# Flexible Variable-Rate Image Feature Compression for Edge-Cloud Systems

Md Adnan Faisal Hossain, Zhihao Duan, Yuning Huang, Fengqing Zhu  Elmore
Family School of Electrical and Computer Engineering
Purdue University
West Lafayette, IN, 47906, USA
{hossai34, duan90, huan1781<EMAIL_ADDRESS>
###### Abstract

Feature compression is a promising direction for coding for machines. Existing
methods have made substantial progress, but they require designing and
training separate neural network models to meet different specifications of
compression rate, performance accuracy and computational complexity. In this
paper, a flexible variable-rate feature compression method is presented that
can operate on a range of rates by introducing a rate control parameter as an
input to the neural network model. By compressing different intermediate
features of a pre-trained vision task model, the proposed method can scale the
encoding complexity without changing the overall size of the model. The
proposed method is more flexible than existing baselines, at the same time
outperforming them in terms of the three-way trade-off between feature
compression rate, vision task accuracy, and encoding complexity. We have made
the source code available at github.com/adnan-hossain/var-feat-comp.

###### Index Terms:

Variable-rate feature compression, coding for machines, image classification,
edge-cloud system

##  I Introduction

The use of Artificial Intelligence (AI) for applications such as the Internet
of Things (IoT), visual surveillance, and autonomous drones often calls for a
need to deploy mobile or wearable devices that can process data in real-time.
With the advent of light-weight machine learning models [1, 2] and small
graphical processing units(GPUs) specifically designed for mobile devices
there has been a general trend towards the use of machine learning models on
these mobile devices. However, often it is necessary to deploy more powerful
deep neural networks to perform complex tasks (e.g., object detection) or to
enhance task performance. This raises an issue, as computational and energy
constraints on small mobile devices limit their ability to host these powerful
models.

A common approach to solve this dilemma is to transfer data from the edge
(mobile or wearable devices) to the cloud (a local server or a data center)
and execute the AI models on the cloud instead, forming an edge-cloud system.
Typically, for visual data such as images and videos, a pre-designed
image/video codec is adopted on the edge device to compress the data before
transmission. However, most existing codecs(either handcrafted [3, 4] or
learned [5, 6, 7, 8]) are generally designed to reconstruct images with good
visual quality and hence are not optimized for vision tasks [9]. This work
instead focuses on feature compression [10, 11, 12, 13, 9], which has the main
advantage of being able to improve the compression performance for the
specific vision task the compression model is trained on.

(a) Model architecture during training

(b) Model architecture during deployment

Figure 1:  Overview of our method. We compress the intermediate features of a
pre-trained vision task model using a variable-rate compressive autoencoder,
by which we achieve flexible feature compression for an edge-cloud system.

(a) Partition of $Convx$ blocks in the ResNet-50 model.

(b) Partition of $Convx$ blocks in the ConvNext-T model.

Figure 2: The general framework for partitioning a classifier model into
subsequent $Convx$ blocks. The encoding latency is varied by changing the
number of $Convx$ blocks at the front end. Note that the network architectures
of the Residual Blocks are different for the ResNet-50 and ConvNext-T models.

Developing feature compression methods for edge-cloud systems requires
consideration of the resource-constrained nature of these systems. First, the
bandwidth-limited channel between the edge and the cloud posits a constraint
on the transmission bit rate. On the other hand, low bit rates are associated
with performance degradation for the vision task performed on the cloud.
Finally, due to the variation in computational resources of edge devices,
encoders with varying complexity are desired to adapt to different edge
devices. Taking these factors into consideration gives rise to a three-way
trade-off between rate, accuracy, and complexity (RAC). This trade-off is
addressed in [9] by introducing a flexible encoder strategy that can vary in
encoding latency by changing the encoder depth. Although the method in [9] can
be adapted to different edge devices considering their computational
resources, the bandwidth limit is not elegantly addressed since separate
models need to be trained for operating on different compression rates. In
this work, we tackle the problem by taking inspiration from the variable-rate
image compression model of [14] to develop a method to perform variable-rate
feature compression in edge-cloud systems.

We present a novel method, overviewed in Fig. 1, to address the RAC trade-off
in feature compression for edge-cloud systems. Our contributions can be
summarized as:

  * •

We propose a variable-rate feature compression method for edge-cloud systems.

  * •

We design a dynamic method for distributing a feature compression model
between an edge and a cloud device based on specific application scenarios.

  * •

We show improved rate-accuracy-complexity trade-off performance compared to
previous methods.

##  II Preliminaries

In this section, we discuss a general framework of learned feature compression
for edge cloud systems based on previous works on split computing [11, 15] and
learned feature compression [12, 13, 9]. As shown in Fig. 1, a network that
has already been trained to perform a certain vision task is split up into a
front-end that resides on the edge and a back-end that resides on the cloud
using an autoencoder. An entropy model learns to compress the latent feature
generated by the encoder network of the autoencoder while the autoencoder is
simultaneously trained on a machine vision task, e.g., image classification in
our case. The overall training loss is adopted from [10] and is described as
follows:

$L=l_{CE}(\hat{y},y)+\lambda l_{R}(\hat{z})$ (1)

where $\hat{y}$ is the output prediction, $y$ is the true label and
$l_{CE}(\hat{y},y)$ is the cross-entropy loss for image classification. In
general, this loss can be replaced by the objective loss function for other
machine vision tasks. $z$ is the output of the encoder which is also referred
to as the bottleneck or latent feature, and $\hat{z}$ is its quantized
version. The second component of the loss function is the data-rate or entropy
loss $l_{R}(\hat{z})$. $\lambda$ is the Lagrangian multiplier that controls
the trade-off between rate and accuracy [10].

We choose the factorized entropy model from [5] as our entropy model and use
it to estimate the entropy loss $l_{R}(\hat{z})$. As quantization is a non-
differentiable process, during training $\hat{z}$ is estimated by sampling
from a uniform distribution of width $1$ centered around the value of $z$.
$l_{R}(\hat{z})$ is defined as:

$l_{R}(\hat{z})=\mathbb{E}\left[-\log_{2}p(\hat{z};\phi)\right]$ (2)

where $\phi$ represents the parameters of the entropy model, and the
expectation is w.r.t. the marginal distribution of $\hat{z}$, which is
determined by the training dataset and the encoder.

The goal of the model optimization process is to minimize the loss $L$ to
obtain the optimum $\theta^{*}$ and $\phi^{*}$, where $\theta$ denotes the
parameters of the autoencoder. This is formulated as:

$\theta^{*},\phi^{*}=\operatorname*{arg\,min}_{\theta,\phi}\sum_{x,y\in
D}l_{CE}(\hat{y},y)+\lambda l_{R}(\hat{z})$ (3)

Here, $x$ and $y$ are the input and labels sampled from the training set $D$.
For image compression, $x$ consists of training images, but for feature
compression $x$ consists of features generated by a learned feature generator
such as a neural network. $x$ is passed on to the encoder where it is used to
generate the latent feature $z$ according to (4). The entropy model uses $z$
to create its quantized version $\hat{z}$ and the prior probability
distribution $p(\hat{z})$. $\hat{z}$ is passed to the decoder to generate the
reconstructed feature $\hat{x}$ according to (5).

$\displaystyle z$ $\displaystyle=$ $\displaystyle g_{z}(x;\theta_{enc})$ (4)
$\displaystyle\hat{x}$ $\displaystyle=$ $\displaystyle
f_{\hat{x}}(\hat{z};\theta_{dec})$ (5)

During inference, the entropy model is shared between the edge and cloud
device. On the edge, the entropy model uses the distribution $p(\hat{z})$ to
encode the latent feature into bits using the range based asymmetric numeral
system (rANS) algorithm [16] and then transmits the bits to the cloud. On the
cloud, the entropy model transforms the bits back to $\hat{x}$.

##  III Method

In order to develop a flexible method for distributing the feature compression
model between the edge and the cloud, we first analyze the constraints that
govern the decision of choosing the split point (the point in the network
where subsequent layers are sent to the cloud and preceding layers are kept in
the edge) in a typical neural network-based image classifier, e.g., ResNet-50
[17] in Sec. III-A. Based on these constraints, a general strategy for
splitting up an image classification model between the front-end and the back-
end as shown in Fig. 1 is described in Sec. III-B. Finally, we describe the
design of our autoencoder in Sec. III-C and explain our variable-rate feature
compression method in Sec. III-D

###  III-A Constraints on Choosing the Split Point

Choosing the split point depends on factors such as transmission bandwidth,
machine vision task performance and encoding latency. Limited transmission
bandwidth requires that a split point be chosen such that the transmitted
features are coded with the minimum possible bit rate. However, improving
performance accuracy often comes at the cost of coding the data using higher
bit rates to reduce information loss. In general for a given depth of the
front-end architecture, this trade-off can be tuned by varying the value of
the parameter $\lambda$. At the same time, for a fixed $\lambda$ and accuracy,
increasing the depth of the front-end architecture should improve bit rate
performance as deeper layers are more compressible [18] and contain more
usable information [19]. On the other hand, in many real-time application
scenarios, it is often desirable to have minimum encoding latency on the edge
device which can be achieved by selecting a shallower split point.

###  III-B Strategy for Splitting the Network

Based on the discussion in Sec. III-A, we propose a general strategy for
splitting the classification network. As most modern classification networks
consist of groups of similar residual blocks cascaded together [17, 20, 21],
the proposed strategy can be generalized. The process is described as follows:

  1. 1.

The classification network is split into groups of similar residual layers
[17]. These groups of similar residual layers are referred to as bottleneck
blocks.

  2. 2.

The residual layers of each bottleneck block are split into groups of two or
three depending on the total number of layers in the block. Each of these
groups is called a Convx block.

  3. 3.

The number of Convx blocks in the front-end can be varied. More Convx blocks
leads to better rate-accuracy performance due to a deeper front-end
architecture while fewer Convx blocks reduce the encoding latency.

In Fig. 2(a), we apply our strategy to split up the ResNet-50 model from [17]
which we use to show the RAC performance of our proposed method in Sec. IV. We
also apply our strategy to split up the ConvNext-Tiny model from [20] in Fig.
2(b) to demonstrate the generalizability of our method.

In order to achieve a rate-accuracy-complexity (RAC) trade-off, $\lambda$ is
varied to tune between bit-rate and classification accuracy, and the encoding
complexity is varied by changing the number of Convx blocks in the front-end.
In the latter part of this discussion, three model configurations are tested
and each configuration is referred to as Config.$k$ where $k$ is the number of
Convx blocks in the front-end. The general model configuration is shown in
Fig. 3.

Figure 3: The proposed model configuration Config.$k$, where $k$ is the number
of $Convx$ blocks in the encoder. $N$ is the total number of $Convx$ blocks
the classifier can be divided into using the strategy proposed in III-B. For
the ResNet-50 model, $N=7$. The solid boxes represent layers that are fixed
and the dashed boxes represent layers that can be moved between the
encoder/decoder for different model configurations.

(a) The feed-forward neural network used to generate vector embeddings from
$\lambda$

(b) Autoencoder architecture used for variable-rate feature compression.
Conv($N$, $k$, $s$) represents a convolution layer with $N$ kernels of size
($k$, $k$) and a stride of $s$ while ConvT($N$, $k$, $s$) represents its
transposed convolution. GeLU is the Gelu activation function.
ConvNextAdaLN($N$, $k$, $s$) is the network shown in Fig. 5.

Figure 4: (a) and (b) illustrate the network architectures used to perform
varaible-rate feature compression. $\lambda$ is passed as the input to the
network in (a) and its output $\bar{\lambda}$ is used as the input to the
network in (b) that is responsible for tuning the bit-rate of the model. The
number of filters $\beta$ in the last convolution layer of the autoencoder is
varied between model configurations so that the product $\beta*w/2*h/2$ is
always a constant.

###  III-C Design of Autoencoder

An autoencoder is inserted at the split point to generate latent features at
the encoder which are compressed and transmitted to the decoder. At the
decoder, the received feature is decompressed and used to reconstruct the
original input feature. The autoencoder is designed in a manner so that the
same design principles can be applied to all the model configurations. The
encoder and decoder have symmetrical structures with an initial convolution
layer for downsampling or upsampling, respectively. The last layer of both the
encoder and decoder are also convolution layers, but they are used for channel
reduction or expansion only. In between the two convolution layers there are
three ConvNext Adaptive Layer Normalization (ConvNextAdaLN) blocks adopted
from [22].

The encoding latency primarily depends on two factors: the depth of the edge
model architecture that controls the transmission time of the input image data
through the network, and the total number of elements in the bottleneck
feature $z$ that dictates the time for the entropy model to perform feature
compression. In order to analyze the change in latency due to the depth of the
edge network only, the encoder is designed so that the total number of
elements in the bottleneck feature $z$ is constant for all model
configurations. This is achieved by tuning the number of kernels in the last
convolution layer of the encoder as shown in Fig. 4.

###  III-D Variable Rate Feature Compression

In this work, we propose a novel variable rate feature compression model. By
employing a variable-rate scheme, the proposed model only has to train one
single dynamic network which can be deployed in a range of bit rate settings
by tuning the rate control parameter $\lambda$.

In fixed rate compression models, $\lambda$ is a hyperparameter that is added
to the loss function and remains fixed throughout training. However, to
operate at variable bit rates, our model incorporates $\lambda$ as an input to
the network rather than a hyperparameter. This is achieved by using
ConvNextAdaLN blocks from [22] in place of regular convolution layers in the
autoencoder network. The network architecture of the ConvNextAdaLN block is
shown in Fig. 5. The block serves as a conditional convolution layer
conditioned on $\lambda$ [14, 23]. Vector embeddings of $\lambda$, referred to
as $\bar{\lambda}\in\mathbb{R}^{\text{embed\\_dim}}$ are passed as the input
to the ConvNextAdaLN block along with the output from the previous layer.
$\bar{\lambda}$ is generated from $\lambda$ using a feedforward neural network
as shown in Fig. 4(a). Vector embeddings are commonly used to enhance the
performance of neural networks by creating low-dimensional representations of
the input, which allows the network to learn the relationships between inputs
more efficiently [24, 25, 22].

As $\lambda$ is now an input to the model that is varied during training
within the interval $(0,\lambda_{\text{max}}]$ (referred to as $\Lambda$), the
training loss function is modified to (6) and the goal of the model is to find
the optimum $\theta$ and $\phi$ according to (7):

$L=\sum_{\lambda\in\Lambda}l_{CE}(\hat{y},y;\lambda)+\lambda
l_{R}(\hat{z};\lambda)\\\ $ (6)
$\theta^{*},\phi^{*}=\operatorname*{arg\,min}_{\theta,\phi}\sum_{x,y\in
D}\sum_{\lambda\in\Lambda}l_{CE}(\hat{y},y;\lambda)+\lambda
l_{R}(\hat{z};\lambda)$ (7)

Figure 5: ConvNext Adaptive Layer Normalization block. Conv($N$, $k$, $s$)
represents a convolution layer with $N$ kernels, a kernel of size ($k$, $k$)
and stride of $s$.

To achieve optimum performance, instead of sampling $\lambda$ from a uniform
distribution in the range $(0,\lambda_{\text{max}}]$ where
$\lambda_{\text{max}}>>0.01$, $\lambda$ is sampled from a log distribution in
the same range. This is done to train the model evenly over high bit-rate and
low bit-rate regions as the bit-rate is more evenly distributed in the log
space over the range $(0,\lambda_{\text{max}}]$.

##  IV Experiments

Our method is compared to two previous methods for feature compression in an
edge-cloud setting: the Entropic Student method [12] and the method by Duan
and Zhu, 2022 [9]. Note that neither of these methods tackle variable-rate
feature compression, which is addressed in our method.

###  IV-A Experiment Settings

We use the 1000-class Image Classification dataset of ImageNet (ILSVRC 2012)
[26] containing $1.28$ million training and $50,000$ validation images for
training and evaluating our method. All the images are resized to $224\times
224$. We train each model for $60$ epochs with a stochastic gradient descent
optimizer and an initial learning rate of $0.01$. A cosine learning rate decay
is also applied. To enable a fair comparison with the previous methods, we use
the ResNet-50 model [17] as the classification network in all our experiments.

We measure classification accuracy using the Top-1 accuracy, bit rate as the
bits per pixel (bpp), and encoding complexity in terms of GPU encoding
latency. In order to visualize a rate-accuracy-complexity trade-off, we use
the Delta-accuracy metric introduced in [9]. The Delta-accuracy of a
compression method is a measure of its average accuracy improvement (following
the convention of BD-rate and BD-PSNR [27]) over all bit rates with respect to
a baseline method.

###  IV-B RAC Comparison between Proposed Model Configurations

We design three network architectures with varying encoder complexity to
perform a quantitative comparison between our proposed model configurations:
Config.1, Config.2 and Config.3. The general architecture for the model
configurations is shown in Fig. 3. The Delta-accuracy (with respect to
Config.1) and encoding latency of the three model configurations are shown in
Table I. From the table, it can be seen that as the depth of the edge network
is increased (moving from Config.1 to Config.3), the Delta-accuracy of the
model also increases. This is expected, as deeper layers of a classification
network have more condensed features which contain more useful information
relevant to the classification task [19]. Hence, it requires fewer bits to
compress features from deeper layers. However, this increase in rate-accuracy
performance comes at the cost of higher encoding latency.

In Table I, classifier latency is defined as the time in milliseconds required
for data to pass through the classifier front-end, compression time refers to
the time needed by the entropy model to compress the latent feature, and
encoding latency is the total time required for the data to pass through the
edge device and get compressed into bits. It can be seen that the compression
time is roughly the same between all model configurations due to the design
principles adopted for the autoencoder as explained in Sec. III-C. We can also
see from the table that the change in encoding latency between configurations
(+0.78, +0.59) ms is almost equal to the change in classifier latency (+0.64,
+0.59) ms. Therefore, the change in encoding latency between the model
configurations is primarily due to the proposed tuning of the number of
$Convx$ blocks in the edge network. By adopting this scheme for varying
complexity we can ensure that the latency always increases from shallower
configurations to deeper ones.

Figure 6: Rate-Accuracy comparison of three different proposed model
configurations

Fig. 6 shows the rate-accuracy curves of the three model configurations. The
expected trend between the networks is seen mostly in the low and medium bit
rate regions [0.05, 0.4] bpp. During the training of these model
configurations, $\lambda$ is sampled from a log-uniform distribution. It is
important to note that the final rate-accuracy curve relies on the range and
technique used for sampling $\lambda$. In this work, the optimal sampling
range is empirically found to be [0.0001, 5.12].

TABLE I: Rate-Accuracy-Complexity comparison of three proposed model configurations Configuration | Config.1 | Config.2 | Config.3
---|---|---|---
Delta-Acc. (%) | 0.0 | 0.46 | 1.50
Classifier Latency (ms) | 0.92 | 1.56 | 2.15
Compression Time (ms) | 2.17 | 2.25 | 2.25
Encoding Latency (ms) | 4.42 | 5.20 | 5.79

###  IV-C RAC Comparison of Proposed Model Configurations with other Baselines

In this section, we compare the proposed method with the Entropic Student
method [12] and the method of Duan and Zhu, 2022 [9]. The Delta-accuracy value
for each method with respect to the Entropic Student method is computed using
their individual rate-accuracy curves. The plot of Delta-accuracy vs. Encoding
latency for each of the outlined methods is shown in Fig. 7. The encoding
complexity of the proposed model increases from Config.1 to Config.2 to
Config.3. In Duan and Zhu, 2022, the encoding complexity is varied by changing
the number of residual blocks $N$ in their variable encoder architecture. The
three architectures used in this study correspond to N = $[0,4,8]$ and are
referred to as n0, n4 and n8. The Entropic Student model does not include a
method to change its complexity. Thus, it only operates in one latency
setting.

We can see from Fig. 7 that Config.1 has a better rate-accuracy performance
(higher Delta-accuracy) compared to n0 (+0.70 %) and the Entropic Student
model (+0.53 %). It also operates at a lower encoding latency than all the
other models. However, it has a lower Delta-accuracy than n4 (-0.41 %) and n8
(-0.46 %). On the other hand, both Config.2 and Config.3 outperform all the
other models in terms of Delta-accuracy although at the cost of a higher
encoding complexity. Config.2 has a Delta-accuracy of 2.49 % compared to the
Entropic Student model and Config.3 has a Delta-accuracy of 4.67 %.

Figure 7: Rate-Accuracy-Complexity performance of our proposed model against
other baselines measured in terms of Delta-accuracy(with respect to the
Entropic Student model) and encoding latency

##  V Conclusion

In this paper, we propose a learned variable rate feature compression method
for edge-cloud systems. Due to the variable rate nature of the method, it is
superior to the existing feature compression methods which require different
networks to be trained for every rate-accuracy pair. This work also proposes a
method for changing the encoder complexity while keeping the overall size of
the system roughly constant. We perform experiments to demonstrate the ability
of our proposed method to adapt to different rate-accuracy-complexity
requirements depending on the application scenario. We also compare our
proposed method with existing baseline methods, and show that our approach
outperforms them in terms of RAC trade-off. A drawback of the proposed method
is that a new model has to be trained to operate at each encoder complexity.
It is left as future work to develop a single dynamic model that can operate
at a range of encoder complexities.

## References

  * [1] Andrew Howard, Mark Sandler, Grace Chu, Liang-Chieh Chen, Bo Chen, Mingxing Tan, Weijun Wang, Yukun Zhu, Ruoming Pang, Vijay Vasudevan, et al.,  “Searching for mobilenetv3,”  in Proceedings of the IEEE/CVF international conference on computer vision, 2019, pp. 1314–1324.
  * [2] Mark Sandler, Andrew Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh Chen,  “Mobilenetv2: Inverted residuals and linear bottlenecks,”  in Proceedings of the IEEE conference on computer vision and pattern recognition, 2018, pp. 4510–4520.
  * [3] David S Taubman, Michael W Marcellin, and Majid Rabbani,  “Jpeg2000: Image compression fundamentals, standards and practice,”  Journal of Electronic Imaging, vol. 11, no. 2, pp. 286–287, 2002.
  * [4] Jani Lainema, Frank Bossen, Woo-Jin Han, Junghye Min, and Kemal Ugur,  “Intra coding of the hevc standard,”  IEEE transactions on circuits and systems for video technology, vol. 22, no. 12, pp. 1792–1801, 2012.
  * [5] Johannes Ballé, Valero Laparra, and Eero P Simoncelli,  “End-to-end optimized image compression,”  arXiv preprint arXiv:1611.01704, 2016.
  * [6] Eirikur Agustsson, Fabian Mentzer, Michael Tschannen, Lukas Cavigelli, Radu Timofte, Luca Benini, and Luc V Gool,  “Soft-to-hard vector quantization for end-to-end learning compressible representations,”  Advances in neural information processing systems, vol. 30, 2017.
  * [7] Johannes Ballé, David Minnen, Saurabh Singh, Sung Jin Hwang, and Nick Johnston,  “Variational image compression with a scale hyperprior,”  arXiv preprint arXiv:1802.01436, 2018.
  * [8] Jooyoung Lee, Seunghyun Cho, and Seung-Kwon Beack,  “Context-adaptive entropy model for end-to-end optimized image compression,”  arXiv preprint arXiv:1809.10452, 2018.
  * [9] Zhihao Duan and Fengqing Zhu,  “Efficient feature compression for edge-cloud systems,”  in 2022 Picture Coding Symposium (PCS). IEEE, 2022, pp. 187–191.
  * [10] Saurabh Singh, Sami Abu-El-Haija, Nick Johnston, Johannes Ballé, Abhinav Shrivastava, and George Toderici,  “End-to-end learning of compressible features,”  in 2020 IEEE International Conference on Image Processing (ICIP). IEEE, 2020, pp. 3349–3353.
  * [11] Robert A. Cohen, Hyomin Choi, and Ivan V. Bajić,  “Lightweight compression of neural network feature tensors for collaborative intelligence,”  in IEEE International Conference on Multimedia and Expo, July 2020, pp. 1–6.
  * [12] Yoshitomo Matsubara, Ruihan Yang, Marco Levorato, and Stephan Mandt,  “Supervised compression for resource-constrained edge computing systems,”  in Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, 2022, pp. 2685–2695.
  * [13] Parual Datta, Nilesh Ahuja, V Srinivasa Somayazulu, and Omesh Tickoo,  “A low-complexity approach to rate-distortion optimized variable bit-rate compression for split dnn computing,”  in 2022 26th International Conference on Pattern Recognition (ICPR). IEEE, 2022, pp. 182–188.
  * [14] Yoojin Choi, Mostafa El-Khamy, and Jungwon Lee,  “Variable rate deep image compression with a conditional autoencoder,”  in Proceedings of the IEEE/CVF International Conference on Computer Vision, 2019, pp. 3146–3154.
  * [15] Robert A. Cohen, Hyomin Choi, and Ivan V. Bajić,  “Lightweight compression of intermediate neural network features for collaborative intelligence,”  IEEE Open Journal of Circuits and Systems, vol. 2, pp. 350–362, May 2021.
  * [16] Jarek Duda,  “Asymmetric numeral systems: entropy coding combining speed of huffman coding with compression rate of arithmetic coding,”  arXiv preprint arXiv:1311.2540, 2013.
  * [17] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun,  “Deep residual learning for image recognition,”  in Proceedings of the IEEE conference on computer vision and pattern recognition, 2016, pp. 770–778.
  * [18] Hyomin Choi and Ivan V Bajić,  “Scalable image coding for humans and machines,”  IEEE Transactions on Image Processing, vol. 31, pp. 2739–2754, 2022.
  * [19] Yilun Xu, Shengjia Zhao, Jiaming Song, Russell Stewart, and Stefano Ermon,  “A theory of usable information under computational constraints,”  arXiv preprint arXiv:2002.10689, 2020.
  * [20] Zhuang Liu, Hanzi Mao, Chao-Yuan Wu, Christoph Feichtenhofer, Trevor Darrell, and Saining Xie,  “A convnet for the 2020s,”  in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp. 11976–11986.
  * [21] Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna,  “Rethinking the inception architecture for computer vision,”  in Proceedings of the IEEE conference on computer vision and pattern recognition, 2016, pp. 2818–2826.
  * [22] Zhihao Duan, Ming Lu, Jack Ma, Zhan Ma, and Fengqing Zhu,  “Qarv: Quantization-aware resnet vae for lossy image compression,”  arXiv preprint arXiv:2302.08899, 2023.
  * [23] Aaron Van den Oord, Nal Kalchbrenner, Lasse Espeholt, Oriol Vinyals, Alex Graves, et al.,  “Conditional image generation with pixelcnn decoders,”  Advances in neural information processing systems, vol. 29, 2016.
  * [24] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin,  “Attention is all you need,”  Advances in neural information processing systems, vol. 30, 2017.
  * [25] David Tellez, Geert Litjens, Jeroen van der Laak, and Francesco Ciompi,  “Neural image compression for gigapixel histopathology image analysis,”  IEEE transactions on pattern analysis and machine intelligence, vol. 43, no. 2, pp. 567–578, 2019.
  * [26] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei,  “Imagenet: A large-scale hierarchical image database,”  in 2009 IEEE conference on computer vision and pattern recognition. Ieee, 2009, pp. 248–255.
  * [27] Gisle Bjontegaard,  “Calculation of average psnr differences between rd-curves,”  ITU SG16 Doc. VCEG-M33, 2001.


# On the cone conjecture for log Calabi-Yau mirrors of Fano 3-folds

Jennifer Li

###### Abstract.

Let $Y$ be a smooth projective $3$-fold admitting a K3 fibration
$f:Y\rightarrow\mathbb{P}^{1}$ with $-K_{Y}=f^{*}\mathcal{O}(1)$. We show that
the pseudoautomorphism group of $Y$ acts with finitely many orbits on the
codimension one faces of the movable cone if $H^{3}(Y,\mathbb{C})=0$,
confirming a special case of the Kawamata–Morrison–Totaro cone conjecture. In
[CCGK16] [P18], and [CP18], the authors construct log Calabi-Yau 3-folds with
K3 fibrations satisfying the hypotheses of our theorem as the mirrors of Fano
3-folds.

##  1\. Introduction

Morrison’s cone conjecture is stated as follows:

######  Conjecture 1.1.

(The Morrison cone conjecture) Let $X$ be a Calabi-Yau 3-fold. Then,

  1. (1)

The automorphism group of $X$ acts on the nef cone of $X$ with a rational
polyhedral fundamental domain; and

  2. (2)

The pseudoautomorphism group of $X$ (see Definition 1.7 below) acts on the
movable cone of $X$ with a rational polyhedral fundamental domain.

In particular, Morrison’s conjecture implies the following:

  1. (1)

The automorphism group of $X$ acts on the faces of the nef cone of $X$ with
finitely many orbits; and

  2. (2)

The pseudoautomorphism group of $X$ acts on the faces of the movable cone of
$X$ with finitely many orbits.

The Morrison cone conjecture was generalized by Totaro to klt pairs (see
[T10]). In this paper, we prove a version of the cone conjecture for certain
types of log Calabi-Yau 3-folds, namely, for smooth projective 3-folds $Y$
that admit a K3 fibration $f:Y\rightarrow\mathbb{P}^{1}$ such that
$-K_{Y}=f^{\ast}\mathcal{O}(1)$.

######  Remark 1.2.

If $f:Y\rightarrow\mathbb{P}^{1}$ is a K3 fibration such that $K_{Y}$ is
relatively nef, $-K_{Y}$ is effective and nonzero, $f$ is not isotrivial, and
$f$ has no multiple fibers, then $-K_{Y}=f^{*}\mathcal{O}(1)$. This follows
from positivity of the pushforward of the relative canonical bundle, see e.g.
[F78].

######  Remark 1.3.

If $D$ is any smooth (or reduced normal crossing) fiber of $f$, then $(Y,D)$
is log Calabi-Yau.

######  Definition 1.4.

The cone of curves
$\operatorname{\text{$\overline{\mbox{Curv}\raisebox{8.53581pt}{}}$}}(Y)$ is
defined as:

$\operatorname{\text{$\overline{\mbox{Curv}\raisebox{8.53581pt}{}}$}}(Y)=\mathop{\overline{\\{\sum
a_{i}[C_{i}]\;|\;a_{i}\in\mathbb{R}_{\geq 0}\text{ and }C_{i}\subset Y\text{
is a curve }\\}}}$.

######  Definition 1.5.

The nef cone $\operatorname{Nef}(Y)$ is the dual cone of the cone of curves.
The nef effective cone is defined as
$\operatorname{\operatorname{Nef}^{\;e}}(Y):=\operatorname{Nef}(Y)\cap\operatorname{Eff}(Y)$.

######  Definition 1.6.

The movable cone is defined as:

$\operatorname{Mov}(Y):=\mathop{\overline{\langle B\;|\;B\text{ is a divisor
with }|B|\text{ having no fixed part}\rangle}}_{\mathbb{R}\geq 0}$.

The movable effective cone is defined as
$\operatorname{\operatorname{Mov}^{\;e}}(Y):=\operatorname{Mov}(Y)\cap\operatorname{Eff}(Y)$.

######  Definition 1.7.

A pseudoautomorphism of a variety $Y$ is a birational map
$\theta:Y\dashrightarrow Y$ that is an isomorphism outside of codimension 2
subsets of the domain and codomain.

######  Definition 1.8.

A K3 fibration is a morphism $f:Y\rightarrow B$ such that the general fiber
$F\subset Y$ of $f$ is a K3 surface.

Here we briefly lay out some key results used in this paper. First of all, we
note that our setup is a special case of the Kawamata-Morrison-Totaro cone
conjecture (see Theorem 2.1). This allows us to conclude that
$\operatorname{Mov}(Y)$ is the union of the nef cones of all flops of $Y$ (see
Theorem 2.2). Two important results that we will use next are Mori’s cone
theorem ([KM98], Theorem 1.24) and Mori’s classification of extremal rays of
the cone of curves for smooth 3-folds ([M82]). Given a smooth projective
variety $Y$, Mori’s cone theorem describes the cone of curves of $Y$ in the
region $K_{Y}<0$. Mori gives a classification of the contractions
$\mathrm{cont}_{R}:Y\rightarrow X$ associated to extremal rays
$R\subset\operatorname{\text{$\overline{\mbox{Curv}\raisebox{8.53581pt}{}}$}}(Y)$
in the region $K_{Y}<0$ when $Y$ is a smooth projective 3-fold over
$\mathbb{C}$. There are a total of eight possibilities (see [M82] or [KM98]).

Now suppose that $Y$ is a smooth projective 3-fold and
$f:Y\rightarrow\mathbb{P}^{1}$ is a K3 fibration such that
$-K_{Y}=f^{\ast}\mathcal{O}(1)$. In this setting, we show (Proposition 2.5)
that for any $Z$ obtained by a sequence of flops of $Y$, six of the eight
possible types of extremal rays of
$\operatorname{\text{$\overline{\mbox{Curv}\raisebox{8.53581pt}{}}$}}(Z)$ in
the region $K_{Z}<0$ can be eliminated, leaving two remaining types:

  1. (1)

Type (1): the blowup of a smooth curve $\Gamma$; or

  2. (2)

Type (6): a conic bundle.

We prove the following main results.

######  Theorem 1.9.

Let $Y$ be a smooth projective 3-fold and $f:Y\rightarrow\mathbb{P}^{1}$ a K3
fibration such that $-K_{Y}=f^{\ast}\mathcal{O}(1)$. Then the
pseudoautomorphism group $\operatorname{PsAut}(Y)$ of $Y$ acts on

  1. (1)

the Type (6) faces of $\operatorname{\operatorname{Mov}^{\;e}}(Y)$ with
finitely many orbits;

  2. (2)

the Type (1) faces of $\operatorname{\operatorname{Mov}^{\;e}}(Y)$ with
finitely many orbits if $H^{3}(Y,\mathbb{C})=0$.

######  Remark 1.10.

If $H^{3}(Y,\mathbb{C})=0$, then any contraction of Type (1) has genus $g=0$.

By a result of Kawamata, we can also take care of all curves on $K^{\bot}$:

######  Theorem 1.11.

(Kawamata [K97]) $\operatorname{PsAut}(Y)$ acts on the faces of
$\operatorname{\operatorname{Mov}^{\;e}}(Y)$ containing $-K_{Y}$ with finitely
many orbits.

######  Remark 1.12.

Because $-K_{Y}$ (a fiber) is nef, there are no curves in the region $K>0$.

######  Theorem 1.13.

$\operatorname{PsAut}(Y)$ acts on the codimension 1 faces of
$\operatorname{\operatorname{Mov}^{\;e}}(Y)$ with finitely many orbits if
$H^{3}(Y,\mathbb{C})=0$.

######  Conjecture 1.14.

$\operatorname{PsAut}(Y)$ acts with finitely many orbits on the faces of
$\operatorname{\operatorname{Mov}^{\;e}}(Y)$ of Type (1) for any genus $g\geq
0$.

######  Theorem 1.15.

Assume that Conjecture 1.14 holds. Then $\operatorname{PsAut}(Y)$ acts with
finitely many orbits on the codimension 1 faces of
$\operatorname{\operatorname{Mov}^{\;e}}(Y)$.

In Section 3, we show that there are many examples of smooth projective
3-folds $Y$ admitting a K3 fibration $f:Y\rightarrow\mathbb{P}^{1}$ such that
$-K_{Y}=f^{\ast}\mathcal{O}(1)$ and $Y$ has infinite pseudoautomorphism group.

Acknowledgements. I thank Paul Hacking, János Kollár, and Chenyang Xu for very
helpful discussions and suggestions.

##  2\. Proof of main theorem

Here we prove Theorem 1.9. Let $Y$ be a smooth projective 3-fold admitting a
K3 fibration $f:Y\rightarrow\mathbb{P}^{1}$ such that
$-K_{Y}=f^{\ast}\mathcal{O}(1)$. Choose

$\Delta=\displaystyle{\tfrac{1}{2}}F_{1}+\displaystyle{\tfrac{1}{2}}F_{2}$,

where $F_{1}$ and $F_{2}$ are smooth, distinct fibers. Then the pair
$(Y,\Delta)$ is klt log Calabi-Yau, since the coefficients of $\Delta$ are
strictly less than one, and $K_{Y}+\Delta\sim_{\mathbb{Q}}0$. Note that we are
in the setting that Totaro considered in [T10]. Thus, Theorem 1.9 confirms the
special case of the Kawamata-Morrison-Totaro cone conjecture.

######  Theorem 2.1.

(The Kawamata-Morrison-Totaro cone conjecture, [T10], Conjecture 2.1). Let
$(Y,\Delta)$ be a klt log Calabi-Yau pair. Then

  1. (1)

$\operatorname{\text{Aut}}(Y,\Delta)$ acts on
$\operatorname{\operatorname{Nef}^{\;e}}(Y)$ with a rational polyhedral
fundamental domain; and

  2. (2)

$\operatorname{PsAut}(Y,\Delta)$ acts on
$\operatorname{\operatorname{Mov}^{\;e}}(Y)$ with a rational polyhedral
fundamental domain.

Because we are in the klt log Calabi-Yau setting, we may use a result implied
by Birkar-Cascini-Hacon-McKernan [BCHM10], stated as Theorem 2.2 below. This
allows us to describe the movable cone of $Y$ as the union of the nef cones of
$Z$ for $Y\dashrightarrow Z$ an SQM (small $\mathbb{Q}$-factorial
modification).

######  Theorem 2.2.

(cf. [BCHM10], Corollary 1.1.5) Let $(Y,\Delta)$ be klt where $\Delta$ is an
effective $\mathbb{Q}$-divisor on $Y$ and $K_{Y}+\Delta\sim_{\mathbb{Q}}0$.
Then we have a decomposition

$\operatorname{\operatorname{Mov}^{\;e}}(Y)=\bigcup_{Y\dashrightarrow
Z}\operatorname{\operatorname{Nef}^{\;e}}(Z)$

that is locally rational polyhedral in the interior of
$\operatorname{Mov}(Y)$, where the union is taken over SQMs $Y\dashrightarrow
Z$.

In our setting, the only type of SQM that appears is a flop (or a composition
of finitely many flops). The reason is because
$-K_{Y}=f^{\ast}\mathcal{O}(1)$, a fiber, and thus $-K_{Y}$ is nef, that is,
$-K_{Y}\cdot C\geq 0$ for any curve $C\subset Y$. If $C\subset Y$ is an
exceptional curve on an SQM, then $-K_{Y}\cdot C\geq 0$ because a smooth
3-fold has no flips ([M82]). So $K_{Y}\cdot C=0$, which corresponds to a flop.
Also note that a flop of a smooth 3-fold is again smooth ([K90]).

Let $S$ denote the following set:

$S:=\displaystyle{\bigcup_{Y\dashrightarrow Z}}\\{\text{codimension 1 faces of
}\operatorname{Nef}(Z)\text{ in the boundary of }\operatorname{Mov}(Y)\\}$,

where the union is taken over all flops $Y\dashrightarrow Z$. There is a
surjection from $S$ to the collection of all codimension 1 faces of
$\operatorname{Mov}^{e}(Y)$. Moreover, the elements of $S$ are in one-to-one
correspondence with the elements of

$\displaystyle{\bigcup_{Y\dashrightarrow Z}}\\{\text{extremal rays of
}\operatorname{\text{$\overline{\mbox{Curv}\raisebox{8.53581pt}{}}$}}(Z)=\operatorname{Nef}(Z)^{\ast}\text{
not corresponding to small contractions}\\}$.

The goal is to narrow down these extremal rays and the main tools used to
accomplish this are the following:

  1. (1)

Mori’s cone theorem ([KM98], Theorem 1.24); and

  2. (2)

Mori’s classification of extremal rays of the cone of curves for smooth
3-folds ([M82]).

We state Mori’s classification of extremal rays below for the reader’s
convenience.

######  Remark 2.3.

Because $-K_{Z}$ is nef, there are no curves in the $K_{Z}>0$ region.
Therefore we only need to consider the $K_{Z}<0$ region and the hyperplane
$-K_{Z}^{\bot}$.

######  Theorem 2.4.

(Mori’s classification of extremal rays of
$\operatorname{\text{$\overline{\mbox{Curv}\raisebox{8.53581pt}{}}$}}(Y)$,
[KM98], Theorem 1.32) Let $X$ be a nonsingular projective 3-fold over
$\mathbb{C}$ and $\mathrm{\operatorname{\text{cont}}_{R}}:Y\rightarrow X$ the
contraction of a $K_{Y}$-negative extremal ray
$R\subset\operatorname{\text{$\overline{\mbox{Curv}\raisebox{8.53581pt}{}}$}}(Y)$.
Then we have the following possibilities:

  1. (1)

$\mathrm{\operatorname{\text{cont}}_{R}}$ is the (inverse of the) blowup of a
smooth curve in the smooth 3-fold $X$.

  2. (2)

$\mathrm{\operatorname{\text{cont}}_{R}}$ is the (inverse of the) blowup of a
smooth point of the smooth 3-fold $X$.

  3. (3)

$\mathrm{\operatorname{\text{cont}}_{R}}$ is the (inverse of the) blowup of an
ordinary double point of $X$ (locally analytically, the point is given by
$x^{2}+y^{2}+z^{2}+w^{2}=0$).

  4. (4)

$\mathrm{\operatorname{\text{cont}}_{R}}$ is the (inverse of the) blowup of a
point of $X$ that is locally analytically given by
$x^{2}+y^{2}+z^{2}+w^{3}=0$.

  5. (5)

$\mathrm{\operatorname{\text{cont}}_{R}}$ contracts a smooth
$\mathbb{P}^{2}_{\mathbb{C}}$ with normal bundle $\mathcal{O}(-2)$ to a point
on $Y$ which is locally analytically the quotient of $\mathbb{C}^{3}$ by the
involution $(x,y,z)\mapsto(-x,-y,-z)$.

  6. (6)

$\dim(X)=2$ and $\mathrm{\operatorname{\text{cont}}_{R}}$ is a fibration with
plane conics as fibers (that is, $\mathrm{\operatorname{\text{cont}}_{R}}$ is
a conic bundle).

  7. (7)

$\dim(X)=1$ and the general fibers of $\operatorname{\text{cont}}_{R}$ are del
Pezzo surfaces.

  8. (8)

$\dim(X)=0$ and $-K_{Y}$ is ample so that $Y$ is a Fano manifold.

######  Proposition 2.5.

Of the eight possible types of extremal rays of
$\operatorname{\text{$\overline{\mbox{Curv}\raisebox{8.53581pt}{}}$}}(Y)$ from
Theorem 2.4, we can eliminate all but (1) and (6) above:
Type (1). The blowup of a smooth curve; and
Type (6). A conic bundle $g:Y\rightarrow S$ (meaning that $g$ is a flat
morphism and its generic fiber is a rational curve).

The main idea of the proof of the Theorem 1.9 is as follows. We start with a
smooth projective 3-fold $Z$ such that $Z$ and $Y$ are related by a flop (or
by a composition of flops). First, we show that the $K_{Z}<0$ region only
contains extremal rays of the cone
$\operatorname{\text{$\overline{\mbox{Curv}\raisebox{8.53581pt}{}}$}}(Z)$ that
are of Type (1) or Type (6) (this is Proposition 2.5). Then we prove Theorem
1.9 (2), followed by Theorem 1.9 (1), which takes care of the $K_{Z}<0$ region
(meaning that the pseudoautomorphism group acts on all codimension 1 faces of
$\operatorname{Mov}(Y)$ corresponding to extremal rays of
$\operatorname{\text{$\overline{\mbox{Curv}\raisebox{8.53581pt}{}}$}}(Z)$ for
some $Z$ in the region $K_{Z}<0$ with finitely many orbits). Finally, we
explain how faces corresponding to extremal rays in $-K_{Z}^{\bot}$ are taken
care of by results of Kawamata (Theorem 2.9). Because no part of
$\operatorname{\text{$\overline{\mbox{Curv}\raisebox{8.53581pt}{}}$}}(Z)$ lies
in the $K_{Z}>0$ region, this completes the proof of Theorem 1.9.

###### Proof.

(Proof of Proposition 2.5). Let $f:Y\rightarrow X$ be the contraction of a
divisor $E\subset Y$ to a point $p\in X$ (so this covers cases (2), (3), (4),
and (5)). If we have any curve $C\subset E$, then by definition of the
contraction $\mathrm{cont}_{R}$ of an extremal ray
$R\subset\operatorname{\text{$\overline{\mbox{Curv}\raisebox{8.53581pt}{}}$}}(Y)$,
the class $[C]\in
R\subset\operatorname{\text{$\overline{\mbox{Curv}\raisebox{8.53581pt}{}}$}}(Y)$.
Since $R$ is in the $-K_{Y}>0$ region, $-K_{Y}\cdot
C=f^{\ast}\mathcal{O}(1)\cdot C>0$. So the induced map
$C\rightarrow\mathbb{P}^{1}$ is nonconstant. Now we have a map
$f|_{E}:E\rightarrow\mathbb{P}^{1}$ from a surface to a curve that does not
contract any curves, a contradiction. Therefore this eliminates cases (2),
(3), (4), and (5). We can also eliminate (7) by the same argument (replacing
$E$ by a fiber of the del Pezzo fibration $\mathrm{cont}_{R}$).

(8) We cannot have $Y\rightarrow\\{p\\}$, a point, and $Y$ is a Fano manifold.
This is because $-K_{Y}$ defines a fibration $Y\rightarrow\mathbb{P}^{1}$, so
$-K_{Y}$ is not ample. This leaves (1) and (6) as the only two possibilities,
completing the proof. ∎

We prove the second claim of Theorem 1.9.

###### Proof.

(Proof of (2) of Theorem 1.9). Suppose that the contraction is the blowup of a
smooth curve, which may be considered as the contraction of a ruled surface
$E$. Then, we have a $\mathbb{P}^{1}$-bundle $g:E\rightarrow\Gamma$, and we
claim that it is a trivial $\mathbb{P}^{1}$-bundle in our case. Let $l$ be a
fiber of $g$. Because $l\cdot(-K_{Y})=1$, and

$\displaystyle l\cdot(-K_{Y})$ $\displaystyle=l\cdot
f^{\ast}\mathcal{O}(1)\text{ \; by our assumptions}$
$\displaystyle=\deg(f|_{l}^{\ast}\mathcal{O}(1)),$

so $f|_{l}:l\rightarrow\mathbb{P}^{1}$ is an isomorphism. Then, the map
$E\rightarrow\Gamma\times\mathbb{P}^{1}$ given by $(g,f|_{E})$ is an
isomorphism, so that $g:E\rightarrow\Gamma$ is a trivial
$\mathbb{P}^{1}$-bundle.

It follows from above that $\Gamma$ is contained in each general fiber $F$ of
the K3 fibration $f$. Now we assume that $H^{3}(Y,\mathbb{C})=0$, and we show
that this implies that $\Gamma\cong\mathbb{P}^{1}$. If $Y\rightarrow
Y^{\prime}$ is the blowup of $\Gamma\subset Y^{\prime}$, then $H^{1}(\Gamma)$
injects into $H^{3}(Y,\mathbb{C})$. Since $H^{3}(Y,\mathbb{C})=0$, we have
$H^{1}(\Gamma)=0$, and $\Gamma\cong\mathbb{P}^{1}$. Using the facts that
$\Gamma\cong\mathbb{P}^{1}$ and $F$ is a K3 surface, the adjunction formula
shows that $\Gamma$ is a $(-2)$-curve. Therefore every general fiber of the
K3-fibration contains a $(-2)$-curve.

Theorem 2.6 below (due to Sterk and Kawamata) asserts that the automorphism
group of the generic fiber $Y_{\eta}$ of the K3 fibration
$f:Y\rightarrow\mathbb{P}^{1}$ acts on the set of all $(-2)$-curves in
$Y_{\eta}$ with finitely many orbits. Because
$\operatorname{\text{Aut}}(Y_{\eta})\subset\operatorname{PsAut}(Y)$ (see
Proposition 2.7 below), the larger group $\operatorname{PsAut}(Y)$ must also
act on the set of all $(-2)$-curves in $Y_{\eta}$ with finitely many orbits.
There is an injection

$\\{E\subset Y\;|\;E\text{ an exceptional divisor of Type (1) on
}Y\\}\hookrightarrow\\{(-2)\text{-curves in }Y_{\eta}\\}$

which is given by sending any exceptional divisor $E$ of Type (1) to
$E_{\eta}$ (the generic fiber of the restriction
$f|_{E}:E\rightarrow\mathbb{P}^{1}$), and therefore $\operatorname{PsAut}(Y)$
acts on the extremal rays of Type (1) with finitely many orbits, proving (2)
of Theorem 1.9. ∎

Before proving the first claim of Theorem 1.9, we first list some important
results used in this part of the proof.

######  Theorem 2.6.

([S85], Proposition 2.5 and [K97] Remark 2.2 (2)). Let $S$ be a K3 surface
over a field $k$ of characteristic zero (not necessarily algebraically
closed). Then $\operatorname{\text{Aut}}(S)$ acts on
$\operatorname{\operatorname{Nef}^{\;e}}(S)$ with a rational polyhedral
fundamental domain, and in particular, $\operatorname{\text{Aut}}(S)$ acts
with finitely many orbits on the set of all $(-2)$-curves $C$ in $S$.

Theorem 2.6 follows from Morrison’s cone conjecture for K3 surfaces over
$\mathbb{C}$ proved by Sterk ([S85], Proposition 2.5) together with a
generalization by Oguiso-Sakurai in [OS01] and Kawamata in [K97].

######  Proposition 2.7.

Let $Y$ be a smooth projective 3-fold with a K3-fibration
$f:Y\rightarrow\mathbb{P}^{1}$ where $-K_{Y}=f^{\ast}\mathcal{O}(1)$. Let
$Y_{\eta}$ denote the generic fiber of the fibration. Then
$\operatorname{\text{Aut}}(Y_{\eta})\subset{\it PsAut(Y)}$.

###### Proof.

(Proof of Proposition 2.7). This follows from the following two facts:

  1. (1)

$Y\rightarrow\mathbb{P}^{1}$ is a relative minimal model, since $Y$ is smooth
and $K_{Y}$ is nef (in fact, trivial) on the fibers of $f$.

  2. (2)

Let $Y\rightarrow S$ and $Z\rightarrow S$ be relative minimal models and
$\alpha:Y\dashrightarrow Z$ a birational map over $S$. Then $\alpha$ is an
isomorphism in codimension 1 ([K97], Theorem 3.52 (2)).

∎

Another result of Sterk that we need is a consequence of the Morrison cone
conjecture for K3 surfaces.

######  Theorem 2.8.

(Sterk, [S85], Proposition 2.6). Let $F$ be a K3 surface. Consider line
bundles $L$ such that $L$ is nef and $L^{2}=2k$ for some fixed
$k\in\mathbb{N}$. Then $\operatorname{\text{Aut}}(F)$ acts on the set of all
such $L$ with finitely many orbits.

Now we prove the first claim of Theorem 1.9.

###### Proof.

(Proof of (1) of Theorem 1.9). Suppose that the contraction is a conic bundle
$g:Z\rightarrow S$, where $Z$ is a 3-fold. We note that $Z$ may not be the
3-fold $Y$ that we start with, but $Y$ and $Z$ are related by flops
$Y\dashrightarrow Z$. We continue to denote the K3 fibration by
$f:Z\rightarrow\mathbb{P}^{1}$.

${Y}$${Z}$${S}$${\mathbb{P}^{1}}$$\scriptstyle{flops}$$\scriptstyle{f}$$\scriptstyle{g}$

Let $F=Z_{\eta}$ be the generic fiber of the K3 fibration $f$. Denote the
extension of scalars
$S\times\eta=S\otimes_{\mathbb{C}}\mathbb{C}(\mathbb{P}^{1})$ again by $S$.
Define $h:=g|_{F}:F\rightarrow S$, a generically finite morphism of degree 2.
We prove that $h$ is finite (meaning that no curves $\Gamma\subset F$ are
contracted by $h$) by contradiction. Suppose the claim is false, that is,
suppose that $h(\Gamma)$ is a point for some $\Gamma\subset F$. Then since
$h:=g|_{F}$, we know that $g(\Gamma)$ is a point. Because $g$ is the
contraction of an extremal ray in the region $K_{Z}<0$, we have
$K_{Z}\cdot\Gamma<0$. But
$K_{Z}\cdot\Gamma=f^{\ast}\mathcal{O}(1)\cdot\Gamma=0$. Therefore $h$ is
finite of degree two.

There is an involution $i$ acting on $F$, associated to $h:F\rightarrow S$.
Because $h$ is a degree two cover, it is given by a quotient by a
$\mathbb{Z}/2\mathbb{Z}$-action. Let $i:F\rightarrow F$ be the involution
corresponding to $h$.

Since the conic bundle $g$ maps the smooth 3-fold $Y$ to a base $S$, it
follows ([M82]) that $S$ is also smooth. By the classification of involutions
of K3 surfaces, there are two possibilities for the fixed locus
$\operatorname{Fix}(i)$ of the involution $i$: either $\operatorname{Fix}(i)$
is empty, or $\operatorname{Fix}(i)$ is the union of smooth curves. (Note that
there also exist examples of involutions of K3 surfaces where
$\operatorname{Fix}(i)$ has isolated fixed points. But then $S$ would be
singular, and therefore this is not a possibility in our setting.) In the
first case, the base $S$ must be an Enriques surface, and in the second case
$S$ is a rational surface.

If $S$ is an Enriques surface, then we can explicitly describe the movable
cone of $Y$ and show that the cone conjecture holds in this case (see Example
3.6).

This leaves the second case, where $S$ is a rational surface. First, Riemann-
Hurwitz gives $K_{F}=h^{\ast}(K_{S}+\displaystyle{\frac{1}{2}B})$. Since $F$
is K3, it follows that $K_{F}\equiv 0$, so
$K_{S}+\displaystyle{\frac{1}{2}}B\equiv 0$, and thus $B\in|-2K_{S}|$. Next,
run MMP on $S$ to obtain $\pi:S\rightarrow\bar{S}$, where
$\bar{S}=\mathbb{P}^{2}$ or $\mathbb{F}_{n}$ with $n\neq 1$. We claim that
$n\leq 4$. The idea is that when we have a K3 cover, the complete linear
system $|-2K_{\bar{S}}|$ does not have a multiple fixed component, and thus
$n\leq 4$.

We show this in more detail here. Suppose that $n>4$. The diagram below shows
the relevant maps.

${F}$${\bar{F}}$${S}$${\bar{S}}$${B}$${\bar{B}}$$\scriptstyle{h}$$\scriptstyle{\bar{h}}$$\scriptstyle{MMP}$

We write $\bar{B}=Im(B)\subset\bar{S}$. On $\bar{S}=\mathbb{F}_{n}$, let $f$
denote a fiber and $C_{1}$ the positive section and $C_{0}$ the negative
section. Observe that $C_{1}=C_{0}+nf$. We show that $|-2K_{\bar{S}}|$
contains $2C_{0}$. Since $-K_{\mathbb{F}_{n}}=D$, the toric boundary, we can
write

$\displaystyle-K_{\mathbb{F}_{n}}$ $\displaystyle=C_{0}+f+C_{1}+f$
$\displaystyle=2C_{0}+(n+2)f,$

using the observation above. Then,

$-K_{\mathbb{F}_{n}}\cdot C_{0}=-(n-2)<0\text{ for }n>2$.

Thus $C_{0}$ is fixed in $|-2K_{\mathbb{F}_{n}}|$ for $n>4$ and
$C_{0}\subset\bar{B}$. On the other hand,

$\displaystyle(|-2K_{\mathbb{F}_{n}}-C_{0}|)\cdot C_{0}$
$\displaystyle=-2(K_{\mathbb{F}_{n}}\cdot C_{0})-C_{0}^{2}$
$\displaystyle=-2(n-2)+n$ $\displaystyle=-2n+4+n$ $\displaystyle-(n-4)$
$\displaystyle<0\text{ for }n>4.$

Then for $n>4$, the curve $C_{0}$ is fixed in $|-2K_{\mathbb{F}_{n}}-C_{0}|$.
Therefore $2C_{0}\subset\bar{B}$. But this is a contradiction because we see
that the double cover is singular along the curve $\bar{h}^{-1}(C_{0})$, but
the cover should be smooth there. Therefore $n\leq 4$.

Next, we construct a birational morphism $\phi:\bar{S}\rightarrow S$ by taking
$\phi$ to be the identity if $\bar{S}=\mathbb{P}^{2}$, or in the latter case
(that is if $\bar{S}=\mathbb{F}_{n}$ with $0\leq n\leq 4$ and $n\neq 1$), by
letting $\phi:\mathbb{F}_{n}\rightarrow\mathbb{P}(1,1,n)$ be the contraction
of the negative section of $\mathbb{F}_{n}$. Now we have the sequence of maps

$F\xrightarrow{h}S\xrightarrow{\pi}\bar{S}\xrightarrow{\phi}\hat{S}$.

Let $\theta=(\pi\circ h):F\rightarrow\bar{S}$. If $\bar{S}=\mathbb{P}^{2}$,
take $L=\theta^{\ast}M$, where $M=H$ is a hyperplane class. Then $M^{2}=1$ and
$L^{2}=2$. Suppose that $\bar{S}=\mathbb{F}_{n}$ (with $0\leq n\leq 4$ and
$n\neq 1$). Take $L=\theta^{\ast}M$, where $M$ is the positive section of
$\mathbb{F}_{n}$. Then $M^{2}=n$ and therefore $L^{2}=2n\leq 8$ (because
$n\leq 4$). This proves that in any case (if $S=\mathbb{P}^{2}$ or
$\mathbb{F}_{n}$, where $n\leq 4$ and $n\neq 1$), there is a line bundle $L$
where $L^{2}$ is bounded, and thus we may apply Theorem 2.8 to conclude that
$\operatorname{\text{Aut}}(F)$ acts with finitely many orbits on such $L$.

Now we have $\theta:F\rightarrow\bar{S}$ where $\theta=\pi\circ h$ and
$L=\theta^{\ast}M$. We want to show that we can recover the 2-to-1 map
$F\rightarrow S$ from $L$. It suffices to show that $L$ defines the morphism
$F\rightarrow\hat{S}$. First of all, there is always an injection

$H^{0}(F,L)\hookleftarrow H^{0}(\hat{S},M)$,

given by $\theta^{\ast}$. We need to show that this map is also surjective. To
do this, it suffices to show that the dimensions of the left and right sides
are equal, that is,

$\dim H^{0}(F,L)=\dim(H^{0}(\hat{S},M))$.

Recall that $L$ and $M$ are nef and big. Moreover, recall that $F$ is smooth
and $\bar{S}$ is log terminal. First of all,

$\displaystyle h^{0}(L)$ $\displaystyle=\chi(F,L)\text{ by Kawamata Viehweg
vanishing}$
$\displaystyle=\chi(\mathcal{O})+\displaystyle{\frac{1}{2}}L\cdot(L-K_{F})\text{
by Riemann-Roch}$ $\displaystyle=2+\displaystyle{\frac{1}{2}}L^{2}$
$\displaystyle=2+\displaystyle{\frac{1}{2}}(2\cdot M^{2})$
$\displaystyle=2+M^{2}$ $\displaystyle=2+n,$

where $\bar{S}=\mathbb{P}(1,1,n)$ for $n\in\mathbb{N}$. Then on $\bar{S}$, we
can compute directly that $h^{0}(M)=n+2$. Therefore $\dim
H^{0}(\bar{D},L)=\dim(H^{0}(\bar{S},M))$. ∎

What remains is to prove Theorem 1.9 for the extremal rays on $-K_{Z}^{\bot}$.
We use the following theorem of Kawamata:

######  Theorem 2.9.

(Kawamata, [K97]). Let $Y$ be a 3-fold and $f:Y\rightarrow S$ a K3 fibration.
Then $\operatorname{PsAut}(Y/S)$ acts on $\operatorname{Mov}^{e}(Y/S)$ with
finitely many orbits on faces.

Kawamata’s cone lives in
$N^{1}(Y/S)=N^{1}(Y)\times\mathbb{R}/f^{\ast}N^{1}(S)\otimes\mathbb{R}$, which
for us is

$N^{1}(Y)\otimes\mathbb{R}/\mathbb{R}\cdot[-K_{Y}]$

(this is because in our setting, $S=\mathbb{P}^{1}$ so that
$N^{1}(S)=\mathbb{Z}\cdot[-K_{Y}]$). We know that
$\operatorname{PsAut}(Y/\mathbb{P}^{1})$ acts on faces of the chamber
decomposition containing $[-K_{Y}]$ with finitely many orbits. In our picture,
every chamber $\operatorname{Nef}(Z)$ contains $[-K_{Y}]$ because every model
has this fibration (the models are related by flops). Then
$\operatorname{PsAut}(Y)$ acts on the codimension 1 faces of $Mov(Y)$ with
finitely many orbits, taking care of the extremal rays on $-K_{Z}^{\bot}$ and
completing the proof of Theorem 1.9.

Theorem 1.13 follows from Proposition 2.5, Theorem 1.9, and Theorem 2.9.

##  3\. Examples

Let $S$ be a K3 surface over a field $K$ of characteristic zero. Assume that
$K$ is the function field of a complex curve. Then the Brauer group of $K$ is
trivial (Tsen’s theorem). Then, writing
$G=\operatorname{\text{Gal}}(\bar{K}/K)$ for the absolute Galois group of $K$,
$\operatorname{Pic}(S)=\operatorname{Pic}(S\otimes_{K}\bar{K})^{G}$, see
[H16], Chapter 17 §2.2. Assume in addition that $G$ acts trivially on
$\operatorname{Pic}(S\otimes_{K}\bar{K})$, so that
$\operatorname{Pic}(S)=\operatorname{Pic}(S\otimes_{K}\bar{K})$. Let
$W\subset\operatorname{\text{Aut}}(\operatorname{Pic}(S))$ be the Weyl group
generated by the reflections

$s_{\alpha}(x)=x+(x.\alpha)\alpha$

for $\alpha\in\operatorname{Pic}(S)$ such that $\alpha^{2}=-2$. Then
$\operatorname{\text{Aut}}(S)$ is commensurable with
$\operatorname{\text{Aut}}(\operatorname{Pic}(S))/W$. This follows from the
Torelli theorem over $K$ (Theorem 2.6), cf. [H16], Chapter 15 Corollary 2.7.
In particular:

######  Proposition 3.1.

With notation as above, $\operatorname{\text{Aut}}(S)$ is finite if and only
if $W\subset\operatorname{\text{Aut}}(\operatorname{Pic}(S))$ has finite
index. Then the lattice $\operatorname{Pic}(S)$ is called 2-reflective in
[D83]. ∎

######  Remark 3.2.

If $G$ does not act trivially on $\operatorname{Pic}(S\otimes_{K}\bar{K})$,
then one can define a reflection group $W^{\prime}$ such that
$\operatorname{\text{Aut}}(S)$ is commensurable with
$\operatorname{\text{Aut}}(\operatorname{Pic}(S))/W^{\prime}$ (cf. [OS01],
Definition 1.6), but $W^{\prime}\nsubseteq W$ in general.

Many examples are obtained via mirror symmetry for Fano 3-folds. Let $X$ be a
Fano 3-fold and $E\subset X$ a smooth divisor such that $K_{X}+E=0$. Then
there exists an associated smooth projective 3-fold $Y$ and a K3-fibration
$f:Y\rightarrow\mathbb{P}^{1}$ such that $-K_{Y}=f^{\ast}\mathcal{O}(1)$ and
$H^{3}(Y,\mathbb{Z})=0$ and there is a fiber $D$ of $f$ which is a normal
crossing divisor with a 0-dimensional stratum ([CCGK16], [CP18], [P18]
[DHKOP23]). In particular, $f:Y\rightarrow\mathbb{P}^{1}$ satisfies the
hypotheses of Theorem 1.13.

######  Remark 3.3.

The Fano 3-fold $X$ is mirror to the K3-fibration $f:Y\setminus
D\rightarrow\mathbb{A}^{1}$ (a so-called Landau-Ginzburg model).

Moreover, the Picard lattice $\operatorname{Pic}(Y_{\bar{\eta}})$ of the
geometric generic fiber
$Y_{\bar{\eta}}=Y_{\eta}\otimes_{\mathbb{C}(\mathbb{P}^{1})}\overline{\mathbb{C}(\mathbb{P}^{1})}$
of $f:Y\rightarrow\mathbb{P}^{1}$ and the action of the Galois group on it
were computed ([P18], [DHKOP23]). There are 105 deformation types of Fano
3-folds. We will use the notation of Mori-Mukai (as in [CCGK16]) for the Fano
3-folds of Picard rank at least 2. The Galois group action on
$\operatorname{Pic}(Y_{\bar{\eta}})$ is trivial except in cases 2.12, 4.3,
6.1, 7.1, 8.1, 9.1, and 10.1. In the remaining cases, the lattice
$\operatorname{Pic}(Y_{\eta})=\operatorname{Pic}(Y_{\bar{\eta}})$ is not
2-reflective except in cases 2.1, 3.2, 3.6, 4.9, and the Picard rank 1 cases
$X_{6}\subset\mathbb{P}(1,1,1,1,3)$ and $X_{6}\subset\mathbb{P}(1,1,1,2,3)$.
Here we use Nikulin’s classification of 2-reflective lattices (cf. [D83],
Theorem 2.2.2). In the remaining $105-13=92$ cases,
$\operatorname{\text{Aut}}(Y_{\eta})$ is infinite by Proposition 3.1, so
$\operatorname{PsAut}(Y)$ is infinite (and so $\operatorname{Mov}(Y)$ is not
rational polyhedral).

######  Remark 3.4.

The K3 fibration $f:(Y,D)\rightarrow(\mathbb{P}^{1},\infty)$ mirror to a Fano
3-fold $X$ deforms in a family of dimension $\rho(X)-1$ with the same
properties ([DHKOP23], Remark 1.8).

######  Example 3.5.

(Mirror of quartic 3-fold $X_{4}\subset\mathbb{P}^{4}$.) Consider a pencil of
quartic K3 surfaces in $\mathbb{P}^{3}$:

$((X+Y+Z+T)^{4}+t(XYZT)=0)\subset\mathbb{P}^{3}$, where
$t\in\mathbb{P}^{1}=\mathbb{C}\cup\\{\infty\\}$.

Let $Y\rightarrow\mathbb{P}^{3}$ be a crepant resolution $Y\rightarrow\hat{Y}$
of the blowup $\hat{Y}\rightarrow\mathbb{P}^{3}$ of the base locus of the
pencil. Then $f$ is a K3 fibration, $-K_{Y}=f^{\ast}\mathcal{O}(1)$,
$D=f^{\ast}\infty$ is a type III K3, and

$\operatorname{Pic}(Y_{\eta})\cong\langle-4\rangle\oplus
U\oplus(-E_{8})^{\oplus 2}$

is a non 2-reflective lattice with trivial Galois action. So
$\operatorname{PsAut}(Y)$ is infinite.

######  Example 3.6.

Here we describe all examples of Type (6) with base $S$ an Enriques surface.
Recall that in this case, we have a conic bundle $g:Y\rightarrow S$. The
finite 2-to-1 morphism $(g,f):Y\rightarrow S\times\mathbb{P}^{1}$ has branch
locus
$\mathcal{B}\sim\mathrm{pr}^{\ast}_{1}(-2K_{S})+\mathrm{pr}^{\ast}_{2}\mathcal{O}(2)$
by Riemann-Hurwitz and our assumption that $-K_{Y}=f^{\ast}\mathcal{O}(1)$.
Since $S$ is an Enriques surface, we have $2K_{S}\sim 0$ and
$\mathcal{B}\sim\mathrm{pr}^{\ast}_{2}\mathcal{O}(2)$. So the double cover
$(g,f)$ is branched over two fibers of the trivial family
$S\times\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$ and restricts to the
universal cover $F\rightarrow S$ of $S$ on the remaining fibers. It follows
that

$Y=(F\times\mathbb{P}^{1})/(\mathbb{Z}/2\mathbb{Z})$,

where the $\mathbb{Z}/2\mathbb{Z}$ action is given by the involution $i\times
j$, where $i$ is the fixed point free involution on $F$ with quotient $S$ and
$j:\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$ with $j(t)=-t$. We consider the
conic bundle $g:Y\rightarrow S=F/\langle i\rangle$ as the first projection map
and the K3 fibration $f:Y\rightarrow\mathbb{P}^{1}/\langle
j\rangle\cong\mathbb{P}^{1}$ as the second projection.

In general if $G$ is a finite group acting on $X$ and $Y=X/G$, then the nef
cone of $Y$ may be described as

$\operatorname{Nef}(Y)=\operatorname{Nef}(X)\cap N^{1}(Y)$,

where $N^{1}(Y)=N^{1}(X)^{G}\subset N^{1}(X)$. Therefore, to understand
$\operatorname{Nef}(Y)$, it is enough to understand $\operatorname{Nef}(X)$.

In our case, we have $X=F\times\mathbb{P}^{1}$ and $G=\mathbb{Z}/2\mathbb{Z}$.
So

$\operatorname{Nef}(X)=\langle\mathrm{pr}^{\ast}_{1}\operatorname{Nef}(F),\mathrm{pr}^{\ast}_{2}\operatorname{Nef}\mathcal{O}(1)\rangle_{\mathbb{R}\geq
0}$.

Now,

$\displaystyle\operatorname{Nef}(Y)$
$\displaystyle=\operatorname{Nef}((F\times\mathbb{P}^{1})/(\mathbb{Z}/2\mathbb{Z}))$
$\displaystyle=\operatorname{Nef}(F\times\mathbb{P}^{1})\cap
N^{1}(F\times\mathbb{P}^{1})^{G}$
$\displaystyle=\langle\mathrm{pr}^{\ast}_{1}(\operatorname{Nef}(F)),\mathrm{pr}^{\ast}_{2}(\mathcal{O}(1))\rangle\cap
N^{1}(F\times\mathbb{P}^{1})^{G}$ $\displaystyle=\langle
g^{\ast}\operatorname{Nef}(S),f^{\ast}\mathcal{O}(1)\rangle_{\mathbb{R}\geq
0}.$

In particular, there are no faces of $\operatorname{Nef}(Y)$ corresponding to
small contractions, so $\operatorname{Mov}(Y)=\operatorname{Nef}(Y)$. Finally,
$\operatorname{PsAut}(Y)=\operatorname{\text{Aut}}(Y)$ acts on
$N^{1}(Y)=N^{1}(S)\times N^{1}(\mathbb{P}^{1})$ via the surjection
$\operatorname{\text{Aut}}(Y)\twoheadrightarrow\operatorname{\text{Aut}}(S)$.
Now the cone conjecture for Enriques surfaces (see [K97], Theorem 2.1) implies
the cone conjecture for $Y$. In particular, Theorem 1.8 holds.

## References

  * [BCHM10] C. Birkar, P. Cascini, C. Hacon, J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405-468.
  * [CCGK16] T. Coates, A. Corti, S. Galkin, A. Kasprzyk, Quantum periods for 3-dimensional Fano manifolds (English summary) Geom. Topol. 20 (2016), no. 1, 103-256.
  * [CP18] I. Cheltsov, V. Przyjalkowski, Katzarkov-Kontsevich-Pantev conjecture for Fano threefolds, arXiv:1809.09218 [math.AG].
  * [D83] I. Dolgachev, Integral quadratic forms: applications to algebraic geometry, Séminaire N. Bourbaki exp. no.611, p.251-278 (1982-1983).
  * [DHKOP23] D. Charles, A. Harder, L. Katzarkov, M. Ovcharenko, V. Przyjalkowski, Modularity of Landau-Ginzburg models, arXiv:2307.15607 [math.AG]
  * [F78] T. Fujita, On Kähler fiber spaces over curves, J. Math. Soc. Japan 30 (1978), no. 4, 779–794.
  * [FM00] O. Fujino and S. Mori, A canonical bundle formula, J. Differential Geom. 56 (2000), no. 1, 167–188.
  * [H16] D. Huybrechts, Lectures on K3 surfaces, Cambridge Stud. Adv. Math. 158, C.U.P. (2016).
  * [K90] J. Kollár, Flips, Flops, Minimal Models etc., Surveys in Diff. Geom. 1(1991), 113–199.
  * [K97] Y. Kawamata, On the Cone of divisors of Calabi-Yau fiber spaces, International Journal of Mathematics 08 (1997): 665-687.
  * [KM98] J. Kollár, S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998, with the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. MR MR1658959 (2000b:14018).
  * [M82] S. Mori, Threefolds whose canonical bundles are not numerically effective, Annals of Mathematics, 116(1), 133–176. https://doi.org/10.2307/2007050 (1982).
  * [M87] S. Mori, Classification of higher-dimensional varieties, in Algebraic geometry, Bowdoin, 1985, 269–331, Proc. Sympos. Pure Math., 46, Part 1, A.M.S., (1987).
  * [OS01] K. Oguiso, Sakurai, Calabi-Yau threefolds of quotient type, Asian J. Math. Vol. 5, 43-77 (2001)
  * [P18] V. Przyjalkowski, Toric Landau-Ginzburg models. Russian Math. Surveys 73 (2018), no. 6, 1033-1118.
  * [S85] H. Sterk, Finiteness results for algebraic K3 surfaces, Math. Z. 189, 507–513 (1985).
  * [T10] B. Totaro, The cone conjecture for Calabi–Yau pairs in dimension 2, Duke Math. J. 154 (2010), no. 2, 241–263.