11institutetext: Gordon Center for Medical Imaging, Department of Radiology,
Massachusetts General Hospital and Harvard Medical School, Boston, MA, 02114,
USA22institutetext: Facebook Artificial Intelligence, Boston, MA,
0214233institutetext: Ming Hsieh Department of Electrical and Computer
Engineering, University of Southern California, Los Angeles, CA, 90007, USA
$\textdagger$ Contribute Equally.

# Symmetric-Constrained Irregular Structure Inpainting for Brain MRI
Registration with Tumor Pathology

Xiaofeng Liu 1†1† Fangxu Xing 1†1† Chao Yang 22 C.-C. Jay Kuo 33 Georges El
Fakhri 11 Jonghye Woo 11

###### Abstract

Deformable registration of magnetic resonance images between patients with
brain tumors and healthy subjects has been an important tool to specify tumor
geometry through location alignment and facilitate pathological analysis.
Since tumor region does not match with any ordinary brain tissue, it has been
difficult to deformably register a patient’s brain to a normal one. Many
patient images are associated with irregularly distributed lesions, resulting
in further distortion of normal tissue structures and complicating
registration’s similarity measure. In this work, we follow a multi-step
context-aware image inpainting framework to generate synthetic tissue
intensities in the tumor region. The coarse image-to-image translation is
applied to make a rough inference of the missing parts. Then, a feature-level
patch-match refinement module is applied to refine the details by modeling the
semantic relevance between patch-wise features. A symmetry constraint
reflecting a large degree of anatomical symmetry in the brain is further
proposed to achieve better structure understanding. Deformable registration is
applied between inpainted patient images and normal brains, and the resulting
deformation field is eventually used to deform original patient data for the
final alignment. The method was applied to the Multimodal Brain Tumor
Segmentation (BraTS) 2018 challenge database and compared against three
existing inpainting methods. The proposed method yielded results with
increased peak signal-to-noise ratio, structural similarity index, inception
score, and reduced L1 error, leading to successful patient-to-normal brain
image registration.

###### Keywords:

Brain Tumor Registration Image Inpainting Irregular Structure Symmetry
Contextual Learning Deep Learning

##  1 Introduction

In brain imaging studies, magnetic resonance imaging (MRI) as a noninvasive
tool is widely used to provide information on the brain’s clinical structure,
tissue anatomy, and functional behaviors [28, 4]. When multiple datasets from
a population of interest are involved, to establish a comparable framework in
which similarity and variability in the tissue structure can be evaluated,
deformable image registration between subjects are often used to achieve
inter-subject alignment [37]. Brain tumor is a common type of disorder
diagnosed using medical imaging [35]. However, tumors in MRI tend to cause
difficulties with deformable registration: 1) Tumor regions have no matching
structure in a normal brain, nullifying the basic mathematical assumptions
made for regular image registration methods and subsiding their performance;
2) Expansion of tumor regions often alters its peripheral structure, causing
the whole image to become asymmetric with distorted hemispheres or ventricles;
and 3) The locations of tumors are sometimes irregularly scattered around the
whole brain, causing inconsistencies when matching multiple tumor spots [10].

There has been a great deal of work that tackles patient-to-normal tissue
registration in a traditional way [38, 19]. Especially, for small tumor cases,
Dawant et al. [9] introduced a tumor seed and Cuadra et al. [8] extended it
with a tumor growth model to drive the registration process. For larger
tumors, Mohamed et al. [27] used a biomechanical model of tumor-induced
deformation to generate a similar tumor image from the normal image. Since
then many methods have been focusing on tumor growth simulations to facilitate
symmetry computation [14, 40]. More traditional methods are summarized in
[37]. In this work, we propose a new image inpainting method—i.e., a
restorative method that treats tumor as defective holes in an ideal image and
reconstructs them with synthetic normal tissue. The synthesized brain can be
processed with regular deformable registration and the tumor region will
eventually be re-applied after being mapped to the new space.

Traditional inpainting methods are either diffusion-based or patch-based with
low-level features [3, 2, 5, 7, 12, 32]. These prior approaches usually
perform poorly in generating semantically meaningful contents and filling in
large missing regions [21]. Recently developed learning-based inpainting
methods usually use generative adversarial networks (GANs) to learn image
semantics and infer contents in the missing region [15, 36, 31, 39]. In the
brain tumor application, difficulties 2) and 3) need to be addressed
specifically. Starting from the initial context encoder deep learning method
[31], Liu et al. [20] updated the mask and convolution weights in each layer
to handle irregular holes. However, it is challenging for these 1-step
inpainting solutions to address the large holes with complicated texture. Song
et al. [36] proposed a multi-step framework to refine the results with patch-
swap, but its coarse inpainting module does not fit for multiple irregular
holes. Moreover, the above methods are designed for general image cases and do
not involve priors such as brain anatomy and physiology.

In this work, we propose a novel multi-step inpainting method capable of
making fine-grained prediction within irregular holes with feature patch-wise
conditional refinement. It also incorporates a symmetry constraint to
explicitly exploit the quasi-symmetry property of the human brain for better
structure understanding. Deformable registration is applied between inpainted
patient images and normal controls whose deformation field is then used to
deform original patient data into the target space, achieving patient-to-
normal registration.

##  2 Methods

Given a brain MRI slice $I_{0}$ with tumor, the goal is to replace the
pathological regions with normal brain appearances. The incomplete input
$I_{0}$ is composed of $R$ and $\overline{R}$, representing the removed
pathological region (the hole) and the remaining normal region (boundary or
context), respectively. Mathematically, the task is to generate a new,
complete image $I$ with plausible contents in $\overline{R}$.

Following the basic idea of contextual-based image inpainting [36], our
framework consists of three sequential modules: global perception inference
(GPI), context-aware patch swapping (CPS), and feature-to-image translator
(F2I). The intuition behind the multi-step operation is that direct learning
of the distribution of high dimensional image data is challenging. Thus using
a coarse generation followed by a refinement scheme can increase the
inpainting performance [36]. Our network architecture is shown in Fig. 1.

Figure 1: Overview of the proposed network architecture. GPI is used for
coarse inference and VGG is used for extracting the feature map. The patch-
swap layer propagates high frequency information from the boundary to the
hole. F2I translates to a complete, high-resolution image further constrained
with symmetric loss.

###  2.1 Global Perception Inference

The input to the GPI network $I_{0}$ is a 1$\times$240$\times$240 image with
irregular holes. Its output is a coarse prediction $I_{1}$. Considering the
potential irregular distribution of tumor locations, the rectangular hole
generation module used in [36] is not applicable. Therefore, we first adopt
the GPI network structure from the image-to-image translation network proposed
in [17], which consists of 4$\times$4 convolutions with skip connections in
order to concatenate different features from each encoder layer and the
corresponding decoder layer. We slightly modify the size of each layer since
only single channel T1-weighted MRI is used in this task.

The GPI module is explicitly trained using the $L_{1}$ reconstruction loss,
which is important for stabilizing the adversarial training [23]. It can be
formulated as

$\displaystyle\mathcal{L}_{1}(I_{1},I_{gt})$ $\displaystyle=$
$\displaystyle\parallel I_{1}-I_{gt}\parallel_{1},\vspace{-5pt}$ (1)

where $I_{1}$ and $I_{gt}$ are the rough inpainting result of GPI and the
ground truth, respectively.

The second objective is the adversarial loss based on GANs [24], which can be
defined as:

$\displaystyle\mathcal{L}_{adv}=\max_{D_{1}}\mathbb{E}[\log(D_{1}(I_{0},I_{gt}))+\log(1-D_{1}(I_{0},I_{1}))].\vspace{-5pt}$
(2)

Here, a pair of images are input to the discriminator $D_{1}$ as is the
setting of adversarial training. The incomplete image $I_{0}$ and the original
image $I_{gt}$ are the real pair, and the incomplete image $I_{0}$ and the
prediction $I_{1}$ are the fake pair.

During training, the overall loss function is given by
$\mathcal{L}_{GPI}=\lambda_{1}\mathcal{L}_{1}+\lambda_{2}\mathcal{L}_{adv}$,
where $\lambda_{1}$ and $\lambda_{1}$ are the balancing hyperparameters for
the two losses.

###  2.2 Context-aware Patch Swapping

We use $I_{1}$ as input to the CPS network which is implemented in two phases.
First, $I_{1}$ is encoded as $F_{1}$ by a fully convolutional network (FCN)
using the pre-trained VGG network as in [36]. Then the patch-swap operation is
applied to propagate the texture from $\overline{R}$ to $R$ while maintaining
the high frequency information in $R$ [22].

$r$ and $\bar{r}$ denote the regions in $F_{1}$ corresponding to $R$ and
$\bar{R}$ in $I_{1}$, respectively. For each 1$\times$1 neural patch111In the
inpainting community, the 1$\times$1 patch (in a feature map) is a widely used
concept. The output of F1 $\in\mathbb{R}^{256\times 60\times 60}$, while the
original image is 240$\times$240$\times$1; therefore a 1$\times$1 area in a
feature map is not considered as a pixel. $p_{i}$ of $F_{1}$ overlapping with
$r$, the closest-matching neural patch in $\bar{r}$, indexed by $q_{i}$, is
found using the following cross-correlation metric

$\displaystyle d(p,q)=\frac{<p,q>}{\parallel p\parallel\cdot\parallel
q\parallel}\vspace{-5pt},$ (3)

where $p_{i}$ is replaced by $q_{i}$. We first swap each patch in $r$ with its
most similar patch in $\bar{r}$, followed by averaging overlapping patches.
The output is then a new feature map $F^{\prime}_{1}$. This process is
illustrated in Fig. 2 left.

Figure 2: Illustration of the patch-swap operation (left) and symmetry
constraint (right). Patch-swap is implemented in the FCN-based VGG’s feature
space to search for the most similar boundary $1\times 1$ feature patch with
minimum $d(p,q)$.

###  2.3 Feature-to-image Translator

Next, we use the F2I network to learn the mapping from the swapped feature map
to a complete and vivid image, which has a U-Net style generator. The input to
the U-Net is a feature map extracted by the FCN-based VGG network. The
generator consists of seven convolution layers and eight deconvolution layers,
where the first six corresponding deconvolutional and convolutional layers are
connected through skip connections. The output is a complete
1$\times$240$\times$240 image. In addition, the F2I network comprises a patch-
GAN based discriminator $D_{2}$ for adversarial training. However, the input
to $D_{2}$ is a pair of an image and its feature map in contrast to the GPI
network.

In practice, we follow [36] that uses the ground truth as training input.
Specifically, the feature map $F_{gt}=\mbox{vgg}(I_{gt})$ is the input to the
patch-swap layer followed by using the swapped feature
$F^{\prime}_{gt}=\mbox{patch\\_swap}(F_{gt})$ to train the F2I model.
$F^{\prime}_{1}=\mbox{patch\\_swap}(F_{1})$ is still used as input for
inference, since $I_{gt}$ is not accessible at test time. Of note, using
different types of input for both training and testing is not a common
practice in training a machine learning model. However, its effectiveness in
inpainting has been demonstrated in [36]. Similar to [42], the robustness can
be further improved by sampling from both the ground truth and the GPI
prediction.

The first objective is the perceptual loss defined on the entire image between
the final output $I$ and the ground truth $I_{gt}$:

$\displaystyle\mathcal{L}_{perceptual}(I,I_{gt})=\parallel
vgg(I)-vgg(I_{gt})\parallel_{2}.$ (4)

This perceptual loss has been widely used in many tasks [13, 18, 11, 6] as it
corresponds better with human perception of similarity [41].

The adversarial loss is defined by the discriminator $D_{2}$, which can be
expressed as:

$\displaystyle\mathcal{L}_{adv}=\max_{D_{2}}\mathbb{E}[\log(D_{2}(F^{\prime}_{gt},I_{gt}))+\log(1-D_{2}(F^{\prime}_{gt},I))],$
(5)

where the real and fake pairs for adversarial training are
($F^{\prime}_{gt},I_{gt}$) and ($F^{\prime}_{gt},I$), respectively.

###  2.4 Quasi-symmetry Constraint

While the brain is not exactly symmetrical w.r.t. the mid-sagittal plane,
there is a large degree of symmetry between left and right hemispheres in the
brain which we call the “quasi-symmetry property” [33, 29]. As such, using
this anatomical symmetry constraint on the generated images can mitigate the
ill-posed inpainting task and further improve performance especially for large
hole cases. The symmetry loss is given by

$\displaystyle\mathcal{L}_{sym}(I)=\mathbb{E}\parallel
I_{R}-I_{\hat{R}}\parallel_{2},$ (6)

where $R$ and $\hat{R}$ are the hole and its mirrored region as shown in Fig.
2 right.

Therefore, we can easily transfer the appearance of the normal brain tissue to
the corresponding tumor part by teaching the network to recover the lost
information from the mirrored side. Note that the brains used in our
experiments are coarsely aligned on their mid-sagittal planes. More
importantly, our technique is robust against any potential misalignments,
since the resolution of the feature space is 60$\times$60, while the input is
240$\times$240, with the down-sampling of the maxpooling operation, the deep
neural networks, in general, are robust against small rotation [25]. Besides,
the deep neural network can tackle this simple rotation. With the symmetry
constraint, the overall loss for the F2I translation network is defined as:

$\displaystyle\mathcal{L}_{F2I}=\lambda_{3}\mathcal{L}_{perceptual}+\lambda_{4}\mathcal{L}_{adv}+\lambda_{5}\mathcal{L}_{sym},$
(7)

where $\lambda_{3}$, $\lambda_{4}$, and $\lambda_{5}$ are the balancing
hyperparameters for different losses. Considering the brain is not strictly
symmetrical w.r.t. the mid-sagittal plane, we usually choose a relatively
small weight $\lambda_{5}$ for $\mathcal{L}_{sym}$.

##  3 Experiments and Results

Figure 3: An ablation study of our symmetry constraint and the comparison with
the other inpainting methods.

The proposed method was validated both qualitatively and quantitatively on the
T1 modality of Brain Tumor Segmentation (BraTS) 2018
database222https://www.med.upenn.edu/sbia/brats2018/data.html. From a total of
210 patients each with ~150 slices, we randomly selected 16 patients for
testing and the remaining subjects were used for training in a subject
independent manner. Training was performed on four NVIDIA TITAN Xp GPUs with
the PyTorch deep learning toolbox [30], which took about 5 hours.

The normal slices without tumors in the training set were selected to train
our network. Since tumors in BraTS data can occur in different spatial
locations, our network is capable of familiarizing with the normal appearance
in different slices. We randomly chose the irregular tumor segmentation labels
in our training set as training masks.

The process of computing cross-correlation for all the neural patch pairs
between the hole and the remaining region (e.g., boundary) is computationally
prohibitive. To alleviate this, the strategy in [6, 36] was used to speed up
computation via paralleled convolution. In practice, processing one feature
map only took about 0.1 seconds.

In order to match the absolute value of each loss, we set different weights
for each part. For the training of GPI, we set weight $\lambda_{1}=10$ and
$\lambda_{2}=1$. Adam optimizer was used for training. The learning rate was
set at $lr_{GPI}=1\mathrm{e}{-3}$ and $lr_{D_{1}}=1\mathrm{e}{-4}$ and the
momentum was set at 0.5. When training the F2I network, we set
$\lambda_{3}=10$, $\lambda_{4}=3$ and $\lambda_{5}=1$. For the learning rate,
we set $lr_{F2I}=2\mathrm{e}{-4}$ and $lr_{D_{2}}=2\mathrm{e}{-4}$. Same as
the GPI module, the momentum was set as 0.5.

The inpainting results of various cases are shown in Figs. 3, 4, and 5. The
proposed network can deal with incomplete data from different unseen patients,
different slice positions, and arbitrary shape and number of holes.

Comparisons with the other inpainting methods are shown in Fig. 3. Our
proposed method using context-aware inpainting [36] shows superior performance
over the other methods as visually assessed. In addition, an ablation study to
evaluate the contribution of the symmetry constraint is illustrated in Fig. 3.
Of note, the inpainting quality was further improved using the symmetry
constraint plus marginal training cost without additional testing cost. This
is partly attributed to the use of the context and quasi-symmetry property of
the brain.

Figure 4: Inpainting results comparing to the ground truth. Table 1: Numerical comparison of four methods using BraTS 2018 testing set. Note that smaller mean L1 error and larger SSIM mean error indicate higher similarity. Methods |  mean L1 error $\downarrow$ |  SSIM $\uparrow$ |  PSNR $\uparrow$ | Inception Score $\uparrow$
---|---|---|---|---
Patch-match [3] | 445.8 | 0.9460 | 29.55 | 9.13
GLC [16] | 432.6 | 0.9506 | 30.34 | 9.68
Partial Conv [20] | 373.2 | 0.9512 | 33.57 | 9.77
Proposed | 292.5 | 0.9667 | 34.26 | 10.26
Proposed+symmetry | 254.8 | 0.9682 | 34.52 | 10.58
Figure 5: Deformable registration of two brain tumor subjects to a brain
atlas: direct registration vs. inpainted registration. Tumors are marked in
red boxes. Table 2: Mutual information between registered brain volumes and
the brain atlas on ten test subjects using direct patient registration and
inpainted volume registration.

Methods | Sub1 | Sub2 | Sub3 | Sub4 | Sub5 | Sub6 | Sub7 | Sub8 | Sub9 | Sub10
---|---|---|---|---|---|---|---|---|---|---
Direct registration | 0.303 | 0.311 | 0.308 | 0.324 | 0.315 | 0.299 | 0.309 | 0.303 | 0.317 | 0.308
Inpainted registration | 0.309 | 0.311 | 0.309 | 0.324 | 0.316 | 0.304 | 0.312 | 0.313 | 0.320 | 0.312

For quantitative evaluation, we manually generated holes with random size and
positions on normal slices of the testing subjects. Therefore, the ground
truth is known. The inpainted images were expected to have sharp and realistic
looking textures, be coherent with $\bar{R}$, and look similar to its
corresponding ground truth. Our results are illustrated in Fig. 4. The
proposed method generated visually satisfying results. Table 1 lists numerical
comparisons between the proposed approach, Patch-match [3], GLC [16], and
Partial Conv [20]. We note that the compared inpainting baselines [16, 20] are
based on the 1-step framework. We used four quality measurements to assess the
performance: mean L1 error, structural similarity index (SSIM), peak signal-
to-noise ratio (PSNR), and inception score [34]. We directly computed the mean
L1 error and SSIM over the holes, while the incepetion score is measured on
the completed $I$.

Finally, Fig. 5 and Table 2 show the results of deformable registration using
the ANTs SyN method [1] with normalized cross-correlation as a similarity
metric. As for the target atlas, we used a T1-weighted brain atlas constructed
using healthy subjects from the OASIS database [26]. The result was evaluated
using mutual information (MI) computed only in normal tissues to achieve a
fair comparison (tumor masks were used to exclude the tumor region). Direct
patient-to-normal registration was affected by the existence of tumor, thus
reducing the MI score even in normal tissues. This was corrected by using the
inpainted volume as registration input, yielding improved or equal MI scores
on every subject tested. The mean of MI was improved from 0.3097 to 0.3129.

##  4 Conclusion

This paper presented an inpainting network that replaces the pathological
tumor regions with normal brain appearances, targeting patient-to-normal
deformable registration. The challenges lie in irregular brain tumor
distribution. The two-stage inpainting scheme utilized both the complete and
segmented samples, producing the refined results based on pixel-wise semantic
relevance. Our experimental results demonstrate that the proposed method
surpassed the comparison methods, which can be used for the registration
between healthy subjects and tumor patients.

##  5 Acknowledgements

This work was supported by NIH R01DE027989, R01DC018511, R01AG061445, and
P41EB022544.

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# Analysis of account behaviors in Ethereum during an economic impact event

Pedro Henrique F. S. Oliveira, Daniel Muller Rezende, Heder Soares Bernardino,
Saulo Moraes Villela∗, Alex Borges Vieira  ∗Corresponding author:
<EMAIL_ADDRESS><EMAIL_ADDRESS>Computer Science Department, Federal
University of Juiz de Fora (UFJF), Juiz de Fora, Minas Gerais, Brazil  Glauber
Dias Gonçalves<EMAIL_ADDRESS>Computer Science Department, Federal
University of Piauí (UFPI), Picos, Piauí, Brazil

###### Abstract

One of the main events that involve the world economy in 2022 is the conflict
between Russia and Ukraine. This event offers a rare opportunity to analyze
how events of this magnitude can reflect the use of cryptocurrencies. This
work aims to investigate the behavior of accounts and their transactions on
the Ethereum cryptocurrency during this event. To this end, we collected all
transactions that occurred two weeks before and two weeks after the beginning
of the conflict, organized into two groups: the collection of the accounts
involved in these transactions and the subset of these ones that interacted
with a service in Ethereum, called Flashbots Auction. We modeled temporal
graphs where each node represents an account, and each edge represents a
transaction between two accounts. Then, we analyzed the behavior of these
accounts with graph metrics for both groups during each observed week. The
results showed changes in the behavior and activity of users and their
accounts, as well as variations in the daily volume of transactions. Ethereum,
Flashbots, World Economy, Temporal Graphs

##  1 Introduction

In April 2021, Russian armed forces carried out a military mobilization on the
border of Ukraine, compared to the mobilization for the annexation of the
Republic of Crimea to the Russian Federation, which took place in
2014111https://www.reuters.com/article/us-ukraine-crisis-usa-idUSKBN2BV2Z3.
Almost a year later, on February 24, 2022, Russia carried out a new military
mobilization on the territory of Ukraine, starting a conflict between these
two countries222https://www.cnbc.com/2022/02/24/russian-forces-invade-
ukraine.html. To avoid a further escalation of global tensions, the way that
other countries around the world have sought to interfere is through sanctions
on Russia and economic support for Ukraine333https://www.bbc.com/news/world-
europe-60125659. Events of great magnitude like this tend to involve financial
transactions, which may reflect the use and popularization of
cryptocurrencies, based on blockchain technology, for transfers and
acquisition of digital assets in a decentralized way.

Among the various existing cryptocurrencies, the Ethereum stands out as one of
the leading platforms for trading cryptocurrencies, currently having the
second largest market cap with over 363 billion
dollars444https://coinmarketcap.com/. Additionally, Ethereum is the main
cryptocurrency that includes the functionality of storing and processing
program codes – called smart contracts – within its blockchain.

Faced with this situation, there are still many uncertainties about how such
an armed conflict with impacts on the world economy and financial market can
reflect on crypto-assets. An important question arises in this context: How to
monitor large crypto platforms like Ethereum to observe the possible impact of
events external to these platforms on the behavior of their users in the
shortest possible period of time?

To begin to elucidate this issue, in this work we propose a characterization
of the behavior of Ethereum accounts within a short period of time,
specifically, less than a month. In this regard, we collect all transactions
on Ethereum between two weeks before and two weeks after the conflict started.
Next, we contrast two different views about the behavior of the accounts
involved in these transactions in this period. The first view represents a
general context that includes all accounts that performed transactions in the
four weeks collected. The second view represents a more specific context,
contemplating only accounts with transactions on Ethereum via the service
Flashbots Auction, which is a new communication mechanism between platform
maintainers (miners) and users555https://docs.flashbots.net/.

The choice of the Flashbots service is due to its relevance as a mediating
mechanism for transactions aimed at extracting MEV revenues, that is, the
maximum profits obtained by trading crypto-assets. A relevant example is a
scenario of buying and selling non-fungible tokens (NFTs). In it, users use
techniques to monitor the network and profit from the price differences of
these assets between exchange institutions or exchanges. The analysis of
Flashbots allows us to obtain an approximation of the behavioral
characteristics of this aspect of the platform that has grown considerably in
recent years.

In our characterization of Ethereum accounts with general and specific views,
we explored properties of graphs, where vertices represent accounts and edges
characterize transactions between two accounts. This approach allowed us to
analyze the temporal dynamics of the degree of accounts, considering the value
and number of transactions, as well as the identification of central accounts
with anomalous behavior in a period that involves an event of impact on the
economy. To sum things up, this work has two main contributions: (i) a
strategy to characterize the behavior of users on cryptocurrency platforms
such as Ethereum, which allows the use of various services to trade
cryptocurrencies within the platform, and (ii) the characterization of the
behavior of the accounts due to an event external to the platform of relevance
for analysis. We used the Flashbots service and the current conflict between
Russia and Ukraine as a case study.

The next sections of this article are organized as follows. We present related
works in the Section 2. In Section 3, we describe the collection of data used,
while in Section 4 we describe the methodology for the analysis of these data.
The results obtained are explored in Section 5 and, the conclusions and
discussion of future work are presented in Section 6.

##  2 Related works

Graphs have become an important way of modeling and characterizing
transactions on cryptocurrency platforms. The first graph-based analyzes were
conducted on Bitcoin, which was the first crypto-asset platform conceived,
based on blockchain technology, and is still considered the largest crypto-
asset currently in terms of the number of transactions and value of invested
capital. In [6, 7] graphs were used to model the structure of the Bitcoin
network to identify profiles of influential or suspicious accounts. However,
Bitcoin is considered a first-generation crypto-asset that does not support
smart contracts and the various contract-based services for trading crypto-
assets.

Ethereum, on the other hand, is a second-generation cryptocurrency platform,
in which the use of smart contracts was first conceived. Given the variability
of services for contract-based cryptocurrencies, graph models have been widely
developed for Ethereum. The model proposed in [4] makes use of graphs to
represent creating smart contracts, invoking smart contracts, and performing
external transactions. Through these graphs, the authors are able to make use
of centrality metrics in complex networks to detect important and anomalous
vertices in Ethereum.

Techniques that exploit temporal graphs have also been proposed in the
literature for the analysis of the Ethereum platform. In [1], for example,
features extracted from targeted temporal graphs were used to develop a
predictive model based on machine learning trained to identify profiles of
malicious accounts within this platform. Furthermore, in [9, 2, 8] graph-based
models were used to analyze the evolutionary behavior of Ethereum over time
since its origin in 2014.

Unlike the above-mentioned graph-based works, in this work, we propose
temporal graph models to analyze the Ethereum platform in a short period
before and after a reference event. To our knowledge, our work is the first to
explore graph metrics with this approach. Additionally, we contrast metrics
extracted from a complete temporal graph and its subgraph that respectively
represent the general Ethereum accounts in the period of interest and the
accounts that interacted with the Flashbots service.

It is also important to mention that our work uses Flashbots as an
approximation for the behavior of users on Ethereum that aim to exploit
maximum profits from the speculation of digital assets. In [5, 3] the risks
that the use of crypto-asset arbitrage services such as Flashbots pose to the
security of Ethereum are analyzed, which may negatively affect the functioning
of the platform and the experience of users. However, these works do not
explore graph metrics and temporal analysis with reference to an event of an
impact on the economy.

##  3 Data Description

###  3.1 Data characterization

Russian troops invaded eastern Ukraine on 24/02. From this, we chose to
subdivide our dataset into four weeks. Weeks 1 and 2 are defined by the
initial and final blocks, respectively with the identifiers 14174989 and
14265470, characterizing the pre-invasion period. Moreover, weeks 3 and 4
composed of the initial and final blocks, respectively with the identifiers
14265812 and 14355747, define the post-invasion period.

For the analyses, we chose to characterize the behavior of the network from
two points of view. The first refers to the general context, in which all
transactions that occurred on the network during the collection period are
used as a database. The second addresses a specific context in which we focus
on transactions that interact with the Flashbots Auction service.

Transactions that interact with this mechanism, after undergoing an error
check in a gateway denominated MEV-Relay, are sent directly to miners, making
them invisible on the network until mined. This, in addition to preventing the
occurrence of attacks that try to steal these revenues, contributes to a
better user experience in relation to the expense of fees for failed
transactions, making Flashbots a relevant transaction intermediary mechanism
for our analysis.

Amidst the collected data, we define as transaction volume all the trades
successfully processed on the Ethereum network during the collection period.
Therefore, we define accounts as all individual addresses that participate in
a transaction, whether they are senders or recipients. The set of accounts
that make up our database encompasses both smart contracts and user wallet
addresses, not being necessary to differentiate them for the construction of
account volume analyses. The distinction between contract and user was made by
identifying addresses that do not have a public tag on Etherscan in the
section 5.3.

###  3.2 Form of collection

To form our database, we collected all transactions carried out on the
Ethereum blockchain between 02/10/2022 and 03/10/2022, which belong to the
closed interval defined between blocks 14174989 and 14355747. This time
interval was chosen in order to obtain behavioral information from the network
before and after the Russian invasion of Ukraine. Transactions were collected
through APIs Etherscan.io666https://etherscan.io/ and
blocks.flashbots.net777https://blocks.flashbots.net/. The first was used to
obtain all transactions that took place on the Ethereum network during the
collection period, being the source for obtaining the main characteristics of
each transaction. The second was used to identify Flashbot-type transactions.

Both APIs have different purposes, while Etherscan.io provides all public data
of Ethereum transactions, blocks.flashbots.net provides information aimed at
extracting MEV revenues, such as the amount of ether transferred to coinbase,
transaction fees, and total miner profits. For the purpose of this work, we
focused our efforts on obtaining general data that allows the characterization
of the network without delving into the field of arbitration and extraction of
MEV revenues. Both datasets: general and Flashbots-type transactions, have the
same collected attributes. Table 1 shows the data used for the analyses.

Table 1: Characteristics extracted from Ethereum transactions. Attribute | Description
---|---
hash | Hash code that identifies each transaction
blockNumber | Block number (required for sorting)
timestamp |  Timestamp setting the date the transaction was processed
to | Transaction recipient account
from | Transaction sender account

##  4 Modeling

To perform a network characterization, we initially model our problem using
directed temporal graphs to represent two sets within the main Ethereum
network: the set $C_{1}$, with all external transactions collected in the
period from 02/10 to 03/10 ; and the $C_{2}$ set, with all transactions that
interact with Flashbots Auction within the same time frame.

Let a static graph $G$ be defined as a tuple $(V,A)$, with $V$ as a set of
vertices and $A$ a set of edges, where $A\subseteq\ ${$(u,v)\ |\ u,v\in V$}.
Since $(u,v)$ is an ordered pair, G is also considered a directed graph.
Therefore, a directed temporal graph is defined as a sequence of directed
static graphs, as a function of a discrete variable $t$. Figure 1 shows a
diagram representing a directed temporal graph. The circles represent the
vertices of the graph, being filled in as they are added to the previous
graph.

Figure 1: Example of a directed temporal graph.

In the case of this work, temporal graphs will be used in two different
instances: Analysis of main accounts and Analysis of new accounts.

###  4.1 Analysis of main accounts

In this situation, the discrete variable $t$ represents different weeks on
which we focus our analyses. Therefore, $G_{1}$ represents a graph in the
first week within the analysis interval; that is the week that includes the
first block of the 10/02 to the last block of the 16/02. Likewise, $G_{2}$
forms a range from the first block on 02/17 to the last block on 02/23, and so
on.

To assess the relevance of a given account in a period we focus on the degree
metric. A degree represents in this problem the number of transactions that an
account receives or sends, that is, the number of edges that connect with the
vertex in question. We also separate the indegree metric, which considers only
transactions with the account in question being the destination of the
transaction, and outdegree, considering only transactions with the account in
question being the origin of the transaction.

We understand that high degree accounts or accounts that show high degree
growth from one week to the next show up as relevant accounts in the
short/medium term, and therefore, will be the object of greatest interest in
this work.

###  4.2 Analysis of new accounts

For this scenario, the variable $t$ represents days that have passed since the
first day. Thus, $G_{1}$ represents a graph containing all blocks and
transactions that entered the blockchain on 02/10, $G_{2}$ all blocks and
transactions that entered the blockchain on 02/11, and so on.

The objective of this approach is to analyze the volume of accounts involved
in transactions each day and how many of these accounts had already been
observed with recent activity on the network. To define a starting point, we
collected all accounts that had appeared in transactions on the network
between 01/10 and 02/10 and inserted them as a starting point in $G_{1}$,
adding accounts that had already participated to the temporal graph. of recent
transactions.

Let $N_{t}$ be the number of vertices that the temporal graph has on the day
$t$. For each day $t$, we have that the daily volume of new accounts is
defined by $N_{t}$ – $N_{t-1}$. For day 1, the number of new accounts is
arbitrarily set to $N_{1}$. In this context, the volume of active accounts in
a day $t$ is defined by the number of vertices of $G_{t}$ that have at least
one transaction with timestamp which corresponds to the day $t$.

##  5 Characterization

In this section, we present the results found for the analyzes carried out on
the Ethereum network of transactions in the period described in Section 3. We
used the modeling techniques presented above to analyze the volume of
transactions, and the activity of the accounts and later elaborate on the
ranking of the nodes based on the change of degree between the weeks. As this
is a characterization work, we focused on presenting the results without
worrying about understanding the causes and motivations for them.

###  5.1 Transaction Volume

In Figure 2, we present the results obtained for the volume of network
transactions. For this, we build time charts relating each transaction to the
date on which it occurred to obtain the daily total of trades. As previously
described, we have separated the analyzes into two points of view, the first
as an overview of the network and the second in the context of Flashbots. With
this in mind, it is worth noting that the graphs in Figure 2 are on different
scales, so that the curves of each must be analyzed separately.

(a) Ethereum network transactions

(b) Flashbot Transactions

Figure 2: Daily volume of transactions.

First, we analyze the set $C_{1}$, referring to the general behavior of the
system, represented by Figure 2(a). In it, we can observe a clear difference
between the periods before and after the invasion, in which in mid-February,
the network had large peaks in transactions, the most significant being
approximately 1.35 million close to February 20th. On the other hand, it is
possible to highlight an evident drop in this number after the 20th, causing
the total transactions in one day to reach a value of less than 1.10 million
at the beginning of March.

Subsequently, we performed the same analysis for the set $C_{2}$, referring to
Flashbots transactions, which are represented by Figure 2(b). Unlike the
general context, the set $C_{2}$ shows a growth behavior throughout the
analyzed period, having a more intense fall only between the 10th and 15th of
February, causing the total number of negotiations to reach a value of less
than 20 thousand. It is also worth noting that after the invasion, although
the volume of transactions continued to grow, the fluctuations were more
significant than before February 24, with variations of more than 6 thousand
transactions.

###  5.2 Account Activity

In this topic, we present the results for the analysis of account activity. As
discussed in Section 4, we model the network as a oriented graph in which each
account corresponds to a node. From this, we built time charts to relate the
activity of these nodes over the days of the analyzed period, contrasting the
volume of active accounts with the occurrence of new accounts. All results are
shown in Figure 3.

(a) Ethereum network accounts

(b) Flashbots accounts

Figure 3: Active accounts and new accounts.

For the set $C_{1}$, Figure 3(a), we observed that the activity of the
accounts remained stable, even though we observed a decrease in active
addresses between the 10th and 20th of February, keeping the volume of active
nodes between 450 thousand and 550 thousand. After that, the curve starts to
have a growth trend that started close to the invasion that extends until the
beginning of March. Similarly, the volume of new accounts follows the same
pattern, varying between 200,000 and 300,000 new accounts in mid-February and
later being characterized by growth until the end of the analysis period, with
a more accentuated variation close to the 10th of February. March when the
volume of new nodes approaches 350 thousand.

Regarding the $C_{2}$ set, Figure 3(b), it is noted that despite the
occurrence of a fall of about 5 thousand accounts in relation to the number of
active nodes between February 10th and 15th, the graph maintains the trend of
growth over time, with more intense variations from the day of the invasion.
In this context, it is noted that, as for the $C_{1}$ set, the new accounts
curve follows a similar pattern to the active addresses curve, with the former
showing a significantly smoother growth than the latter, maintaining the
variation at approximately a thousand addresses.

###  5.3 Main Accounts

As discussed in Section 4.1, we model the network through directed graphs.
Figures 4 and 5 show the distribution of values of different inflows and
outflows between accounts during each week. The higher volume of transactions
in the main network database decreases uncertainty within this distribution.

Figure 4: Relation of degree between weeks for Flashbots Figure 5: Relation of
degree between weeks for mainnet Ethereum

Note that the behavior of the blue and orange curves in the graphs in Figure 4
show that some accounts, in a small quantity, have higher outdegree, while the
indegree curve does not reach such high values. Thus, the profile of this
database in this period indicates the presence of accounts distributing many
transactions to different accounts and many accounts receiving few
transactions. This behavior does not change significantly over the weeks.

Observing Figure 5, this pattern between the curves is reversed. In this case,
there appears to be a greater number of accounts with a high indegree, which
reflects slightly different behavior, with more accounts receiving
transactions from many different users. It is important to emphasize that in
this case, the curves are closer to each other, which indicates a more
balanced behavior in this sense. Likewise, the passing of weeks did not
significantly change the behavior of the network.

In addition to a general observation about the grades of the accounts, we
create a ranking in order to find accounts considered relevant for the period
in question. The relevance of an account, in this work, is defined by how much
the degree of an account increases from one week to another, that is, with how
many accounts it started to send and receive transactions.

That said, we performed a four-week comparative analysis, comparing the
accounts that increased their grade the most between one week and the
following week. In order to compress the results, we will only show the 10
most relevant accounts in the period. Additionally, hash addresses for each
account will be shortened to the last seven characters of the address. For
example, the account 0xa090e606e30bd747d4e6245a1517ebe430f0057e will only be
shown as 0f0057e. To facilitate the identification of accounts, we label each
account according to publicly available tags on Etherscan.

Within this collection period, the first two weeks describe the period before
the Russian invasion. Possible effects of pre-conflict tension may be
reflected within this period. Weeks 3 and 4 describe the period after the
invasion, which may contain the effects and consequences of the new situation.

The first analyzed interval was the interval of the first two weeks. It is
observed that, as Table 2 demonstrates, a large number of accounts related to
token trading stand out. X2Y2888https://x2y2.io/, for example, is a
marketplace for NFT exchanges and trades. This also occurs in the context of
Flashbots. As shown in Table 3, the account with the highest degree growth was
OpenSea999https://opensea.io/, which also deals with a market focused on
trading tokens. In addition, the presence of a wallet address without a public
tag on Etherscan is noted, which obtained a degree increase of 1200 units, as
well as the occurrence of other assets such as USDT Stablecoin with
Tether101010https://tether.to/ and DEX aggregator with
1inch111111https://app.1inch.io/. It is also worth noting the appearance of
MEV extraction bots as relevant addresses, given the purpose of Flashbots as a
safe and transparent environment for the extraction of these recipes.

Table 2: Comparasion of accounts between the first and second weeks in the mainnet of Ethereum. Address | Tag | Degree Growth
---|---|---
1abebc9 | X2Y2: X2Y2 Token | 44774
49a41fa | X2Y2: X2Y2 Drop | 27772
a67eed3 | X2Y2: Exchange | 24353
b23c98a | Cat Blox: CATBLOXGEN Token | 17192
c756cc2 | Wrapped Ether | 15267
76a1b85 | X2Y2: Fee Sharing System | 14792
147ea85 | ENS: Base Registrar Implementation | 12849
903a5d0 | The Sandbox: SAND Token | 12720
5fd74c0 | Contract: No public tag | 12403
def3c7a | Contract: No public tag | 11541
Table 3: Comparasion of accounts between the first and second weeks in the Flashbots subset. Address | Tag | Degree Growth
---|---|---
bB538E5 | OpenSea: Wyvern Exchange v2 | 3787
5843dE9 | Usuário: Sem tag pública | 1200
991e2D4 | MEV Bot: 0x000…2D4 | 833
7455580 | MEV Bot: 0x8aF…580 | 768
57Fb793 | Wintermute 1 | 597
643097d | 1inch v4: Router | 335
140449c | Contract: No public tag | 325
5206C3E | Contract: No public tag | 287
3f7c613 | Contract: No public tag | 282
D831ec7 | Tether: USDT Stablecoin | 259

Then, a comparison was made between weeks 2 and 3 of the collection period.
This analysis allows us to observe transformations the Ethereum network had
during the period of the Russian invasion, comparing the week immediately
before and the first week after the fact. It can be noted, comparing Table 2
with Table 4, that the previously high quantity of tokens gave rise to a
greater variety of account profiles. Coinbase exchange starts to stand out a
little more. There is also a greater presence of contracts related to other
assets, such as USDT Stablecoin or USD Coin. Finally, the account 7455580 also
displays a bot aimed at MEV extraction activities.

Regarding Flashbots, comparing the Tables 3 and 5, it is possible to highlight
the increase in the number of MEV bots among the most relevant nodes, as well
as the appearance of addresses without public tags on the network.
Furthermore, it is worth noting that Ethermine121212https://ethermine.org/
appears in the first position in Table 5 as the node with the highest degree
growth, with this growth being almost 55 thousand units higher than the
account that occupies the second position in the table.

Table 4: Comparasion of accounts between the second and third weeks in the mainnet of Ethereum. Address | Tag | Degree Growth
---|---|---
0f0057e | Coinbase: Miscellaneous | 106853
bb538e5 | OpenSea: Wyvern Exchange v2 | 101837
d831ec7 | Tether: USDT Stablecoin | 48732
606eb48 | Centre: USD Coin | 35541
8d01279 | Celsius Network: Wallet 5 | 23776
7975ef8 | OogaVerse: MekaApes Game Contract | 16276
b898ec8 | Ethermine | 15246
7455580 | MEV Bot: 0x8aF…580 | 13650
dc49699 | Coinbase 4 | 13600
bf77006 | Contract: No public tag | 12533
Table 5: Comparasion of accounts between the second and third weeks in the Flashbots subset. Address | Tag | Degree Growth
---|---|---
B898ec8 | Ethermine | 56624
991e2D4 | MEV Bot: 0x000…2D4 | 1881
0a3f8e7 | MEV Bot: 0x4Cb…8e7 | 1568
C378B9F | SushiSwap: Router | 1324
5645718 | User: No public tag | 1195
e416B40 | MEV Bot: 0x000…B40 | 1191
0fDd6CF | MEV Bot: 0xa57…6CF | 1117
7455580 | MEV Bot: 0x8aF…580 | 724
01Db497 | User: No public tag | 703
08850aF | User: No public tag | 666

The two last weeks form the last analysis window, comparing weeks after the
invasion. Looking at Table 6, some characteristics remain compared with Table
4, such as the presence of MEV extraction bots and the still high relevance of
accounts such as Ethermine and Coinbase: Miscellaneous. Token accounts
reappear with relevance, as in the first two weeks, which can be seen, for
example, with the growth of the Uniswap protocol, which can be used to
exchange and acquire tokens131313https://uniswap.org/faq. Another new
observation in this window is the presence of accounts with greater relevance
that do not have public identification tags on Etherscan. The same behavior is
observed with Flashbots which, as shown in Table 7, keep MEV bots among the
nodes with the highest degree growth, in addition to the greater relevance of
nodes focused on token trading, such as the previously mentioned Uniswap.

Table 6: Comparasion of accounts between the third and fourth weeks in the mainnet of Ethereum. Address | Tag | Degree Growth
---|---|---
33ab239 | Livepeer: LPT Token | 95535
0f0057e | Coinbase: Miscellaneous | 76216
97b3fca | Contract: No public tag | 56225
665fc45 | Uniswap V3: Router 2 | 52794
b898ec8 | Ethermine | 21609
9f2488d | Uniswap V2: Router 2 | 19930
500fb08 | Contract: No public tag | 17267
d831ec7 | Tether: USDT Stablecoin | 16330
0a3f8e7 | MEV Bot: 0x4Cb…8e7 | 12988
6155ba9 | KPOP CTzen: KCPT1 Token | 11093
Table 7: Comparasion of accounts between the third and fourth weeks in the Flashbots subset. Address | Tag | Degree Growth
---|---|---
B898ec8 | Ethermine | 3950
0a3f8e7 | MEV Bot: 0x4Cb…8e7 | 2986
40f594e | MEV Bot: 0x000…94e | 1398
9F2488D | Uniswap V2: Router 2 | 1306
3923BcA | User: No public tag | 922
9e5D3Fa | Contract: No public tag | 636
80E83b6 | Gelato Network: Gelato | 444
665Fc45 | Uniswap V3: Router 2 | 421
643097d | 1inch v4: Router | 409
A7fAc82 | User: No public tag | 333

##  6 Conclusion

In this work, we characterize the Ethereum network in a general and specific
way, building an evolutionary model of the network and performing quantitative
and qualitative analyzes on its accounts. As a result, we sought to quantify
the active nodes on the network and, finally, isolated and analyzed the most
relevant accounts during that period.

Our investigations indicate an initial path for monitoring the possible
effects of important external events on a large cryptocurrency platform. It
was possible to observe short-term changes concerning the behavior of accounts
and transactions on the network, in addition to the contrast between the
behavior of transactions on the general network and transactions carried out
with Flashbots Auction.

It is important to emphasize that this work seeks only to make a
characterization concentrated in a period and that these analyses verify a
short-term behavior. Therefore, it is not possible to make cause-and-effect
inferences regarding the invasion just by looking at these specific details.

For future works, we intend to use wider temporal windows and analyze in more
detail the most influential nodes to enable more assertive causal inferences.
Another focal point is to expand the investigation of the impact of external
events on other past events that have economic relevance.

## Acknowledgments

The authors acknowledge the financial support of CNPq, FAPEMIG, Capes and
UFJF.

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y\otimes x]$ and $[y\otimes x\otimes x]-[x\otimes y\otimes x]$ are algebraic.
Since these two elements are in the same orbit under the action of
$\textnormal{Aut}(A)$, it suffices to show that one of them is algebraic. This
holds, because we have

$\displaystyle x\otimes y-y\otimes
x=c_{1}({\mathcal{P}}_{E})\in{\mathrm{H}}^{1}(E(\mathbb{C}),\mathbb{Z})\otimes{\mathrm{H}}^{1}(E(\mathbb{C}),\mathbb{Z})\subset{\mathrm{H}}^{2}(E^{2}(\mathbb{C}),\mathbb{Z}),$

where ${\mathcal{P}}_{E}$ is the Poincaré bundle of $E$ viewed as a line
bundle on $E\times E$ via the principal polarization
$E\xrightarrow{\sim}E^{\vee}$, and because
$[x]\in{\mathrm{H}}^{1}(G,{\mathrm{H}}^{1}(E(\mathbb{C}),\mathbb{Z}(1)))$ is
algebraic. ∎

##  10 Connections with the Griffiths group of an abelian threefold

For a smooth projective variety $X$ over $\mathbb{C}$, let
${\mathrm{Griff}}^{2}(X)=\textnormal{CH}^{2}(X)_{{\textnormal{hom}}}/\textnormal{CH}^{2}(X)_{{\textnormal{alg}}}$
be the group of homologically trivial codimension-two cycles on $X$ modulo the
algebraically trivial ones. Non-zero elements in ${\mathrm{Griff}}^{2}(X)$ are
difficult to find. For a very general complex abelian threefold $A$, the
groups ${\mathrm{Griff}}^{2}(A)\otimes\mathbb{Q}$ and
${\mathrm{Griff}}^{2}(A)\otimes\mathbb{Z}/\ell$ are infinite for every prime
number $\ell$ [Nor89, Sch02, RS10, Tot16]. These statements rely on non-
vanishing of Ceresa cycles on Jacobians of very general curves of genus three
[Cer83, Hai95, RS10, Tot16]. In this section, we use different methods to
study the subgroup ${\mathrm{Griff}}^{2}(A)^{G}\subset{\mathrm{Griff}}^{2}(A)$
when $A$ is defined over $\mathbb{R}$, relating this group to the real
integral Hodge conjecture.

For the Jacobian $J(C)$ of a smooth genus three curve $C$ over $\mathbb{R}$
with $p\in C(\mathbb{R})$, one can consider the _Ceresa cycle_
$\Xi_{p}=\iota_{p}(C)-[-1]^{\ast}\iota_{p}(C)$. Here, $\iota_{p}\colon C\to
J(C)$ is the Abel-Jacobi embedding induced by $p$. The resulting class
$[\Xi]\in{\mathrm{Griff}}^{2}(J(C)_{\mathbb{C}})^{G}$ is independent of $p$.

###### Proof of Theorem 1.11.

Let $A$ be a real abelian threefold, and consider the second intermediate
Jacobian $J^{2}(A_{\mathbb{C}})$ of $A_{\mathbb{C}}$ (see Section 7). Let
$J^{2}(A_{\mathbb{C}})_{{\textnormal{alg}}}$ be the algebraic part of
$J^{2}(A_{\mathbb{C}})$ and
$J^{2}(A_{\mathbb{C}})_{tr}=J^{2}(A_{\mathbb{C}})/J^{2}(A_{\mathbb{C}})_{{\textnormal{alg}}}$
its transcendental part [Voi02, §2.2.2]. We get an exact sequence of weight
$-1$ Hodge structures
$0\to\Lambda_{\textnormal{alg}}\to\Lambda\to\Lambda_{{\textnormal{tr}}}\to 0$;
by Theorem 7.2 and [Voi02, Corollaire 12.19], we obtain a commutative diagram
with exact rows:

$\displaystyle\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
31.09042pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\left(\textnormal{CH}^{2}(A_{\mathbb{C}})_{\textnormal{alg}}\right)^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
55.09042pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-19.62119pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
55.09042pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\left(\textnormal{CH}^{2}(A_{\mathbb{C}})_{{\textnormal{hom}}}\right)^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
87.89197pt\raise-19.62119pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
144.69351pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
144.69351pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathrm{Griff}}^{2}(A_{\mathbb{C}})^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
171.28255pt\raise-19.62119pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathrm{H}}^{1}(G,\Lambda_{{\textnormal{alg}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
66.22739pt\raise-31.25894pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
66.22739pt\raise-31.25894pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathrm{H}}^{1}(G,\Lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
146.04352pt\raise-31.25894pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
146.04352pt\raise-31.25894pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathrm{H}}^{1}(G,\Lambda_{{\textnormal{tr}}}).}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(65)

Define ${\mathrm{H}}^{1}(G,\Lambda)_{0}\subset{\mathrm{H}}^{1}(G,\Lambda)$ as
in Definition 6.1. The real integral Hodge conjecture for $A$ would imply that
${\mathrm{Griff}}^{2}(A_{\mathbb{C}})^{G}\otimes\mathbb{Z}/2\to{\mathrm{H}}^{1}(G,\Lambda_{{\textnormal{tr}}})$
surjects onto the image of

$\displaystyle{\mathrm{H}}^{1}(G,\Lambda)_{0}\to{\mathrm{H}}^{1}(G,\Lambda_{{\textnormal{tr}}}).$
(66)

Our goal is to show that if $A$ is an abelian threefold as in the statement of
the theorem, then (66) is non-zero, whereas if $A=J(C)$ is the Jacobian of a
real genus three curve with $p\in C(\mathbb{R})$ then the Ceresa cycle
$[\Xi_{p}]\in\textnormal{CH}^{2}(J(C))_{{\textnormal{hom}}}$ maps to zero in
${\mathrm{H}}^{4}_{G}(J(C)(\mathbb{C}),\mathbb{Z}(2))$.

Let us first prove the latter statement. Assume that $C$ is hyperelliptic and
let $p\in C(\mathbb{R})$ be a real Weierstrass point. We have
$[\Xi_{p}]=0\in\textnormal{CH}^{2}(A)$, and for arbitrary $q\in C(\mathbb{R})$
there exists $Q\in J(C)(\mathbb{R})$ such that the translation by $Q$ map
$\tau_{Q}\colon J(C)\to J(C)$ satisfies $\tau_{Q}\circ\iota_{p}=\iota_{q}$.
For any abelian threefold $B$ over $\mathbb{R}$, any cycle
$\Gamma\in\textnormal{CH}^{2}(B)$ and any $Q\in B(\mathbb{R})^{0}$, the cycle
$\tau_{Q}^{\ast}(\Gamma)-\Gamma\in\textnormal{CH}^{2}(B)$ is real
algebraically equivalent to zero (see [Ham97, Definition 1.5]). In particular,
$[\tau_{Q}^{\ast}(\Gamma)]=[\Gamma]\in{\mathrm{H}}^{4}_{G}(B(\mathbb{C}),\mathbb{Z}(2))$
by [Ham97, Lemma 1.2]. In our case, this gives

$0=(\tau_{Q})_{\ast}[\Xi_{p}]=(\iota_{q})_{\ast}(C)-(\tau_{2Q})_{\ast}(-1)^{\ast}(\iota_{q})_{\ast}(C)=[\Xi_{q}]\in{\mathrm{H}}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2)).$

Next, suppose that $C$ is a non-hyperelliptic curve of genus three over
$\mathbb{R}$ equipped with $p\in C(\mathbb{R})$. Let
$K\subset{\mathcal{M}}_{3}(\mathbb{R})$ be the connected component containing
$[C]\in{\mathcal{M}}_{3}(\mathbb{R})$ (see Section 5.4). Curves in $K$ have
non-empty real locus, and $K$ contains hyperelliptic curves by [Cir06,
Proposition 3.4]. In particular (see [SS89], [GF22a, Proof of Theorem
1.3(B)]), there exists a contractible complex manifold $B^{\prime}$, a family
of genus three curves ${\mathscr{C}}^{\prime}\to B^{\prime}$ with a real
structure, and two points $a,b\in B^{\prime}(\mathbb{R})$ such that
$B^{\prime}(\mathbb{R})$ is connected, ${\mathscr{C}}^{\prime}_{a}\cong C$ and
${\mathscr{C}}^{\prime}_{b}\eqqcolon C_{2}$ is hyperelliptic.

Consider the family of genus three curves
$\pi\colon{\mathscr{C}}^{\prime}\times_{B^{\prime}}{\mathscr{C}}^{\prime}\eqqcolon{\mathscr{C}}\to
B\coloneqq{\mathscr{C}}^{\prime}$. This family $\pi$ has a canonical real
structure and equivariant section $\Delta\colon B\to{\mathscr{C}}$. By [GF22a,
Lemma 5.1], the real structure on $\pi$ extends to a real structure on the
relative Jacobian $J({\mathscr{C}})$ of ${\mathscr{C}}$ over $B$; moreover,
$\Delta$ induces a $G$-equivariant map $\Psi\colon{\mathscr{C}}\to
J({\mathscr{C}})$ over $B$.

Note that ${\mathscr{C}}^{\prime}(\mathbb{R})=B(\mathbb{R})\neq\emptyset$
because $C(\mathbb{R})\neq\emptyset$. Let $f\colon[0,1]\to B(\mathbb{R})$ be a
path whose image in $B^{\prime}(\mathbb{R})$ connects $a$ and $b$, and let
$q=f(1)\in B(\mathbb{R})$. Then $p$ and $q\in B(\mathbb{R})$ lie in the same
component $K\subset B(\mathbb{R})$. The family of Abel-Jacobi maps
$\Psi|_{K}\colon{\mathscr{C}}|_{K}\to J({\mathscr{C}})|_{K}\to K$ is locally
trivial over $K$, so there are $G$-equivariant Lie group diffeomorphisms

$\rho\colon J(C)\cong J({\mathscr{C}}_{p})\cong J({\mathscr{C}}_{q})\cong
J(C_{2})\;{\textnormal{ with
}}\;0=\rho^{\ast}[\iota_{q}(C_{2})]=[\iota_{p}(C)]\in{\mathrm{H}}^{4}_{G}(J(C)(\mathbb{C}),\mathbb{Z}(2)).$

We conclude that Ceresa cycles map to zero in equivariant cohomology as
desired.

Let $(A,\lambda)$ be any polarized complex abelian threefold. The embedding of
weight $-1$ Hodge structures
${\mathrm{H}}^{1}(A(\mathbb{C}),\mathbb{Z}(1))\to{\mathrm{H}}^{3}(A(\mathbb{C}),\mathbb{Z}(2))$
induced by the polarization $\lambda$ has saturated image, thus gives an
embedding of complex tori
${\widehat{A}}(\mathbb{C})=\textnormal{Pic}^{0}(A)(\mathbb{C})\to J^{2}(A)$.

######  Lemma 10.1.

If $(A,\lambda)$ is very general, then
$J^{2}(A)_{\textnormal{alg}}={\widehat{A}}$.

###### Proof.

By [PS03, Theorem 17], it suffices to show that the representation of
${\textnormal{GSp}}_{6}(\mathbb{Q})$ on
${\mathrm{H}}^{3}(A(\mathbb{C}),\mathbb{Q})_{{\textnormal{prim}}}$ is
irreducible. This follows from [FH91, Theorem 17.5]. ∎

Let ${\mathcal{A}}_{3,\delta}$ be the moduli stack of polarized abelian
threefolds of polarization type $\delta$, with coarse moduli space
$\pi\colon{\mathcal{A}}_{3,\delta}\to{\mathsf{A}}_{3,\delta}$. By Lemma 10.1,
there is a countable union
$\cup_{i}Z_{i}\subsetneq{\mathsf{A}}_{3,\delta}(\mathbb{C})$ of complex
analytic subvarieties $Z_{i}\subset{\mathsf{A}}_{3,\mathbb{C}}$ with the
following property: if the image of a point
$[(A,\lambda)]\in{\mathcal{A}}_{3,\delta}(\mathbb{R})$ under the map
$\pi_{\mathbb{R}}\colon{\mathcal{A}}_{3,\delta}(\mathbb{R})\to{\mathsf{A}}_{3,\delta}(\mathbb{R})\subset{\mathsf{A}}_{3,\delta}(\mathbb{C})$
does not lie in $\cup_{i}Z_{i}$, then
$J^{2}(A_{\mathbb{C}})_{{\textnormal{alg}}}={\widehat{A}}(\mathbb{C})$. Let
$(A,\lambda)$ be such a polarized abelian threefold over $\mathbb{R}$, and
assume $A(\mathbb{R})$ is not connected. Via the spectral sequence (6),
Corollary 9.2 and Claim 9.6 imply that the rank of
${\mathrm{H}}^{1}(G,{\mathrm{H}}^{1}(A(\mathbb{C}),\mathbb{Z}(1)))$ is smaller
than the rank of
${\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(A(\mathbb{C}),\mathbb{Z}(2)))_{0}$. Thus,
there are elements in ${\mathrm{H}}^{1}(G,\Lambda)_{0}$ which are not in the
image of
${\mathrm{H}}^{1}(G,\Lambda_{{\textnormal{alg}}})\to{\mathrm{H}}^{1}(G,\Lambda)$,
see diagram (65). In particular, the map (66) is non-zero, and Theorem 1.11
follows. ∎

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Olivier de Gaay Fortman, Institute of Algebraic Geometry, Leibniz University
Hannover, Welfengarten 1, 30167 Hannover, Germany

E-mail address<EMAIL_ADDRESS>