}_{X/S}\otimes\mathrm{ad}(E^{H})\,\longrightarrow\,\Omega^{1}_{X/S}\otimes\text{At}_{S}(E^{H})\,\stackrel{{\scriptstyle
q}}{{\longrightarrow}}\,\Omega^{1}_{X/S}\otimes
T_{X/S}\,\longrightarrow\,0\,,$ (2.29)

where $q=\mathbf{1}_{\Omega^{1}_{X/S}}\otimes\widetilde{d\varpi}$.

Note that
$\mathcal{O}_{X}\cdot\mathbf{1}_{T_{X/S}}\,\subset\,\text{End}(T_{X/S})\,=\,\Omega^{1}_{X/S}\otimes
T_{X/S}$. Define

$\Omega^{1}_{X/S}(\text{At}^{\prime}_{S}(E^{H}))\,:=\,q^{-1}(\mathcal{O}_{X}\cdot\mathbf{1}_{T_{X/S}})\,\subset\,\Omega^{1}_{X/S}\otimes\text{At}_{S}(E^{H})\,,$

where $q$ is the projection in (2.29). So we have the short exact sequence of
sheaves

$0\,\longrightarrow\,\Omega^{1}_{X/S}\otimes\mathrm{ad}(E^{H})\,\longrightarrow\,\Omega^{1}_{X/S}(\text{At}^{\prime}_{S}(E^{H}))\,\stackrel{{\scriptstyle
q}}{{\longrightarrow}}\,\mathcal{O}_{X}\,\longrightarrow\,0$ (2.30)

on $X$, where $\Omega^{1}_{X/S}(\text{At}^{\prime}_{S}(E^{H}))$ is constructed
above. Let

$\Phi\,:\,{\rm H}^{0}(X,\,\mathcal{O}_{X})\,\longrightarrow\,{\rm
H}^{1}(X,\,\Omega^{1}_{X/S}\otimes\mathrm{ad}(E^{H}))$ (2.31)

be the connecting homomorphism in the long exact sequence of cohomologies
associated to the exact sequence in (2.30). The relative Atiyah class
$\mathrm{at}_{S}(E^{H})$ (see (2.18)) coincides with $\Phi(1)\,\in\,{\rm
H}^{1}(X,\,\Omega^{1}_{X/S}\otimes\mathrm{ad}(E^{H}))$. Therefore, from Lemma
2.1 it follows that $E^{H}$ admits a relative holomorphic connection if and
only if

$\Phi(1)\,=\,0\,.$ (2.32)

To prove the vanishing statement in (2.32), first note that ${\rm
H}^{1}(X,\,\Omega^{1}_{X/S}\otimes\mathrm{ad}(E^{H}))$ fits in the five terms
exact sequence (Leray spectral sequence in low degrees)

$0\to{\rm
H}^{1}(S,\,\pi_{*}(\Omega^{1}_{X/S}\otimes\mathrm{ad}(E^{H})))\,\stackrel{{\scriptstyle\beta_{1}}}{{\longrightarrow}}\,{\rm
H}^{1}(X,\,\Omega^{1}_{X/S}\otimes\mathrm{ad}(E^{H}))\,$
$\stackrel{{\scriptstyle q_{1}}}{{\longrightarrow}}\,{\rm
H}^{0}(S,\,R^{1}\pi_{*}(\Omega^{1}_{X/S}\otimes\mathrm{ad}(E^{H})))\,\stackrel{{\scriptstyle
q_{2}}}{{\longrightarrow}}\,\mathrm{H}^{2}(S,\,\pi_{*}\Omega^{1}_{X/S}\otimes\mathrm{ad}(E^{H}))\stackrel{{\scriptstyle
q_{3}}}{{\longrightarrow}}\,\mathrm{H}^{2}(X,\,\Omega^{1}_{X/S}\otimes\mathrm{ad}(E^{H})),$
(2.33)

where $\pi$ is the projection of $X$ to $S$. We use only first three terms in
the above exact sequence.

The given condition that for every $s\in S$, there is a holomorphic connection
on the holomorphic principal $H$-bundle
$\varpi|_{E^{H}_{s}}\,:\,E^{H}_{s}\,\longrightarrow\,X_{s}$, implies that

$q_{1}(\Phi(1))\,=\,0\,,$

where $q_{1}$ is the homomorphism in (2.33). Therefore, from the exact
sequence in (2.33) we conclude that

$\Phi(1)\,\in\,\beta_{1}({\rm
H}^{1}(S,\,\pi_{*}(\Omega^{1}_{X/S}\otimes\mathrm{ad}(E^{H}))))\,.$

Finally, the given condition that ${\rm
H}^{1}(S,\,\pi_{*}(\Omega^{1}_{X/S}\otimes\mathrm{ad}(E^{H})))\,=\,0$ implies
that $\Phi(1)\,=\,0$. Since (2.32) holds, the principal $H$-bundle $E^{H}$
admits a relative holomorphic connection. ∎

Next, consider the case of relative holomorphic $G$-connections on principal
$H$-bundle $E^{H}\xrightarrow{\varpi}X\xrightarrow{\pi}S$. Under the
assumption that the holomorphic map $\pi:X\to S$ is $G$-invariant, from
Proposition 2.5, a relative holomorphic $G$-connection on $E^{H}$ gives a
family of holomorphic $G$-connections on $\\{E^{H}_{s}\\}_{s\in S}$. Again,
the converse of the Proposition 2.5 need not be true.

######  Theorem 2.7.

Let
$E^{H}\,\stackrel{{\scriptstyle\varpi}}{{\longrightarrow}}\,X\xrightarrow{\pi}S$
be a holomorphic principal $H$-bundle. Suppose that $\pi:X\to S$ is
$G$-invariant. Let $V$ denote the trivial vector bundle $X\times\mathfrak{g}$
over $X$, where $\mathfrak{g}$ is the Lie algebra of $G$. Suppose that for
every $s\in S$, there is a holomorphic $G$-connection on the principal
$H$-bundle $\varpi|_{E^{H}_{s}}\,:\,E^{H}_{s}\,\longrightarrow\,X_{s}$, and

$\mathrm{H}^{1}(S,\,\pi_{*}(V^{*}\otimes\mathrm{ad}(E^{H})))\,=\,0\,,$

where $V^{*}$ denotes the dual of $V$. Then, $E^{H}$ admits a relative
holomorphic $G$-connection.

###### Proof.

The proof is exactly similar to the proof of Theorem 2.6. ∎

######  Example 2.8.

Let $S$ be a Stein manifold. Let $\pi:X\to S$ be an analytic family of compact
connected Riemann surfaces. Let $H$ be a connected reductive linear algebraic
group over $\mathbb{C}$. Let $E^{H}$ be a holomorphic principal $H$-bundle
over $X$. Suppose that for every $s\in S$, the principal $H$-bundle
$E^{H}_{s}\to X_{s}$ admits a holomorphic connection (see [2, Theorem 4.1] for
the criterion of existence of holomorphic connection on $E^{H}_{s}$). Since
$S$ is Stein, and $\pi_{*}(\Omega^{1}_{X/S}\otimes\mathrm{ad}(E^{H}))$ is a
coherent analytic sheaf, we have
$\mathrm{H}^{1}(S,\,\pi_{*}(\Omega^{1}_{X/S}\otimes\mathrm{ad}(E^{H})))\,=\,0\,.$
Hence by Theorem 2.6 , $E^{H}$ admits a relative holomorphic connection.

A similar reasoning can be given for the existence of relative holomorphic
$G$-connection on $E^{H}$. In particular, let $G$ be the complex torus of
dimension two, $X_{s}$ be a fixed smooth toric surface under $G$-action, and
$H=GL(r,\mathbb{C})$. Then it follows from [6, Section 2.3, Example 4] that
$G$-equivariant principal $H$-bundles over $X_{s}$ are classified by families
of filtrations of $\mathbb{C}^{r}$, each family being indexed by the torus
invariant divisors of $X_{s}$. One can take $S$ to be a suitable parameter
space of such families isomorphic to the affine space with trivial $G$-action,
and $X=X_{s}\times S$. This produces a nontrivial family $E^{H}$ of
$G$-equivariant principal $H$-bundles over $X/S$. By Theorem 2.7, and [3,
Theorem 3.1] or [4, Lemma 4.1], $E^{H}$ admits a relative $G$-connection.

######  Example 2.9.

We give an example of existence of relative holomorphic $G$-connection on
$E^{H}$, where the complex Lie group $H$ is abelian. Let $F$ be a rank $n+1$
holomorphic vector bundle over $\mathbb{P}^{k}$. Then, in the notations of
Theorem 2.7,

$\pi:X=\mathbb{P}(F)\to\mathbb{P}^{k}=S$

is a holomorphic flat morphism, which is a $\mathbb{P}^{n}$-bundle. Assume
that the complex Lie group $H$ is abelian. Then, the adjoint vector bundle
$\mathrm{ad}(E^{H})$ is a trivial vector bundle over $X=\mathbb{P}(F)$. For
every $s\in S=\mathbb{P}^{k}$, $E^{H}_{s}$ admits a holomorphic $G$-connection
(follows from the criterion [4, Lemma 2.2]), because (see [9, p. 5])

$\mathrm{H}^{1}(\mathbb{P}^{n},\,\mathrm{ad}(E^{H}_{s}))=0,$

which follows from the fact that $\mathrm{ad}(E^{H}_{s})$ is trivial.

Next, since $\pi$ is flat, the sheaf $\pi_{*}(V^{*}\otimes\mathrm{ad}(E^{H}))$
is a trivial vector bundle over $\mathbb{P}^{k}$, and hence we get

$\mathrm{H}^{1}(\mathbb{P}^{k},\,\pi_{*}(V^{*}\otimes\mathrm{ad}(E^{H})))=0.$

Therefore, from Theorem 2.7, $E^{H}$ admits a relative holomorphic
$G$-connection.

###  2.D. Curvature of relative holomorphic $G$-connection

Let
$E^{H}\,\stackrel{{\scriptstyle\varpi}}{{\longrightarrow}}\,X\xrightarrow{\pi}S$
be a holomorphic principal $H$-bundle and as in section 2.B, $G$ acts on $X$
and $S$ such that $\pi$ is $G$-invariant. Note that the sheaf of holomorphic
sections of the vector bundle $\text{At}_{S}(E^{H})$ has the Lie algebra
structure. Therefore, we get a Lie algebra structure on the sheaf of
holomorphic sections of the vector bundle $\text{At}^{\tau}_{S}(E^{H})$,
because

$\text{At}_{S}^{\tau}(E^{H}):=\mu^{-1}(0)\subset\text{At}_{S}(E^{H})\oplus(X\times\mathfrak{g})$

and $\mathfrak{g}$ is the Lie algebra of $G$. Also, the morphisms $\iota_{0}$
and $q$ in (2.27) are compatible with the Lie bracket operations on the
sections of $\mathrm{ad}(E^{H})$ and $\text{At}^{\tau}_{S}(E^{H})$,
respectively.

Let $\nabla:X\times\mathfrak{g}\longrightarrow\text{At}^{\tau}_{S}(E^{H})$ be
a splitting of the short exact sequence in (2.27), that is, $\nabla$ is a
relative holomorphic $G$-connection on $E^{H}$. Let $U\subset X$ be an open
subset and let $\alpha$ and $\beta$ be any two sections of
$X\times\mathfrak{g}$ over $U$. Consider

$\mathcal{R}(\nabla)(\alpha,\beta):=[\nabla(\alpha),\nabla(\beta)]-\nabla([\alpha,\beta])\in\Gamma(U,\text{At}^{\tau}_{S}(E^{H})).$

Note that $q(\mathcal{R}(\nabla)(\alpha,\beta))=0$, because $q$ in (2.27) is
compatible with the Lie algebra structures. Hence
$\mathcal{R}(\nabla)(\alpha,\beta)$ lies in the image of $\mathrm{ad}(E^{H})$
over $U$. We also have following equalities

  1. (1)

$\mathcal{R}(\nabla)(f\alpha,\beta)=f\mathcal{R}(\nabla)(\alpha,\beta),$ where
$f$ is a holomorphic function on $U$.

  2. (2)

$\mathcal{R}(\nabla)(\alpha,\beta)=-\mathcal{R}(\nabla)(\beta,\alpha).$

Altogether, we get that

$\mathcal{R}(\nabla)\in\mathrm{H}^{0}(X,\,\mathrm{ad}(E^{H})\otimes\bigwedge^{2}V^{*})=\mathrm{H}^{0}(X,\,\mathrm{ad}(E^{H}))\otimes\bigwedge^{2}\mathfrak{g}^{*},$
(2.34)

where $V=X\times\mathfrak{g}$.

The section $\mathcal{R}(\nabla)$ is called the relative curvature of the
relative holomorphic $G$-connection $\nabla$ on $E^{H}$.

Now, we describe the induced relative connection and curvature for the
holomorphic homomorphism of complex Lie groups.

Let $\phi:G_{1}\longrightarrow G$ be a holomorphic homomorphism of complex Lie
groups. Then, $G_{1}$ acts on $X$ as follows

$\tau_{1}:G_{1}\times X\longrightarrow X,\,\,\,\,(g,x)\mapsto\tau(\phi(g),x)$
(2.35)

where $\tau$ is the holomorphic action of $G$ on $X$ in (2.20). Let
$\mathfrak{g}_{1}$ denote the Lie algebra of $G_{1}$. The differential of the
morphism $\phi$ gives a homomorphism

$d\phi:\mathfrak{g}_{1}\longrightarrow\mathfrak{g}$ (2.36)

of Lie algebras.

Note that for $G$ action on $X$ and $S$ such that $\pi:X\to S$ is
$G$-invariant, the action $\tau_{1}$ in (2.35) induces an action of $G_{1}$ on
$S$ such that $\pi$ is $G_{1}$-invariant. Hence, we have a relative Atiyah
bundle $\text{At}^{\tau_{1}}_{S}(E^{H})$ over $X$ as in (2.24). Since the
action $G_{1}$ on $X$ and $S$ are given in terms of action of $G$ using the
map $\phi$, from the construction of $\text{At}^{\tau}_{S}(E^{H})$ we have

$\text{At}^{\tau_{1}}_{S}(E^{H})\,=\,\\{(u,v)\in\text{At}^{\tau}_{S}(E^{H})\oplus(X\times\mathfrak{g}_{1})\,\,|\,\,q(u)=(\mathbf{1}_{X}\times
d\phi)(v)\\},$ (2.37)

where $q$ is given in (2.27).

An easy observation is stated as follows.

######  Proposition 2.10.

A relative holomorphic $G$-connection $\nabla$ on $E^{H}$ induces a relative
holomorphic $G_{1}$-connection $\nabla_{1}$ on $E^{H}$. The relative curvature
$\mathcal{R}(\nabla_{1})$ coincides with the image of $\mathcal{R}(\nabla)$
under the homomorphism

$\mathrm{H}^{0}(X,\,\mathrm{ad}(E^{H}))\otimes\bigwedge^{2}\mathfrak{g}^{*}\longrightarrow\mathrm{H}^{0}(X,\,\mathrm{ad}(E^{H}))\otimes\bigwedge^{2}\mathfrak{g}_{1}^{*}$
(2.38)

induced by the dual homomorphism
$(d\phi)^{*}:\mathfrak{g}^{*}\longrightarrow\mathfrak{g}^{*}_{1}$.

##  3\. Relative connection and lifting of an action

As in the previous section, let $G$ be a complex Lie group,
$\pi:X\longrightarrow S$ equipped with the $G$-equivariant action, and
$\varpi:E^{H}\to X$ be the family of principal $H$-bundles parametrised by
$S$.

Henceforth, we will also assume that $X$ is compact.

Let $\text{Aut}(E^{H})$ be the group of all automorphisms of $E^{H}$ over the
identity map of $X/S$. Because of the commutativity of the desired diagram,
given any $\theta\in\text{Aut}(E^{H})$, we have an automorphism
$\theta_{s}:E^{H}_{s}\to E^{H}_{s}$ over the identity map of $X_{s}$ for every
$s\in S$.

Let

$\nu:G\times S\to S$ (3.1)

denote the action of $G$ on $S$ such that for any $g\in G$ we have

$\pi\circ\tau_{g}=\nu_{g}\circ\pi,$

where $\nu_{g}:S\longrightarrow S$ is an automorphism. Then for every $s\in
S$, we have an isomorphism

$\tau_{g}:X_{s}\longrightarrow X_{\nu_{g}(s)}.$ (3.2)

Let $\text{Aut}_{S}(E^{H})$ be the set of relative automorphisms, that is,
$\text{Aut}_{S}(E_{H})$ consists of those holomorphism automorphisms
$\theta:E^{H}\longrightarrow E^{H}$ such that for every $s\in S$,
$\theta_{s}:E^{H}_{s}\to E^{H}_{\nu_{g}(s)}$ is an isomorphism over
$\tau_{g}:X_{s}\to X_{\nu_{g}(s)}$.

For any $g\in G$, we have an automorphism

$\tau_{g}:X\to X.$ (3.3)

Let

$G_{S}\subset G$ (3.4)

be the subset consisting of all $g\in G$ such that for every $s\in S$ the
pulled back principal $H$-bundle $\tau_{g}^{*}E^{H}_{\nu_{g}(s)}$ is
isomorphic to $E^{H}_{s}$.

Note that if $\pi:X\to S$ is $G$-invariant, then for every $s\in S$, we get a
subset $G_{s}$ of $G$, consisting of all $g\in G$ such that
$\tau^{*}_{g}E^{H}_{s}\cong E^{H}_{s}$.

Let $\mathcal{G}_{S}$ denote the space of all pairs of the form $(\theta,g)$
where $g\in G_{S}$, and $\theta:E^{H}\longrightarrow E^{H}$ is a holomorphic
automorphism such that for every $s\in S$, $\theta_{s}:E^{H}_{s}\to
E^{H}_{\nu_{g}(s)}$ is an isomorphism over $\tau_{g}:X_{s}\to X_{\nu_{g}(s)}$.

Again, if $\pi:X\to S$ is $G$-invariant, then for every $s\in S$, we get a
space $\mathcal{G}_{s}$ consisting of all pairs of the form $(\theta,g)$,
where $g\in G_{s}$ and $\theta:E^{H}_{s}\to E^{H}_{s}$ is a holomorphic
automorphism over the automorphism $\tau_{g}:X_{s}\to X_{s}$.

Note that $\mathcal{G}_{S}$ is equipped with the group operation defined as
follows

$(\theta^{\prime},g^{\prime})\cdot(\theta,g)=(\theta^{\prime}\circ\theta,g^{\prime}g)$
(3.5)

while the inverse is the map $(\theta,g)\mapsto(\theta^{-1},g^{-1})$. Thus,
$\mathcal{G}_{S}$ fits into the following short exact sequence of groups

$0\to\text{Aut}(E^{H})\,\xrightarrow{\alpha}\,\mathcal{G}_{S}\xrightarrow{\beta}G_{S}\to
0,$ (3.6)

where $\beta(\theta,g)=g$ and $\alpha(\theta)=(\theta,e)$, where $e$ is the
identity of $G_{S}$.

There is a complex Lie group structure on $\mathcal{G}_{S}$ which is uniquely
determined by the condition that (3.6) is a sequence of complex Lie groups.

We already seen that the sheaf of sections of $\text{At}_{S}^{\tau}(E^{H})$
admits a Lie algebra structure, and hence induces a Lie algebra structure on
$\mathrm{H}^{0}(X,\,\text{At}_{S}^{\tau}(E^{H}))$.

######  Proposition 3.1.

Suppose that $\pi:X\to S$ is $G$-invariant. Then, the Lie algebra of
$\mathcal{G}_{S}$ is canonically identified with the above Lie algebra
$\mathrm{H}^{0}(X,\,\text{At}_{S}^{\tau}(E^{H}))$.

###### Proof.

Let $\mathfrak{g}_{S}$ denote the Lie algebra of $\mathcal{G}_{S}$. We shall
produce a natural homomorphism from $\mathfrak{g}_{S}$ to
$\mathrm{H}^{0}(X,\,\text{At}_{S}^{\tau}(E^{H}))$. Note that the group
$\mathcal{G}_{S}$ acts on $E^{H}$ naturally which commutes with the action $H$
on $E^{H}$. In fact, this action gives a map from $\mathcal{G}_{S}$ to
$\text{Aut}_{S}(E^{H})$. Consequently, we get a homomorphism of complex Lie
algebras

$\eta:\mathfrak{g}_{S}\longrightarrow\mathrm{H}^{0}(X,\,\text{At}_{S}(E^{H})).$
(3.7)

Next define

$\eta_{1}:\mathfrak{g}_{S}\to\mathrm{H}^{0}(X,\,\text{At}_{S}^{\tau}(E^{H})),\,\,\,v\mapsto(\eta(v),d\beta(v))\in\mathrm{H}^{0}(X,\,\text{At}_{S}(E^{H}))\oplus\mathfrak{g},$
(3.8)

where $d\beta:\mathfrak{g}_{S}\to\text{Lie}(G_{S})\hookrightarrow\mathfrak{g}$
is the homomorphism of Lie algebras associated to $\beta$ in (3.6). It is easy
to verify that
$(\eta(v),d\beta(v))\in\mathrm{H}^{0}(X,\,\text{At}_{S}^{\tau}(E^{H}))\subset\mathrm{H}^{0}(X,\,\text{At}_{S}(E^{H}))\oplus\mathfrak{g}$
(see (2.24) for the definition of $\text{At}_{S}^{\tau}(E^{H})$). Clearly,
$\eta_{1}$ in (3.8) is an injective homomorphism of complex Lie algebras.
Since $X$ is compact, using the similar statements as in [4, Proposition 3.1],
we can show that $\eta_{1}$ is surjective.

∎

Observe that there is a natural action of $\mathcal{G}_{S}$ on $X$ defined as
follows

$\chi:\mathcal{G}_{S}\times X\to
X\,\,\,\,\,((\theta,g),x)\mapsto\tau(g,x),x\in X,$ (3.9)

where $\tau$ is the action in (2.20). Then, we have a vector bundle
$\text{At}^{\chi}_{S}(E^{H})$ over $X$ as constructed in (2.24).

######  Proposition 3.2.

There is a natural isomorphism of vector bundles

$\text{At}^{\chi}_{S}(E^{H})\longrightarrow\mathrm{ad}(E^{H})\oplus(X\times\mathrm{H}^{0}(X,\,\text{At}^{\tau}_{S}(E^{H})))$
(3.10)

where $X\times\mathrm{H}^{0}(X,\,\text{At}^{\tau}_{S}(E^{H}))$ is a trivial
vector bundle on $X$ with fibre
$\mathrm{H}^{0}(X,\,\text{At}^{\tau}_{S}(E^{H}))$.

###### Proof.

Note that we have a natural projection $p_{1}:\text{At}^{\chi}_{S}(E^{H})\to
X\times\mathfrak{g}_{S}$ from the short exact sequence as in (2.27) for the
group $\mathcal{G}_{S}$. From previous Proposition 3.1, $\mathfrak{g}_{S}$ is
identified with $\mathrm{H}^{0}(X,\,\text{At}^{\tau}_{S}(E^{H}))$, therefore
we have

$p_{1}:\text{At}^{\chi}_{S}(E^{H})\longrightarrow
X\times\mathrm{H}^{0}(X,\,\text{At}^{\tau}_{S}(E^{H})).$

Further, the action of $\mathcal{G}_{S}$ on $X$ factors through the action of
$G$ on $X$, therefore from (2.37) we have another description of
$\text{At}^{\chi}_{S}(E^{H})$ as subbundle of
$\text{At}^{\tau}_{S}(E^{H})\oplus(X\times\mathfrak{g}_{S})$. Thus, we have a
natural projection

$p^{\prime}:\text{At}^{\tau}_{S}(E^{H})\oplus(X\times\mathfrak{g}_{S})=\text{At}^{\tau}_{S}(E^{H})\oplus(X\times\mathrm{H}^{0}(X,\,\text{At}^{\tau}_{S}(E^{H})))\longrightarrow\text{At}^{\tau}_{S}(E^{H})$

which sends $(a,(x,\eta))\mapsto a-\eta(x)$, where $x\in
X,a\in\text{At}^{\tau}_{S}(E^{H})_{x}$ and
$\eta\in\mathrm{H}^{0}(X,\,\text{At}^{\tau}_{S}(E^{H})).$ From (2.37), it
follows that

$q\circ(p^{\prime}|_{\text{At}^{\chi}_{S}(E^{H})})=0,$

where $q$ is the projection in (2.27). Therefore, the restriction
$p^{\prime}|_{\text{At}^{\chi}_{S}(E^{H})}$ produces a homomorphism of vector
bundles

$p_{2}:\text{At}^{\chi}_{S}(E^{H})\longrightarrow\ker{(q)}=\mathrm{ad}(E^{H}).$
(3.11)

From $p_{1}$ and $p_{2}$, we get a homomorphism

$p_{1}\oplus
p_{2}:\text{At}^{\chi}_{S}(E^{H})\longrightarrow\mathrm{ad}(E^{H})\oplus(X\times\mathrm{H}^{0}(X,\,\text{At}^{\tau}_{S}(E^{H}))),$

which is an isomorphism.

∎

######  Proposition 3.3.

The holomorphic principal $H$-bundle $E^{H}$ admits a tautological relative
holomorphic $\mathcal{G}_{S}$-connection. The relative curvature of this
relative holomorphic $\mathcal{G}_{S}$-connection on $E^{H}$ vanishes
identically.

###### Proof.

Using the morphism $p_{2}$ in (3.11), Proposition easily follows. ∎

##  4\. Relative equivariant bundles and relative connections

In this section, we introduce the notion of relative equivariance structure
(see [3] for the absolute case). Let $\pi:X\to S$ is $G$-equivariant map,
where $G$ acts on $X$ and $S$ via the actions $\tau$ in (2.20) and $\nu$ in
(3.1) respectively.

Let
$E^{H}\,\stackrel{{\scriptstyle\varpi}}{{\longrightarrow}}\,X\xrightarrow{\pi}S$
be a holomorphic principal $H$-bundle over $X/S$.

A relative equivariance structure on the principal $H$-bundle $E^{H}$ is a
holomorphic action of $G$ on the total space $E^{H}$

$\sigma^{E}:G\times E^{H}\longrightarrow E^{H}$ (4.1)

such that the following diagrams commute.

  1. (1)

$\textstyle{G\times
E^{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{1}_{G}\times\varpi}$$\scriptstyle{\sigma^{E}}$$\textstyle{E^{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varpi}$$\textstyle{G\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{1}_{G}\times\pi}$$\scriptstyle{\tau}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{G\times
S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{S}$

where $\varpi$ and $\tau$ are maps defined in (2.2) and (2.11)

  2. (2)

$\textstyle{G\times E^{H}\times
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{1}_{G}\times\sigma}$$\scriptstyle{\sigma^{E}\times\mathbf{1}_{H}}$$\textstyle{E^{H}\times
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{G\times
E^{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{1}_{G}\times\varpi}$$\scriptstyle{\sigma^{E}}$$\textstyle{E^{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varpi}$$\textstyle{G\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{1}_{G}\times\pi}$$\scriptstyle{\tau}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{G\times
S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{S}$

where $\sigma$ is defined in (2.11).

A relative equivariant principal $H$-bundle is a principal $H$-bundle over
$X/S$ with a relative equivariant structure.

Further, if $\pi:X\to S$ is $G$-invariant, then for every $s\in S$, the
principal $H$-bundle $E^{H}_{s}$ over $X_{s}$ has a equivariant structure,
that is, we have a family $\\{E^{H}_{s}\\}_{s\in S}$ of equivariant principal
$H$-bundle parametrised by $S$.

######  Proposition 4.1.

Suppose $\pi:X\to S$ is $G$-invariant. Let $(E^{H},\sigma^{E})$ be a relative
equivariant principal $H$-bundle over $X/S$. Then, $E^{H}$ has a tautological
relative holomorphic $G$-connection. The relative curvature of this relative
holomorphic $G$-connection vanishes identically.

###### Proof.

Observe that for every $g\in G$, we have a holomorphic automorphism

$\sigma^{E}_{g}:E^{H}\longrightarrow E^{H}$

such that for every $s\in S$, the map

$\sigma^{E}_{g,s}:E^{H}_{s}\longrightarrow
E^{H}_{\nu_{g}(s)},\,\,\,\,\,\,z\mapsto\sigma^{E}(g,z),\,\,\,\,z\in E^{H}_{s}$

is an isomorphism over $\tau_{g}:X_{s}\longrightarrow X_{\nu_{g}(s)}$ in
(3.2). Thus, in this case

$G_{S}=G,$

where $G_{S}$ is defined in (3.4). In fact, we get a group homomorphism

$\beta^{E}:G_{S}=G\longrightarrow\mathcal{G}_{S}$

defined by $g\mapsto(g,\sigma^{E}_{g})$ such that

$\beta\circ\beta^{E}=\mathbf{1}_{G},$

where $\beta$ is the homomorphism in (3.6). In view of Proposition 3.3, we
have a tautological relative holomorphic $\mathcal{G}_{S}$-connection and we
consider this. Next, using the above group homomorphism $\beta^{E}$ and
Proposition 2.10, we get a relative holomorphic $G$-connection on $E^{H}$.
Again from Proposition 3.3 and Proposition 2.10, the relative curvature of
this relative holomorphic $G$-connection vanishes identically. This completes
the proof. ∎

The following is the converse of Proposition 4.1.

######  Proposition 4.2.

Suppose that $\pi:X\to S$ is $G$-invariant. Let
$h:X\times\mathfrak{g}\longrightarrow\text{At}^{\tau}_{S}(E^{H})$ be a
relative holomorphic $G$-connection on $E^{H}$ such that the relative
curvature vanishes identically. Assume that $G$ is simply connected. Then,
there exists a relative equivariant structure

$\sigma^{E}:G\times E^{H}\longrightarrow E^{H}$

such that the relative holomorphic $G$-connection associated to it as in
Proposition 4.1 coincides with $h$.

###### Proof.

Since $\pi:X\to S$ is $G$-invariant, from Proposition 3.1, we have

$\mathrm{Lie}(\mathcal{G}_{S})=\mathfrak{g}_{S}=\mathrm{H}^{0}(X,\,\text{At}_{S}^{\tau}(E^{H})).$

Let

$h_{*}:\mathfrak{g}=\mathrm{H}^{0}(X,\,X\times\mathfrak{g})\longrightarrow\mathrm{H}^{0}(X,\,\text{At}_{S}^{\tau}(E^{H}))=\mathfrak{g}_{S}$

be the $\mathbb{C}$-linear map induced by $h$. Since, the relative curvature
of the relative holomorphic $G$-connection $h$ vanishes identically, it
follows that $h_{*}$ is a homomorphism of Lie algebras. Further, since $G$ is
simply connected, there is a unique holomorphic homomorphism of complex Lie
groups

$\epsilon:G\longrightarrow\mathcal{G}_{S}$

such that the differential

$\text{d}\epsilon(1):\mathfrak{g}\longrightarrow\mathfrak{g}_{S}$

coincides with $h_{*}$, where $1$ denotes the identity element of $G$. Recall
that $\mathcal{G}_{S}$ acts naturally on $E^{H}$, and using the above
holomorphic homomorphism $\epsilon$ of complex Lie groups, we produce a
relative equivariant structure $\sigma^{E}$ on $E^{H}$. Now, observe that the
corresponding relative holomorphic $G$-connection given by Proposition 4.1
coincides with $h$.

∎

######  Theorem 4.3.

Suppose that $\pi:X\to S$ is $G$-invariant. Assume that $G$ is a semisimple
and simply connected affine algebraic group defined over $\mathbb{C}$. Let
$E^{H}\,\stackrel{{\scriptstyle\varpi}}{{\longrightarrow}}\,X\xrightarrow{\pi}S$
be a holomorphic principal $H$-bundle that admits a relative holomorphic
$G$-connection $h$. Then, $E^{H}$ admits a relative equivariant structure

$\sigma^{E}:G\times E^{H}\longrightarrow E^{H}.$

###### Proof.

In view of Proposition 4.2, it is enough to show that $E^{H}$ admits a
relative holomorphic $G$-connection such that the relative curvature vanishes.
Now, consider a part of long exact sequence

$\delta:\mathfrak{g}_{S}=\mathrm{H}^{0}(X,\,\text{At}_{S}^{\tau}(E^{H}))\longrightarrow\mathrm{H}^{0}(X,\,X\times\mathfrak{g})=\mathfrak{g}$
(4.2)

associated to the short exact sequence (2.27). Now, consider the homomorphism

$h_{*}:\mathfrak{g}=\mathrm{H}^{0}(X,\,X\times\mathfrak{g})\longrightarrow\mathrm{H}^{0}(X,\,\text{At}_{S}^{\tau}(E^{H}))=\mathfrak{g}_{S}$
(4.3)

associated to the relative holomorphic $G$-connection $h$ on $E^{H}$. Since
$E^{H}$ admits a relative holomorphic $G$-connection $h$, we have

$\delta\circ h_{*}=\mathbf{1}_{\mathfrak{g}},$

and hence the Lie algebra homomorphism $\delta$ is surjective. As $G$ is
semisimple, there exists a Lie subalgebra

$\mathfrak{h}\subset\mathrm{H}^{0}(X,\,\text{At}_{S}^{\tau}(E^{H}))$

such that the restriction

$\hat{\delta}:=\delta|_{\mathfrak{h}}:\mathfrak{h}\longrightarrow\mathfrak{g}=\mathrm{H}^{0}(X,\,X\times\mathfrak{g})$

is an isomorphism [7, p. 91, Corollaire 3]. We fix a subspace $\mathfrak{h}$
as above. Define $\widetilde{h_{*}}$ to be the following composition

$\mathfrak{g}=\mathrm{H}^{0}(X,\,X\times\mathfrak{g})\xrightarrow{\hat{\delta}^{-1}}\mathfrak{h}\hookrightarrow\mathrm{H}^{0}(X,\,\text{At}_{S}^{\tau}(E^{H}))=\mathfrak{g}_{S},$

which is a Lie algebra homomorphism, and hence the relative curvature of the
relative holomorphic $G$-connection induced from $\widetilde{h_{*}}$ on
$E^{H}$ vanishes identically. This completes the proof.

∎

## References

  * [1] M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207.
  * [2] H. Azad and I. Biswas, On holomorphic principal bundles over a compact Riemann surface admitting a flat connection, Math. Ann. 322 (2002), 333–346.
  * [3] I. Biswas, S. S. Kannan, D. S. Nagaraj, Equivariant principal bundles for G-actions and G-connections. Complex Manifolds 2 (2015), no. 1, 178–185.
  * [4] I. Biswas and A. Paul, Equivariant bundles and connections. Ann. Global Anal. Geom. 51 (2017), no. 4, 347–-358.
  * [5] I. Biswas and A. Singh, On the relative connections, Comm. Algebra 48 (2020), no. 4, 1452–1475
  * [6] A. A. Klyachko: Equivariant bundles over toric varieties, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 1001–1039, 1135; translation in Math. USSR-Izv. 35 (1990), 337–375.
  * [7] N. Bourbaki, Éléments de mathématique. XXVI. Groupes et algèbres de Lie. Chapitre 1: Algèebres de Lie, Actualités Sci. Ind. No. 1285, Hermann, Paris (1960)
  * [8] K. Kodaira and D. C. Spencer, On deformations of complex analytic structures, I, II, Annals of Math. 67(1958), 328–466.
  * [9] C. Okonek, M. Schneider, and H. Spindler, Vector bundles on complex projective spaces. Corrected reprint of the 1988 edition. With an appendix by S. I. Gelfand. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2011. viii+239 pp.
  * [10] A. Weil, Généralisation des fonctions abéliennes, J. Math. Pures Appl. 17 (1938), 47–87.