# Author Commitment and Social Power:
Automatic Belief Tagging to Infer the Social Context of Interactions

Vinodkumar Prabhakaran
Stanford University
<EMAIL_ADDRESS>
&Premkumar Ganeshkumar
Agolo, Inc.
<EMAIL_ADDRESS>&Owen Rambow
Elemental Cognition, Inc.
<EMAIL_ADDRESS>

###### Abstract

Understanding how social power structures affect the way we interact with one
another is of great interest to social scientists who want to answer
fundamental questions about human behavior, as well as to computer scientists
who want to build automatic methods to infer the social contexts of
interactions. In this paper, we employ advancements in extra-propositional
semantics extraction within NLP to study how author commitment reflects the
social context of an interactions. Specifically, we investigate whether the
level of commitment expressed by individuals in an organizational interaction
reflects the hierarchical power structures they are part of. We find that
subordinates use significantly more instances of non-commitment than
superiors. More importantly, we also find that subordinates attribute
propositions to other agents more often than superiors do — an aspect that has
not been studied before. Finally, we show that enriching lexical features with
commitment labels captures important distinctions in social meanings.

##  1 Introduction

Social power is a difficult concept to define, but is often manifested in how
we interact with one another. Understanding these manifestations is important
not only to answer fundamental questions in social sciences about power and
social interactions, but also to build computational models that can
automatically infer social power structures from interactions. The
availability and access to large digital repositories of naturally occurring
social interactions and the advancements in natural language processing
techniques in recent years have enabled researchers to perform large scale
studies on linguistic correlates of power, such as words and phrases Bramsen
et al. (2011); Gilbert (2012), linguistic coordination Danescu-Niculescu-Mizil
et al. (2012), agenda control Taylor et al. (2012), and dialog structure
Prabhakaran and Rambow (2014).

Another area of research that has recently garnered interest within the NLP
community is the modeling of author commitment in text. Initial studies in
this area were done in processing hedges, uncertainty and lack of commitment,
specifically focused on scientific text Mercer et al. (2004); Di Marco et al.
(2006); Farkas et al. (2010). More recently, researchers have also looked into
capturing author commitment in non-scientific text, e.g., levels of factuality
in newswire Saurí and Pustejovsky (2009), types of commitment of beliefs in a
variety of genres including conversational text Diab et al. (2009);
Prabhakaran et al. (2015). These approaches are motivated from an information
extraction perspective, for instance in aiding tasks such as knowledge base
population.111The BeSt track of the 2017 TAC-KBP evaluation aimed at detecting
the “belief and sentiment of an entity toward another entity, relation, or
event” (http://www.cs.columbia.edu/~rambow/best-eval-2017/). However, it has
not been studied whether such sophisticated author commitment analysis can go
beyond what is expressed in language and reveal the underlying social contexts
in which language is exchanged.

In this paper, we bring together these two lines of research; we study how
power relations correlate with the levels of commitment authors express in
interactions. We use the power analysis framework built by Prabhakaran and
Rambow (2014) to perform this study, and measure author commitment using the
committed belief tagging framework introduced by Diab et al. (2009) that
distinguishes different types of beliefs expressed in text. Our contributions
are two-fold — statistical analysis of author commitment in relation with
power, and enrichment of lexical features with commitment labels to aid in
computational prediction of power relations. In the first part, we find that
author commitment is significantly correlated with the social power relations
between their participants — subordinates use more instances of non-
commitment, a finding that is in line with sociolinguistics studies in this
area. We also find that subordinates use significantly more reported beliefs
(i.e., attributing beliefs to other agents) than superiors. This is a new
finding; to our knowledge, there has not been any sociolinguistics studies
investigating this aspect of interaction in relation with power. In the second
part, we present novel ways of incorporating the author commitment information
into lexical features that can capture important distinctions in word meanings
conveyed through the belief contexts in which they occur; distinctions that
are lost in a model that conflates all occurrences of a word into one unit.

We first describe the related work in computational power analysis and
computational modeling of cognitive states in Section 2. In Section 3, we
describe the power analysis framework we use. Section 4 formally defines the
research questions we are investigating, and describes how we obtain the
belief information. In Section 5, we present the statistical analysis of
author commitment and power. Section 6 presents the utility of enriching
lexical features with belief labels in the context of automatic power
prediction. Section 7 concludes the paper and summarizes the results.

##  2 Related Work

The notion of belief that we use in this paper Diab et al. (2009); Prabhakaran
et al. (2015) is closely related to the notion of factuality that is captured
in FactBank Saurí and Pustejovsky (2009). They capture three levels of
factuality, certain (CT), probable (PB), and possible (PS), as well as the
underspecified factuality (Uu). They also record the corresponding polarity
values, and the source of the factuality assertions to distinguish between
factuality assertions by the author and those by the agents/sources introduced
by the author. While FactBank offers a finer granularity, they are annotated
on newswire text. Hence, we use the corpus of belief annotations Prabhakaran
et al. (2015) that is obtained on online discussion forums, which is closer to
our genre.

Automatic hedge/uncertainty detection is a very closely related task to belief
detection. The belief tagging framework we use aims to capture the cognitive
states of authors, whereas hedges are linguistic expressions that convey one
of those cognitive states — non-committed beliefs. Automatic hedge/uncertainty
detection has generated active research in recent years within the NLP
community. Early work in this area focused on detecting speculative language
in scientific text Mercer et al. (2004); Di Marco et al. (2006); Kilicoglu and
Bergler (2008). The open evaluation as part of the CoNLL shared task in 2010
to detect uncertainty and hedging in biomedical and Wikipedia text Farkas et
al. (2010) triggered further research on this problem in the general domain
Agarwal and Yu (2010); Morante et al. (2010); Velldal et al. (2012); Choi et
al. (2012). Most of this work was aimed at formal scientific text in English.
More recent work has tried to extend this work to other genres Wei et al.
(2013); Sanchez and Vogel (2015) and languages Velupillai (2012); Vincze
(2014), as well as building general purpose hedge lexicons Prokofieva and
Hirschberg (2014). In our work, we use the lexicons from Prokofieva and
Hirschberg (2014) to capture hedges in text.

Sociolinguists have long studied the association between level of commitment
and social contexts Lakoff (1973); O’Barr and Atkins (1980); Hyland (1998). A
majority of this work studies gender differences in the use of hedges,
triggered by the influential work by Robin Lakoff Lakoff (1973). She argued
that women use linguistic strategies such as hedging and hesitations in order
to adopt an unassertive communication style, which she terms “women’s
language”. While many studies have found evidence to support Lakoff’s theory
(e.g., Crosby and Nyquist (1977); Preisler (1986); Carli (1990)), there have
also been contradictory findings (e.g., O’Barr and Atkins (1980)) that link
the difference in the use of hedges to other social factors (e.g., power).
O’Barr and Atkins (1980) argue that the use of hedges is linked more to the
social positions rather than gender, suggesting to rename “women’s language”
to “powerless language”. In later work, O’Barr (1982) formalized the notion of
powerless language, which formed the basis of many sociolinguistics studies on
social power and communication. O’Barr (1982) analyzed courtroom interactions
and identified hedges and hesitations as some of the linguistic markers of
“powerless” speech. However, there has not been any computational work which
has looked into how power relations relate to the level of commitment
expressed in text. In this paper, we use computational power analysis to
perform a large scale data-oriented study on how author commitment in text
reveals the underlying power relations.

There is a large body of literature in the social sciences that studies power
as a social construct (e.g., French and Raven (1959); Dahl (1957); Emerson
(1962); Pfeffer (1981); Wartenberg (1990)) and how it relates to the ways
people use language in social situations (e.g., Bales et al. (1951); Bales
(1970); O’Barr (1982); Van Dijk (1989); Bourdieu and Thompson (1991); Ng and
Bradac (1993); Fairclough (2001); Locher (2004)). Recent years have seen
growing interest in computationally analyzing and detecting power and
influence from interactions. Early work in computational power analysis used
social network analysis based approaches Diesner and Carley (2005); Shetty and
Adibi (2005); Creamer et al. (2009) or email traffic patterns Namata et al.
(2007). Using NLP to deduce social relations from online communication is a
relatively new area of active research.

Bramsen et al. (2011) and Gilbert (2012) first applied NLP based techniques to
predict power relations in Enron emails, approaching this task as a text
classification problem using bag of words or ngram features. More recently,
our work has used dialog structure features derived from deeper dialog act
analysis for the task of power prediction in Enron emails Prabhakaran and
Rambow (2014); Prabhakaran et al. (2012); Prabhakaran and Rambow (2013). In
this paper, We use the framework of Prabhakaran and Rambow (2014), but we
analyze a novel aspect of interaction that has not been studied before — what
level of commitment do the authors express in language.

There has also been work on analyzing power in other genres of interactions.
Strzalkowski et al. (2010) and Taylor et al. (2012) concentrate on lower-level
constructs called Language Uses such as agenda control to predict power in
Wikipedia talk pages. Danescu-Niculescu-Mizil et al. (2012) study how social
power and linguistic coordination are correlated in Wikipedia interactions as
well as Supreme Court hearings. Bracewell et al. (2012) and Swayamdipta and
Rambow (2012) try to identify pursuit of power in discussion forums. Biran et
al. (2012) and Rosenthal (2014) study the problem of predicting influence in
Wikipedia talk pages, blogs, and other online forums. Prabhakaran et al.
(2013) study manifestations of power of confidence in presidential debates.

##  3 Power in Workplace Email: Data and Analysis Framework

The focus of our study is to investigate whether the level of commitment
participants express in their contributions in an interaction is related to
the power relations they have with other participants, and how it can help in
the problem of predicting social power. In this section, we introduce the
power analysis framework as well as the data we use in this study.

###  3.1 Problem

In order to model manifestations of power relations in interactions, we use
our interaction analysis framework from Prabhakaran and Rambow (2014), where
we introduced the problem of predicting organizational power relations between
pairs of participants based on single email threads. The problem is formally
defined as follows: given an email thread $\mathit{t}$, and a related
interacting participant pair $\mathit{(p_{1},p_{2})}$ in the thread, predict
whether $\mathit{p_{1}}$ is the superior or subordinate of $\mathit{p_{2}}$.
In this formulation, a related interacting participant pair (RIPP) is a pair
of participants of the thread such that there is at least one message
exchanged within the thread between them (in either direction) and that they
are hierarchically related with a superior/subordinate relation.

###  3.2 Data

We use the same dataset we used in Prabhakaran and Rambow (2014), which is a
version of the Enron email corpus in which the thread structure of email
messages is reconstructed Yeh and Harnly (2006), and enriched by Agarwal et
al. (2012) with gold organizational power relations, manually determined using
information from Enron organizational charts. The corpus captures dominance
relations between 13,724 pairs of Enron employees. As in Prabhakaran and
Rambow (2014), we use these dominance relation tuples to obtain gold labels
for the superior or subordinate relationships between pairs of participants.
We use the same train-test-dev split as in Prabhakaran and Rambow (2014). We
summarize the number of threads and related interacting participant pairs in
each subset of the data in Table 1.

Description | Train | Dev | Test
---|---|---|---
Email threads | 18079 | 8973 | 9144
# of RIPPs | 7510 | 3578 | 3920
Table 1:  Data Statistics. Row 1: number of threads in subsets of the corpus.
Row 2: number of related interacting participant pairs in those subsets. RIPP:
Related interacting participant pairs

##  4 Research Hypotheses

Our first objective in this paper is to perform a large scale computational
analysis of author commitment and power relations. Specifically, we want to
investigate whether the commitment authors express towards their contributions
in organizational interactions is correlated with the power relations they
have with other participants. Sociolinguistics studies have found some
evidence to suggest that lack of commitment expressed through hedges and
hesitations is associated with lower power status O’Barr (1982). However, in
our study, we go beyond hedge word lists, and analyze different cognitive
belief states expressed by authors using a belief tagging framework that takes
into account the syntactic contexts within which propositions are expressed.

###  4.1 Obtaining Belief Labels

We use the committed belief analysis framework introduced by Diab et al.
(2009); Prabhakaran et al. (2015) to model different levels of beliefs
expressed in text. Specifically, in this paper, we use the 4-way belief
distinction — CommittedBelief, NonCommittedBelief, ReportedBelief, and
NonApplicable— introduced in Prabhakaran et al. (2015).222We also performed
analysis and experiments using an earlier 3-way belief distinction proposed by
Diab et al. (2009), which also yielded similar findings. We do not report the
details of those analyses in this paper. Prabhakaran et al. (2015) presented a
corpus of online discussion forums with over 850K words, annotating each
propositional head in text with one of the four belief labels. The paper also
presented an automatic belief tagger trained on this data, which we use to
obtain belief labels in our data. We describe each belief label and our
associated hypotheses below.

#### Committed belief (CB):

the writer strongly believes that the proposition is true, and wants the
reader/hearer to believe that. E.g.:

  * a.

John will submit the report.

  * b.

I know that John is capable.

As discussed earlier, lack of commitment in one’s writing/speech is identified
as markers of powerless language. We thus hypothesize:

######  H. 1.

Superiors use more instances of committed belief in their messages than
subordinates.

#### Non-committed belief (NCB):

the writer explicitly identifies the proposition as something which he or she
could believe, but he or she happens not to have a strong belief in, for
example by using an epistemic modal auxiliary. E.g.:

  * a.

John may submit the report.

  * b.

I guess John is capable.

This class captures a more semantic notion of non-commitment than hedges,
since the belief annotation attempts to model the underlying meaning rather
than language uses, and hence captures other linguistic means of expressing
non-committedness. Following O’Barr (1982), we formulate the below hypothesis:

######  H. 2.

Subordinates use more instances of non committed belief in their messages than
superiors.

#### Reported belief (ROB):

the writer attributes belief (either committed or non-committed) to another
person or group. E.g.:

  * a.

Sara says John will submit the report.

  * b.

Sara thinks John may be capable.

Note that this label is only applied when the writer’s own belief in the
proposition is unclear. For instance, if the first example above was Sara
knows John will submit the report on-time, the writer is expressing commitment
toward the proposition that John will submit the report and it will be labeled
as committed belief rather than reported belief. Reported belief captures
instances where the writer is in effect limiting his/her commitment towards
what is stated by attributing the belief to someone else. So, in line with our
hypotheses for non-committed beliefs, we formulate the following hypothesis:

######  H. 3.

Subordinates use more instances of reported beliefs in their messages than
superiors.

#### Non-belief propositions (NA):

– the writer expresses some other cognitive attitude toward the proposition,
such as desire or intention (4.1a), or expressly states that he/she has no
belief about the proposition (e.g., asking a question (4.1b)). E.g.:

  * a.

I need John to submit the report.

  * b.

Will John be capable?

As per the above definition, requests for information (i.e., questions) and
requests for actions are cases where the author is not expressing a belief
about the proposition, but rather expressing the desire that some action be
done. In the study correlating power with dialog act tags Prabhakaran and
Rambow (2014), we found that superiors issue significantly more requests than
subordinates. Hence, we expect the superiors to have significantly more non
belief expressions in their messages, and formulate the following hypothesis:

######  H. 4.

Superiors use more instances of non beliefs in their messages than
subordinates.

###  4.2 Testing Belief Tagger Bias

NLP tools are imperfect and may produce errors, which poses a problem when
using any NLP tool for sociolinguistic analysis. More than the magnitude of
error, we believe that whether the error is correlated with the social
variable of interest (i.e., power) is more important; e.g., is the belief-
tagger more likely to find ROB false-positives in subordinates’ text? To test
whether this is the case, we performed manual belief annotation on around 500
propositional heads in our corpus. Logistic regression test revealed that the
belief-tagger is equally likely to make errors (both false-positives and
false-negatives, for all four belief-labels) in sentences written by
subordinates as superiors (the null hypothesis accepted at $p>0.05$ for all
eight tests).

##  5 Statistical Analysis

Now that we have set up the analysis framework and research hypotheses, we
present the statistical analysis of how superiors and subordinates differ in
their relative use of expressions of commitment.

###  5.1 Features

For each participant of each pair of related interacting participants in our
corpus, we aggregate each of the four belief tags:

  * •

CBCount: number of propositional heads tagged as Committed Belief (CB)

  * •

NCBCount: number of propositional heads tagged as Non Committed Belief (NCB)

  * •

ROBCount: number of propositional heads tagged as Reported Belief (ROB)

  * •

NACount: number of propositional heads tagged as Non Belief (NA)

###  5.2 Hypotheses Testing

Our general hypothesis is that power relations do correlate with the level of
commitment people express in their messages; i.e., at least one of H.1 \- H.4
is true. In this analysis, each participant of the pair
$\mathit{(p_{1},p_{2})}$ is a data instance. We exclude the instances for
which a feature value is undefined.333These are instances corresponding to
participants who did not send any messages in the thread (some of the pairs in
the set of related interacting participant pairs only had one-way
communication) or whose messages were empty (e.g., forwarding messages).

In order to test whether superiors and subordinates use different types of
beliefs, we used a linear regression based analysis. For each feature, we
built a linear regression model predicting the feature value using power
(i.e., superior vs. subordinate) as the independent variable. Since verbosity
of a participant can be highly correlated with each of these feature values
(we found it to be highly correlated with subordinates Prabhakaran and Rambow
(2014)), we added token count as a control variable to the linear regression.

Our linear regression test revealed significant differences in NCB (b=-.095,
t(-8.09), p$<$.001), ROB (b=-.083, t(-7.162), p$<$.001) and NA (b=.125,
t(4.351), p$<$.001), and no significant difference in CB (b=.007, t(0.227),
p=0.821). Figure 1 pictorially demonstrates these results by plotting the
difference between the mean values of each commitment feature (here normalized
by token count) of superiors vs. subordinates, as a percentage of mean feature
value of the corresponding commitment feature for superiors. Dark bars denote
statistically significant differences.

Figure 1: Relative difference (RD) between subordinates and superiors in their
use of different types of commitment (counts normalized by word count of
contributions). Dark bars: statistical significance at $p<0.05$. (RD =
$\frac{(Mean(\textit{Subordinates})-Mean(\textit{Superiors}))*100}{Mean(\textit{Superiors})}$).

###  5.3 Interpretation of Findings

The results from our statistical analysis validate our original hypothesis
that power relations do correlate with the level of commitment people express
in their messages. This finding remains statistically significant ($p<0.001$)
even after applying the Bonferroni correction for multiple testing.

The results on NCB confirm our hypothesis that subordinates use more non-
committedness in their language. Subordinates’ messages contain 48% more
instances of non-committed belief than superiors’ messages, even after
normalizing for the length of messages. This is in line with prior
sociolinguistics literature suggesting that people with less power tend to use
less commitment, previously measured in terms of hedges. However, in our work,
we go beyond hedge dictionaries and use expressions of non-committedness that
takes into account the syntactic configurations in which the words appear.

Another important finding is in terms of reported belief (ROB). Our results
strongly verify the hypothesis H.3 that subordinates use significantly more
reported beliefs than superiors. In fact, it obtained the largest magnitude of
relative difference (65.3% more) of all features we analyzed. To our
knowledge, ours is the first study that analyzed the manifestation of power in
authors attributing beliefs to others. Our results are in line with the
finding in Agarwal et al. (2014) that “if many more people get mentioned to a
person then that person is the boss”, because as subordinates report other
people’s beliefs to superiors, they are also likely to mention them.

The finding that superiors use more NAs confirms our hypothesis H.4. As
discussed earlier, this is expected since superiors issue more requests (as
found by Prabhakaran and Rambow (2014)), the propositional heads of which
would be tagged as NA by the belief tagger. However, our hypothesis H.1 is
proven false. Being a superior or subordinate does not affect how often their
messages contain CB, which suggests that power differences are manifested only
in terms of lack of commitment.

##  6 Commitment in Power Prediction

Our next step is to explore whether we can utilize the hedge and belief labels
to improve the performance of an automatic power prediction system. For this
purpose, we use our PowerPredictor system Prabhakaran and Rambow (2014) that
predicts the direction of power between a pair of related interacting
participants in an email thread. It uses a variety of linguistic and dialog
structural features consisting of verbosity features (message count, message
ratio, token count, token ratio, and tokens per message), positional features
(initiator, first message position, last message position), thread structure
features (number of all recipients and those in the To and CC fields of the
email, reply rate, binary features denoting the adding and removing of other
participants), dialog act features (request for action, request for
information, providing information, and conventional), and overt displays of
power, and lexical features (lemma ngrams, part-of-speech ngrams, and mixed
ngrams, a version of lemma ngrams with open class words replaced with their
part-of-speech tags). The feature sets are summarized in Table 2 (Prabhakaran
and Rambow (2014) has a detailed description of these features).

Set | Description
---|---
VRB | Verbosity (e.g., message count)
PST | Positional (e.g., thread initiator?)
THR | Thread structure (e.g., reply rate)
DIA | Dialog act tagging (e.g., request count)
ODP | Overt displays of power
LEX | Lexical ngrams (lemma, POS, mixed ngrams)
Table 2: PowerPredictor system: Features used

None of the features used in PowerPredictor use information from the parse
trees of sentences in the text However, in order to accurately obtain the
belief labels, deep dependency parse based features are critical Prabhakaran
et al. (2010). We use the ClearTk wrapper for the Stanford CoreNLP pipeline to
obtain the dependency parses of sentences in the email text. To ensure an
unified analysis framework, we also use the Stanford CoreNLP for tokenization,
part-of-speech tagging, and lemmatization steps, instead of OpenNLP. This
change affects our analysis in two ways. First, the source of part-of-speech
tags and word lemmas is different from what was presented in the original
system, which might affect the performance of the dialog act tagger and overt
display of power tagger (DIA and ODP features). Second, we had to exclude 117
threads (0.3%) from the corpus for which the Stanford CoreNLP failed to parse
some sentences, resulting in the removal of 11 data points (0.2%), only one of
which was in the test set. On randomly checking, we found that they contained
non-parsable text such as dumps of large tables, system logs, or unedited
dumps of large legal documents.

In order to better interpret how the commitment features help in power
prediction, we use a linear kernel SVM in our experiments. Linear kernel SVMs
are significantly faster than higher order SVMs, and our preliminary
experiments revealed the performance gain by using a higher order SVM to be
only marginal. We use the best performing feature set from Prabhakaran and
Rambow (2014) as a strong baseline for our experiments. This baseline feature
set is the combination of thread structure features (THR) and lexical features
(LEX). This baseline system obtained an accuracy of 68.8% in the development
set.

###  6.1 Belief Label Enriched Lexical Features

Adding the belief label counts into the SVM directly as features will not
yield much performance improvements, as signal in the aggregate counts would
be minimal given the effect sizes of differences we find in Section 5. In this
section, we investigate a more sophisticated way of incorporating the belief
tags into the power prediction framework. Lexical features are very useful for
the task of power prediction. However, it is often hard to capture deeper
syntactic/semantic contexts of words and phrases using ngram features. We
hypothesize that incorporating belief tags into the ngrams will enrich the
representation and will help disambiguate different usages of same
words/phrases. For example, let us consider two sentences: I need the report
by tomorrow vs. If I need the report, I will let you know. The former is
likely coming from a person who has power, whereas the latter does not give
any such indication. Applying the belief tagger to these two sentences will
result in I need(CB) the report … and If I need(NA) the report …. Capturing
the difference between need(CB) vs. need(NA) will help the machine learning
system to make the distinction between these two usages and in turn improve
the power prediction performance.

In building the ngram features, whenever we encounter a token that is assigned
a belief tag, we append the belief tag to the corresponding lemma or part-of-
speech tag in the ngram. We call it the Append version of corresponding ngram
feature. We summarize the different versions of each type of ngram features
below:

  * •

LN: the original word lemma ngram; e.g., i_need_the.

  * •

LNCBApnd: word lemma ngram with appended belief tags; e.g., i_need(CB)_the.

  * •

PN: the original part-of-speech ngram; e.g., PRP_VB_DT.

  * •

PNCBApnd: part-of-speech ngram with appended belief tags; e.g., PRP_VB(CB)_DT.

  * •

MN: the original mixed ngram; e.g., i_VB_the.

  * •

MNCBApnd: mixed ngram with appended belief tags; e.g., i_VB(CB)_the.

Feature Configuration in Lexical | Accuracy
---|---
LN +PN +MN (BaseLine) | 68.8
LNCBApnd+PN +MN | 69.3
LN +PNCBApnd+MN | 68.6
LN +PN +MNCBApnd | 69.0
LNCBApnd\+ PN \+ MNCBApnd | 69.2
Table 3: Power prediction results using different configurations of LEX
features. (The full feature set also includes THR.)

In Table 3, we show the results obtained by incorporating the belief tags in
this manner to the Lexical features of the original baseline feature set. The
first row indicates the baseline results and the following rows show the
impact of incorporating belief tags using the Append method. While the Append
version of both lemma ngrams and mixed ngrams improved the results, the Append
version of part of speech ngrams reduced the results. The combination of best
performing version of each type of ngram obtained slightly lower result than
using the Append version of word ngram alone, which posted the overall best
performance of 69.3%, a significant improvement (p$<$0.05) over not using any
belief information. We use the approximate randomization test Yeh (2000) for
testing statistical significance of the improvement.

Finally, we verified that our best performing feature sets obtain similar
improvements in the unseen test set. The baseline system obtained 70.2%
accuracy in the test set. The best performing configuration from Table 3
significantly improved this accuracy to 70.8%. The second best performing
configuration of using the Append version of both word and mixed ngrams
obtained only a small improvement upon the baseline in the test set.

Figure 2: Feature weights of different belief appended versions of 25
propositional heads whose lemma unigrams had the highest standard deviation.
Y-axis denotes the propositional heads in decreasing order of standard
deviation from bottom to top. X-axis denotes the feature weights.

###  6.2 Word NGram Feature Analysis

We inspect the feature weights assigned to the LNCBApnd version of lemma
ngrams in our best performing model. Each lemma ngram that contains a
propositional head (e.g., need) has four possible LNCBApnd ngram versions:
need(CB), need(NCB), need(ROB), and need(NA). For each lemma ngram, we
calculate the standard deviation of weights assigned to different LNCBApnd
versions in the learned model as a measure of variation captured by
incorporating belief tags into that ngram.444Not all lemma ngrams have all
four versions; we calculated standard deviation using the versions present.

Figure 2 shows the feature weights of different LNCBApnd versions of twenty
five propositional heads whose lemma unigrams had the highest standard
deviation. The y-axis lists propositional heads arranged in the decreasing
order of standard deviation from bottom to top, while the x-axis denotes the
feature weights. The markers distinguish the different LNCBApnd versions of
each propositional head — square denotes CommittedBelief, circle denotes
NonCommittedBelief, triangle denotes ReportedBelief, and diamond denotes
NonApplicable. The feature versions with negative weights are associated more
with subordinates’ messages, whereas those with positive weights are
associated more with superiors’ messages. Since NCB and ROB versions are rare,
they rarely get high weights in the model.

We find that by incorporating belief labels into lexical features, we capture
important distinctions in social meanings expressed through words that are
lost in the regular lemma ngram formulation. For example, propositional heads
such as know, need, hold, mean and want are indicators of power when they
occur in CB contexts (e.g., i need …), whereas their usages in NA contexts
(e.g., do you need?, if i need…, etc.) are indicators of lack of power. In
contrast, the CB version of attend, let, plan, could, check, discuss, and feel
(e.g., i will attend/check/plan …) are strongly associated with lack of power,
while their NA versions (e.g., can you attend/check/plan?) are indicators of
power.

##  7 Conclusion

In this paper, we made two major contributions. First, we presented a large-
scale data oriented analysis of how social power relations between
participants of an interaction correlate with different types of author
commitment in terms of their relative usage of hedges and different levels of
beliefs — committed belief, non-committed belief, reported belief, and non-
belief. We found evidence that subordinates use significantly more
propositional hedges than superiors, and that superiors and subordinates use
significantly different proportions of different types of beliefs in their
messages. In particular, subordinates use significantly more non-committed
beliefs than superiors. They also report others’ beliefs more often than
superiors. Second, we investigated different ways of incorporating the belief
tag information into the machine learning system that automatically detects
the direction of power between pairs of participants in an interaction. We
devised a sophisticated way of incorporating this information into the machine
learning framework by appending the heads of propositions in lexical features
with corresponding belief tags, demonstrating its utility in distinguishing
social meanings expressed through the different belief contexts.

This study is based on emails from a single corporation, at the beginning of
the 21st century. Our findings on the correlation between author commitment
and power may be reflective of the work culture that prevailed in that
organization at the time when the emails were exchanged. It is important to
replicate this study on emails from multiple organizations in order to assess
whether these results generalize across board. It is likely that behavior
patterns are affected by factors such as ethnic culture Cox et al. (1991) of
the organization, and the kinds of conversations interactants engage in (for
instance, co-operative vs. competitive behavior Hill et al. (1992)). We intend
to explore this line of inquiry in future work.

## Acknowledgments

This paper is partially based upon work supported by the DARPA DEFT program
under a grant to Columbia University; all three co-authors were at Columbia
University when portions of this work were performed. The views expressed here
are those of the author(s) and do not reflect the official policy or position
of the Department of Defense or the U.S. Government. We thank Dan Jurafsky and
the anonymous reviewers for their helpful feedback.

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that is closed under taking intersections and faces. We impose moreover the
following conditions:

  * •

no cone of $\Delta$ contain a linear subspace of $N_{\mathbb{R}}$ (i.e.,
$\sigma$ is strongly convex);

  * •

each cone $\sigma\in\Delta$ is simplicial (but not necessarily regular666 a
simplicial cone $\sigma$ of dimension $k$ is regular if the semi-group
$\sigma\cap\tilde{N}$ is generated by $k$ elements);

  * •

no cone in contained in $N_{\mathbb{R}}\times\\{0\\}$.

The last two conditions respectively correspond to the fact that we want the
resulting toric variety $\mathcal{X}$ to be smooth, and the generic fiber over
$\mathbb{D}$ to be the split torus.

A ray of $\Delta$ is a one-dimensional cone.

For each cone $\sigma$, we define the semi-group $S_{\sigma}=\\{\tilde{m }\in
M\times\mathbb{Z},\,\langle\tilde{m},\tilde{n}\rangle\geq 0\text{ for all }
\tilde{n}\in\sigma\\}$. Following, e.g., [Oda88, §1.2], we associate to each
cone $\sigma$ the relative (affine) toric manifold of dimension $g$ over
$\mathbb{D}$ by setting

$X_{\sigma}=\\{x\colon
S_{\sigma}\to\mathbb{C},\,x(\tilde{m}_{1}+\tilde{m}_{2})=x(\tilde{m}_{1})\cdot
x(\tilde{m}_{2}),\,x(0)=1\ \text{ and } |x(0,1)|<1\\}$

which we endowed with the topology of the pointwise convergence, and the
unique structure of complex manifold for which any $\tilde{m}\in S_{\sigma}$
induces a holomorphic function.

The manifold $X_{\sigma}$ is equipped with a canonical holomorphic map
$\pi_{\sigma}(x):=x(0,1)\in\mathbb{D}$, and $\mathbb{C}[\tilde{M}]$ forms a
subalgebra of the field of meromorphic functions on $X_{\sigma}$ that are
regular on $X_{\sigma}^{*}=\pi_{\sigma}^{-1}(\mathbb{D}^{*})$ (the point
$(0,1)$ corresponds to $\pi_{\Delta}$).

Note that by construction an element in $\mathbb{C}[\tilde{M}]$ is holomorphic
in $X_{\sigma}$ if and only if it belongs to $\mathbb{C}[S_{\sigma}]$. And
this algebra generates the structure sheaf $\mathcal{O}_{X_{\sigma}}$ as a
$\mathcal{O}_{X_{\sigma}}$-module. It is customary to write
$\mathbb{e}(\tilde{m})$ for the holomorphic function induced by $\tilde{m}\in
S_{\sigma}$ on $X_{\sigma}$.

The torus $\mathbb{G}_{m}^{g}$ acts on itself by multiplication, hence on the
$\mathbb{C}$-algebra $\mathbb{C}[M]$ and the trivial extension to
$\mathbb{C}[\tilde{M}]$ (leaving the vector $(0,1)$ fixed) turns $X_{\sigma}$
into a relative toric manifold over $\mathbb{D}$.

######  Remark 6.1.

As $\sigma$ is regular, we may suppose that
$\tilde{e}_{0}=(e_{0},1),\cdots,\tilde{e}_{k}=(e_{k},1)\in N\times\mathbb{N}$
such that $\sigma\cap\tilde{N}$ is generated by
$(\tilde{e}_{0},\cdots,\tilde{e}_{k})$. We may suppose $e_{0}=0$ and complete
the family of vectors $e_{i}$ such that $e_{1},\cdots,e_{g}$ form a basis of
$N$. In the canonical basis $(0,1),(e_{i}^{*},0)$ of $M\times\mathbb{Z}$,
$S_{\sigma}$ is equal to $\\{(a_{1},\cdots,a_{g},b),\in\mathbb{Z}^{g+1},b\geq
0,b+a_{j}\geq 0\text{ for all }j=1,\cdots,k\\}$ which is generated by
$(e_{i}^{*},0),(-e_{i}^{*},1)$ for $j=1,\cdots,k$, and $\pm(e_{i}^{*},0)$ for
$j=k+1,\cdots,g$. Hence $X_{\sigma}$ is biholomorphic to

$\\{(z_{1},w_{1},\cdots,z_{k},w_{k},y_{k+1},\cdots,y_{g},t)\in(\mathbb{C}^{*})^{2k}\times\mathbb{C}^{g-k}\times\mathbb{D},\,z_{1}w_{1}=t,\cdots,z_{k}w_{k}=t\\}~{}.$

Any inclusion of cones $\sigma\subset\sigma^{\prime}$ yields an inclusion
$S_{\sigma}\supset S_{\sigma^{\prime}}$ hence induces by restriction an
embedding $X_{\sigma}\subset X_{\sigma^{\prime}}$ preserving the action by
$\mathbb{G}_{m}^{g}$ and the projection to $\mathbb{D}$. We define $X(\Delta)$
as the disjoint union of all $X_{\sigma}$ where $\sigma$ ranges over all cones
of the fan $\Delta$, and $X_{\sigma}$ and $X_{\sigma^{\prime}}$ are patched
along $X_{\sigma\cap\sigma^{\prime}}$ (interpreted as an open subset of both).

Any relative toric manifold over $\mathbb{D}$ is isomorphic to $X(\Delta)$ for
some fan satisfying the conditions above. We shall denote by
$\pi_{\Delta}\colon X(\Delta)\to\mathbb{D}$ the canonical projection map. As
before, $\mathbb{C}[\tilde{M}]$ forms a subalgebra of the field of meromorphic
functions on $X(\Delta)$ that are regular on $X(\Delta)^{*}$, and the action
of $\mathbb{G}_{m}^{g}$ on $X(\Delta)$ is the dual to its natural action on
$\mathbb{C}[\tilde{M}]$.

######  Lemma 6.2.

Let $\phi\colon\mathbb{D}^{*}\to\mathbb{G}_{m}^{g}$ be any holomorphic map
that is meromorphic at $0$, and let $n_{\phi}\in N$ be the unique co-character
satisfying

$\langle n_{\phi},m\rangle:=\operatorname{ord}_{t}\mathrm{e}(m)(\phi(t))\text{
for all } m\in M~{}.$

Then for any fan $\Delta$, the map $t\mapsto(\phi(t),t)$ extends as a
holomorphic function $\mathbb{D}\to X(\Delta)$ if and only if
$\mathbb{R}_{+}\cdot(n_{\phi},1)$ belongs to a cone in $\Delta$.

###### Proof.

Let $\Delta$ be any admissible fan. Then $\tilde{\phi}(t):=(\phi(t),t)$
extends as a holomorphic function $\mathbb{D}\to X(\Delta)$ if and only if
there exists a cone $\sigma\in\Delta$ such that
$\tilde{\phi}\colon\mathbb{D}\to X_{\sigma}$ is holomorphic.

Since $\mathbb{C}[S_{\sigma}]$ generates the structure sheaf
$\mathcal{O}_{X_{\sigma}}$ as a $\mathcal{O}_{X_{\sigma}}$-module,
$\tilde{\phi}\colon\mathbb{D}\to X_{\sigma}$ is holomorphic if and only if
$\mathrm{e}(\tilde{m})(\tilde{\phi})$ is holomorphic at $0$ for any
$\tilde{m}\in S_{\sigma}$. The latter is equivalent to the condition
$\operatorname{ord}_{t}(\mathrm{e}(\tilde{m})(\tilde{\phi}(t)))\geq 0$ for all
$\tilde{m}\in S_{\sigma}$.

If $(n_{\phi},1)$ does not belong to $\sigma$, then one can find
$\tilde{m}=(m,k)\in S_{\sigma}\subset M\times\mathbb{Z}$ such that
$\langle(n_{\phi},1),\tilde{m}\rangle<0$, and

$\displaystyle\operatorname{ord}_{t}(\mathrm{e}(\tilde{m})(\tilde{\phi}(t)))$
$\displaystyle=\operatorname{ord}_{t}(\mathrm{e}(m,k)(\tilde{\phi}(t)))$
$\displaystyle=\operatorname{ord}_{t}(\mathrm{e}(m)(\phi(t)))+k=\langle(n_{\phi},1),\tilde{m}\rangle<0$

so that $\tilde{\phi}$ does not extend holomorphically.

If $(n_{\phi},1)$ belongs to $\sigma$, then the same computation shows that
$\tilde{\phi}$ extends holomorphically. ∎

###  6.2. Analytic Mumford’s construction following Nakamura

We describe in this section the construction of a proper model of a family of
polarized abelian varieties over an open disc as done in [Nak77]. Recall our
notation for the upper half-plane $\mathfrak{H}=\\{l\in\mathbb{C}|\
\Im(l)>0\\}$ and for the Siegel domain
$\operatorname{\mathfrak{H}}_{g}=\\{M\in\operatorname{Mat}_{g\times
g}(\mathbb{C})|\ {}^{t}\\!M=M\text{ and }\Im(M)>0\\}$ for any $g\geq 1$.

Let $\pi\colon\mathcal{X}\to\mathbb{D}$, $\mathcal{L}\to\mathcal{X}$ be any
family of polarized abelian varieties. Fix any $t_{\star}\in\mathbb{D}^{*}$,
write $X_{\star}=X_{t_{\star}}$ as the quotient of some complex vector space
$V$ of dimension $g$ by a co-compact lattice $\Lambda_{\star}$. The
polarization induces a positive-definite hermitian form $H$ on $V$ whose
imaginary part $E=\Im H$ takes integral values on $\Lambda_{\star}$ (note that
$H(v,w)=E(iv,w)+iE(v,w)$). Since $E$ is a skew-symmetric form on $V$, we may
choose a symplectic basis $f_{1},\cdots,f_{g},e_{1},\cdots,e_{g}$ of the
lattice $\Lambda_{\star}$ so that

$E=\begin{pmatrix}0&D\\\ -D&0\end{pmatrix}$

for some diagonal integral matrix $D=\operatorname{diag}(d_{1},\cdots,d_{g})$
with $d_{1}|\cdots|d_{g}$. In the basis $\tilde{e}_{i}=\frac{1}{d_{i}}e_{i}$
of $V$, the lattice is generated by the vectors given by the columns of the
following period matrix

$\Pi_{\star}=\begin{pmatrix}a_{11}&\dots&a_{1g}&d_{1}&0&\dots&0&\\\
a_{21}&\dots&a_{2g}&0&d_{2}&\dots&0&\\\ &\ddots&&&&\ddots&\\\
a_{g1}&\dots&a_{gg}&0&0&\dots&d_{g}\end{pmatrix}=(Z_{\star}~{}D)$

where $Z_{\star}$ belongs to $\operatorname{\mathfrak{H}}_{g}$. Now consider
the universal covering map $\mathbb{e}\colon\mathfrak{H}\ \to\mathbb{D}^{*}$
with $\mathbb{e}(s)=\exp(2\pi is)$. Identifying the universal covers of
$X_{t}$ to $V$ so that the polarization $\mathcal{L}_{t}\to X_{t}$ defines the
same hermitian form $H$ for all $t$, we may locally follow holomorphically the
lattices $\Lambda_{t}=\ker(V\to X_{t})$ such that $e_{1},\cdots,e_{g}$ belongs
to $\Lambda_{t}$, and since $\mathfrak{H}$ is simply connected, we get a
holomorphic function $s\mapsto
Z(s)=[a_{ij}(s)]\in\operatorname{\mathfrak{H}}_{g}$ so that
$\Lambda_{\mathbb{e}(s)}$ is generated by the column of $\Pi(s)=(Z(s)~{}D)$.
Here is a diagram summarizing the construction to follow:

${{\mathbb{G}_{m}^{g}\times\mathbb{D}\circlearrowleft\Gamma}}$${{Z_{\star}\in\mathfrak{H}_{g}}}$${{\mathcal{X}\supset
X_{\star}}}$${{s\in\mathfrak{H}}}$${{\mathbb{D}\ni
t}}$$\scriptstyle{\pi}$$\scriptstyle{\exp}$$\scriptstyle{Z}$$\scriptstyle{Z_{0}}$

In the rest of the discussion, we shall replace our original family by the
family $X^{\prime}_{t}:=V/\Lambda^{\prime}_{t}$ where
$\Lambda^{\prime}_{t}:=\Lambda_{t}+\mathbb{Z}\tilde{e}_{1}+\cdots+\mathbb{Z}\tilde{e}_{g}$
is generated by the columns of $\Pi^{\prime}(s)=(Z(s)\operatorname{Id}_{g})$.
Then $\mathcal{X}^{\prime}$ forms a family of abelian varieties which is
principally polarized (i.e., the canonical map
$X^{\prime}_{t}\to(X^{\prime})^{\vee}_{t}$ is an isomophism), and we thus have
a family of isogenies $\phi_{t}\colon X_{t}\to X^{\prime}_{t}$.

Since $(Z(s+1)~{}\operatorname{Id}_{g})$ and $(Z(s)~{}\operatorname{Id}_{g})$
define the same polarized abelian variety, there exists an element
$M\in\mathrm{Sp}(2g,\mathbb{Z})$ such that $M\cdot Z(s+1)=Z(s)$ for all $s$,
see [BL04, Proposition 8.1.3] (the action is by generalized Möbius
transformations). The matrix $M$ encodes the monodromy action of
$\pi_{1}(\mathbb{D}^{*})$ on $H^{1}(X_{\star},\mathbb{Z})$ which fixes
$\tilde{e}_{1},\cdots,\tilde{e}_{g}$, hence is upper block-triangular. By the
Griffiths-Landman-Grothendieck’s monodromy theorem (see, e.g., [CMSP17,
Theorem 13.7.3]) $M$ is moreover quasi-unipotent. Up to base change we may
(and shall) suppose that $M$ is unipotent. This means that we can find an
integral-valued matrix $B\in M(\mathbb{Z},g)$ such that $Z(s+1)=Z(s)+B$.

######  Lemma 6.3.

[Nak77, Lemma 2.3] The matrix $B$ is symmetric positive semi-definite, and
there exists a holomorphic map
$Z_{0}\colon\mathbb{D}\to\operatorname{\mathfrak{H}}_{g}$ such that
$Z(s)=Z_{0}(\mathbb{e}(s))+Bs$.

Denote by $r^{\prime}$ the rank of the matrix $B$. After changing the basis
$e_{i}$, we may suppose that

(6.4) $\displaystyle B=\begin{pmatrix}0&0\\\ 0&B^{\prime}\end{pmatrix};$
$\displaystyle Z_{0}(t)=\begin{pmatrix}Z_{1}(t)&Z_{2}(t)\\\
{}^{t}\\!Z_{2}(t)&Z_{3}(t)\end{pmatrix},$

where $B^{\prime}\in\operatorname{Mat}(r^{\prime},\mathbb{Z})$ is a positive
definite integral matrix of size $r^{\prime}$, $g^{\prime}:=g-r^{\prime}$, and
$Z_{1}\colon\mathbb{D}\to\operatorname{Sym}(g^{\prime},\mathbb{C})$,
$Z_{2}\colon\mathbb{D}\to\operatorname{Mat}(g^{\prime}\times
r^{\prime},\mathbb{C})$,
$Z_{3}\colon\mathbb{D}\to\operatorname{Sym}(r^{\prime},\mathbb{C})$, are
holomorphic. Here $\operatorname{Sym}(h,\mathbb{C})$ denotes the space of
complex symmetric square matrices of size $h$.

######  Remark 6.5.

In order to be consistent with the notation of the rest of the paper, we
insist on writing $r^{\prime}$ for the rank of $B$. This is in conflict with
the notation used by Nakamura who used $g^{\prime}$ for this rank.

######  Remark 6.6.

The number $r^{\prime}$ equals the dimension of maximal torus in a semi-
abelian component of the Néron model of $\mathcal{X}$ in the case when
$\mathcal{X}$ has semi-abelian reduction. In particular, $r^{\prime}=0$ iff
$X_{t}$ is not degenerating.

Let us explain now how to recover the family $\mathcal{X}^{*}$ from (6.4).
Observe that for a fixed $t\in\mathbb{D}^{*}$ with $t=\mathbb{e}(s)$, then
$X_{t}$ is the quotient of $V\simeq\mathbb{C}^{g}$ by the lattice generated by
the column of $\Pi(s)$. Taking the quotient by
$\mathbb{Z}e_{1}\oplus\cdots\oplus\mathbb{Z}e_{g}$ first, we get
$X_{t}=\mathbb{G}_{m}^{g}/\Gamma$ where $\Gamma=\mathbb{Z}^{g}$ acts on the
split torus by multiplication

(6.7)
$\gamma\cdot(z^{\prime}_{1},\cdots,z^{\prime}_{g})=\left(z^{\prime}_{1}\,\prod_{j=1}^{g}\mathbb{e}(a_{1j}(s)\gamma_{j}),\cdots,z^{\prime}_{g}\,\prod_{j=1}^{g}\mathbb{e}(a_{gj}(s)\gamma_{j})\right).$

The family $\mathcal{X}^{*}$ can be thus obtained as the quotient of
$\mathbb{G}_{m}^{g}\times\mathbb{D}^{*}$ by an action of $\Gamma$ which we now
describe in dual terms. Recall from the previous section that $M$ is the set
of characters of $\mathbb{G}_{m}^{g}$. The universal covering map
$V\to\mathbb{G}_{m}^{g}$ is given in the coordinates $e_{i}$ by the map
$v=\sum v_{i}{e}_{i}\mapsto(\mathbb{e}(v_{1}),\cdots,\mathbb{e}(v_{g}))$. To
exploit the block-diagonal form of $B$ as in (6.4), we write the coordinates
on $\mathbb{G}_{m}^{g}$ as
$(z_{1},\cdots,z_{g^{\prime}},w_{1},\cdots,w_{r^{\prime}})=(\mathbb{e}(v_{1}),\cdots,\mathbb{e}(v_{g}))$.
A character $m\in M$ can thus be written as $m=z^{p}w^{q}$ with
$p\in\mathbb{Z}^{g^{\prime}}$ (resp. $q\in\mathbb{Z}^{r^{\prime}}$) a
character of $\mathbb{G}_{m}^{g^{\prime}}$ (resp. of
$\mathbb{G}_{m}^{r^{\prime}}$).

The action of $\Gamma$ can then be described on $M\times\mathbb{N}$ as
follows. For any $\gamma=(\alpha,\beta)\in\Gamma$, where
$\alpha\in\mathbb{Z}^{g^{\prime}}$ and $\beta\in\mathbb{Z}^{r^{\prime}}$ we
have (with $t=\mathbb{e}(s)$):

$\displaystyle{\gamma}^{*}(z^{p})$ $\displaystyle=\mathbb{e}(\alpha\cdot
Z_{1}(t)\cdot{}^{t}\\!p+\beta\cdot{}^{t}\\!Z_{2}(t)\cdot{}^{t}\\!p)\,z^{p};$
$\displaystyle{\gamma}^{*}(w^{q}t^{l})$ $\displaystyle=\mathbb{e}(\alpha\cdot
Z_{2}(t)\cdot{}^{t}\\!q+\beta\cdot Z_{3}(t)\cdot{}^{t}\\!q)\,t^{l+\beta\cdot
B\cdot{}^{t}\\!q}w^{q};$

Note that this action induces a dual action of $\Gamma$ on the lattice
$N\times\mathbb{N}$ so that
$(\gamma\cdot\tilde{n})(\tilde{m})=n(\gamma^{*}\tilde{m})$ for all $\tilde{n}
\in N\times\mathbb{N}$ and $\tilde{m}\in M\times\mathbb{N}$. In the dual basis
$N=\\{(a,b)\in\mathbb{Z}^{g^{\prime}}\times\mathbb{Z}^{r^{\prime}}\\}$ so that
$\langle(a,b),(p,q)\rangle=(a\cdot{}^{t}\\!p)+(b\cdot{}^{t}\\!q)$, we have
$(\alpha,\beta)\cdot(a,b,k)(p,q,l)=(a,b,k)(p,q,l+\beta\cdot
B^{\prime}\cdot{}^{t}\\!q)$, hence

(6.8) $(\alpha,\beta)\cdot(a,b,k)=(a,b+k\,\beta\cdot B^{\prime},k)~{}.$

It follows that $\Gamma$ preserves the affine hyperplanes $N\times\\{k\\}$ for
all $k$, acts trivially on $N\times\\{0\\}$, and the action of the subgroup
$\\{0\\}\times\mathbb{Z}^{g^{\prime}}$ is free on $N\times\\{k\\}$ for all
$k\in\mathbb{N}^{*}$.

Since $\Gamma$ acts by multiplication on each element of $M\times\mathbb{N}$,
for any $\gamma\in\Gamma$, the map
$(z^{\prime},t)\mapsto\gamma\cdot(z^{\prime},t)$ on
$\mathbb{G}_{m}^{g}\times\mathbb{D}^{*}$ extends as a holomorphic map from
$X_{\sigma}$ to $X_{\gamma\cdot\sigma}$ for any (rational polyhedral strongly
convex) cone $\sigma$ (in $\tilde{N}_{\mathbb{R}}$).

It follows that the action of $\Gamma$ extends to the relative toric variety
$X(\Delta)$ over $\mathbb{D}$ if and only if the fan $\Delta$ is
$\Gamma$-invariant.

######  Theorem 6.9.

There exists a $\Gamma$-invariant fan $\Delta$ such that the following holds:

  1. (1)

the set of rays of $\Delta$ is exactly given by $\mathbb{R}_{+}\cdot(n,1)$ for
all $n\in N$ such that $(n,1)$ belongs to the span of the $\Gamma$-orbit of
$(0,1)$;

  2. (2)

the action of $\Gamma$ on $X(\Delta)$ is free and $\pi_{\Delta}$-equivariant;

  3. (3)

the restriction of the action of $\Gamma$ to any fiber of $\pi_{\Delta}$ is
co-compact.

######  Remark 6.10.

In order for the $\Gamma$-action to be co-compact it is necessary for the
support of $\Delta$ to intersect $N\times\\{1\\}$ along an affine space. One
can show that the Néron model is obtained by taking the subfan of $\Delta$
having only one dimensional faces generated by the $\Gamma$-orbit of $(0,1)$.
In general $\Delta$ contains more rays. This happens exactly when the central
fiber of the Néron model is reducible.

###### Sketch of proof.

The proof is given in [Nak77, Theorem 2.6]. Let $\Pi$ be the real affine
subspace generated by the $\Gamma$-orbit of $(0,1)$. Observe that
$N_{\Pi}:=N\times\\{1\\}\cap\Pi$ is a co-compact lattice in $\Pi$. Let
$\Delta_{0}$ be the fan whose cones are $(0)$ and the $1$-dimensional cones
$\mathbb{R}_{+}(n,1)$ with $n\in\Pi\cap N\times\\{1\\}$.

Since $\Gamma$ acts by translation on $\tilde{N}_{\mathbb{R}}$ preserving
$N\times\mathbb{N}$, we may find a scalar product (hence a metric $d$ on
$\tilde{N}_{\mathbb{R}}$) which is $\Gamma$-invariant. Following [AN99, §1],
we define a Delaunay cell $\sigma$ to be the closed convex hull of all
elements $\tilde{n}\in N_{\Pi}$ for which there exists $\alpha\in\Pi$
satisfying $\inf_{N_{\Pi}}d(\alpha,\cdot)=d(\alpha,\tilde{n})$. If the metric
is sufficiently general, then we obtain a $\Gamma$-invariant cell-
decomposition of $\Pi$ by simplices whose vertices are all contained in
$N_{\Pi}$. Observe that this fan is not necessarily regular if $g\geq 5$ (see
[AN99, §1.14]).

We define $\Delta$ to be the fan whose cones are generated by cells of this
triangulation. The fact that $\Gamma$ acts co-compactly on $X(\Delta)$ is
proved in [Nak77] for any dimension $g$. ∎

A crucial observation is the following.

######  Proposition 6.11.

Let $\pi\colon\mathcal{X}\to\mathbb{D}$ be any principally polarized family of
abelian varieties. Then any holomorphic section over $\mathbb{D}^{*}$ which is
meromorphic at $0$ extends holomorphically to $\mathbb{D}$.

If moreover $\mathcal{X}=X(\Delta)/\Gamma$ for some discrete group
$\Gamma\simeq\mathbb{Z}^{g}$ acting on $\mathbb{G}_{m}^{g}\times\mathbb{D}$ as
in (6.7), then any meromorphic section
$\psi\colon\mathbb{D}^{*}\to\mathcal{X}$ lifts to a holomorphic section
$\tilde{\psi}\colon\mathbb{D}\to X(\Delta)$ having the following property.

Since $X(\Delta)\supset\mathbb{G}_{m}^{g}\times\mathbb{D}^{*}$, post-composing
$\tilde{\psi}|_{\mathbb{D}^{*}}$ with the first projection yields a
meromorphic map $\phi\colon\mathbb{D}^{*}\to\mathbb{G}_{m}^{g}$. Then
$(n_{\phi},1)$ as defined in Lemma 6.2 belongs to the linear span of
$\Gamma\cdot(0,1)\subset\tilde{N}_{\mathbb{R}}$.

Note that $\Gamma\cdot(0,1)$ is the translate by $(0,1)$ of a discrete group
(i.e., $\\{(0,\beta\cdot B^{\prime}),\beta\in\mathbb{Z}¨{r^{\prime}}\\}$)
whose rank equals the one of $B$ (i.e., $r^{\prime}$), see (6.8).

###### Proof.

We first claim that $\psi$ extends holomorphically through $0$. Since $\psi$
is meromorphic, there exists a closed analytic subset $C$ of dimension $1$ in
$\mathbb{D}\times\mathcal{X}$ that contains
$\\{(t,\psi(t)),t\in\mathbb{D}^{*}\\}$. We observe that $C$ is necessarily
analytically irreducible at the point $p=C\cap X_{0}$, hence $\psi$ extends
continuously (hence holomorphically) at $0$.

Choose coordinates $z=(z_{1},\cdots,z_{g})$ centered at $p$, and write
$\pi(z)=\sum_{I}a_{I}z^{I}$. Since $\pi(\psi(t))=t$, we infer that at least
one $a_{i}\neq 0$. It follows that $\pi^{-1}(0)$ is smooth and reduced at $p$.

Suppose now that $\mathcal{X}=X(\Delta)/\Gamma$ as given by the previous
theorem. Then pick a point $q\in X(\Delta)$ which is mapped to $p$ in
$\mathcal{X}$. Since $\Gamma$ acts freely, the canonical map
$X(\Delta)\to\mathcal{X}$ is a covering map, hence we can lift $\psi$ to a
holomorphic map $\tilde{\psi}\colon\mathbb{D}\to\mathcal{X}(\Delta)$ such that
$\tilde{\phi}(0)=q$. The projection of $\tilde{\psi}$ to $\mathbb{G}_{m}^{g}$
induces a holomorphic map $\phi\colon\mathbb{D}^{*}\to\mathbb{G}_{m}^{g}$,
which is meromorphic at $0$, hence by Lemma 6.2 we deduce that
$\mathbb{R}_{+}\cdot(n_{\phi},1)$ forms a one-dimensional cone of $\Delta$.
This condition is equivalent to say that $(n_{\phi},1)$ belongs to the linear
span of $\Gamma\cdot(0,1)\subset\tilde{N}_{\mathbb{R}}$. ∎

######  Remark 6.12.

The proof shows a weak version of the Néron property for $X(\Delta)/\Gamma$.
In fact the Néron model is given by the smooth locus of $\pi\colon
X(\Delta)/\Gamma\to\mathbb{D}$ whose complement has codimension $2$ in
$X(\Delta)/\Gamma$.

###  6.3. Proof of Theorem E

Let $f\colon\mathcal{X}\dashrightarrow\mathcal{X}$ be a meromorphic map of a
polarized family $\pi\colon\mathcal{X}\to\mathbb{D}$ of $g$-dimensional
abelian varieties satisfying $\pi\circ f=\pi$ such that for any
$t\in\mathbb{D}^{*}$, the map $f^{*}_{t}\colon H^{1}(X_{t},\mathbb{Z})\to
H^{1}(X_{t},\mathbb{Z})$ has finite order.

Replacing $f$ by a suitable iterate and applying Proposition 2.12, we may
suppose that $f^{*}_{t}=\operatorname{Id}$ for all $t$ so that the induced map
$f_{t}\colon\mathcal{X}_{t}\to\mathcal{X}_{t}$ is a translation of the abelian
variety $\mathcal{X}_{t}$.

By Proposition 4.10 and the discussion before Lemma 6.3, we may assume that
$X_{t}$ is principally polarized. We may also replace the family $\mathcal{X}$
by any base change by Proposition 4.9, and by the discussion of the previous
section, we may suppose that $\mathcal{X}=X(\Delta)/\Gamma$ where
$\Gamma\simeq\mathbb{Z}^{g}$ acts on
$X(\Delta)\subset\mathbb{G}_{m}^{g}\times\mathbb{D}$ as in (6.7), and $\Delta$
is a $\Gamma$-invariant fan containing $(0,1)$.

Let $\phi(t)=f_{t}(0)$. This is a holomorphic section over $\mathbb{D}^{*}$
which is meromorphic at $0$ since $f$ is meromorphic. It follows from Theorem
6.9 (1) and Proposition 6.11 that $\phi$ lifts to a section
$\tilde{\phi}\colon\mathbb{D}\to X(\Delta)$, hence $(n_{\phi},1)$ lies in the
linear span of the $\Gamma$-orbit of $(0,1)$ and we can find an integer $N$
such that $(Nn_{\phi},1)=\gamma\cdot(0,1)$ for some $\gamma\in\Gamma$.

The map $F(z,t)=(\phi(t)\cdot z,t)$ defines a meromorphic self-map of
$\mathbb{G}_{m}^{g}\times\mathbb{D}$ lifting $f$. Let
$\psi(t)=\widetilde{\phi}(t)^{N}/\exp(\gamma)$ Then the vector $n_{\psi}$
defined in Lemma 6.2 equals $0$. By Lemma 6.2, $\psi$ extends as a holomorphic
map $\psi\colon\mathbb{D}\to\mathbb{G}_{m}^{g}$, and $g$ defines an
automorphism of $X(\Delta)$ since any (relative) toric manifold carries an
action by multiplication by $\mathbb{G}_{m}^{g}$.

Since $F^{N}$ is the composition of $g$ and the action of $\gamma$ it also
induces a biholomorphism on $X(\Delta)$ which commutes with the action of
$\Gamma$. Therefore $f^{N}$ is a biholomorphism on $X(\Delta)/\Gamma$, hence
$f$ is regularizable by Propositions 2.11 and 2.12.

###  6.4. Orbits of translations on families of abelian varieties

We now explore the closure of the orbit of a general point under a family of
translations on a family of abelian varieties. The setting is as follows:
$f\colon\mathcal{X}\to\mathcal{X}$ is an automorphism of a polarized family of
abelian varieties $\mathcal{L}\to\mathcal{X}$,
$\pi\colon\mathcal{X}\to\mathbb{D}$ given by the translation along a section
$\alpha\colon\mathbb{D} \to\mathcal{X}$ of $\pi$ so that
$f_{t}(z)=z+\alpha(t)$ for all $z\in X_{t}$ and all $t\in\mathbb{D}^{*}$.

For each $t$, we denote by
$\tilde{P}(t):=\overline{\\{n\cdot\alpha(t)\\}}_{n\in\mathbb{Z}}\subset X_{t}$
the closure (for the euclidean topology) of the orbit of $0$, and by $P(t)$
its connected component containing $0$. Then $P(t)$ is a closed connected
abelian real-subgroup of $X_{t}$ isomorphic to
$(\mathbb{R}/\mathbb{Z})^{h(t)}$ for some $h(t)\in\\{0,\cdots,2g\\}$, and
$\tilde{P}(t)$ is a finite union of translates of $P(t)$. The group $P(t)$
also contains a maximal complex Lie group $A(t)$ of (complex) dimension
$s(t)$. Write $r(t):=h(t)-2s(t)$.

Recall that a $F_{\sigma}$-set is a countable union of closed subset, and that
any countable union of proper analytic subsets is an $F_{\sigma}$-set having
zero Lebesgue measure (hence has empty interior).

We start with the following observation.

######  Proposition 6.13.

Let $f\colon\mathcal{X}\dashrightarrow\mathcal{X}$ be a family of
translations. Set $h(f)=\max_{t\in\mathbb{D}^{*}}h(t)$.

  1. (1)

The set $\mathcal{E}_{0}:=\\{t\in\mathbb{D}^{*},h(t)<h(f)\\}$ is included in a
countable union of proper analytic subsets.

  2. (2)

There exists a real analytic family
$\\{P^{\prime}_{t}\\}_{t\in\mathbb{D}^{*}}$ of real subtori of $X_{t}$, such
that $P^{\prime}_{t}\supset P(t)$ for all $t$ and $P^{\prime}_{t}=P(t)$ for
all $t\notin\mathcal{E}_{0}$.

  3. (3)

There exists a real analytic family
$\\{\tilde{P}^{\prime}_{t}\\}_{t\in\mathbb{D}^{*}}$ of real closed subgroups
of $X_{t}$, such that $\tilde{P}^{\prime}_{t}\supset\tilde{P}(t)$ for all $t$
and $\tilde{P}^{\prime}_{t}=\tilde{P}(t)$ for all $t\notin\mathcal{E}_{0}$.

  4. (4)

There exists a countable union of proper analytic subsets
$\mathcal{E}\supset\mathcal{E}_{0}$ such that $s(t)=s(f)\in\\{0,\cdots,g\\}$
for all $t\notin\mathcal{E}$.

  5. (5)

There exists a subfamily $A^{\prime}_{t}$ of abelian subvarieties of $X_{t}$,
such that $A^{\prime}_{t}=A(t)$ for all $t\notin\mathcal{E}$.

###### Proof.

We may argue locally near $t_{0}\in\mathbb{D}^{*}$, and suppose that
$X_{t}=V/\Lambda_{t}$ where $V\simeq\mathbb{C}^{g}$, and $\Lambda_{t}$ is a
family of rank $2g$ sublattices of $V$ moving holomorphically with $t$. Denote
by $\pi_{t}\colon V\to X_{t}$ the canonical projection.

Fix a lattice basis $e_{1},\cdots,e_{2g}$ of $\Lambda_{t_{0}}$ and denote by
$e_{i}\colon\mathbb{D}\to V$ the holomorphic map such that
$e_{i}(t)\in\Lambda_{t}$ for all $t$, and $e_{i}(t_{0})=e_{i}$. Let
$e_{i,t}^{\vee}$ be the dual $\mathbb{R}$-basis so that
$e_{i,t}^{\vee}(e_{j}(t))=\delta_{ij}$. Observe that $e_{i,t}^{\vee}$ is a
real linear form on $V$ which varies real analytically in $t$ (it does not
vary holomorphically).

Consider the $\mathbb{Q}$-vector space
$L_{t}=\\{q\in\mathbb{Q}^{2g},\,\sum_{j=1}^{2g}
q_{j}e^{\vee}_{j,t}(\alpha(t))\in\mathbb{Q}\\}$ so that

$P(t):=\pi_{t}\left(\bigcap_{q\in L_{t}} \left\\{v\in V,\sum_{j=1}^{2g}
q_{j}e^{\vee}_{j,t}(v)=0\right\\}\right)$

and $h(t)=2g-\dim_{\mathbb{Q}}(L_{t})$. For fixed
$q,q^{\prime}\in\mathbb{Q}^{2g}\times\mathbb{Q}$, the set
$\\{t\in\mathbb{D}^{*},\sum_{j=1}^{2g}
q_{j}e^{\vee}_{j,t}(\alpha(t))=q^{\prime}\\}$ is real-analytic. It follows
that $L_{t}$ is fixed, say equal to $L_{\star}$, for all $t$ outside a
countable union of proper real-analytic subsets $\mathcal{E}_{0}$. And
$L_{\star}\subset L_{t}$ for all $t\in\mathbb{D}^{*}$ which proves $h(t)\leq
h(f)$ for all $t$. This proves (1). Observe also that
$P^{\prime}_{t}=\pi_{t}(\bigcap_{q\in L_{\star}} \\{\sum_{j=1}^{2g}
q_{j}e^{\vee}_{j,t}=0\\})$ is a real-analytic family of real subtori for which
(2) holds. In the same way,
$\tilde{L}_{t}=\\{(q,q^{\prime})\in\mathbb{Q}^{2g+1},\,\sum_{j=1}^{2g}
q_{j}e^{\vee}_{j,t}(\alpha(t))=q^{\prime}\\}$ is constant equal to
$\tilde{L}_{\star}$ off $\mathcal{E}_{0}$, hence
$\tilde{P}^{\prime}_{t}=\pi_{t}(\bigcap_{(q,q^{\prime})\in\tilde{L}_{\star}}
\\{\sum_{j=1}^{2g} q_{j}e^{\vee}_{j,t}=q^{\prime}\\})$ is a real-analytic
family of closed subgroups of $X_{t}$ equal to $\tilde{P}(t)$ for all
$t\notin\mathcal{E}_{0}$, proving (3).

To any $\mathbb{R}$-linear form $\ell_{q,t}=\sum_{j=1}^{2g}
q_{j}e^{\vee}_{j,t}$, we attach the $\mathbb{C}$-linear form
$u_{q,t}(v)=\ell_{q,t}(v)-i\ell_{q,t}(iv)\in V^{*}$ so that $\ker(u_{q,t})$ is
the complex linear hypersurface included in $\ker(\ell_{q,t})$, and

$A(t):=\pi_{t}\left(\bigcap_{q\in L_{t}} \ker(u_{q,t})\right)~{}.$

As $u_{q,t}$ varies analytically in $t$, it follows that
$t\mapsto\dim_{\mathbb{C}}\left(\bigcap_{q\in L_{\star}} \ker(u_{q,t})\right)$
is again constant off a countable union of proper analytic subsets
$\mathcal{E}_{1}$ (and is upper-semicontinuous). It follows that
$s(t):=\dim_{\mathbb{C}}A(t)$ is constant on
$\mathcal{E}:=\mathcal{E}_{0}\cup\mathcal{E}_{1}$ equal to some constant
$s_{\star}$, hence (4) holds.

We now prove that $\\{A(t)\\}_{t\notin\mathcal{E}}$ glue to a holomorphic
family over $\mathbb{D}$. Suppose that $t_{0}\notin\mathcal{E}$, and pick a
basis of lattice points $f_{1},\cdots,f_{2s(f)}$ in $\bigcap_{q\in L_{\star}}
\ker(u_{q,t})$. Then $A(t)$ is the image under $\pi_{t}$ of the complex vector
space generated by the holomorphically varying points
$f_{1,t},\cdots,f_{2s(f),t}$. It follows that
$A^{\prime}_{t}=\pi_{t}(\mathrm{Span}\\{f_{1,t},\cdots,f_{2s(f),t}\\})$ forms
a family of subabelian varieties over $\mathbb{D}^{*}$. Since the family
$X_{t}$ is polarized, $A^{\prime}_{t}$ is also polarized and extend through
$0$. This proves (5). ∎

In the remaining of the section, $\mathcal{E}$ will always denote the
$F_{\sigma}$-set defined by the previous lemma. Our next objective is the
following decomposition result.

######  Proposition 6.14.

Let $f\in\operatorname{Aut}(\mathcal{X})$ be a family of translations on a
family $\pi\colon\mathcal{X}\to\mathbb{D}$ of polarized abelian varieties.
Then there exist two families of sub-abelian varieties
$\pi_{\mathcal{A}}\colon\mathcal{A}\to\mathbb{D}$,
$\pi_{\mathcal{B}}\colon\mathcal{B}\to\mathbb{D}$ of $\mathcal{X}$, two
families of translations $f_{\mathcal{A}}\in\operatorname{Aut}(\mathcal{A})$,
and $f_{\mathcal{B}}\in\operatorname{Aut}(\mathcal{B})$, and a meromorphic map
$\Phi\colon\mathcal{A}\times\mathcal{B}\dashrightarrow\mathcal{X}$ such that:

  1. (1)

$\Phi$ induces an isogeny $A_{t}\times B_{t}\to X_{t}$ for all
$t\in\mathbb{D}^{*}$;

  2. (2)

the orbit of $0$ under $f_{\mathcal{A},t}$ is dense in $A_{t}$ for all
$t\notin\mathcal{E}$;

  3. (3)

the closure of the orbit of $0$ under $f_{\mathcal{B},t}$ is totally real for
all $t\notin\mathcal{E}$.

###### Proof.

We let $\mathcal{A}$ be the family of subabelian varieties given by
Proposition 6.13. Recall the definition of the dual family
$\mathcal{X}^{\vee}$ and of the canonical meromorphic map
$\phi\colon\mathcal{X}\dashrightarrow\mathcal{X}^{\vee}$ from the proof of
Proposition 4.10.

Define a family of subabelian varieties $B_{t}$ as the connected components
containing $0$ of the kernel of the composition map
$X_{t}\mathop{\longrightarrow}X^{\vee}_{t}\to
A^{\vee}_{t}\mathop{\longrightarrow}A_{t}\to X_{t}$. We obtain a family of
polarized subvarieties over $\mathbb{D}^{*}$ hence over $\mathbb{D}$ and the
first property is clear, see [BL04, Chapter 5, §3].

Let us fix any parameter $t\in\mathbb{D}^{*}$. We have a canonical splitting
$V=E_{t}\oplus F_{t}$ so that $\pi_{t}(E_{t})=A_{t}$ and
$\pi_{t}(F_{t})=B_{t}$ and since $A_{t},B_{t}$ are varying holomorphically,
the same holds for the vector spaces $E_{t}$ and $F_{t}$. We may thus write
$\alpha(t)=a(t)\oplus b(t)$ with $a,b$ holomorphic. Denote by
$f_{\mathcal{A},t}$ the translation by $a(t)$ on $A_{t}$ and by
$f_{\mathcal{B},t}$ the translation by $b(t)$ on $B_{t}$.

Suppose that $t\notin\mathcal{E}$. Replacing $\alpha(t)$ by a suitable
multiple, we may suppose that the closure of the orbit of $0$ is a real torus
so that $\widetilde{P}(t)=P(t)$. Denote by $\Pi_{t}$ its lift to $V$, and pick
$\widetilde{\alpha}(t)$ (resp. $\widetilde{a}(t)$) a lift of $\alpha(t)$
(resp. of $a(t)$) to $V$. Observe that $\Pi_{t}$ is the closure of
$\mathbb{Z}\widetilde{\alpha}(t)+\Lambda_{t}$, and that
$\Pi_{t}\cap\Lambda_{t}$ is a co-compact lattice in $\Pi_{t}$. Since $A_{t}$
is the largest subabelian variety in $P(t)$, we have $E_{t}=\Pi_{t}\cap
i\Pi_{t}$, and $F_{t}\cap\Pi_{t}$ cannot contain any complex linear subspace,
hence is totally real.

The linear projection $p_{E,t} \colon V\to E_{t}$ parallel to $F_{t}$ semi-
conjugates the translation by $\alpha(t)$ and the translation by $a(t)$. In
particular, the image of $\mathbb{Z}\widetilde{\alpha}(t)+\Lambda_{t}$ under
$p_{E,t}$ equals $\mathbb{Z}a(t)+p_{E,t}(\Lambda_{t})$, so that the latter is
dense in $E_{t}$. Since the addition map $A_{t}\times B_{t}\to X_{t}$ is an
isogeny, the co-compact lattice $\Lambda_{t}\cap E_{t}$ has finite index in
$p_{E,t}(\Lambda_{t})$, and we conclude that
$\mathbb{Z}\widetilde{a}(t)+(\Lambda_{t}\cap E_{t})$ is dense in $E_{t}$. This
proves (2).

The proof of (3) is completely analogous, once one observes that the image of
$\Pi_{t}$ under the projection $p_{F,t} \colon V\to F_{t}$ parallel to $E_{t}$
is totally real. ∎

The proof of Theorem F will be complete after we prove:

######  Proposition 6.15.

Let $\pi\colon\mathcal{X}\to\mathbb{D}$ be a family of polarized abelian
varieties. Let $f\in\operatorname{Aut}(\mathcal{X})$ be a family of
translations such that the closure of the orbit of $0$ under
$f_{\mathcal{X},t}$ is totally real for Lebesgue-almost every $t$.

Then the closure of the orbit of $0$ under $f_{\mathcal{X},t}$ is totally real
for all $t$, and there exist a proper model $\mathcal{X}^{\prime}$ and
sequence of automorphisms $f_{n}\in\operatorname{Aut}(\mathcal{X}^{\prime})$
of finite order such that $f_{n}\to f$ locally uniformly on
$\mathcal{X}^{\prime}$.

###### Proof.

Replacing $\alpha$ by a suitable multiple, we may (and shall) assume that
$\tilde{P}(t)=P(t)$ is connected.

We shall first argue locally near a fixed point $t_{0}\notin\mathcal{E}$,
where $\mathcal{E}$ is the set defined in Proposition 6.13. We use the same
notation as in the proof of the previous proposition. Fix a basis
$\tilde{e}_{1},\cdots,\tilde{e}_{k}$ for the lattice
$\Pi_{t_{0}}\cap\Lambda_{t_{0}}$, and complete it as a maximal set of
$\mathbb{Z}$-independent elements $\tilde{e}_{1},\cdots,\tilde{e}_{2g}$ in
$\Lambda_{t_{0}}$. Observe that it may happen that
$\tilde{e}_{1},\cdots,\tilde{e}_{2g}$ generate only a finite index subgroup of
$\Lambda_{t_{0}}$. Observe also, that the holomorphically varying vectors
$\tilde{e}_{1,t},\cdots,\tilde{e}_{k,t}$ are $\mathbb{C}$-linearly independent
because $\Pi_{t}$ is totally real. We may thus suppose that
$\tilde{e}_{1},\cdots,\tilde{e}_{g}$ are $\mathbb{C}$-linearly independent.

Since the closure of the orbit of $0$ is connected,
$\widetilde{\alpha}(t)\in\Pi_{t}$ for all $t$, and we may find
$\alpha_{j}(t)\in\mathbb{R}$ such that
$\widetilde{\alpha}(t)=\sum_{j=1}^{k}\alpha_{j}(t)\tilde{e}_{j,t}$. The map
$t\mapsto\widetilde{\alpha}(t)$ is holomorphic map from $\mathbb{D}^{*}$ to
the dual space $V^{*}$ hence $\alpha_{1},\cdots,\alpha_{k}$ are all constant.

We now replace $\mathcal{X}$ by an isogenous family which is principally
polarized, and choose the model $\mathcal{X}^{\prime}=X(\Delta)/\Gamma$
described in §6.2. Pick any $\beta\in\mathbb{R}^{k}$, and set
$\beta(t)=\sum_{j=1}^{k}\beta_{j}\tilde{e}_{j,t}$ for $t$ close to $t_{0}$. We
claim that $\beta(t)$ extends to a holomorphic section of
$\mathcal{X}^{\prime}\to\mathbb{D}$ over $\mathbb{D}$.

To see this, write $\pi\colon X(\Delta)\to\mathcal{X}^{\prime}$ for the
canonical projection. By analytic continuation, we may find a lift
$s\mapsto\hat{\beta}(s)$ defined over $s\in\mathfrak{H}$ such that
$\pi(\hat{\beta}(s))=\beta(e^{2i\pi s})$. We now write $\hat{\beta}$ in terms
of a holomorphic varying symplectic basis $e_{1},\cdots,e_{2g}$ as in §6.2:
there exists some real numbers $\theta_{1},\cdots,\theta_{g}$ and
$b_{1},\cdots,b_{g}$ such that

$\hat{\beta}(s)=\left(e^{i\pi\theta_{1}}\,\prod_{j=1}^{g}\mathbb{e}(a_{1j}(s)b_{j}),\cdots,e^{i\pi\theta_{g}}\,\prod_{j=1}^{g}\mathbb{e}(a_{gj}(s)b_{j})\right)$

(see (6.7)). Doing the same for $\alpha$, we get some
$\vartheta_{j},a_{j}\in\mathbb{R}$ such that

$\hat{\alpha}(s)=\left(e^{i\pi\vartheta_{1}}\,\prod_{j=1}^{g}\mathbb{e}(a_{1j}(s)a_{j}),\cdots,e^{i\pi\vartheta_{g}}\,\prod_{j=1}^{g}\mathbb{e}(a_{gj}(s)a_{j})\right)$

But $\alpha$ is a section over $\mathbb{D}$ so the vector $Ba$ belongs to
$\mathbb{Z}^{g}$.

Observe that the function $v\in\mathbb{R}^{g}/\mathbb{Z}^{g}\mapsto
Bv\in\mathbb{R}^{g}/\mathbb{Z}^{g}$ is continuous, and vanishes at all vectors
$\\{na\\}_{n\in\mathbb{N}}$. Since $P(t)=\pi_{t}(\Pi_{t})$ is the closure of
the orbit of $0$ under the translation by $\alpha(t)$ we may find a sequence
$q^{(n)}\in\mathbb{N}$ such that $q^{(n)}\alpha(t)\to\beta(t)$ in a
neighborhood of $t_{0}$. This implies $q^{(n)}a\to b$ in
$\mathbb{R}^{g}/\mathbb{Z}^{g}$, hence $Bb=0$ in
$\mathbb{R}^{g}/\mathbb{Z}^{g}$, and $\beta$ defines a holomorphic section
over $\mathbb{D}^{*}$. By Lemma 6.3 this section is meromorphic at $0$. By
Proposition 6.11 it extends to a holomorphic section over $\mathbb{D}$.

Now choose any sequence $\beta^{(n)}\in\mathbb{Q}^{k}$ converging to
$(\alpha_{1},\cdots,\alpha_{k})$. Then
$\beta^{(n)}(t)=\sum_{j=1}^{k}\beta^{(n)}_{j}e_{j,t}$ converges locally
uniformly to $\alpha(t)$ on $\mathbb{D}$, and the translation $f_{n}$ by
$\beta^{(n)}(t)$ has finite order, which concludes the proof. ∎

######  Remark 6.16.

The previous proof implies that the family of real subtori $P^{\prime}_{t}$
actually extends to a family over $\mathbb{D}$. Recall that the smooth locus
of the central fiber $\mathcal{X}^{\prime}_{0}$ is a finite union of semi-
abelian varieties which are principal $\mathbb{G}_{m}^{r^{\prime}}$-bundles
over an abelian variety (where $r^{\prime}$ is the rank of the monodromy, see
the discussion after Lemma 6.3).

One can also prove that $P^{\prime}_{0}$ is transversal to the fibers of the
$\mathbb{G}_{m}^{r^{\prime}}$-fibration, hence $s(f)\leq r^{\prime}$.

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