# Building a Dialogue Corpus Annotated with
Expressed and Experienced Emotions

Tatsuya Ide  Daisuke Kawahara
Department of Computer Science and Communications Engineering, Waseda
University
<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

In communication, a human would recognize the emotion of an interlocutor and
respond with an appropriate emotion, such as empathy and comfort. Toward
developing a dialogue system with such a human-like ability, we propose a
method to build a dialogue corpus annotated with two kinds of emotions. We
collect dialogues from Twitter and annotate each utterance with the emotion
that a speaker put into the utterance (expressed emotion) and the emotion that
a listener felt after listening to the utterance (experienced emotion). We
built a dialogue corpus in Japanese using this method, and its statistical
analysis revealed the differences between expressed and experienced emotions.
We conducted experiments on recognition of the two kinds of emotions. The
experimental results indicated the difficulty in recognizing experienced
emotions and the effectiveness of multi-task learning of the two kinds of
emotions. We hope that the constructed corpus will facilitate the study on
emotion recognition in a dialogue and emotion-aware dialogue response
generation.

##  1 Introduction

Text-based communication has become indispensable as society accelerates
online. In natural language processing, communication between humans and
machines has attracted attention, and the development of dialogue systems has
been a hot topic. Through the invention of Transformer (Vaswani et al., 2017)
and the success of transfer learning (e.g., Radford et al. (2018); Devlin et
al. (2019)), the performance of natural language understanding models and
dialogue systems continues to improve. In recent years, there have been
studies toward building open-domain neural chatbots that can generate a human-
like response (Zhou et al., 2020; Adiwardana et al., 2020; Roller et al.,
2021).

One of the keys to building more human-like chatbots is to generate a response
that takes into account the emotion of the interlocutor. A human would
recognize the emotion of the interlocutor and respond with an appropriate
emotion, such as empathy and comfort, or give a response that promotes
positive emotion of the interlocutor. Accordingly, developing a chatbot with
such a human-like ability (Rashkin et al., 2019; Lubis et al., 2018, 2019) is
essential. Although several dialogue corpora with emotion annotation have been
proposed, an utterance is annotated only with a speaker’s emotion (Li et al.,
2017; Hsu et al., 2018) or a dialogue as a whole is annotated Rashkin et al.
(2019), all of which are not appropriate for enabling the above ability.

In this paper, we propose a method to build an emotion-annotated multi-turn
dialogue corpus, which is necessary for developing a dialogue system that can
recognize the emotion of an interlocutor and generate a response with an
appropriate emotion. We annotate each utterance in a dialogue with an
expressed emotion, which a speaker put into the utterance, and an experienced
emotion, which a listener felt when listening to the utterance.

Figure 1: An example dialogue with expressed and experienced emotions.

To construct a multi-turn dialogue corpus annotated with these emotions, we
collect dialogues from Twitter and crowdsource their emotion annotation. As a
dialogue corpus, we extract tweet sequences where two people speak
alternately. For the emotion annotation, we adopt Plutchik’s wheel of emotions
(Plutchik, 1980) as emotion labels and ask crowdworkers whether an utterance
indicates each emotion label for expressed and experienced emotion categories.
Each utterance is allowed to have multiple emotion labels and has an
intensity, strong and weak, according to the number of crowdworkers’ votes. We
build a Japanese dialogue corpus as a testbed in this paper, but our proposed
method can be applied to any language.

Using the above method, we constructed a Japanese emotion-tagged dialogue
corpus consisting of 3,828 dialogues and 13,806 utterances.111We will release
our corpus and code at https://github.com/nlp-waseda/expr-exper-emo.
Statistical analysis of the constructed corpus revealed the characteristics of
words for each emotion and the relationship between expressed and experienced
emotions. We further conducted experiments to recognize expressed and
experienced emotions using BERT (Devlin et al., 2019). We defined the task of
emotion recognition as regression and evaluated BERT’s performance using
correlation coefficients. The experimental results showed that it was more
difficult to infer experienced emotions than expressed emotions, and that
multi-task learning of both emotion categories improved the overall
performance of emotion recognition. From these results, we can see that
expressed and experienced emotions are different, and that it is meaningful to
annotate both. We expect that the constructed corpus will facilitate the study
on emotion recognition in dialogue and emotion-aware response generation.

##  2 Related Work

###  2.1 Emotion-Tagged Corpora

Many non-dialogue corpora annotated with emotions have been constructed.
EmoBank (Buechel and Hahn, 2017) is a corpus of social media or reviews with
emotion annotation. They annotate sentences with the emotions of a person who
read them and a person who wrote them. WRIME (Kajiwara et al., 2021) is an
emotion-annotated corpus in Japanese, where SNS posts are tagged with both
subjective and objective emotions. The concept of this corpus is similar to
EmoBank. However, they emphasize the subjectivity of annotation and ask
writers to annotate their own sentences with emotions. Furthermore, EmoInt
(Mohammad and Bravo-Marquez, 2017) aims at the task of detecting emotion
intensity. They annotate Twitter posts with anger, fear, joy, and sadness and
give each emotion a real value between 0 and 1 as the intensity level.

Some corpora are tagged with non-emotional factors, along with emotions.
EmotionStimulus (Ghazi et al., 2015) and GroundedEmotions (Liu et al., 2017)
are corpora that focus on the reason for an expressed emotion. The former uses
FrameNet to detect a cause, while the latter treats weather and news as
external emotion factors. In terms of emotion labels, the two corpora adopt
seven emotions (Ekman’s six emotions (Ekman, 1992) and shame) and two emotions
(only happiness and sadness), respectively. In StoryCommonsense (Rashkin et
al., 2018), a series of sentences comprising of a short story is tagged with
motivation and emotional reaction for each character. For emotion labels, they
use some theories of psychology, including Plutchik’s wheel of emotions
(Plutchik, 1980).

None of the above corpora, however, are relevant to dialogue. StoryCommonsense
is similar to ours but differs in that characters in a story are annotated
instead of speakers’ utterances.

###  2.2 Dialogue Corpora

Several dialogue corpora annotated with emotions are available. DailyDialog
(Li et al., 2017) is one collected from educational websites and tagged with
emotions and intentions. EmotionLines (Hsu et al., 2018) is a multi-turn
dialogue corpus with annotation of emotions. Both of them use seven labels for
tagging: Ekman’s six emotions (Ekman, 1992) and an other/neutral emotion. MELD
(Poria et al., 2019) is an extension of EmotionLines, tagged with not only
emotions but also visual and audio modalities. EmpatheticDialogues (Rashkin et
al., 2019) is a dialogue-level emotion-tagged corpus, considering two
participants as a speaker and a listener, and tagged with the speaker’s
emotion and its context.

In EmpatheticDialogues, not each utterance but each dialogue is annotated,
which is not suitable for recognizing emotional transition throughout a
dialogue. For Japanese, there is a Japanese version of EmpatheticDialogues
called JEmpatheticDialogues (Sugiyama et al., 2021), which suffers from the
same problem. In this work, we conduct utterance-level annotation like
DailyDialog and EmotionLines. Although these corpora contain only the
speaker’s emotion (expressed emotion), we also annotate an utterance with the
emotion of a person who hears it (experienced emotion). Furthermore, while an
utterance has only one emotion label in these corpora, we allow multiple
emotion labels to be tagged per utterance and also consider their strength.

There are also some studies toward developing emotion-aware dialogue systems.
Smith et al. (2020) propose three skills for a human-like dialogue system:
recognizing emotions, using knowledge (Dinan et al., 2019), and considering
personality (Zhang et al., 2018). Furthermore, Roller et al. (2021) build a
dialogue system capable of blending these three skills.

##  3 Corpus Building

Length | # Dialogues | # Utterances
---|---|---
2 | 1,330 | 2,660
3 | 1,071 | 3,213
4 | 509 | 2,036
5 | 310 | 1,550
6 | 225 | 1,350
7 | 158 | 1,106
8 | 134 | 1,072
9 | 91 | 819
2-9 | 3,828 | 13,806
Table 1: The statistics of dialogues and utterances. Label | Expressed | Experienced
---|---|---
Strong | Weak | Strong | Weak
Anger | 430 | 1,349 | 124 | 870
Anticipation | 1,906 | 4,229 | 1,215 | 4,068
Joy | 1,629 | 3,672 | 1.553 | 4,549
Trust | 247 | 1,732 | 520 | 3,455
Fear | 252 | 942 | 123 | 846
Surprise | 602 | 2,018 | 434 | 2,798
Sadness | 1,227 | 2,936 | 889 | 3,037
Disgust | 476 | 1,979 | 186 | 1,535
Any | 6,371 | 12,215 | 4,705 | 12,515
Table 2: The statistics of utterances for each emotion label. Figure 2: An example of the crowdsourced task. Checkboxes allow crowdworkers to select multiple emotions for an utterance. Utterance | Expressed | Experienced
---|---|---
A1: 来週、車で京都行く 普通に観光してきます (I’m driving to Kyoto next week. We’ll just do some sightseeing.) | {Anticipation, Joy} | {Anticipation}
B1: いいなぁ、京都 (I like Kyoto.) | {Anticipation} | {Anticipation, Joy}
A2: 楽しんできます (I’ll enjoy it.) | {Anticipation, Joy} | {Anticipation, Joy}
B2: 行ったことないから純粋に羨ましい (I’ve never been there, so I’m genuinely jealous.) | {Anticipation} | {Anticipation, Joy}
A3: ないんや意外 (I’m surprised you haven’t.) | {Surprise} | {Joy, Surprise}
Table 3: An example dialogue annotated with expressed and experienced emotions by crowdsourcing. The labels in bold indicate strong emotions. Label |  Expressed |  Experienced
---|---|---
Anger |  糞, せる, マジだ (shit, force, serious) |  糞, うるさい, 居る (shit, noisy, exist)
Anticipation |  教える, 願う, 待つ (teach, hope, wait) |  待つ, 楽しみだ, 強い (wait, looking forward to, strong)
Joy |  楽しい, 嬉しい, おもろい (joyful, glad, funny) |  楽しい, 嬉しい, おもろい (joyful, glad, funny)
Trust |  全然, 大丈夫だ, ちゃんと (at all, all right, properly) |  やすみ, 教える, 大事だ (rest, teach, important)
Fear |  怖い, やばい, どう (afraid, serious, how) |  怖い, やばい, 危険だ (afraid, serious, dangerous)
Surprise |  やばい, なんで, ？ (serious, why, ?) |  居る, ビックリ, 年 (exist, surprise, year)
Sadness |  泣く, 痛い, 悲しい (cry, hurt, sad) |  泣く, 辛い, 痛い (cry, hard, hurt)
Disgust |  悪い, 嫌いだ, 嫌だ (bad, hate, dislike) |  悪い, 気持ち, 嫌だ (bad, surprise, dislike)
Table 4: Top-3 frequent words for each emotion label. An IDF filtering is
applied to exclude common words.

(a) Expressed and experienced emotions (for a certain utterance).

(b) Experienced and next expressed emotions (for a certain person).

Figure 3: The confusion matrices of the relationship between expressed and
experienced emotions. In this analysis, we focus on only the strong labels.
Note that the matrices’ elements are normalized in the row direction.

(a) Expressed emotions at the beginning and end of dialogue.

(b) Expressed emotions at the beginning and experienced emotions at the end of
dialogue.

Figure 4: The confusion matrices for the emotion labels at the beginning and
end of dialogue. In this analysis, we consider only the emotions of a person
who begins a dialogue. Note that the targets are limited to the dialogues
containing six to nine utterances, and the elements are normalized in the row
direction.

###  3.1 Dialogue Collection

We collect dialogue texts from Twitter by considering the interaction between
tweets and their replies by two users as a dialogue. To improve the text
quality, we exclude tweets that contain images or hashtags and set the maximum
number of utterances included in a dialogue to nine. We also apply several
filters: excluding dialogues that contain special symbols, emojis, repeated
characters, and utterances that are too short. Note that the reason why we
exclude emojis is that they are relatively explicit emotional factors, and we
intend to analyze emotions implied from usual textual expressions.

We collected Japanese dialogues using this method. The numbers of dialogues
and utterances are shown in Table 1. We obtained 3,828 dialogues that
correspond to 13,806 utterances in total. Regarding the length of dialogues,
the number of dialogues tends to decrease as that of utterances per dialogue
increases.

###  3.2 Emotion Annotation

We adopt Plutchik’s wheel of emotions (Plutchik, 1980) as annotation
labels.222Ekman’s six emotions (Ekman, 1992) and Plutchik’s wheel of emotions
(Plutchik, 1980) are commonly used in emotion-tagged corpora. Preliminary
experiments by crowdsourcing showed that the latter is more appropriate for
our crawled dialogues. In this work, therefore, we use eight emotions by
Plutchik (1980). Specifically, our annotation labels consist of eight
emotions: anger, anticipation, joy, trust, fear, surprise, sadness, and
disgust. We annotate each utterance with two emotion categories: an expressed
emotion, which is expressed by a speaker of the utterance, and an experienced
emotion, which is experienced by a listener of the utterance. In other words,
an utterance is annotated with both subjective and objective emotions, which
is similar to EmoBank (Buechel and Hahn, 2017) for non-dialogue texts. By
annotating expressed and experienced emotions, we can trace the changes in the
emotion surrounding both an utterance and a participant in a dialogue.

As a crowdsourcing platform, we use Yahoo!
Crowdsourcing.333https://crowdsourcing.yahoo.co.jp/ By showing the target
utterance and its context, we ask seven workers whether the target utterance
has a specified emotion or not about each emotion label for expressed and
experienced emotion categories. For the expressed emotions, we ask which
emotion a speaker expressed when saying the utterance. For the experienced
emotions, we ask which emotion a listener experienced when hearing the
utterance. Workers are allowed to select multiple emotion labels or none of
them. An interface of the crowdsourcing task for expressed emotions is shown
in Figure 2.

Because a view of expressed and experienced emotions can vary among
annotators, we employ many workers per an utterance and aggregate their votes
to obtain highly reliable annotations.444For the expressed emotion, we ask
workers to annotate the emotion of the speaker of an utterance. This
annotation, however, is not strictly what a speaker had in mind but what the
workers think a speaker would want to express, which can be considered
objective in some sense. Having truly subjective annotation as the expressed
emotion, like Kajiwara et al. (2021), is our future work. We consider strength
for each emotion according to the number of workers’ votes; emotions selected
by more than half of the workers are regarded as strong, and ones selected by
more than a quarter are regarded as weak. Note that the set of strong emotions
is a subset of the set of weak ones. We expect that providing the emotions
with intensity enables us to handle their granularity.

We applied the above emotion annotation method to our dialogue corpus. The
number of utterances for each emotion is shown in Table 2. For the expressed
emotion, 46.15% and 88.48% of the utterances are tagged with at least one
strong and weak emotion, respectively. For the experienced emotion, the
percentages are 34.08% and 90.65%, respectively. Approximately 90% of the
utterances are accompanied by one or more emotion labels, and thus our corpus
is consequently suitable for recognizing emotions in dialogues and analyzing
their changes. In contrast to ours, for example, less than 20% of utterances
are tagged with a specific emotion in DailyDialog (Li et al., 2017). Hence it
is difficult to analyze emotion changes using such corpora with a small amount
of emotion annotation. In addition, we can see a bias among the emotion labels
for both expressed and experienced emotions, with more instances of
anticipation and joy and fewer instances of trust and fear. An example of a
dialogue with the annotation is shown in Table 3.

##  4 Corpus Analysis

###  4.1 Frequent Words for Emotion Categories and Labels

To investigate the characteristics of utterances with different emotions, we
count words for each strong emotion label in our corpus. In this analysis, we
identify words by the Japanese morphological analyzer Juman++ (Tolmachev et
al., 2018). To exclude common words likely to appear for all emotions, we
apply an IDF filtering. Specifically, words with IDF less than half of the
maximum are ignored.

Top-3 words appearing for strong emotion labels are shown in Table 4. The same
words tend to appear in the two emotion categories for joy and sadness. In
contrast, the frequent words in the two categories are different for
anticipation, trust, and surprise.

###  4.2 Relationship Between Expressed and Experienced Emotions

We annotated utterances with the expressed and experienced emotions. Here, we
focus on the relationship between these two emotion categories. Specifically,
we investigate the following two relationships:

  1. a.

The expressed emotion and the experienced emotion for the same utterance
(different persons).

  2. b.

The experienced emotion for an utterance and the expressed emotion for the
next utterance (the same person).

The confusion matrices for the strong emotion labels are shown in Figure 3,
where the elements are normalized in the row direction. First, diagonal
components of the two confusion matrices have large values, indicating that
the same emotions are likely to occur both for the same utterance and for the
same person. Figure 3(a) shows that people are likely to experience joy for an
utterance of anticipation, trust, and surprise in addition to the same
emotion. People also tend to experience disgust and sadness for anger and
disgust, respectively. Figure 3(b) shows that after experiencing trust, people
are more likely to express joy than trust. For an anger experience, people are
more likely to express disgust than anger. Figures 3(a) and 3(b) reveal that
the relationship of sadness is particularly different. For a certain
utterance, sadness makes the other person feel sad in most cases, but for a
certain person, anticipation in addition to sadness can be expressed after
experiencing sadness. We speculate that when a person experiences sadness from
the interlocutor, the person brings an utterance with anticipation to comfort
them.

###  4.3 Emotions at the Beginning and End of a Dialogue

To analyze the emotion changes through a dialogue, we compare emotions at the
beginning and end of a dialogue. In other words, we see how the emotions of a
person who starts the dialogue change through the dialogue. In this analysis,
we focus on the following two relationships:

  1. a.

The emotion expressed first and the emotion expressed last by the same person.

  2. b.

The emotion expressed first and the emotion experienced last by the same
person.

The confusion matrices for the strong emotion labels are shown in Figure 4.
The targets are limited to dialogues containing six to nine utterances to
analyze the emotion changes in long dialogues. Figure 4(a) shows that a
speaker of the first utterance is likely to finally express anticipation and
joy regardless of the first emotion. A speaker who first expresses surprise
can express sadness through the dialogue. Figure 4(b) also shows that the
first speaker can experience anticipation at the end of a dialogue. A person
who first expresses anger and disgust tends to finally experience trust. From
these two figures, we can see that a dialogue causes a person who first
expresses fear to finally feel either a positive or negative emotion.

##  5 Experiments

###  5.1 Model Setup

We conduct experiments on expressed and experienced emotion recognition using
our corpus. We solve a regression task of each emotion intensity for an
utterance with its context for the emotion recognition task. We assign 0, 1,
and 2 for none, weak, and strong emotion labels, respectively, and let a model
regress these values for each emotion. As such, we train two separate models
for expressed and experienced emotions with the mean squared error loss:

$\displaystyle\mathcal{L}=\frac{1}{NK}\sum_{i=1}^{N}\sum_{j=1}^{K}(y_{ij}-t_{ij})^{2},$
(1)

where $N$ is the number of samples and $K$ is the number of emotion labels.
$y_{ij}$ is the output from the model for the $j$th label of the $i$th sample,
and $t_{ij}$ is its gold label.

We adopt a Japanese pre-trained BERT model and fine-tune it. We compare two
pre-trained models from Kyoto
University555https://nlp.ist.i.kyoto-u.ac.jp/?ku_bert_japanese and one from
NICT666https://alaginrc.nict.go.jp/nict-bert/index.html. We use the WWM and
BPE versions for Kyoto University’s and NICT’s BERT models, respectively.

Input utterances are segmented into words with Juman++ (Tolmachev et al.,
2018) and tokenized into subwords by applying BPE. We join utterances with
[SEP] and append [CLS] and [SEP] to the beginning and end, respectively. As
there are two participants in a dialogue, we give each utterance a segment ID
of 0 or 1. It provides the models with the information about the speaker of an
utterance. Based on a series of utterances joined with [SEP], we predict an
emotion label for the last utterance. The vector corresponding to [CLS] is
passed to a fully-connected layer, and an eight-dimensional vector
representing the eight emotions is obtained. Each of the elements is supposed
to regress the intensity of each emotion.

Since we are dealing with a regression task, Pearson’s and Spearman’s
correlation coefficients are used as evaluation metrics. The dialogues in our
corpus are split into 8:1:1, corresponding to training, validation, and test
sets. We fine-tune our models for three epochs and evaluate them on the test
set. The implementation of the models is based on HuggingFace
Transformers777https://huggingface.co/transformers/. The models are trained
using NVIDIA Tesla V100 SXM2 GPU.

###  5.2 Results

Model | Expressed | Experienced
---|---|---
Kyoto (base) | 58.84/44.33 | 53.60/41.84
Kyoto (large) | 60.85/45.16 | 55.09/42.94
NICT |  61.50/46.05 |  56.23/43.88
Table 5: The results of regression for expressed and experienced emotions. The metrics are Pearson’s and Spearman’s correlation coefficients. Label | Expressed | Experienced
---|---|---
Anger | 50.21/33.80 | 38.11/23.80
Anticipation | 62.76/55.55 | 57.46/51.22
Joy |  67.25/55.22 |  61.92/54.47
Trust | 41.15/36.69 | 43.91/40.48
Fear | 59.09/31.47 | 49.60/24.90
Surprise | 49.86/39.58 | 40.58/33.86
Sadness | 63.70/51.50 | 55.48/43.88
Disgust | 47.76/38.18 | 37.32/28.13
Table 6: The correlation coefficients for each emotion label. The metrics are Pearson’s and Spearman’s correlation coefficients. The scores are from the NICT model that achieved the highest performance in Table 5. Dialogue |  Predicted |  Gold
---|---|---
A1: ゲームの検証してる人が検証してほしいことあれば言ってください的なこと言ってたから依頼したら無視されて悲しくなったのはいい思い出 (I have a good memory of a guy who was verifying a game and said if there was anything he wanted verified, please let him know, so I made a request and he ignored it, which made me sad.) B1: それは悲しいね (That’s sad.) |  Strong sadness |  Strong sadness
A1: youtubeでバーのマスターが氷砕いてる動画見てボーッとしてる (I’ve been watching videos of bar masters crushing ice on youtube and I’m in a daze.) B1: なんかしてよ (Do something.) A2: そのうちこういうときにツイキャスをしようかなと思っておる (One of these days I’m going to do a tweak for this.) B2: 天才の発想 スマホでも見やすいから助かる (It’s a genius idea, and it’s easy to watch on my phone.) |  Weak anticipation and joy |  Strong joy and weak trust
A1: 今、部活終わって帰るとこやけど 雨やばいしかっぱ持ってきてないし 最悪 (I’m on my way home after club activities, but it’s raining and I didn’t bring my hat, so that sucks.) B1: わたしも学校出た瞬間大雨降ってきた (I’m going back to school now, but it’s raining really hard and I didn’t bring my jacket.) |  Strong surprise |  Strong sadness
Table 7: Example dialogues with predicted and gold expressed emotions. The predicted emotion labels are taken from the predictions of the NICT model, which predicted an emotion label for the last utterance of each dialogue. Train\Test | Expressed | Experienced
---|---|---
Expressed | 61.50/46.05 | 52.89/40.91
Experienced | 55.49/43.34 | 56.23/43.88
Multi-Task |  62.20/46.63 |  57.35/45.01
Table 8: The results of multi-task learning with expressed and experienced emotions. The metrics are Pearson’s and Spearman’s correlation coefficients. Train\Test | Experienced | Next Expressed
---|---|---
Experienced | 54.62/43.47 | 29.53/25.46
Next Expressed | 43.32/35.27 | 33.91/28.31
Multi-Task |  55.75/49.50 |  35.17/30.49
Table 9: The results of multi-task learning with experienced and next
expressed emotions. The metrics are Pearson’s and Spearman’s correlation
coefficients.

For the regression task defined in Section 5.1, the correlation coefficients
for each model are shown in Table 5. In terms of performance, the NICT model
achieved the best score across all values. For the values regarding expressed
and experienced emotions, the performance of the experienced emotion is
inferior to that of expressed emotion in all models. This result indicated
that it is more difficult to recognize the experienced emotion than the
expressed emotion.

The correlation coefficients for each emotion inferred by the NICT model are
shown in Table 6. For both the expressed and experienced emotions, the highest
scores were achieved for anticipation and joy. In contrast, the emotions with
lower values were trust and fear for the expressed emotion and anger and
disgust for the experienced emotion. From Tables 6 and 2, we can see that the
larger the number of the samples for an emotion is, the higher the correlation
coefficient becomes. As a case study, we show example dialogues and their
emotions predicted by the NICT model in Table 7.

###  5.3 Multi-Task Learning

Our analysis in Section 4.2 indicated that there is a correlation between
expressed and experienced emotions. Therefore, we consider training a single
model for recognizing both the emotion categories. The information for solving
the two similar tasks is expected to allow a model to improve the performance
of each other (Liu et al., 2019). We provide a model with two separate fully-
connected layers for the tasks and train them simultaneously, where the inputs
are the same as those in Section 5.1. Here, the mean of the losses for
expressed and experienced emotions is optimized:

$\displaystyle\mathcal{L}_{\text{multi-
task}}=\frac{\mathcal{L}_{\text{expressed}}+\mathcal{L}_{\text{experienced}}}{2}.$
(2)

Based on Figures 3(a) and 3(b), we consider multi-task learning of expressed
and experienced emotions for a certain utterance and a certain person. For the
relationship in a certain person, we use the experienced emotion of an
utterance and the expressed emotion of the following utterance. We also
conduct experiments on the cases where the training and test sets are
different from each other. In such a case, for example, expressed emotions are
used for training, but experienced emotions are used for testing.

The correlation coefficients for an utterance and a speaker by multi-task
learning are shown in Tables 8 and 9, respectively. First, the scores when the
training and test sets are different from each other are lower than those when
they are the same. This gap indicates the significance of annotating
utterances with expressed and experienced emotions separately. In all columns,
the multi-task models achieved higher performance than the single-task models.
Especially, in Table 9, the multi-task scores for both the two tasks are
higher than the single-task baselines by one point. In other words, expressed,
experienced, and next expressed emotions have the information for helping the
recognition of each other.

##  6 Conclusion

We proposed a method to build an emotion-tagged multi-turn dialogue corpus to
help machines recognize emotional transition in a dialogue. Dialogues between
two speakers are collected from Twitter, and each utterance is annotated with
emotions by crowdsourcing. In the annotation process, we consider the emotions
expressed by a speaker who said the utterance and the emotion experienced by a
listener who heard the utterance. In addition, the labels are provided with
their intensity, representing the granularity of emotions.

We built a Japanese emotion-tagged dialogue corpus and analyzed it. The
results showed the characteristics of words for each emotion, the correlation
between the emotions about a certain utterance and speaker, and the tendency
for speakers to become positive through a dialogue. We also developed emotion
recognition models for expressed and experienced emotions based on the
Japanese pre-trained BERT models. The experimental results indicated that it
is more difficult to recognize a listener’s emotion than a speaker’s emotion.
Multi-task learning of expressed and experienced emotions improved the
performance of the two emotion recognition tasks about an utterance and a
speaker.

For our future work, we will tackle response generation based on predicted
emotions. With our corpus, a dialogue system is expected to predict which
emotion it experiences from a given utterance and which emotion it should
express for the next utterance. Once such emotions are recognized, the
dialogue system should be able to generate an appropriate response depending
on the predicted expressed emotion.

The corpus in this work is annotated only with expressed and experienced
emotions about an utterance. In addition to the emotion annotation, we should
also consider dialogue situations Rashkin et al. (2019). The cause of a
dialogue or an utterance helps recognize a speaker’s emotion and how it
changes. We can also consider non-emotional annotation, such as a dialogue’s
topic and an utterance’s intention Li et al. (2017). The relationship between
emotions and non-emotional factors is also important for machines to better
recognize a speaker’s emotion.

## Acknowledgements

This work was supported by a joint research grant from LINE Corporation.

## Ethical Considerations

We built the dataset by collecting texts from Twitter and annotating them by
crowdsourcing. For crowdsourcing, we employed 3,847 workers. It took
approximately five minutes for a task of annotating 10 utterances. Every
worker was paid 4 JPY per 10 utterances, and in total, the built dataset costs
195,700 JPY. Since the dataset was collected from Twitter, it may include
contents that are harmful for some of the dataset or its application users.
For building the dataset through the Twitter API and crowdsourcing, we did not
include any sensitive information that allows personal identification.

The dataset or models trained on it enable downstream applications to infer
the emotions of their users, resulting in facilitating communication between
the users and the applications. In terms of dialogue systems, this ability is
considered valuable for both task-oriented and non-task-oriented dialogue
systems. For example, it assists the user in decision-making and solves the
user’s worry and trouble. In contrast to such benefit, it is difficult for the
model to infer the emotion accurately, with the relatively small dataset.
Therefore, prediction errors by the model, especially for sensitive utterances
or negative emotions, may bring harmful experiences on the users.

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Comment on “Origin of correlated isolated flat bands in copper-substituted
lead phosphate apatite” by Michael W. Swift and John L. Lyons, Center for
Computational Materials Science, US Naval Research Laboratory

Griffin [1] presented first-principles calculations using PBE+U showing that
in Pb10(PO4)6(OH)2, copper substitution on the Pb(1) site gives rise to a pair
of bands approximately 160 meV above the host material’s valence-band maximum
(VBM). This two-band manifold is half occupied and very flat, resulting in a
large density of states at the Fermi level. Such an electronic structure is a
characteristic of many high-$T_{\mathrm{c}}$ superconductors, and was
suggested as a possible explanation for recent experimental reports of room-
temperature superconductivity in so-called LK-99 [2, 3, 4, 5, 6].

Our calculations show that copper substituting on the Pb(1) site (which we
will call CuPb(1)) does not give rise to a half-occupied flat band. Instead,
the two copper bands are split into a lower, fully occupied band (which forms
the VBM) and an upper, unoccupied band (see Figure 1). The difference is due
to our use of the HSE hybrid functional [7], which provides more accurate
band-gap energies and generally provides a better description of dopants in
semiconductors. Other calculation parameters were similar, and our tests using
PBE+U reproduce the results of Ref. 1 111Our HSE calculations used VASP, PAW
pseudopotentials, a 500 eV plane-wave energy cutoff, a 2$\times$2$\times$3
k-point mesh, and standard “HSE06” parameters. Relaxations achieved force
convergence of 0.03 eV/Å. PBE+U calculations used $U=4$ eV on the copper d
states. HSE-relaxed and PBE+U-relaxed structures were very similar, and HSE
calculation of the PBE+U-relaxed structure yielded nearly identical electronic
structure to the full HSE calculation.. Semilocal functionals like PBE are
known to underestimate band gaps, in oxides largely by predicting oxygen 2p
bands that are too high in energy. PBE can therefore predict defect and
impurity states to be resonant with the valence band when they should be
localized in the band gap [9]. In Ref.1 PBEsol+U was used, thus taking
electron correlations into account by incorporating an onsite Hubbard
interaction $U$ for the copper impurity. While this approach can correct the
band gap and defect levels in some circumstances [10], it would provide no
correction for the energy of the oxygen p states. We find that PBE
underestimates the HSE-calculated band gap of the parent compound, lead
phosphate apatite, by almost 1.3 eV 222The PBE gap of Pb10(PO4)6(OH)2 is 3.61
eV compared to an HSE gap of 4.89 eV.

Our results suggest that the half-occupied flat band is an artifact of the PBE
overestimation of the valence band energy. The occupied copper state (which we
observe using HSE) is hybridized with oxygen, so its energy is overestimated
in PBE+U along with the valence band. This pushes the state up in the band
structure and brings it close in energy to the unoccupied copper state,
causing the two copper states to mix, and resulting in a half-occupied flat
band. Within HSE, the two copper states are separate. This also explains why
varying the copper $U$ value between 2 and 8 eV did not significantly shift
the relative positions of the flat band and the valence band for PBE+U in Ref.
1: the flat-band energy is not dictated by the applied $U$, but rather by
hybridization with the oxygen states that are artificially high in energy.

Griffin also pointed out that CuPb(2), the lower-energy substitutional site,
does not lead to a half-occupied flat band, but instead exhibits oxygen 2p
character at the VBM and a single unoccupied copper band at mid-gap. Our
results are qualitatively similar, both in PBE+U and HSE. We find CuPb(2) is
lower in energy than CuPb(1) by 0.83 eV. For CuPb(2), we calculate the gap
between the VBM and copper state to be 0.47 eV in PBE+U and 2.10 eV in HSE;
PBE+U underestimates the gap between the VBM and the copper states in CuPb(2)
by 1.6 eV. Therefore it is not surprising that PBE+U closes the 0.9 eV gap
between copper states in CuPb(1).

Figure 1: Band structure of copper-substituted lead phosphate apatite with
copper on the Pb(1) site (CuPb(1)). Bands are color-coded according to the
fraction of the charge density that projects onto the copper orbitals. Results
using HSE are on the left and PBE+U results are on the right. The top plots
show details near the Fermi level (set to zero and indicated with a dotted
line). The bottom row shows the full gap, vertically aligned based on the
conduction band. HSE predicts an insulating state: the VBM consists of a fully
occupied copper state, and an unoccupied copper state lies about 0.9 eV higher
in energy. This contrasts with the half-occupied two-band manifold in PBE+U.

Our results predict that stoichiometric Pb9Cu(PO4)6(OH)2 is an insulator, not
a superconductor. But, these results on their own do not rule out the
possibility of superconductivity in LK-99; with degenerate doping, the Fermi
level could be pushed into one of the high density-of-states copper bands.
Given the questions surrounding the quality of the initial LK-99 material,[12]
careful experiments on high-quality samples will be crucial. We have
furthermore shown that density functional theory calculations at the PBE+U
level predict an electronic structure that is qualitatively inconsistent with
results from a hybrid functional, indicating that future first-principles
studies should employ advanced methods to ensure that the oxygen-derived
valence band is correctly described.

## References

  * Griffin [2023] S. M. Griffin, Origin of correlated isolated flat bands in copper-substituted lead phosphate apatite (2023), arXiv:2307.16892 [cond-mat.supr-con] .
  * Lee  _et al._ [2023a] S. Lee, J.-H. Kim, and Y.-W. Kwon, The First Room-Temperature Ambient-Pressure Superconductor (2023a), arXiv:2307.12008 [cond-mat.supr-con] .
  * Lee  _et al._ [2023b] S. Lee, J. Kim, H.-T. Kim, S. Im, S. An, and K. H. Auh, Superconductor Pb10-xCux(PO4)6O showing levitation at room temperature and atmospheric pressure and mechanism (2023b), arXiv:2307.12037 [cond-mat.supr-con] .
  * Si and Held [2023] L. Si and K. Held, Electronic structure of the putative room-temperature superconductor Pb9Cu(PO4)6O (2023), arXiv:2308.00676 [cond-mat.supr-con] .
  * Cabezas-Escares  _et al._ [2023] J. Cabezas-Escares, N. F. Barrera, C. Cardenas, and F. Munoz, Theoretical insight on the LK-99 material (2023), arXiv:2308.01135 [cond-mat.supr-con] .
  * Kurleto  _et al._ [2023] R. Kurleto, S. Lany, D. Pashov, S. Acharya, M. van Schilfgaarde, and D. S. Dessau, Pb-apatite framework as a generator of novel flat-band CuO based physics, including possible room temperature superconductivity (2023), arXiv:2308.00698 [cond-mat.supr-con] .
  * Heyd  _et al._ [2003] J. Heyd, G. E. Scuseria, and M. Ernzerhof, Hybrid functionals based on a screened Coulomb potential, The Journal of Chemical Physics 118, 8207 (2003).
  * Note [1] Our HSE calculations used VASP, PAW pseudopotentials, a 500 eV plane-wave energy cutoff, a 2$\times$2$\times$3 k-point mesh, and standard “HSE06” parameters. Relaxations achieved force convergence of 0.03 eV/Å. PBE+U calculations used $U=4$ eV on the copper d states. HSE-relaxed and PBE+U-relaxed structures were very similar, and HSE calculation of the PBE+U-relaxed structure yielded nearly identical electronic structure to the full HSE calculation.
  * Janotti  _et al._ [2010] A. Janotti, J. B. Varley, P. Rinke, N. Umezawa, G. Kresse, and C. G. Van De Walle, Hybrid functional studies of the oxygen vacancy in TiO 2, Phys. Rev. B 81, 085212 (2010).
  * Lany [2011] S. Lany, Predicting polaronic defect states by means of generalized Koopmans density functional calculations, Phys. Status Solidi B 248, 1052 (2011).
  * Note [2] The PBE gap of Pb10(PO4)6(OH)2 is 3.61 eV compared to an HSE gap of 4.89 eV.
  * Zhu  _et al._ [2023] S. Zhu, W. Wu, Z. Li, and J. Luo, First order transition in Pb10-xCux(PO4)6O ($0.9<x<1.1$) containing Cu2S (2023), arXiv:2308.04353 [cond-mat.supr-con] .


# Flop connections between minimal models for corank 1 foliations over
threefolds

Dongchen Jiao  Pascale Voegtli

###### Abstract

In 2007 Kawamata proved that two different minimal models can be connected by
a sequence of flops. The aim of this paper is to show that the same holds true
for 2 foliated minimal models descending from a common 3-fold pair equipped
with a F-dlt foliation of corank 1.

##  1 Introduction

In recent years the understanding of the fundamental birational geometry of
foliations, especially on 3-folds, has been promoted by several groundbreaking
works. In [3] most parts of the classical MMP have been extended to 3-fold
pairs equipped with a mildly singular corank 1 foliation. In particular, the
existence of log-flips has been established.

The successful establishment of a foliated analogue of the classical MMP in
low dimensions naturally raises the question whether classical results being
closely related to the MMP do find their natural generalizations for foliated
pairs.

One such classical result one might strive to convey to foliations is the
well-known theorem of Kawamata [5], stating that two minimal models with
terminal singularities are related by a sequence of flops.

Building upon the findings of the authors in [3], the aim of this paper is to
generalize the results of [5] to foliated 3-folds equipped with a corank 1
foliation with F-dlt singularities.

In the proof of the main theorem (see below), we closely follow the line of
reasoning found in [5] and deviate only where adaptations or modifications
seem inevitable.

Concretely, in the present publication, the following theorem is proven (For
the precise definitions we refer to section (2)):

######  Theorem 1.1.

Let $(Y_{1},\mathcal{F}_{1})$ and $(Y_{2},\mathcal{F}_{2})$ be two foliated
minimal models descending from a common foliated 3-fold pair
$(X,\mathcal{F})$. We assume that $X$ is klt and $\mathbb{Q}$-factorial.

Assume there is a birational map $\alpha:Y_{1}\dashrightarrow Y_{2}$, such
that $K_{\mathcal{F}_{1}}$ and $K_{\mathcal{F}_{2}}$ are big. Then $\alpha$ is
composed of a sequence of flops.

In detail, there are an effective $\mathbb{Q}$-divisor A on $Y_{1}$ such that
$(\mathcal{F}_{1},A)$ is F-dlt and a factorization

$(Y_{1},\mathcal{F}_{1},A)=(X_{0},\mathcal{G}_{0},A_{0})\dashrightarrow(X_{1},\mathcal{G}_{1},A_{1})\dashrightarrow...\dashrightarrow(X_{r},\mathcal{G}_{r},A_{r})=(Y_{2},\mathcal{F}_{2},A^{{}^{\prime}})$
(1)

satisfying

  1. 1.

$\beta_{i}:X_{i-1}\dashrightarrow X_{i}$ is a flip associated to a
$(K_{\mathcal{G}_{i-1}}+A_{i-1})$-negative extremal ray, where $A_{i}$ is the
strict transform of $A$ on $X_{i}$.

  2. 2.

$\beta_{i}$ is crepant for $K_{\mathcal{G}_{i-1}}$ in the sense that pull-
backs of $K_{\mathcal{G}_{i-1}}$ and $K_{\mathcal{G}_{i}}$ coincide on a
common resolution.

  3. 3.

the flipping contractions $\beta_{i}$ are $K_{\mathcal{G}_{i}}$-trivial.

## Acknowledgements

We would like to express our deep gratitude to the PhD-advisor of the second
author, Paolo Cascini, for the initialization of the project and his
indispensable guidance throughout its further evolvement. Furthermore, are
both authors indebted to Calum Spicer for his extended explanations,
invaluable suggestions and careful revision of the first draft of this
publication.

The first author is supported by EPSRC DTP studentship (EP/V520196/1) Brunel
University London and would like to show appreciation to his supervisor Anne-
Sophie Kaloghiros for useful discussion and invaluable suggestions.

The second author would like to thank the LSGNT for laying the mathematical
foundations of this work by providing a stimulating and supporting working
environment. The second author was supported by the Engineering and Physical
Sciences Research Council [EP/S021590/1]. The EPSRC Centre for Doctoral
Training in Geometry and Number Theory (The London School of Geometry and
Number Theory), University College London.

##  2 Preliminaries

###  2.1 Basic definitions

Throughout, we work over the complex numbers and by 3-fold or variety we mean
a complex analytic space of dimension 3 if not specified otherwise.

######  Definition 2.1.

Let $X$ be a normal variety. A foliation $\mathcal{F}$ on $X$ is a coherent
subsheaf of $\mathcal{T}_{X}$, such that

  * •

$\mathcal{F}$ is saturated, i.e. $\mathcal{T}_{X}/\mathcal{F}$ is torsion free
and

  * •

$\mathcal{F}$ is closed under Lie bracket.

######  Remark.

$\mathcal{F}$ is always locally free on the smooth locus $X_{0}$ of $X$.

######  Definition 2.2.

Suppose $\mathcal{F}$ is a rank $r$ foliation on a normal variety $X$. Notice
that there exists an open embedding $j:X_{0}\hookrightarrow X$ such that
$X_{0}$ is smooth, $\mathrm{codim}(X\backslash X_{0})\geq 2$ and $\mathcal{F}$
is locally free on $X_{0}$, so $\wedge^{r}\mathcal{F}$ is an invertible sheaf
on $X_{0}$. We define the canonical divisor of $\mathcal{F}$ to be any divisor
$K_{\mathcal{F}}$ on $X$ such that $\mathcal{O}_{X}(-K_{\mathcal{F}})\cong
j_{*}(\wedge^{r}\mathcal{F})$.

######  Definition 2.3.

By the notation $(X,\mathcal{F})$ we mean a variety $X$ equipped with a corank
1 foliation $\mathcal{F}$.

######  Definition 2.4.

A triple $(X,D,\mathcal{F})$ consists of a foliation $\mathcal{F}$ on a
variety X and an effective $\mathbb{R}$-divisor D such that
$K_{\mathcal{F}}+D$ is $\mathbb{R}$-Cartier.

For the definition of foliated minimal model we adopt the one found in [[3],
section 10]. For reader’s convenience, we restate it here:

######  Definition 2.5.

A minimal model of $\mathcal{F}$ is a $K_{\mathcal{F}}$-negative birational
map $f:X\dashrightarrow X^{\prime}$ such that if $\mathcal{F}^{\prime}$ is the
transformed foliation on $X^{\prime}$, then

  1. 1.

$X^{\prime}$ is $\mathbb{Q}$-factorial and klt

  2. 2.

$\mathcal{F^{\prime}}$ is F-dlt and $K_{\mathcal{F}^{\prime}}$ is nef.

We now recall the definitions of foliation singularities as well as the
notions of invariance, tangency rsp. transversality of subvarieties:

######  Definition 2.6.

Let X be a normal variety and $\mathcal{F}$ a rank r foliation on X. A
subvariety $S\subset X$ is called $\mathcal{F}$-invariant if for any open
subset $U\subset X$ and any section $\partial\in H^{0}(U,\mathcal{F})$ we
have:

$\partial(I_{S\cap U})\subset I_{S\cap U}$ (2)

where $I_{S\cap U}$ denotes the ideal sheaf of $S\cap U$ in U.
If $\Delta\subset X$ is a prime divisor, one defines $\epsilon(\Delta)=-1$ if
$\Delta$ is invariant in the above sense and $\epsilon(\Delta)=0$ otherwise.

######  Definition 2.7.

Let X be a normal projective variety equipped with a non-dicritical corank 1
foliation $\mathcal{F}$.

We call a subvariety $W\subset X$ tangent to $\mathcal{F}$ if for any
birational morphism ${\pi:\tilde{X}\to X}$ and any divisor E on X such that
$E$ dominates $W$, we have that E is $\mathcal{\tilde{F}}$-invariant, where
$\mathcal{\tilde{F}}$ denotes the pulled back foliation on $\tilde{X}$.

Otherwise, we call $W\subset X$ transverse to $\mathcal{F}$.

######  Definition 2.8.

For a birational morphism $\pi:\tilde{X}\to X$ and a foliated pair
$(\mathcal{F},\Delta)$ on $X$, let $\tilde{\mathcal{F}}$ be the pulled back
foliation on $\tilde{X}$ and $\tilde{\Delta}$ the strict transform of $\Delta$
on $\tilde{X}$. We then write:

$K_{\mathcal{\tilde{F}}}+\tilde{\Delta}=\pi^{*}(K_{\mathcal{F}}+\Delta)+\Sigma
a(E,\mathcal{F},\Delta)E$ (3)

where $\pi_{*}K_{\mathcal{\tilde{F}}}=K_{\mathcal{F}}$ and the sum runs over
all prime exceptional divisors on $\tilde{X}$. The rational numbers
$a(E,\mathcal{F},\Delta)$ are called discrepancies.

Given a normal variety X and a foliated pair $(\mathcal{F},\Delta)$ on X we
call $(\mathcal{F},\Delta)$ terminal (rsp. canonical, log-canonical) if
$a(E,\mathcal{F},\Delta)>0$ (rsp.$\geq 0,\geq-\epsilon(E)$) for every
exceptional prime divisors E.

For a not necessarily closed point $P\in X$ we say $(\mathcal{F},\Delta)$ is
terminal (rsp. canonical, log-canonical) at P if for all birational morphisms
$\pi:\tilde{X}\to X$ and any $\pi$-exceptional divisor E on $\tilde{X}$ whose
center lies in the Zariski closure $\bar{P}$ of P we have that the discrepancy
of E is $>0(\geq 0,rsp.\geq-\epsilon(E))$.

There are yet another two important concepts related to singularities that
will be needed in the following: For reader’s convenience we recall here the
definition of F-dlt-singularities and the notion of log-smoothness.

######  Definition 2.9.

Let X be a normal variety and $\mathcal{F}$ a corank 1 foliation on X. Let A
be a $\mathbb{Q}$-divisor such that $(K_{\mathcal{F}}+A)$ is
$\mathbb{Q}$-Cartier.

We call $(\mathcal{F},A)$ foliated divisorial log terminal (F-dlt) if

  1. 1.

Each irreducible component of A is generically transverse to the foliation
$\mathcal{F}$ and has coefficients at most one

  2. 2.

There is a foliated log resolution $\pi:Y\rightarrow X$ of $(\mathcal{F},A)$
which only extracts divisors E of log discrepancy
$a(E,\mathcal{F},A)>-\epsilon(E)$.

######  Definition 2.10.

Given a germ $p\in X$ with a foliation $\mathcal{F}$ such that $p$ is a
singular point of $\mathcal{F}$ we call a (formal) hypersurface germ $p\in S$
a (formal) separatrix if it is invariant under $\mathcal{F}$.

######  Definition 2.11.

Given $(X,\mathcal{F})$ as in the previous definition, we say that
$(\mathcal{F},A)$ is foliated log smooth if:

  1. 1.

$(X,A)$ is log smooth

  2. 2.

$\mathcal{F}$ has simple singularities

  3. 3.

If S denotes the support of non-$\mathcal{F}$-invariant components of A, $p\in
S$ is a closed point and $\Sigma_{1},...,\Sigma_{k}$ are
$\mathcal{F}$-invariant divisors passing through p, then
$\Sigma_{1}\cup...\cup\Sigma_{k}$ is a normal crossings divisor at p.

For the precise definition of simple singularities, the inaugurated reader may
again wish to consult [[8], Def. 2.13]. Next, we clarify the notions of log
flips and flops: We state them in classical terms, i.e in the way they arise
in the realm of the classical MMP. The respective generalizations to the
foliated situation are though straight forward ( simply replace $K_{X}$ by
$K_{\mathcal{F}}$). The definitions are verbatim taken from [[6], Def. 3.33
rsp 6.10]

######  Definition 2.12.

Let X be a normal scheme and D a $\mathbb{Q}$-divisor on X such that $K_{X}+D$
is $\mathbb{Q}$-Cartier.
A $(K_{X}+D)$-flipping contraction is a proper birational morphism
$f:X\rightarrow Y$ to a normal scheme Y such that $Exc(f)$ has codimension at
least two in X and -$(K_{X}+D)$ is f-ample. A normal scheme $X^{+}$ together
with a proper birational morphism $f^{+}:X^{+}\rightarrow Y$ is called a
$(K_{X}+D)$-flip of f if:

  1. 1.

$K_{X^{+}}+D^{+}$ is $\mathbb{Q}$-Cartier, where $D^{+}$ is the birational
transform of D on $X^{+}$

  2. 2.

$K_{X^{+}}+D^{+}$ is $f^{+}$-ample

  3. 3.

$Exc(f^{+})$ has codimension at least two in $X^{+}$

######  Remark.

By an abuse of notation we also call the map $\phi:X\dashrightarrow X^{+}$ a
$(K_{X}+D)$-flip or simply a D-flip. In a analogous way we can define a
$K_{\mathcal{F}}+D$-flip for a foliation $\mathcal{F}$ on $X$.

######  Definition 2.13.

Let X be a normal scheme with klt singularity. Let $\mathcal{F}$ be a F-dlt
foliation on $X$. A flopping contraction is a proper birational morphism
$f:X\rightarrow Y$ to a normal scheme Y such that $Exc(f)$ has codimension at
least two in X and $K_{\mathcal{F}}$ is numerically trivial. In the above
setup, i.e assuming $K_{\mathcal{F}}$ numerically trivial: if D is a
$\mathbb{Q}$-Cartier divisor on X such that $-(K_{\mathcal{F}}+D)$ is f-ample,
then the D-flip of f is also called the D-flop.

We recall that a divisor F on X is called numerically f-trivial for the
birational contraction morphism $f:X\rightarrow Y$ if for every curve $C$
contracted by $f$, we have $F\cdot C=0$ .

In the sequel of the proof of theorem 1.1 we will make use of the following
two results from [3]. We restate them here verbatim but refer to the original
publication for a proof of the statements:

######  Theorem 2.14 ([3],Theorem 9.4).

Let X be a normal projective 3-fold with klt singularities. Let $\mathcal{F}$
be a corank 1 foliation on X. Let $\Delta$ be a $\mathbb{Q}$-divisor such that
$(\mathcal{F},\Delta)$ is a F-dlt pair. Let $A\geq 0$ and $B\geq 0$ be
$\mathbb{Q}$-divisors such that $\Delta=A+B$ and A ample. Assume
$K_{\mathcal{F}}+\Delta$ is nef.

Then $K_{\mathcal{F}}+\Delta$ is semi-ample.

######  Theorem 2.15 ([3],Lemma 3.24).

Let X be a normal projective 3-fold, $\mathcal{F}$ a corank1 foliation on X .
Let $(\mathcal{F},\Delta)$ be an F-dlt pair such that $\lfloor\Delta\rfloor=0$
and let A be an ample $\mathbb{Q}$-divisor on X.

Then there is an effective $\mathbb{Q}$-divisor $A^{{}^{\prime}}\sim_{Q}A$
such that:

  1. 1.

$(\mathcal{F},\Delta)$ is also F-dlt

  2. 2.

$\lfloor\Delta+A^{\prime}\rfloor=0$

  3. 3.

The support of $A^{\prime}$ does not contain any log canonical center of
$(\mathcal{F},\Delta)$

##  3 Proof of the Theorem 1.1

Sticking to the notation in the statement of Theorem 1.1, we first show that
the birational map $\alpha$ relating the 2 foliated minimal models is a small
map, i.e the codimension of the exceptional locus is at least 2.

We start by setting up the notation: Let $(Y_{1},\mathcal{F}_{1})$ and
$(Y_{2},\mathcal{F}_{2})$ be 2 foliated minimal models of a common log-
canonical foliated threefold pair $(X,\mathcal{F})$, such that $X$ is
$\mathbb{Q}$-factorial with klt singularities. Denote by
$\alpha:(Y_{1},\mathcal{F}_{1})\dashrightarrow(Y_{2},\mathcal{F}_{2})$ the map
connecting the 2 minimal models and by
$\alpha_{i}:(X,\mathcal{F})\dashrightarrow(Y_{i},\mathcal{F}_{i})$ the
sequence of MMP-steps run through in order to obtain the respective minimal
models.

We recall that the steps of the foliated MMP are $K_{\mathcal{F}}$-negative
and hence so are the above defined $\alpha_{i}^{\prime}s$. Thus there is a
common log resolution $W$ of $X$, $Y_{1}$ and $Y_{2}$ such that for the
following commutative diagram we have:

${W}$${(X,\mathcal{F})}$${(Y_{2},\mathcal{F}_{1})}$${(Y_{1},\mathcal{F}_{2})}$$\scriptstyle{q_{2}}$$\scriptstyle{q_{1}}$$\scriptstyle{p}$$\scriptstyle{\alpha_{2}}$$\scriptstyle{\alpha_{1}}$$\scriptstyle{\alpha}$
$p^{*}(K_{\mathcal{F}})=q_{1}^{*}(K_{\mathcal{F}_{1}})+E_{1}=q_{2}^{*}(K_{\mathcal{F}_{2}})+E_{2}$
(4)

where $E_{i}\geq 0$, $q_{i}-exceptional$ and $p_{*}^{-1}Exc(\alpha_{i})\subset
suppE_{i}$

######  Lemma 3.1.

With the above notation, we have that $\alpha$ is small.

###### Proof.

Suppose $E_{1}\neq E_{2}$ then without loss of generality we assume that
$\overline{E_{1}}:=E_{1}-\mathrm{min}\\{E_{1},E_{2}\\}>0$,
$\overline{E_{2}}:=E_{2}-\mathrm{min}\\{E_{1},E_{2}\\}\geq 0$. According to
the negativity lemma we can find a curve $C\subseteq\mathrm{Supp}\
\overline{E_{1}}$ such that $C$ is not contained in $\mathrm{Supp}\
\overline{E_{2}}$ and $C\cdot\overline{E_{1}}<0$. Then we get

$q_{2}^{*}(K_{\mathcal{F}_{2}})=q_{1}^{*}(K_{\mathcal{F}_{1}})+E_{1}-E_{2}=q_{1}^{*}(K_{\mathcal{F}_{1}})+\overline{E_{1}}-\overline{E_{2}}$

and

$q_{2}^{*}(K_{\mathcal{F}_{2}})\cdot C=q_{1}^{*}(K_{\mathcal{F}_{1}})\cdot
C+(\overline{E_{1}}-\overline{E_{2}})\cdot C<0$

contradicting the nefness of $\mathcal{F}_{i}$.

Hence we conclude that $\overline{E_{1}}=\overline{E_{2}}$ which implies that
$q_{1}$ and $q_{2}$ contract the same divisors, thus $\alpha$ is small. ∎

The overall strategy of Kawamata in the analogous classical statement, i.e.
for minimal models of terminal varieties instead of foliated minimal models as
in the present paper, is to choose an ample divisor $L^{\prime}$ on minimal
model $Y_{2}$ and consider its strict transform $L$ on minimal model $Y_{1}$.
As klt-ness is an open property for varieties $(Y_{1},lL)$, for a small number
$l$, is still klt and Kawamata can thus assume that the divisor $K_{Y_{1}}+lL$
is not nef because otherwise $\mathbb{Q}$-factoriality of $Y_{2}$ and the
classical base point free theorem ([6], Theorem 3.3) imply that $\alpha$ is an
isomorphism. This allows Kawamata then to run a $(K_{Y_{1}}+lL)$-MMP on
$Y_{1}$ to finally reach his conclusions.

In the foliated case treated here, although there is as well an analogue of
the classical basepoint-free theorem for corank1-foliations on 3-folds (see
section 2), its invocation is not as straightforward as is in the original
proof. The main difficulties arise from the requirement in the foliated
version of the basepoint-free theorem that $(\mathcal{F}_{1},lL)$ must be
F-dlt. A priori it is not immediate that this condition can be satisfied.

The core of the present paper thus consists in the demonstration that a
careful choice of the ample divisor $L^{\prime}$ on $Y_{2}$ guarantees that
$(\mathcal{F}_{1},lL)$ becomes F-dlt.

Invoking (2.15), we know that we can find an ample $L^{\prime}$ on $Y_{2}$ and
a sufficiently small real number l such that $(\mathcal{F}_{2},lL^{\prime})$
is again F-dlt. The next few Lemmata in the present paper lay the ground for
the proof that $L^{\prime}$ can be chosen such that also its strict transform
on $Y_{1}$ satisfies $(\mathcal{F}_{1},lL)$ F-dlt.

######  Remark.

Notice that the Bertini-type theorem 2.15 does not directly apply to
$(\mathcal{F}_{1},lL)$ as $L$ is no longer ample.

###  3.1 Exceptional curves are foliated-trivial

In this section we are going to prove the following proposition:

######  Proposition 3.2.

For any curve $C$ lying in the exceptional locus of $\alpha$, we have
$K_{\mathcal{F}_{1}}\cdot C=0$.

We will prove this in two steps:

######  Lemma 3.3.

Suppose $K_{\mathcal{F}_{1}}$ is big and nef and $K_{\mathcal{F}_{1}}\cdot
C>0$, then there exists $0<\epsilon\ll 1$, such that

$C\nsubseteq\textup{{B}}_{+}(K_{\mathcal{F}_{1}}+\epsilon L)$

###### Proof.

Fix an ample divisor $A$ on $Y_{1}$. Since $K_{\mathcal{F}_{1}}$ is nef and
$K_{\mathcal{F}_{1}}\cdot C>0$, according to [[1], Theorem 1.3] we have

$C\nsubseteq\mathrm{\textbf{B}}_{+}(K_{\mathcal{F}_{1}})$

i.e. for any fixed $n\gg 0$, we have

$C\nsubseteq\mathrm{\textbf{B}}(K_{\mathcal{F}_{1}}-\frac{1}{n}A)$

Then we can find $0\leq H\sim_{\mathbb{Q}}K_{\mathcal{F}_{1}}-\frac{1}{n}A$
such that $C\nsubseteq H$. We thus have

$K_{\mathcal{F}_{1}}+\epsilon
L=K_{\mathcal{F}_{1}}-\frac{1}{n}A+\frac{1}{n}A+\epsilon
L\sim_{\mathbb{Q}}H+(\frac{1}{n}A+\epsilon L)$

Since $A$ is ample, for $0<\epsilon\ll 1$ we have $\frac{1}{n}A+\epsilon L$ is
also ample, and so is $\frac{1}{n}A+\epsilon L-\delta A$ for some $0<\delta\ll
1$ ,so we may find an effective $\mathbb{Q}$-divisor
$H_{0}\sim_{\mathbb{Q}}K_{\mathcal{F}_{1}}+\epsilon L-\delta A$ such that
$C\nsubseteq H_{0}$. ∎

######  Lemma 3.4.

There exists an ample divisor $L^{\prime}$ such that
$\mathrm{Exc}(\alpha)=\textup{{B}}_{+}(K_{\mathcal{F}_{1}}+\epsilon L)$ for
$L=\alpha_{*}^{-1}L^{\prime}$ and $\epsilon$ from the above proposition.

###### Proof.

First we choose an ample divisor $L^{\prime}$ on $Y_{2}$ with $0<\epsilon\ll
1,\epsilon\in\mathbb{Q}$ such that
$C\nsubseteq\textup{{B}}_{+}(K_{\mathcal{F}_{1}}+\epsilon L)$.

According to [[2], Theorem A], we have

$\mathrm{Exc}(\Phi_{m})=\textup{{B}}_{+}(K_{\mathcal{F}_{1}}+\epsilon L)$

where $\Phi_{m}:Y_{1}\dashrightarrow
H^{0}(Y_{1},m(K_{\mathcal{F}_{1}}+\epsilon L))$ is the map induced by the
complete linear system $|m(K_{\mathcal{F}_{1}}+\epsilon L)|$ on $Y_{1}$ with
$m\gg 0$ and sufficiently divisible.

However, since the $\mathbb{Q}$-divisor $K_{\mathcal{F}_{2}}+\epsilon
L^{\prime}$ is ample, we assume that $n(K_{\mathcal{F}_{2}}+\epsilon
L^{\prime})$ is very ample and since $\alpha$ is small, $\alpha$ is the
rational map induced by the complete linear system

$|n(K_{\mathcal{F}_{1}}+\epsilon
L)|=\alpha_{*}^{-1}|n(K_{\mathcal{F}_{2}}+\epsilon L^{\prime})|$

hence $\Phi_{mn}=p\circ\alpha$ where $p$ is the twisted embedding
$H^{0}(Y_{1},n(K_{\mathcal{F}_{1}}+\epsilon L))\hookrightarrow
H^{0}(Y_{1},mn(K_{\mathcal{F}_{1}}+\epsilon L))$. So
$\mathrm{Exc}(\alpha)=\mathrm{Exc}(\Phi_{mn})$. ∎

Now we combine Lemma 3.3 and Lemma 3.4 and one can easily see that Proposition
3.2 holds.

###  3.2 An exceptional curve is not an lcc for the foliation

######  Lemma 3.5.

Notation as above. There is no curve contained in

${Exc(\alpha):=\\{x\in Y_{1}|\text{$\alpha$ is not an isomorphism around}\
x\\}}$

which is a log canonical center for the foliation $\mathcal{F}_{1}$.

###### Proof.

Assume for sake of contradiction there was a curve
$C\subseteq\mathrm{Exc}(\alpha)$ whose generic point was a log canonical
foliation singularity. By [[8], Definition 2.24], passing to a foliated log
resolution $W$, there is a divisor $E$ such that
$a(E,\mathcal{F}_{1})=-\epsilon(E)$. Notice that the
$K_{\mathcal{F}}$-negativity of the map $\alpha_{1}$ gives rise to the
following equation:

$p^{*}K_{\mathcal{F}}=q_{1}^{*}K_{\mathcal{F}_{1}}+E_{1}$ (5)

such that $p_{*}^{-1}\mathrm{Exc}(\alpha_{1})\subset\mathrm{Supp}E_{1}$ and
$E_{1}\geq 0$.

First notice that as $C\subset\mathrm{Exc}(\alpha)$ we know that any
exceptional divisor on the log resolution $W$ is either $\alpha_{1}$ or
$\alpha_{2}$-exceptional.

We can distinguish two scenarios. Either $E$ descends to $X$ as a divisor, or
$center_{X}(E)$ is contained in the flipping locus of some flip arising in the
decomposition of $\alpha_{2}$ into flips and divisorial contractions.

In the first case, $E$ descends to a divisor $E_{X}$ on $X$ with the property
that $\alpha_{1}(E_{X})=C$ and which is of discrepancy
$a(E,F_{1},\Delta_{1})=-\epsilon(E_{X})$ as the restriction of $p$ to $E$ in
this case is an isomorphism and hence preserves discrepancies.

On the other hand, from equation (5) we infer that $p_{*}^{-1}(E_{X})=E\subset
E_{1}$. This is however impossible as $a(E,\mathcal{F}_{1})=-\epsilon(E)\leq
0$ whereas $E_{1}>0$ by construction.

Consequently, we may assume that $D:=center_{X}(E)$ is not divisorial. But
then $C$ is a flipped curve for the map $\alpha_{2}$ (as $V$ in this case is
an isomorphism by and hence cannot be a log canonical center by ([3], Lemma
2.7)). The preceding comment in brackets though suggests that $X$ and $X_{1}$
are locally isomorpic around the curve $C$, which implies together with the
local nature of flips that the flip of $D$ in the course of the MMP defined by
$\alpha_{2}$ could also be carried out on $X_{1}$ contradicting the fact that
$X_{1}$ is minimal. Hence the claim. ∎

###  3.3 The foliation is terminal along the curve

######  Lemma 3.6.

A curve $C$ in $\mathrm{Exc}(\alpha)$ satisfying $K_{\mathcal{F}}\cdot C=0$ is
tangent to the foliation $\mathcal{F}$.

###### Proof.

As $K_{\mathcal{F}}$ is big there exist an ample divisor $A$ and an effective
divisor E such that $K_{\mathcal{F}}\sim_{Q}A+E$. By assumption we furthermore
have $0=K_{\mathcal{F}}\cdot C=(A+E)\cdot C$, which implies that $E\cdot C<0$
due to ampleness of $A$. Let $L$ be a component of E such that $L\cdot C<0$,
i.e. $C$ is contained in $\mathrm{Supp}(L)$. By the lemma (3.5) we know that
$C$ is not an lcc for $\mathcal{F}$ and hence can find $\epsilon>0$ such that
$C$ defines an lcc for $\mathcal{F}+\epsilon L$. Notice that by construction
we have $(K_{\mathcal{F}}+\epsilon L)\cdot C<0$. We can now invoke [[8],Thm
4.5] to conclude that $C$ needs to be tangent as claimed. ∎

Lemma (3.5) proved that $C$ itself is not a log canonical center of the
foliation $\mathcal{F}$, we next want to show that in addition, there are no
0-dimensional log canonical centers located along $C$. Together with the above
demonstrated tangency of $C$ this will allow us to conclude that the foliation
$\mathcal{F}$ is terminal along $C$.

######  Lemma 3.7.

Let $C$ be a tangent curve contained in $\mathrm{Exc}(\alpha)$, then $C$ does
not contain any lc center of $\mathcal{F}$.

###### Proof.

As $C$ is tangent by assumption, we can invoke ([8], Thm 5.11) to deduce that
there is a germ of an analytic surface $S$ such that $C\subset S$ and S is
foliation invariant.

For sake of contradiction we next assume that there is a 0-dimensional log
canonical center located along $C$. By [[3],thm 3.8] we can deduce that
$\mathcal{F}$ is foliated log smooth around $p$.

Notice that in this situation all log canonical centers are precisely given by
the strata of Sing($\mathcal{F}$) around the simple singularity of the
foliation.

Furthermore, the local description of simple singularities ( for definition
see [[3], Def. 2.8] in order for $p$ to be a log canonical center, we must
have at least two one dimensional strata of Sing($\mathcal{F}$), corresponding
to the intersection lines of two formal coordinate hyperplane germs with the
surface $S$, intersecting $C$ in the point $p$. The scenario is depicted in
the picture (1) below where we labelled the 2 presumptive 1-dimensional log
canonical centers meeting $C$ by $\xi_{1}$ and $\xi_{2}$. We next argue that
the depicted picture cannot occur.

According to Lemma 3.3 we know that we can assume $K_{\mathcal{F}}\cdot C=0$.
Thus in the described scenario, the following calculation would hold true:

$0=K_{\mathcal{F}}\cdot C=(K_{S}+\sum(\xi_{i}))\cdot C$ (6)

where the sum runs over strata of Sing($\mathcal{F}$) intersecting in $C$ ( in
(1) we only drew 2 of them for the sake of clarity). In case that there are at
least two we conclude that $-K_{S}\cdot C\geq 2$. As the following reasoning
displays, this would however imply that $C$ deforms which is impossible as $C$
by assumption is contained in the one dimensional $\mathrm{Exc}(\alpha)$. In
order to get the contradiction, we need to assume that $C$ is rational, an
assumption which will be justified right after.

Here we regard $S$ as an analytic surface. And for simplicity we assume that
$S$ is smooth here. The case when $S$ is singular will be discussed later.

According to standard deformation theory, an estimate on the dimension of the
deformation space of a morphism $f$ from a curve $C$ to a surface $S$ is given
as follows:

$\dim_{[f]}\mathrm{Mor}(C,S)\geq\chi(C,f^{*}T_{S})=-K_{S}.f^{*}C+2(1-g(C))$
(7)

where the equality follows from Hirzebruch-Riemann-Roch [4].

Assuming thus for the moment that $g(C)=0$, our previous calculations show
that

$\mathrm{dim}_{[f]}\mathrm{Mor}(C,S)\geq 4$

Since $\dim\mathrm{Aut}(C)\leq 3$, $C$ would deform, a contradiction.

Now we turn back to the general case when $S$ is singular. In this case we
take a minimal resolution $\varphi:S^{\prime}\to S$. Since $S$ is not smooth
hence not terminal we have

$K_{S^{\prime}}+E=\varphi^{*}K_{S}$

for some effective exceptional divisor $E$ on $S^{\prime}$. According to the
simple singularity condition we have that $S$ is smooth at the generic point
of $C$. So if we take $C^{\prime}=\varphi_{*}^{-1}C$ we have

$K_{S^{\prime}}\cdot C^{\prime}\leq K_{S}\cdot C\leq-2$

Then the argument (7) still works.

We are then left to justify the assumption that $C$ is of arithmetic genus 0.

In order to verify this claim it suffices to show that its normalization is a
smooth rational curve. Using equation (6) and taking advantage of the
projection formula as well as generalized adjunction results (see [[7],
chapter 4]) we obtain the following sequence of equalities

$\deg(\nu)(K_{\mathcal{F}}-\sum(\xi_{i}))|_{C}+\tilde{C}^{2}=K_{S}.\cdot\tilde{C}+\tilde{C}^{2}\sim_{\mathcal{Q}}K_{\tilde{C}}+\mathrm{Diff}_{\tilde{C}}$
(8)

where $\nu:\tilde{C}\mapsto C$ denotes the normalization and
$\mathrm{Diff}_{\tilde{C}}$ the different of the adjunction.

Now notice that the normal bundle of $\tilde{C}$, given by $\tilde{C}^{2}$ has
non-positive degree as $C$ does not deform. Thus, the whole left hand side of
the $\mathbb{Q}$-linear equivalence above is of non-positive degree and in
turn so is the right hand side. Noticing that $\mathrm{Diff}_{\tilde{C}}$ is
effective, we conclude that $K_{\tilde{C}}$ has non-positive degree which
implies that $\tilde{C}$ and hence $C$ are genus 0 curves.

Summing up all the gained insight about the local situation around the
presumptive log canonical centre $p$ located on $C$, the only conceivable
picture could be the one depicted in (2). However, as the log canonical locus
of the foliation is given precisely by strata of Sing($\mathcal{F}$) and
codimensionality of a simple singularity is given by the number of maximal
dimensional strata it is contained in, $p$ being 0-dimensional cannot be a log
canonical center of $\mathcal{F}$.

$S$$C$$\xi_{1}$$\xi_{2}$ Figure 1: No more than 2 log canonical centers
$\xi_{1}$ and $\xi_{2}$ meeting $C$ in a common point $p$ $S$$C$$\xi$ Figure
2: 1 lcc meeting $C$ in a common point $p$

∎

###### proof of Theorem 1.1.

Altogether, we have demonstrated that $\mathcal{F}$ is terminal along any
curve $C\subset\mathrm{Exc}(\alpha)$. We claim that this now allows us to find
$l>0$ and an ample divisor $L^{\prime}$ on $Y_{2}$ satisfying the conditions
in theorem (2.15) such that the strict transform
$L:=\alpha_{*}^{-1}L^{\prime}$ fulfills the desired condition that
$(Y_{1},\mathcal{F}_{1},lL)$ is a F-dlt pair.

Herefore notice, that the proof of (2.15) in [[3], Lemma 3.24] essentially
boils down to the demonstration of bullet points $(i)-(iv)$ in the notation of
the original publication. This in turn implies that the theorem can be applied
to L directly in case one is able to establish these 4 conditions. We now
shortly comment on them individually justifying our claim :

Notice first that outside the exceptional locus $\mathrm{Exc}(\alpha)$, $L$ is
isomorphic to $L^{\prime}$ and the conditions are hence satisfied
automatically. Furthermore, there are no log canonical centers located on
$\mathrm{Exc}(\alpha)$, so (i) is established. Point (ii) is automatically
satisfied as $\dim\mathrm{Exc}(\alpha)=1$. For (iii), again, there are no log
canonical centers located on $\mathrm{Exc}(\alpha)$. Lastly, ad (iv) notice
that as $\mathcal{F}$ is terminal along $C\subset\mathrm{Exc}(\alpha)$ as we
have demonstrated, we can find a real number l small enough such that any
exceptional divisor E with center contained in $\mathrm{Exc}(\alpha)$ will
satisfy $a(E,\mathcal{F},lL)\geq\epsilon(E)$. We thus conclude by theorem
(2.14) that the F-dlt pair $(\mathcal{F},lL)$ would be semiample in case it
was nef. By the same logic as in the original publication by Kawamata this
would however force the small map $\alpha$ to be an isomorphism and hence this
case can be disregarded.

We can thus assume in the sequel that $K_{\mathcal{F}_{1}}+lL$ as constructed
above is not nef and run a $(K_{\mathcal{F}_{1}}+lL)$-MMP in complete analogy
to Kawamata. Where it is to be stressed that by results of the authors of [3],
the F-dlt property is preserved under a $K_{\mathcal{F}}$-MMP. It is also
worth mentioning that this $(K_{\mathcal{F}_{1}}+lL)$-MMP which by
construction consists of a sequence of flips is actually
$K_{\mathcal{F}_{1}}$-trivial as has been demonstrated above and need not, as
opposed to the classical case in Kawamata’s publication, be deduced from a
clever application of the cone theorem.

Last, according to Proposition 3.2, every flip
$\beta_{i}:X_{i-1}\dashrightarrow X_{i}$ is a $K_{\mathcal{G}_{i-1}}$-flop
since every curve we contract is $K_{\mathcal{G}_{i-1}}$-trivial. This
concludes the proof of theorem 1.1. ∎

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