# Modeling Homophone Noise for Robust Neural Machine Translation

###### Abstract

In this paper, we propose a robust neural machine translation (NMT) framework.
The framework consists of a homophone noise detector and a syllable-aware NMT
model to homophone errors. The detector identifies potential homophone errors
in a textual sentence and converts them into syllables to form a mixed
sequence that is then fed into the syllable-aware NMT. Extensive experiments
on Chinese$\rightarrow$English translation demonstrate that our proposed
method not only significantly outperforms baselines on noisy test sets with
homophone noise, but also achieves a substantial improvement on clean text.

Index Terms—  Neural Machine Translation, Speech Translation, Robustness,
Homophone

##  1 Introduction

Despite of remarkable progress made in NMT recently [1, 2, 3, 4, 5, 6], most
NMT systems are still prone to translation errors caused by noisy input
sequences [7, 8]. One common type of input noise is homophone noise, where
words or characters are mis-recognized as others with same or similar
pronunciation in ASR [9, 10] or input systems [11] for non-phonetic languages
(e.g., Chinese), as illustrated by the example in Table 1.

Previous works suggest that incorporating phonetic embeddings into NMT [12]
and augmenting training data with adversarial examples [7] with injected
homophone noise [13] would alleviate this issue. Intuitively, humans usually
have no trouble in disambiguating sentences corrupted with moderate homophone
noise via context and syllable information. We propose a human-inspired robust
NMT framework tailored to homophone noise for Chinese-English translation,
which is composed of a homophone noise detector (hereinafter referred to as
detector for brevity) and a syllable-aware NMT (SANMT) model.

Clean Input | 建一所小学
---|---
Output of NMT | build a primary school
Noisy Input | 建议所小学
Output of NMT | suggest a primary school
Mixed Transcript | 建yi所小学
Output of Ours | build a primary school
Table 1: A translation example with homophone noise. The erroneous character
“议” (“discuss”) in the noisy input is a homophone corresponding to the
original character “一” (“one”) in the clean input. The erroneous character “议”
is replaced with its Chinese Pinyin “yi” in the mixed transcript which enables
NMT to translate correctly.

Due to the lack of data annotated with homophone noise, we propose to train
our detector on monolingual data in a self-supervised manner, where Chinese
characters sequences as input and their corresponding syllables sequence as
label to predict the possibility that a character is homophone noise. The
identified homophone errors from a source sentence are then converted into
corresponding syllables to produce a new source sequence mixed with characters
and syllables. Augmenting bilingual training data with instances where
original source sentences are substituted with their corresponding character-
syllable-mixed sequences, we train the SANMT model to translate such
unconventional inputs. To examine the effectiveness of our proposed model, we
conduct extensive experiments on both artificial noisy test sets and a real-
world noise test set with homophone noise in speech translation (ST) scenario.
The test set will be released soon. Our experimental results on
Chinese$\rightarrow$English translation clearly show that the proposed method
is not only significantly superior to previous approaches in alleviating the
impact of homophone noise on NMT, but also achieves a substantial improvement
on the clean text.

Fig. 1: The diagram of our proposed detector. Each character’s log-likelihood
score (LLS) is computed over the whole syllable sequence. In this example, “议”
is identified as a homophone error and replaced with its syllable “yi”,
resulting in a new mixed sentence.

##  2 Approach

###  2.1 Homophone Error Detector

Instead of feeding the noise input indiscriminately to NMT, we actively detect
homophone errors in the inputs and roll back identified erroneous characters
into syllables (i.e., Chinese Pinyin which consists of 400 syllables without
tones representing about 6,500 characters) to minimise the interference of
these non-negligible textual errors with NMT. In order to avoid the negative
impact from textual noise on the detection, we predict each character $x_{i}$
in an original textual sentence $X=(x_{1},x_{2},...,x_{T})$ based on the
corresponding clean syllable sequence $S=(s_{1},s_{2},...,s_{T})$ with the
same length of $X$, instead of using $X$. Therefore, we first build a
syllable-character parallel training data by converting the text-only corpus
into a syllable corpus using a language-specific character-to-syllable
converter. In this work, we utilize the Transformer [4] encoder to model the
interactions of syllables for our detector. Figure 1 shows the detector
architecture. The detector is trained using the standard maximum-likelihood
criterion, to maximize the sum of log probabilities of the ground-truth
outputs given the corresponding syllable inputs.

During inference, we use the LLS of each syllable token based on the whole
syllable sequence as a confidence score that determines if the corresponding
original textual character is a homophone error as follows:

$LLS(x_{i})=logP(x_{i}|S;\theta),~{}$ (1)

where $\theta$ represents the parameter set of the detector. For a source
sentence probably with homophone noise, we substitute the characters with LLS
smaller than a predefined threshold $\beta$ with their corresponding
syllables, forming a new sequence mixed with characters and syllables to
downstream NMT.

###  2.2 Syllable-aware NMT

The previous works [13, 12] on robust NMT to homophone noise create the noise
adversarial training data by injecting some homophone noise into the source
part of original textual training data. Their resulting NMT models are able to
handle inputs with homophone errors innately. However, our proposed detector
is expected to convert identified erroneous characters in such inputs into
corresponding syllables. Thus, such mixed inputs are not supported by
conventional NMT models solely trained on the original textual training data
or noise adversarial training data. To deal with this issue, we propose a
syllable-aware NMT based on the Transformer [4] encoder-decoder architecture
to handle both conventional textual inputs and unconventional sequences mixed
with syllables generated by the detector. Without introducing any changes
either to the architecture or to the training scheme, we confer robustness on
our SANMT model through augmenting the original training data with syllable-
mixed training data. Specifically, we pick sentence pairs from original
training data, and change the source sentences by randomly substituting some
characters with corresponding syllables to create our syllable-mixed training
data. It is worth noting that injected consecutive syllables are separated by
blank space. Examples can be seen for each kind of training data in Table 2.

Data |  Source |  Target
---|---|---
OTTD |  建一所小学 |  build a primary school
SMTD |  建 yi 所小学 |  build a primary school
NATD |  建议所小学 |  build a primary school
Table 2: Examples of original textual training data (OTTD), syllable-mixed
training data (SMTD) and noise adversarial training data (NATD). The dashed
underlined syllable “yi” and character “议” are the injected noise to
substitute the original clean character “一” in the source part of OTTD.

Model |  | MT02 | MT03 | MT04 | MT05 | MT06 | MT08 | Avg.
---|---|---|---|---|---|---|---|---
Baseline | Clean | 47.23 | 46.82 | 47.69 | 45.81 | 44.84 | 35.82 | 44.20
Noisy | 34.84 | 34.18 | 35.35 | 32.31 | 32.39 | 26.63 | 32.17
CPNMT | Clean | 45.25 | 44.14 | 45.84 | 42.56 | 41.81 | 32.30 | 41.33
Noisy | 45.25 | 44.14 | 45.84 | 42.56 | 41.81 | 32.30 | 41.33
Li et al [13] | Clean | 46.58 | 46.09 | 47.46 | 45.48 | 44.68 | 35.81 | 43.90
Noisy | 46.27 | 45.89 | 46.96 | 44.84 | 44.16 | 34.73 | 43.32
Liu et al [12] | Clean | 46.24 | 44.67 | 46.80 | 44.08 | 43.00 | 34.12 | 42.53
Noisy | 46.00 | 44.79 | 46.55 | 43.66 | 42.80 | 33.88 | 42.34
Ours | Clean | 48.34 | 47.93 | 48.67 | 46.42 | 46.39 | 37.65 | 45.41
Noisy | 48.12 | 46.65 | 48.01 | 45.61 | 46.07 | 36.57 | 44.58

Table 3: Results of different NMT systems trained on LDC. Bold indicates the
highest BLEU scores on ANTs.

##  3 Experiments

Despite its rareness in English and other Indo-European languages, homophone
is a common phenomenon in Chinese. In this section, we conducted experiments
on Chinese$\rightarrow$English translation to evaluate the effectiveness of
our proposed method.

###  3.1 Setup

We used two training data sets for NMT, including (1) 1.2M sentence pairs
extracted from LDC corpora and (2) CWMT
corpus111http://www.statmt.org/wmt19/translation-task.html consisting of 9M
sentence pairs. We chose the NIST 2002 data set as our validation set, and
NIST 2003, 2004, 2005, 2006, and 2008 data sets as our clean test sets (CTs).
We built test sets with artificial noise (ANTs) by randomly substituting a
small fraction of each source sentence in CTs with erroneous homophones.
According to the experimental results reported in [14] that the state-of-the-
art ASR performance is about 6.8%-14.6% word error rate on different test
sets, we set 0.1 as the default noise ratio in our experiments to simulate as
closely as possible the conditions of imperfect ASR. We used a Chinese Pinyin
tool222https://github.com/mozillazg/python-pinyin to convert Chinese
characters into syllables.

We applied the standard Transformer-base architecture [4] with its default
hyper-parameters for NMT. We used the tokenization script provided in
Moses333http://www.statmt.org/moses/ and Jieba
tokenizer444https://github.com/fxsjy/jieba for English and Chinese
tokenizations, respectively. We learned BPE [15] models with 30K merge
operations for both sides. We batched sentence pairs by approximately the same
length and limited the number of source and target tokens per batch to 4,096.
We set a beam size of 4 for decoding. We used case-insensitive tokenized BLEU
[16] score as metric.

We implemented our detector based on the standard BERT-base model [17]. The
Chinese part of CWMT corpus is used as text-only data to train the detector.
We set $\beta$ to 0.1 as the threshold for the detector to identify the
potential homophone errors.

###  3.2 Robustness to Artificial Noise

We used three times the original textual training data to build the syllable-
mixed training data, resulting in an augmented training data 4-fold for
training our SANMT. In detail, each source sentence of the syllable-mixed
training data is constructed by randomly replacing some characters of its
original one with their syllables according to a ratio $p\in(0,1)$. For a fair
comparison, we created the noise adversarial training data based on the
syllable-mixed training data combining with original training data to train
the previous robust NMT models [13, 12]. When constructing noise adversarial
training data, for each injected syllable in the syllable-mixed training data,
we randomly replaced an original character with a different homophonic
character. Following to [12], we set the weights of phonetic embedding and
textual embedding to 0.95 and 0.05, respectively.

Table 3 shows the results. Compared with the performance on CTs, the
conventional NMT baseline suffers a huge drop on ANTs by up to 12 BLEU points.
Among all methods, our approach not only achieves the best result across ANTs,
especially up to an average improvement of 12.41 BLEU points compared to the
NMT baseline, but also obtains a substantial improvement across CTs. Although
[12] also achieve a considerable improvement on ANTs, it is a little worse
than the simple data augmentation method [13] which doesn’t introduce
additional phonetic embeddings.

For a noisy input in which all the characters are homophone errors, the input
can be completely rolled back a sequence only with syllables using our
detector. Thus, we investigated the worst performance of our SANMT by training
another NMT model using Pinyin as the modeling unit, namely character-level
Pinyin NMT (CPNMT). Although CPNMT performs the worst among all the systems on
CTs, it also significantly outperforms the baseline on ANTs. However, it still
lags behind all the robust NMT systems. This demonstrates that both textual
representation and syllable representation play an important role in improving
translation quality and enhancing robustness, respectively. We also reproduced
the experiment on the CWMT corpus and Table 4 shows the main results.

Model |  | newstest17 | newstest18 | newstest19 | Avg.
---|---|---|---|---|---
Baseline | Clean | 22.59 | 21.95 | 24.85 | 23.40
Noisy | 6.60 | 6.28 | 6.03 | 6.16
Li et al [13] | Clean | 22.14 | 21.78 | 24.01 | 22.90
Noisy | 16.51 | 16.56 | 17.48 | 17.02
Ours | Clean | 22.37 | 21.84 | 24.24 | 23.04
Noisy | 20.72 | 20.15 | 22.51 | 21.33

Table 4: Results of different NMT systems trained on CWMT. Bold indicates the
highest BLEU scores on ANTs.

###  3.3 Effect of Noise Ratio

We reported the averaged BLEU score of all systems on ANTs when varying the
noise ratio from 0.1 to 0.5 in Table 5. The results indicate that our method
still performs the best at all settings. Not surprisingly, the baseline NMT
system degrades more severely when the noise fraction becomes larger, but all
robust NMT systems maintains relatively stable performance.

Model | 0.1 | 0.2 | 0.3 | 0.4 | 0.5
---|---|---|---|---|---
Baseline | 32.17 | 22.00 | 14.86 | 10.19 | 7.09
Li et al [13] | 43.32 | 42.96 | 42.51 | 42.07 | 41.62
Liu et al [12] | 42.34 | 41.97 | 41.92 | 41.52 | 41.38
Our SANMT | 44.58 | 44.30 | 44.34 | 43.05 | 42.34
Table 5: The averaged BLEU scores of different NMT systems on ANTs with different noise ratios. Model |  Training Data |  RNT
---|---|---
Baseline |  LDC |  18.85
CWMT |  28.56
Li et al [13] |  LDC |  23.20
CWMT |  37.13
Liu et al [12] |  LDC |  22.13
CWMT |  38.12
Our SANMT |  LDC |  25.36
CWMT |  40.89
Table 6: Results of different NMT systems on RNT.

###  3.4 Robustness to Real-World Noise

There are no publicly available dedicated noisy test sets to examine the
robustness of NMT to homophone errors. And the distribution of homophone noise
in real-world scenarios is usually different from that in our simulated ANTs.
Therefore, we propose a benchmark data set, namely test set with real-world
noise (RNT) to facilitate research in this area. We manually collected noisy
transcriptions with at least one homophone error recognized by human from our
in-house Chinese ASR system, and commissioned a professional translator to
translate them into English, resulting in RNT with 850 test samples. To
further validate the effectiveness of different NMT systems when larger
training data is available, we additionally trained all NMT models on the CWMT
corpus. The results shown in Table 6 clearly show that our method still
significantly outperforms baselines on RNT.

Clean Utterance |  请拼写它
---|---
Noisy Transcript |  请拼写他
Li et al [13] |  Please spell him
Liu et al [12] |  Please spell him
Output of Detector |  请 pin xie ta
Output of SANMT |  Please spell it
Clean Utterance |  听医生的建议
Noisy Transcript |  听一生的建议
Li et al [13] |  Listen to the doctor’s advice
Liu et al [12] |  Suggestions on life
Output of Detector |  听 yi 生的建议
Output of SANMT |  Listen to doctors’ advice
Table 7: Cases with different types of homophone errors in RNT and their
translations. The upper case denotes a third-person pronoun error, which is
common in Chinese ASR where “他”, “她”, and “它” are all pronounced “tā”. The
erroneous homophone word “一生” in the lower case is also a noun which has a
different meaning from the original homophone one “医生”.

Table 7 shows two representative examples with different types of homophone
errors from RNT. Results indicates that our detector is able to distinguish
homophone noise and our SANMT generates correct translations using the
syllable-mixed inputs. Thus, our robust NMT is superior to previous work in
the real-world scenarios.

##  4 Conclusion

In this paper, we have presented a novel framework composed of a homophone
error detector and a SANMT model to cope with homophone noise. Experimental
results show that our method not only achieves substantial improvement over
previous robust NMT baselines both on the test sets with artificial or real-
world noise, but also outperforms the NMT baseline on the clean test sets. We
consider that future studies could modeling noise detection and NMT jointly.

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$\phi_{V^{\prime},H}(E)=\phi_{V^{\prime},H}(L)=\phi_{V^{\prime},H}(F).$

Then we have

(12.15.3)
$(\mu_{H}(E)-\mu_{H}(L))V^{\prime}=\frac{(H\cdot\mathrm{ch}_{1}(E))\mathrm{ch}_{2}(L)-(H\cdot\mathrm{ch}_{1}(L))\mathrm{ch}_{2}(E)}{\mathrm{ch}_{0}(L)\mathrm{ch}_{0}(E)},$

which implies that the right hand side of equation (12.15.3) must be smaller
or equal to zero. Since $L$ is $\sigma_{b}^{L}$-stable for any
$b\in\mathbb{P}^{1}$, we must have $\phi_{b}^{L}(E)\leq\phi_{b}^{L}(L)$, and
so overall we must have

$\frac{(H\cdot\mathrm{ch}_{1}(E))\mathrm{ch}_{2}(L)-(H\cdot\mathrm{ch}_{1}(L))\mathrm{ch}_{2}(E)}{\mathrm{ch}_{0}(L)\mathrm{ch}_{0}(E)}=0.$

Then $\phi_{b}^{L}(E)=\phi_{b}^{L}(L)<\phi_{b}^{L}(F)$ for any
$b\in\mathbb{P}^{1}$. Note that $E$ cannot lie in $\textnormal{ker}\,Z_{H}$
since
$\textnormal{ker}\,(Z_{H})\cap\mathcal{B}^{0}_{H,-2}=\langle\mathcal{O}_{\Theta}(-1)\rangle$
while $\textnormal{Hom}(\mathcal{O}_{\Theta}(-1),L)=0$. So we must have
$F\in\ker(Z_{H})\cap\mathcal{B}^{0}_{H,-2}$ (or else
$\phi_{b}^{L}(E)<\phi_{b}^{L}(L)$, a contradiction). Since
$\textnormal{Hom}(L,\mathcal{O}_{\Theta}(-1))=0$, we have
$F\simeq\oplus\mathcal{O}_{X}[1]$. But $\phi_{V,H}(\mathcal{O}_{X}[1])=1$ for
all $V\geq 0$, contradicting $\phi_{V^{\prime},H}(L)=\phi_{V^{\prime},H}(F)$.

Hence $L$ is $\sigma_{V,H}$-stable for all $V>0$. ◼

Recall that $\Upsilon=\Phi(\cdot)\otimes\mathcal{O}(f)$. We define

$\Upsilon^{\prime}=\Phi^{-1}(\\_\otimes\mathcal{O}(-f))[-1].$

Then by definition we have
$\Upsilon^{\prime}\circ\Upsilon=\Upsilon\circ\Upsilon^{\prime}=\mathrm{id}_{\mathcal{D}}[-1]$.
Let $h\in\text{GL}^{+}(2,\mathbb{R})$ be defined by

(12.15.4) $h:=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$

A quick computation shows

(12.15.5) $Z_{D}(\Upsilon(E))=h\cdot Z_{V,H}(E)$

when $D=V$. To apply Proposition 11.6, we first explore the relation between
$\mathcal{B}^{0}_{H,-2}$ and $\mathrm{Coh}^{\omega,0}$, where
$\omega=\Theta+(e+V)f$. Theorem 12.16. Given $V\in\mathbb{R}_{>0}$, let
$\mathcal{P}_{V,H}$ denote the slicing for the weak stability condition
$\sigma_{V,H}$. Let $\mathcal{Q}=\Upsilon\cdot\mathcal{P}_{V,H}$, and denote
the heart $\mathcal{Q}((\frac{1}{2},\frac{3}{2}])$ by
$\mathcal{B}_{\Upsilon}$, then we have

$\mathcal{B}_{\Upsilon}=\mathrm{Coh}^{\omega,0},$

where $\omega=\Theta+(e+V)f$.

###### Proof.

The proof is similar to the proof of Theorem 12. We only write down the
arguments where the proofs are different.

We have

$\mathbb{C}(x)=\Upsilon(Q[1])$

where $Q$ is a pure sheaf supported on dimension $1$ with
$\mathrm{ch}(Q)=\mathrm{ch}(\mathcal{O}_{f})$. By Proposition 12 and Remark
12, we have $Q[1]$ lies in $\mathcal{P}_{V,H}(\frac{3}{2})$ and is
$\sigma_{V,H}$-stable. Note that
$\ker(Z_{V,H})\cap\mathcal{P}_{V,H}((\frac{1}{2},\frac{3}{2}])=\langle\mathcal{O}_{\Theta}(-1)[1]\rangle$.
Since $\phi_{V,H}(\mathcal{O}_{\Theta}(-1))=\frac{1}{2}$, we have objects in
$\ker(Z_{V,H})\cap\mathcal{P}_{V,H}((\frac{1}{2},\frac{3}{2}])$ always have
phase $\phi_{V,H}=\frac{3}{2}$.

Claim: $Q[1]$ is also stable in
$\mathcal{P}_{V,H}((\frac{1}{2},\frac{3}{2}])$.

By the similar argument as in Theorem 12, we have a short exact sequence in
$\mathcal{P}_{V,H}((\frac{1}{2},\frac{3}{2}])$:

$0\to E^{\prime}\to Q[1]\to F^{\prime}\to 0,$

with $E^{\prime}\in\mathcal{P}_{V,H}(\frac{3}{2})$ and
$Q[1]\in\mathcal{P}_{V,H}(\frac{3}{2})$. Since
$\phi_{V,H}(\mathcal{O}_{\Theta}(-1)[1])=\frac{3}{2}$, we have
$F^{\prime}\in\mathcal{P}_{V,H}(\frac{3}{2})$. Then the short exact sequence

$0\to E^{\prime}[-1]\to Q\to F^{\prime}[-1]\to 0$

implies that $Q$ is strictly semistable in $\sigma_{V,H}$, which leads to a
contradiction.

Let $A$ be an object such that $A=\Upsilon(A^{\prime})$, where $A^{\prime}$ is
a $\sigma_{V,H}$-stable object in $\mathcal{P}_{V,H}(t)$. Note that
$\ker(Z_{V,H})\cap
P_{V,H}(\frac{1}{2},\frac{3}{2}])=\mathcal{P}_{V,H}(\frac{3}{2})$, then a
similar argument in Theorem 12 also implies

$\textnormal{Hom}(\mathbb{C}(x),A)=0.$

Since $\Upsilon(\mathcal{O}_{\Theta}(-1))=\mathcal{O}_{X}$, we have
$\textnormal{Hom}(\mathbb{C}(x),\Upsilon(\mathcal{O}_{\Theta}(-1)))=0$, and
$\textnormal{Hom}(\mathbb{C}(x),\Upsilon(\mathcal{O}_{\Theta}(-1)[1]))=0$. Let
$E$ be an arbitrary object in $\mathcal{B}_{\Upsilon}$, then $E$ is an
extension of objects $\Upsilon(\mathcal{O}_{\Theta}(-1))$,
$\Upsilon(\mathcal{O}_{\Theta}(-1)[1])$ and $A=\Upsilon(A^{\prime})$ where
$A^{\prime}$ is $\sigma_{V,H}$-stable in $\mathcal{P}_{V,H}(t)$ for
$\frac{1}{2}<t\leq\frac{3}{2}$. Hence [9, Proposition 5.4] implies that $E$ is
quasi-isomorphic to a two-term complex of locally free sheaves $F^{\bullet}$
such that $F^{i}=0$ for $i>0$ or $i<-1$. This means we have

$\mathcal{B}_{\Upsilon}\subset\langle\mathrm{Coh}(X),\mathrm{Coh}(X)[1]\rangle.$

We denote $\mathrm{Coh}(X)\cap\mathcal{B}_{\Upsilon}$ by
$\mathcal{T}_{\Upsilon}$, and $\mathrm{Coh}(X)\cap\mathcal{B}_{\Upsilon}[-1]$
by $\mathcal{F}_{\Upsilon}$. We need to show that
$\mathcal{T}_{\omega,0}\subset\mathcal{T}_{\Upsilon}$, and
$\mathcal{F}_{\omega,0}\subset\mathcal{F}_{\Upsilon}$.

Let $E\in\mathrm{Coh}(X)$ be $\mu_{\omega,0}$\- stable. By the same argument
of Parts 1 and 2 of Theorem 12, we have if $E$ is a torsion sheaf, then
$E\in\mathcal{T}_{\Upsilon}$. If $\mu_{\omega,0}(E)>0$, then
$E\in\mathcal{T}_{\Upsilon}$. If $\mu_{\omega,0}(E)<0$, then
$E\in\mathcal{F}_{\Upsilon}$.

Finally we consider $E$ is $\mu_{\omega,0}$-semistable and
$\mu_{\omega,0}(E)=0$. We want to show $E\in\mathcal{F}_{\Upsilon}$. If not,
then there exists a short exact sequence in $\mathrm{Coh}(X)$:

$0\rightarrow E_{T}\rightarrow E\rightarrow E_{F}\rightarrow 0$

where $0\neq E_{T}\in\mathcal{T}_{\Upsilon}$ and
$E_{F}\in\mathcal{F}_{\Upsilon}$. Denote $E_{T}$ by $E^{(0)}$. Similar to the
argument in Theorem 12, consider any nonzero map
$\gamma^{(0)}:E^{(0)}\rightarrow\mathbb{C}(x^{(0)})$ for some $x^{(0)}\in X$.
If $\gamma^{(0)}$ is not surjective in the abelian category
$\mathcal{B}_{\Upsilon}$, then taking its image and cokernel in
$\mathcal{B}_{\Upsilon}$, we can consider the short exact sequence in
$\mathcal{B}_{\Upsilon}$

(12.16.1)
$0\to\textnormal{im}\,\gamma^{(0)}\to\mathbb{C}(x^{(0)})\to\textnormal{coker}\,\gamma^{(0)}\to
0.$

This corresponds to a short exact sequence in
$\mathcal{P}_{V,H}((\frac{1}{2},\frac{3}{2}])$

$0\to M\to Q[1]\to R\to 0,$

where $\mathbb{C}(x^{(0)})=\Upsilon(Q[1])$ and $Q[1]$ is stable in
$\mathcal{P}_{V,H}((\frac{1}{2},\frac{3}{2}])$ with
$\phi_{V,H}(Q[1])=\frac{3}{2}$. The stability of $Q[1]$ in
$\mathcal{P}_{V,H}((\frac{1}{2},\frac{3}{2}])$ then implies that
$\phi_{V,H}(R)=\frac{3}{2}$.

Now suppose $M\notin\textnormal{ker}\,(Z_{V,H})$. If
$R\in\textnormal{ker}\,(Z_{V,H})$, then
$\phi_{V,H}(M)=\phi_{V,H}(Q[1])=\tfrac{3}{2}$ making it impossible for
$\phi_{V,H}(M)<\phi_{V,H}(R)$ to hold. If $R\notin\textnormal{ker}\,(Z_{V,H})$
instead, then we obtain
$\phi_{V,H}(M)<\phi_{V,H}(Q[1])=\tfrac{3}{2}<\phi_{V,H}(R)$ which again is
impossible. Hence we must have $M\in\textnormal{ker}\,(Z_{V,H})$. Since
objects in $\ker(Z_{V,H})\cap\mathcal{P}_{V,H}((\frac{1}{2},\frac{3}{2}])$
always have phase $\frac{3}{2}$, we have $\phi_{V,H}(M)=\frac{3}{2}$. This
contradicts $Q[1]$ is stable in
$\mathcal{P}_{V,H}((\frac{1}{2},\frac{3}{2}])$.

Hence we have $E^{(0)}\rightarrow\mathbb{C}(x^{(0)})$ is also a surjection in
$\mathcal{B}_{\Upsilon}$. Recall that for any $i\geq 0$, we set $E^{(i+1)}$ to
be the kernel of $\gamma^{(i)}$ in $\mathcal{B}_{\Upsilon}$, and define
$\gamma^{(i+1)}$ to be any nonzero map $E^{(i+1)}\to\mathbb{C}(x^{(i+1)})$ in
$\mathrm{Coh}(X)$ for some $x^{(i+1)}\in X$. The same argument as in the last
two paragraphs shows that $\gamma^{(i+1)}$ is surjective for all $i\geq 0$.
Consider the short exact sequences in $\mathcal{B}_{\Upsilon}$

$0\to E^{(i+1)}\to E^{(i)}\to\mathbb{C}(x^{(i)})\to 0.$

We have $Z_{D}(E^{(i+1)})=Z_{D}(E^{(i)})+1$. For $i\gg 0$,
$Z_{V,H}(\Upsilon^{-1}(E^{(i)}))$ does not lie in the half plane

$\\{re^{i\phi\pi}|r\in\mathbb{R}_{\geq 0},\phi\in(\frac{1}{2},\frac{3}{2}]\\}$

which leads to a contradiction. The rest of the argument is the same as
Theorem 12. ◼

Corollary 12.17. $\Upsilon(L)$ is $\sigma_{D}$-stable for all $D>0$.

###### Proof.

Given $D\in\mathbb{R}_{>0}$, let $V=D$. We first check that we are in
Configuration III-w:

(a) We take the autoequivalences $\Upsilon$ and $\Upsilon^{\prime}$, then by
definition we have
$\Upsilon\circ\Upsilon^{\prime}=\Upsilon^{\prime}\circ\Upsilon=\mathrm{id}_{\mathcal{D}}[-1]$.

(b) We take the two abelian categories to be $\mathcal{B}^{0}_{H,-2}$ and
$\mathrm{Coh}^{\omega,0}$, where $\omega=\Theta+(V+e)f$. Denote the slicing of
$\sigma_{V,H}$ by $\mathcal{P}_{V,H}$, we have

$\mathcal{B}^{0}_{H,-2}=\mathcal{P}_{V,H}(0,1])\subset\Big{\langle}\mathcal{P}_{V,H}((\frac{1}{2},\frac{3}{2}]),\mathcal{P}_{V,H}((\frac{1}{2},\frac{3}{2}])[-1]\Big{\rangle}$

Since $\mathcal{B}_{\Upsilon}=\mathrm{Coh}^{\omega,0}$, we have

$\Upsilon(\mathcal{P}_{V,H}(0,1])\subset\langle\mathrm{Coh}^{\omega,0},\mathrm{Coh}^{\omega,0}[-1]\rangle.$

(c) We take $T=h$ where $h$ is as in (12.15.4), then (12.15.5) implies that

$Z_{D}(\Upsilon(E))=T\cdot Z_{V,H}(E).$

Since the action of $T$ decreases the phase of objects by $\pi/2$, we define

$\Gamma_{T}:\phi\to\phi-\frac{1}{2}.$

Using the notation in Configuration III-w, we take $a=0$, $b=0$, then we have

$0<\Gamma_{T}^{-1}(0)<1.$

Now we check the conditions in Proposition 11.6.

By the computation in Section LABEL:para:spcase1, if $D_{\alpha}+1-V>0$, then
$L$ is $\Upsilon_{\mathrm{Coh}^{\omega,0}}\text{-WIT}_{0}$. Since
$\ker(Z_{D})\cap\mathrm{Coh}^{\omega,0}=\langle\mathcal{O}_{X}[1]\rangle$ and
$\Upsilon(L)$ is a line bundle from LABEL:para:spcase1, There is no short
exact sequence in $\mathrm{Coh}^{\omega,0}$

$0\rightarrow M\rightarrow\Upsilon(L)\rightarrow N\rightarrow 0$

with $M\in\ker(Z_{D})\cap\mathrm{Coh}^{\omega,0}$. We denote the phase of
$\sigma_{D}$ by $\phi_{D}$. If $N\in\ker(Z_{D})\cap\mathrm{Coh}^{\omega,0}$,
we have $\phi_{D}(N)=1$; since $\Im Z_{D}(\Upsilon(L))=D_{\alpha}+1-V>0$, it
follows that $\phi_{D}(\Upsilon(L))<1$, and so by the weak seesaw property we
have $\phi_{D}(M)<\phi_{D}(N)$.

If $D_{\alpha}+1-V\leq 0$, then $L$ is
$\Upsilon_{\mathrm{Coh}^{\omega,0}}\text{-WIT}_{1}$. Assume we have a short
exact sequence in $\mathrm{Coh}^{\omega,0}$

$0\rightarrow M\rightarrow\Upsilon(L)[1]\rightarrow N\rightarrow 0.$

Since $\mathrm{ch}_{1}(\Upsilon(L))=-\Theta+(D_{\alpha}+1)f$, any morphism of
sheaves $M[-1]\to\Upsilon(L)$ must vanish on the generic fiber of the elliptic
fibration (by considering fiber degree). Since $\Upsilon(L)$ is a line bundle,
this forces $\textnormal{Hom}(M,\Upsilon(L))=0$, hence
$M\notin\ker(Z_{D})\cap\mathrm{Coh}^{\omega,0}$. Now suppose
$N\in\ker(Z_{D})\cap\mathrm{Coh}^{\omega,0}$. In this case, we have
$\phi_{D}(N)=\phi_{D}(\mathcal{O}_{X}[1])=1$. If $D_{\alpha}+1-V<0$ then $\Im
Z_{D}(\Upsilon(L)[1])>0$ and so, by the weak seesaw property as in the
previous paragraph, we have $\phi_{D}(M)<\phi_{D}(N)$. So suppose
$D_{\alpha}+1-V=0$ from now on. Since $N$ is a direct sum of copies of
$\mathcal{O}_{X}[1]$, the morphism $\Upsilon(L)[1]\to N$ induces a nonzero
morphism of sheaves $\Upsilon(L)\to\mathcal{O}_{X}$ that must be injective
(since $\Upsilon(L)$ is a line bundle), and so its cokernel is a torsion sheaf
with $c_{1}=\Theta-(D_{\alpha}+1)f$ which is not effective, a contradiction.

Since $L$ is $\sigma_{V,H}$-stable for all $V>0$, the stable case of
Proposition 11.6 implies that $\Upsilon(L)$ is $\sigma_{D}$-stable for all
$D>0$. ◼

Consider the central charge given by

$Z_{V,D}(E)=-\mathrm{ch}_{2}(E)+V\mathrm{ch}_{0}(E)+i(\Theta+(D+e)f)\cdot\mathrm{ch}_{1}(E).$

Denote $\Theta+(D+e)f$ by $\omega$, and let
$\sigma_{V,D}=(Z_{V,D},\mathrm{Coh}^{\omega,0})$. Theorem 12.18. $\Upsilon(L)$
is $\sigma_{V,D}$-stable for all $D>0$, $V>0$.

###### Proof.

Fix $D\in\mathbb{R}_{>0}$. Let $\omega=\Theta+(D+e)f$. If $L$ is
$\Upsilon_{\mathrm{Coh}^{\omega,0}}\text{-WIT}_{0}$, then $\Upsilon(L)$ is
$\sigma_{D}$-stable for all $D>0$ by Corollary 12, and the argument of Theorem
12 shows the $\sigma_{V,D}$-stability of $\Upsilon(L)$ for all $V>0$ with only
two minor differences: (1) we don’t actually get case (ii) for the short exact
sequence (12.15.1) since $\sigma_{V,D}$’s are Bridgeland stability conditions
(and so $Z_{V,D}$ has zero kernel). (2) since $L$ is
$\Upsilon_{\mathrm{Coh}^{\omega,0}}\text{-WIT}_{0}$, we have
$\Upsilon(L)\in\mathrm{Coh}^{\omega,0}$. Then $\Im(Z_{V,D}(\Upsilon(L)))>0$,
hence we still get the contradiction in last part of the proof of Theorem 12.

We consider the case when $L$ is
$\Upsilon_{\mathrm{Coh}^{\omega,0}}\text{-WIT}_{1}$.

Note that we have $Z_{V,D}=Z_{D}$ when $V=0$, and the heart for $\sigma_{V,D}$
is $\mathrm{Coh}^{\omega,0}$ for any $V\geq 0$. Consider $\sigma_{V,D}$ for
some $V>0$. Let $\rho_{V,D}=-\frac{\Re(Z_{V,D})}{\Im(Z_{V,D})}$. Then a short
exact sequence in $\mathrm{Coh}^{\omega,0}$

(12.18.1) $0\rightarrow E\rightarrow\Upsilon(L)[1]\rightarrow F\rightarrow 0$

is a destabilizing short exact sequence for $\Upsilon(L)[1]$ if
$\rho_{V,D}(E)>\rho_{V,D}(\Upsilon(L)[1])$. Note that by the same argument in
Theorem 12, we have $\Im(Z_{V,D}(E))\neq 0$. Then the inequality
$\rho_{V,D}(E)>\rho_{V,D}(\Upsilon(L)[1])$ is equivalent to

$\displaystyle(\omega\cdot\mathrm{ch}_{1}(E)\mathrm{ch}_{0}(\Upsilon(L)[1])-\omega\cdot\mathrm{ch}_{1}(\Upsilon(L)[1])$
$\displaystyle\mathrm{ch}_{0}(E))V$ (12.18.2)
$\displaystyle>\omega\cdot\mathrm{ch}_{1}(E)\mathrm{ch}_{2}(\Upsilon(L)[1])-\omega\cdot\mathrm{ch}_{1}(\Upsilon(L)[1])\mathrm{ch}_{2}(E).$

Note that $F\in\mathrm{Coh}(X)[1]$ and $F[-1]$ is a torsion-free sheaf. Since
$\mathrm{ch}_{0}(\Upsilon(L)[1])=-1$, we have
$\mathrm{ch}_{0}(E)=\mathrm{ch}_{0}(\Upsilon(L)[1])-\mathrm{ch}_{0}(F)\geq 0$.
Then $\rho_{V,D}(E)>\rho_{V,D}(\Upsilon(L)[1])$ if either

(i)
$\omega\cdot\mathrm{ch}_{1}(E)\mathrm{ch}_{0}(\Upsilon(L)[1])-\omega\cdot\mathrm{ch}_{1}(\Upsilon(L)[1])\mathrm{ch}_{0}(E)=0$,
in which case (12.18.2) holds for all $V\in\mathbb{R}_{\geq 0}$, or

(ii)
$\omega\cdot\mathrm{ch}_{1}(E)\mathrm{ch}_{0}(\Upsilon(L)[1])-\omega\cdot\mathrm{ch}_{1}(\Upsilon(L)[1])\mathrm{ch}_{0}(E)<0$,
in which case (12.18.2) is in turn equivalent to

$V<\frac{\omega\cdot\mathrm{ch}_{1}(E)\mathrm{ch}_{2}(\Upsilon(L)[1])-\omega\cdot\mathrm{ch}_{1}(\Upsilon(L)[1])\mathrm{ch}_{2}(E)}{\omega\cdot\mathrm{ch}_{1}(E)\mathrm{ch}_{0}(\Upsilon(L)[1])-\omega\cdot\mathrm{ch}_{1}(\Upsilon(L)[1])\mathrm{ch}_{0}(E))}.$

Thus if there exists $E$ destabilizing $\Upsilon(L)[1]$ at $\sigma_{V,D}$,
then $E$ destabilizes $\Upsilon(L)[1]$ at $\sigma_{D}$, which contradicts
Corollary 12.

Hence we have $\Upsilon(L)$ is $\sigma_{V,D}$-semistable for all $V$. Assume
that $\Upsilon(L)$ is strictly $\sigma_{V^{\prime},D}$-semistable for some
$V^{\prime}>0$. Then there is a short exact sequence as in (12.18.1) such that
$\phi_{V^{\prime},D}(E)=\phi_{V^{\prime},D}(\Upsilon(L)[1])=\phi_{V^{\prime},D}(F)$.
By the same argument in Theorem 12, we have
$\phi_{D}(E)=\phi_{D}(\Upsilon(L)[1])$ and
$F\in\ker(Z_{D})\cap\mathrm{Coh}^{\omega,0}$. But since
$\ker(Z_{D})\cap\mathrm{Coh}^{\omega,0}=\langle\mathcal{O}_{X}[1]\rangle$ and
$\textnormal{Hom}(\Upsilon(L),\mathcal{O}_{X})=0$, such a short exact sequence
cannot exist.

Hence $\Upsilon(L)$ is $\sigma_{V,D}$-stable for all $V>0$. ◼

Theorem 12.19. The line bundle $L=\mathcal{O}_{X}(\alpha)$ is
$\sigma_{\omega,0}$-stable for all ample $\omega\in\langle\Theta,f\rangle$.

###### Proof.

By (5.2.2), we have

$\Phi\cdot\sigma_{\omega,f}\cdot\tilde{g}=\sigma_{\omega^{\prime},0},$

where $V_{\omega}=D_{\omega^{\prime}}$ and $D_{\omega}=V_{\omega^{\prime}}$.
This implies that

$\Upsilon\cdot\sigma_{\omega,0}\cdot\tilde{g}=\sigma_{\omega^{\prime},0}.$

Then the result follows from Theorem 12. ◼

##  13\. Bounding the rank of destabilising subobjects

As the argument in Section 7 shows, every time we have a criterion for the
stability of a line bundle $L$, we can potentially ‘transform’ the criterion
using an autoequivalence of the derived category, so as to obtain a larger
region over which $L$ is stable. In this section, we prove another criterion
for the stability of a line bundle that is different from Theorem 6.

Lemma 13.1. Let $X$ be a smooth projective surface, and $L$ a line bundle on
$X$ with first Chern class $\alpha=\mathrm{ch}_{1}(L)$. Suppose there is a
$\sigma_{\omega}$-destabilising sequence

$0\to A\to L\to Q\to 0$

in $\mathrm{Coh}^{\omega}$ for some ample $\mathbb{R}$-divisor $\omega$, such
that

  * •

$\omega\alpha>0$;

  * •

$A$ is $\sigma_{\omega}$-semistable;

  * •

$\phi_{\omega}(A)=\phi_{\omega}(L)$.

Then

$\mathrm{ch}_{0}(A)<\frac{\omega(\alpha-\mathrm{ch}_{1}(Q))}{\omega^{2}(D+\sqrt{D^{2}+1})}$

where

$D:=-\frac{\Re Z_{\omega}(L)}{\Im
Z_{\omega}(L)}=\frac{\alpha^{2}-\omega^{2}}{2\omega\alpha}.$

###### Proof.

From the associated long exact sequence, we know $A$ is a torsion-free sheaf.
Let $F$ be the right-most $\mu_{\omega}$-HN factor of $A$. Then the canonical
surjection $A\to F$ is also a $\mathrm{Coh}^{\omega}$-surjection, and the
$\sigma_{\omega}$-semistability of $A$ gives
$\phi_{\omega}(A)\leq\phi_{\omega}(F)$. Hence we have

$\displaystyle\phi_{\omega}$ $\displaystyle(L)\leq\phi_{\omega}(F)$
$\displaystyle\Leftrightarrow-\frac{\Re Z_{\omega}(L)}{\Im
Z_{\omega}(L)}\leq-\frac{\Re Z_{\omega}(F)}{\Im Z_{\omega}(F)}$
$\displaystyle\Leftrightarrow
D\leq\frac{\mathrm{ch}_{2}(F)-\tfrac{\omega^{2}}{2}\mathrm{ch}_{0}(F)}{\omega\mathrm{ch}_{1}(F)}$

where we set

$D=-\frac{\Re Z_{\omega}(L)}{\Im
Z_{\omega}(L)}=\frac{\alpha^{2}-\omega^{2}}{2\omega\alpha}.$

Since $A$ has nonzero rank, the $\mu_{\omega}$-HN factor $F$ of $A$ also has
nonzero rank. We can then rewrite the above inequality as

(13.1.1)
$D\frac{\omega\mathrm{ch}_{1}(F)}{\mathrm{ch}_{0}(F)}\leq\frac{\mathrm{ch}_{2}(F)}{\mathrm{ch}_{0}(F)}-\frac{\omega^{2}}{2}.$

Note that $\omega\alpha>0$ is equivalent to $\Im Z_{\omega}(L)>0$. Since
$\phi_{\omega}(A)=\phi_{\omega}(L)=\phi_{\omega}(Q)$, we have $\Im
Z_{\omega}=\omega\mathrm{ch}_{1}>0$ for both $L$ and $Q$, implying
$0<\omega\mathrm{ch}_{1}(A)<\omega\alpha$. On the other hand, if we let
$F_{1},\cdots,F_{m}$ denote all the $\mu_{\omega}$-HN factors of $A$ (so
$F=F_{m}$), then $\omega\mathrm{ch}_{1}(F_{i})>0$ for each $i$ and
$\omega\mathrm{ch}_{1}(A)=\sum_{i=1}^{m}\omega\mathrm{ch}_{1}(F_{i})$. As a
result, we have $0<\omega\mathrm{ch}_{1}(F)<\omega\alpha$. We also have
$0<\mathrm{ch}_{0}(F)\leq\mathrm{ch}_{0}(A)$. Now,

$\mathrm{ch}_{2}(F)\leq\frac{\mathrm{ch}_{1}(F)^{2}}{2\mathrm{ch}_{0}(F)}\leq\frac{1}{2\mathrm{ch}_{0}(F)}\frac{(\omega\mathrm{ch}_{1}(F))^{2}}{\omega^{2}}$

where the first inequality follows from the Bogomolov-Giseker inequality for
slope semistable sheaves, and the second inequality follows from the Hodge
Index Theorem. Dividing both sides by $\mathrm{ch}_{0}(F)$, we obtain

$\frac{\mathrm{ch}_{2}(F)}{\mathrm{ch}_{0}(F)}\leq\frac{1}{2\omega^{2}}\mu_{\omega}(F)^{2}.$

Together with (13.1.1), we now have

$\frac{1}{2\omega^{2}}\mu_{\omega}(F)^{2}-D\mu_{\omega}(F)-\frac{\omega^{2}}{2}\geq
0.$

The quadratic polynomial in $\mu_{\omega}(F)$ on the left-hand side has
discriminant

$D^{2}+1>0.$

Since $\mu_{\omega}(F)>0$, this means that

$\mu_{\omega}(A)\geq\mu_{\omega}(F)>\omega^{2}(D+\sqrt{D^{2}+1}).$

Hence

$\mathrm{ch}_{0}(A)<\frac{\omega(\alpha-\mathrm{ch}_{1}(Q))}{\omega^{2}(D+\sqrt{D^{2}+1})}.$

◼

Remark 13.2. In Lemma 13, we have $\omega\mathrm{ch}_{1}(Q)>0$, and so the
bound for $\mathrm{ch}_{0}(A)$ can be slightly weakened to

$\mathrm{ch}_{0}(A)<\frac{\omega\alpha}{\omega^{2}(D+\sqrt{D^{2}+1})}.$

Lemma 13.3. Let $X$ be a smooth projective surface and $\omega$ an ample
$\mathbb{R}$-divisor on $X$. Suppose $L$ is a line bundle on $X$ such that
$\omega c_{1}(L)>0$ and

  * •

Every $\sigma_{\omega}$-destabilising subobject of $L$ in
$\mathrm{Coh}^{\omega}$ is of rank one.

  * •

$L$ is twisted ample with respect to $\omega$.

Then $L$ does not lie on any wall with respect to $\sigma_{\omega}$.

###### Proof.

Consider any $\sigma_{\omega}$-destabilising sequence

(13.3.1) $0\to A\to L\to Q\to 0$

where $\phi_{\omega}(A)=\phi_{\omega}(L)$ and $A$ is of rank 1. From the long
exact sequence of sheaves

$0\to H^{-1}(Q)\to A\overset{\tau}{\to}L\to H^{0}(Q)\to 0$

we see that $H^{-1}(Q)$ must vanish since $\tau$ is nonzero. That is, (13.3.1)
is a short exact sequence of sheaves. From [2, Lemma 4.1], it follows that $Q$
is torsion sheaf and $A=L\otimes I_{Z}(-C)$ for some effective curve $C$ and
some 0-dimensional subscheme $Z$ of $X$. We also know $C$ is a negative curve
from [2, Proposition 5.5]. Writing $n$ to denote the length of $Z$ and
$\alpha$ for $c_{1}(L)$, we have

$\mathrm{ch}(A)=\left(1,\alpha-C,\tfrac{C^{2}}{2}-n-\alpha
C+\tfrac{\alpha^{2}}{2}\right).$

We can now rewrite $\phi_{\omega}(A)=\phi_{\omega}(L)$ as

$\displaystyle-\frac{\Re Z_{\omega}(A)}{\Im Z_{\omega}(A)}=-\frac{\Re
Z_{\omega}(L)}{\Im Z_{\omega}(L)}$
$\displaystyle\Leftrightarrow\left(\frac{\omega^{2}-\alpha^{2}}{2\omega\alpha}\omega+\alpha\right)C=\frac{C^{2}}{2}-n,$

where the left-hand side is positive since $L$ is twisted ample with respect
to, while the right-hand side is negative because $C$ is a negative curve and
$n\geq 0$. That is, we have a contradiction. Hence $L$ cannot lie on any wall
with respect to the stability $\sigma_{\omega}$. ◼

Corollary 13.4. Let $X$ be a smooth projective surface, and suppose $L$ is a
line bundle on $X$. Let $\omega$ be an ample $\mathbb{R}$-divisor on $X$, and
write $\alpha=c_{1}(L)$. Suppose the following inequalities hold

$\displaystyle\omega\alpha>0$ $\displaystyle
2\geq\frac{\omega\alpha}{\omega^{2}(D+\sqrt{D^{2}+1})}$

and that $\alpha$ is twisted ample with respect to $\omega$. Then $L$ is
$\sigma_{\omega}$-stable.

###### Proof.

Let $U$ denote the region in the ample cone consisting of all the ample
divisors $\omega$ satisfying the two inequalities and such that $\alpha$ is
twisted ample with respect to $\omega$. By Lemma 13 and Lemma 13, if
$\omega\in U$, then $L$ does lie on any wall with respect to
$\sigma_{\omega}$.

Now take any $\omega\in U$. It is easy to see that $t\omega$ also lies in $U$
for any $t\geq 1$. That is, the entire half-ray $\\{t\omega\\}_{t\geq 1}$ is
contained in $U$. Since $L$ is $\sigma_{t\omega}$-stable for $t\gg 1$ [30,
Theorem 1.1(ii)], and none of the stability conditions
$\\{\sigma_{t\omega}\\}_{t\geq 1}$ lies on a wall for $L$, it follows that the
entire half-ray $\\{\sigma_{t\omega}\\}_{t\geq 1}$ lies in a chamber of the
Bridgeland stability manifold where $L$ is stable. In particular, $L$ is
$\sigma_{\omega}$-stable. ◼

##  Appendix A Solving a central charge equation

In this section, for given $\mathbb{R}$-divisors $\omega,B$ on a Weirstraß
elliptic K3 surface $X$, we solve the central charge equation

(A.0.1) $Z_{\omega^{\prime},B^{\prime}}^{td}(\Phi(-))=TZ_{\omega,B}(-)$

for some $T\in\mathrm{GL}^{+}\\!(2,\mathbb{R})$ and
$\omega^{\prime},B^{\prime}$. Since $\sqrt{\mathrm{td}(X)}=(1,0,1)$ on a K3
surface,

$\displaystyle Z^{td}_{\omega,B}(F)$
$\displaystyle=-\int_{X}e^{-(B+i\omega)}\mathrm{ch}(F)\sqrt{\mathrm{td}(X)}$
$\displaystyle=-\mathrm{ch}_{2}^{B}(F)+(\tfrac{\omega^{2}}{2}-1)\mathrm{ch}_{0}^{B}(F)+i\omega\mathrm{ch}_{1}^{B}(F)$
$\displaystyle=-\mathrm{ch}_{2}^{B}(F)+(V_{\omega}-1)\mathrm{ch}_{0}^{B}(F)+i\omega\mathrm{ch}_{1}^{B}(F)$

which differs from $Z_{\omega,B}(F)$ by a change a variable in the coefficient
for $\mathrm{ch}_{0}^{B}$. Therefore, to solve the equation (A.0.1), we first
solve the equation

(A.0.2) $Z_{\omega^{\prime},B^{\prime}}(\Phi(-))=TZ_{\omega,B}(-).$

Even though (A.0.2) was solved in [28, Section 10] (see also [29, Appendix
A]), the solution there was written in terms of a coordinate system different
from the RDV coordinates introduced in LABEL:para:RDVcoord. Instead of
applying a change of variables to the solution in [28] (which is cumbersome),
we outline a direct solution which uses the RDV coordinates.

To begin with, note that

$\displaystyle Z_{\omega,B}$
$\displaystyle=-\mathrm{ch}_{2}^{B}+\tfrac{\omega^{2}}{2}\mathrm{ch}_{0}^{B}+i\omega\mathrm{ch}_{1}^{B}$
$\displaystyle=-\mathrm{ch}_{2}^{B}+V_{\omega}\mathrm{ch}_{0}^{B}+iR_{\omega}(\Theta+(D_{\omega}+e)f)\mathrm{ch}_{1}^{B}$
$\displaystyle=-\mathrm{ch}_{2}+R_{B}\Theta\mathrm{ch}_{1}+R_{B}(D_{B}+e)f\mathrm{ch}_{1}+(V_{\omega}-V_{B})\mathrm{ch}_{0}$
$\displaystyle\text{\hskip
85.35826pt}+iR_{\omega}(\Theta\mathrm{ch}_{1}+(D_{\omega}+e)f\mathrm{ch}_{1}-R_{B}(D_{B}+D_{\omega}+e)\mathrm{ch}_{0}).$

Now we set

(A.0.3) $\displaystyle Z^{\prime}_{\omega,B}$
$\displaystyle=\begin{pmatrix}1&-R_{B}/R_{\omega}\\\
0&1/R_{\omega}\end{pmatrix}Z_{\omega,B}$
$\displaystyle=-\mathrm{ch}_{2}+L_{\omega,B}f\mathrm{ch}_{1}+M_{\omega,B}\mathrm{ch}_{0}+i(\Theta\mathrm{ch}_{1}+(D_{\omega}+e)f\mathrm{ch}_{1}+N_{\omega,B}\mathrm{ch}_{0})$

where the terms

$\displaystyle L_{\omega,B}$ $\displaystyle=R_{B}(D_{B}-D_{\omega})$
$\displaystyle M_{\omega,B}$
$\displaystyle=V_{\omega}-V_{B}+R_{B}^{2}(D_{B}+D_{\omega}+e)$ $\displaystyle
N_{\omega,B}$ $\displaystyle=-R_{B}(D_{B}+D_{\omega}+e)$

depend only on $\omega$ and $B$.

Solving (A.0.2) is now in turn equivalent to solving

(A.0.4)
$Z^{\prime}_{\omega^{\prime},B^{\prime}}(\Phi(E))=TZ^{\prime}_{\omega,B}(E)\text{\quad
for all $E\in D^{b}(X)$}.$

For an object $E\in D^{b}(X)$, if we write

$\mathrm{ch}_{0}(E)=n,\,\,\,f\mathrm{ch}_{1}(E)=d,\,\,\,\Theta\mathrm{ch}_{1}(E)=c,\,\,\,\mathrm{ch}_{2}(E)=s,$

then from [3, 6.2.6] we have

$\mathrm{ch}_{0}(\Phi(E)=d,\,\,\,f\mathrm{ch}_{1}(\Phi
E)=-n,\,\,\,\Theta\mathrm{ch}_{1}(\Phi
E)=s-\tfrac{e}{2}d+en,\,\,\,\mathrm{ch}_{2}(\Phi E)=-c-ed+\tfrac{e}{2}n.$

Using this notation for Chern classes, we now have

$Z^{\prime}_{\omega,B}(E)=-s+L_{\omega,B}d+M_{\omega,B}n+i(c+(D_{\omega}+e)d+N_{\omega,B}n)$

while

$Z^{\prime}_{\omega^{\prime},B^{\prime}}(\Phi
E)=-(\tfrac{e}{2}+L_{\omega^{\prime},B^{\prime}})n+(M_{\omega^{\prime},B^{\prime}}+e)d+c+i(-D_{\omega^{\prime}}n+(N_{\omega^{\prime},B^{\prime}}-\tfrac{e}{2})d+s).$

It is then clear that if the following relations are satisfied

(A.0.5)
$-L_{\omega,B}=N_{\omega^{\prime},B^{\prime}}-\tfrac{e}{2},\,\,\,M_{\omega,B}=D_{\omega^{\prime}},\,\,\,D_{\omega}+e=M_{\omega^{\prime},B^{\prime}}+e,\,\,\,N_{\omega,B}=-(\tfrac{e}{2}+L_{\omega^{\prime},B^{\prime}})$

and we take $T=-i=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$, then the equation
(A.0.4) would hold.

The four relations in (A.0.5), when written out in RDV coordinates, correspond
to

(A.0.6) $\displaystyle R_{B}(D_{B}-D_{\omega})$
$\displaystyle=R_{B^{\prime}}(D_{B^{\prime}}+D_{\omega^{\prime}}+e)+\tfrac{e}{2}$
(A.0.7) $\displaystyle V_{\omega}-V_{B}+R_{B}^{2}(D_{B}+D_{\omega}+e)$
$\displaystyle=D_{\omega^{\prime}}$ (A.0.8) $\displaystyle D_{\omega}+e$
$\displaystyle=V_{\omega^{\prime}}-V_{B^{\prime}}+R_{B^{\prime}}^{2}(D_{B^{\prime}}+D_{\omega^{\prime}}+e)+e$
(A.0.9) $\displaystyle-R_{B}(D_{B}+D_{\omega}+e)$
$\displaystyle=-(\tfrac{e}{2}+R_{B^{\prime}}(D_{B^{\prime}}-D_{\omega^{\prime}}))$

respectively. Noting that for any divisor $W$ we have
$V_{W}=\tfrac{1}{2}R_{W}^{2}(2D_{W}+e)$, relation (A.0.7) gives

(A.0.10) $D_{\omega^{\prime}}=V_{\omega}+R_{B}^{2}(D_{\omega}+\tfrac{e}{2}).$

Adding (A.0.6) and (A.0.9) together gives

(A.0.11)
$R_{B^{\prime}}=\frac{R_{B}(-2D_{\omega}-e)}{2D_{\omega^{\prime}}+e}.$

Now (A.0.8) yields

(A.0.12)
$V_{\omega^{\prime}}=D_{\omega}-\frac{R_{B}^{2}(2D_{\omega}+e)^{2}}{2(2D_{\omega^{\prime}}+e)}.$

Also, subtracting (A.0.9) from (A.0.6) (and noting $e=2$) gives

(A.0.13) $R_{B^{\prime}}D_{B^{\prime}}=R_{B}D_{B}+R_{B}-R_{B^{\prime}}-1.$

Finally, we see that when we the RDV coordinates of
$\omega,B,\omega^{\prime},B^{\prime}$ satisfy (A.0.10), (A.0.11), (A.0.13),
and

(A.0.14)
$V_{\omega^{\prime}}=D_{\omega}-\frac{R_{B}^{2}(2D_{\omega}+e)^{2}}{2(2D_{\omega^{\prime}}+e)}+1$

(instead of (A.0.12)), and we choose

$T=\begin{pmatrix}1&-R_{B^{\prime}}/R_{\omega^{\prime}}\\\
0&1/R_{\omega^{\prime}}\end{pmatrix}^{-1}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\begin{pmatrix}1&-R_{B}/R_{\omega}\\\
0&1/R_{\omega}\end{pmatrix},$

the parameters $B,\omega,B^{\prime},\omega^{\prime}$ and $T$ together solve
the central charge equation (A.0.1).

The equations in (9.19.3) are precisely (A.0.10), (A.0.11), (A.0.13), and
(A.0.14).

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