# SAMM (Segment Any Medical Model): A 3D Slicer Integration to SAM

Yihao Liu
Department of Computer Science
Johns Hopkins University
Baltimore, MD
<EMAIL_ADDRESS>
Jiaming Zhang*
Department of Computer Science
Johns Hopkins University
Baltimore, MD
<EMAIL_ADDRESS>
Zhangcong She*
Department of Mechanical Engineering
Johns Hopkins University
Baltimore, MD
<EMAIL_ADDRESS>
Amir Kheradmand
Department of Neurology
Johns Hopkins University
Baltimore, MD
<EMAIL_ADDRESS>
Mehran Armand
Department of Computer Science
Johns Hopkins University
Baltimore, MD
<EMAIL_ADDRESS>
Equal Contribution

###### Abstract

The Segment Anything Model (SAM) is a new image segmentation tool trained with
the largest available segmentation dataset. The model has demonstrated that,
with prompts, it can create high-quality masks for general images. However,
the performance of the model on medical images requires further validation. To
assist with the development, assessment, and application of SAM on medical
images, we introduce Segment Any Medical Model (SAMM), an extension of SAM on
3D Slicer - an image processing and visualization software extensively used by
the medical imaging community. This open-source extension to 3D Slicer and its
demonstrations are posted on GitHub (https://github.com/bingogome/samm). SAMM
achieves 0.6-second latency of a complete cycle and can infer image masks in
nearly real-time.

_Keywords_ SAM $\cdot$ Medical Image $\cdot$ 3D Slicer

##  1 Introduction

The advent of foundation models has led to significant progress in image
analysis with potential for future advancements. SAM (Kirillov et al., 2023)
is a revolutionary foundation model for image segmentation and has already
shown the capability of handling diverse segmentation tasks. SAM especially
prevails in its generalization capability compared with the existing fine-
tuned models that are trained on specific domains. Thus, SAM holds significant
promise for application in medical image segmentation: The advantage lies in
adapting it to address the inherent inter-subject variations and low signal-
to-noise ratio commonly found in medical images.

Medical image segmentation is the task to separate different structures within
an image. The segmentation results can then be used to detect the region of
interest or reconstruct 3-dimensional anatomical models (Sinha and Dolz,
2021). The existing AI-based segmentation methods, however, do not fully
bridge the domain gap among different anatomies and different imaging
techniques (Wang et al., 2020). This introduces challenges for training, as AI
systems need to be trained on anatomy- and task-specific datasets. Therefore,
a universal tool would be valuable if it can be applied across all image
modalities and various anatomical structures. However, the universal tool must
also overcome a series of critical challenges including (but not limited to)
data privacy, ethics, expenses, scalability, data integrity, and validation
(Gao et al., 2023). In contrast, pretrained SAM can perform a new segmentation
task without being fine-tuned on task-specific datasets (Kirillov et al.,
2023). This feature makes SAM a promising segmentation tool for various image
modalities with less effort.

Despite the extensive use of AI in medical imaging, the application of
foundation models, such as SAM, remains largely unexplored. The application of
these models would require resolving differences in the structure between
medical images and non-medical images. 3D Slicer Fedorov et al. (2012), as a
widely-used open-source software, provides off-the-shelf functionalities to
manipulate 2D and 3D medical images, using consistent and user-friendly
interfaces and visualization tools. Therefore, we introduce the Segment Any
Medical Model (SAMM) that incorporates 3D Slicer and SAM, to assist the
investigation of applying general foundation models to medical images.

##  2 Methodology

###  2.1 Overall Architecture

Figure 1 presents the overall architecture of SAMM, which consists of a SAM
Server and an interactive prompt plugin for 3D Slicer (Slicer-IPP). SAM Server
first loads the pretrained SAM. It runs in parallel with Slicer-IPP and keeps
monitoring the requests sent from 3D Slicer. On the other side, Slicer-IPP
handles all the image slices with the built-in interfaces of 3D Slicer. Then,
it processes all the slices and send them to SAM Server to compute their
embeddings. Subsequently, the embeddings of the slices are stored in a format
of binary files in Random Access Memory (RAM) for efficient retrieval at the
inference stage. Once the embeddings of all slices are ready, the user may
start the segmentation using prompts. These prompts are the fiducial points
placed on the 2D slice to indicate adding or removing the region. The prompt
points are transmitted to the prompt encoder of SAM, and the inference stage
starts synchronously. Here the prompt transmission and the SAM inference are
synchronized. In such way, the mask generation responds to user action in
real-time.

Note that the image encoders run once per volume, rather than per prompt,
which allows the users to segment the same image multiple times with different
prompts in real-time. Given that the initialization of image embedding occurs
in advance, the subsequent mask generation process can be performed with small
latency (Section 3).

The Slicer-IPP handles the transformations from the volume coordinates to 2D
slice numbers and pixel coordinates. It can work out with discrepancies
between the RAS (right, anterior, superior), IJK (slice ID in different
views), and 2D image pixel coordinate systems by providing proper conversion
functionalities. For instance, at an inference request, Slicer-IPP converts
the coordinates of RAS to IJK to identify the image IDs. Slice-IPP then
transmits the IDs along with the prompts to the SAM Server. With the
coordinates converted, the masks of a 2D slice can be generated by SAM Server
in the inference step. The coordinates of the generated masks are transformed
back from the pixel to the RAS system, and are sent to Slicer-IPP for the
visualization of the segmentation results.

Figure 1: Overall architecture of the integration of 3D Slicer and SAM. Images
from different modalities, such as computed tomography (CT), magnetic
resonance imaging (MRI), or ultrasound (US), are contained in a scalar volume
node. The node (vtkMRMLScalarVolumeNode) is a class in the Visualization
Toolkit (VTK) that represents a volume of the scalar data within the Medical
Reality Markup Language (MRML) framework. Figure 2:  The affiliation of five
tasks, namely are “send inference request” (SND_ INF), “receive inference
request” (RCV_ INF), “complete SAM inference” (CPL_ INF), “receive mask
transmission” (RCV_ MSK), and “apply mask” (APL_ MSK). Slicer-IPP and SAM
Server use ZMQ/Numpy memory mapping package to enable real-time communication.

###  2.2 Slicer-IPP

The Slicer-IPP is composed of a data communication module, a prompt labeling
module, and a visualization module. The data communication module accepts any
volumetric image format, and it can pack them as image files used by SAM. The
Slicer-IPP and SAM Server are designed to run five parallel tasks denoted as
“send inference request” (SND_ INF), “receive inference request” (RCV_ INF),
“complete SAM inference” (CPL_ INF), “receive mask transmission” (RCV_ MSK),
and “apply mask” (APL_ MSK). The affiliation of tasks is shown in Figure 2.
The Slicer-IPP hosts SND_ INF, RCV_ MSK, and CPL_ INF, while the server end
hosts RCV_ MSK and APL_ MSK. Each task is executed synchronously as an
independent loop. All tasks run with the “best-effort” mode (opposite to
“guaranteed delivery” mode), as the realtime-ness is the priority. Since 3D
Slicer is a single-threaded software, each loop in the Slicer-IPP is set to
have a 60 ms gap to process other tasks. A complete inference cycle starts
from SND_ INF and ends with APL_ MSK. The time latency of one inference cycle
is discussed in Section 3.

Figure 3: Example results for different image formats (CT, MRI, and US). The
prompts with green points are for the regions to be selected, whereas the red
points are for the regions to be removed.

To facilitate communication between the 3D Slicer and external tools or
services, the platform uses ZeroMQ (ZMQ) (Hintjens, 2013) messaging library
and Numpy (Harris et al., 2020) memory mapping. ZMQ is a lightweight messaging
library that enables high-performance, asynchronous communication between
applications. In SAMM, ZMQ and Numpy are employed to transfer images, prompts,
and requests between the Slicer-IPP and the SAM Server. The segmentation task
is real-time by applying these two packages. This integration enables
researchers to take advantage of SAM’s cutting-edge segmentation capabilities
within the familiar 3D Slicer platform, expanding the range of tasks that can
be performed on images. The use of ZMQ and Numpy memory mapping also provides
the flexibility to customize the communication protocol to fit the user’s
specific needs, further enhancing the versatility of the 3D Slicer platform.

Figure 4:  (a) is the event plot of the 5 tasks (SND_ INF, RCV_ INF, CPL_ INF,
RCV_ MSK, and APL_ MSK). Each row represents one task and each bar represents
one event. A 16-second period (covers 60 cycles) from the initialization phase
of SAMM (when all tasks are initially launched) to the stable phase (when all
tasks run at a steady frequency) is shown. Events executed within the same
segmentation cycle are marked with the same number on the bottom two panels.
The time interval of five tasks between event 1 and 2 is marked on (b) while
the time interval of five tasks between event 59 and 60 is marked on (c). The
time intervals between two cycles are shown in the green boxes whereas the
time latency of one complete cycle is highlighted in the purple box.

##  3 Experiments and Results

We evaluated the integration with SA-1B pre-trained model 111SA-1B is a model
for SAM that is trained on 11 million diverse and high-resolution images and
1.1 billion high-quality segmentation masks (Kirillov et al., 2023) for
different formats including CT, MRI, and US, using images with different
anatomies. Figure 3 shows the manually placed prompts and the segmentation
masks generated from the sample datasets of 3D Slicer. The Slicer-IPP provides
a built-in markup tool for placing the prompt points. The segmentation mask is
overlaid on the original image once the CPL_ INF task ends. All cases are
tested in the same environment (Ubuntu 20.04, AMD Ryzen 9 3900X, Nvidia
GeForce RTX 3090). The result demonstrates that SAM, although not specifically
trained on medical image datasets, can generate masks for zero-shot
segmentation tasks across different image formats.

Here we use the term “event” to represent a task is completed. A complete
cycle of the image segmentation consists of five events (Figure 4). For each
time instance, the task owner logs the timestamp to a Python data storage
object. The log data, generated in chronological order, is the output to a
Python pickle file once 1000 segmentation cycles are completed. In Figure 4,
only 60 complete cycles are shown. We evaluated the performance of the system
using the end-to-end latency, defined as the time between a request of
inference and the application of the inferred mask for the same image slice.
In addition, the time intervals between every two consecutive tasks were
measured.

In order to facilitate real-time interactive segmentation, the Slicer-IPP
first computes the image embeddings and subsequently enables users to place
prompt points. Therefore, the total execution time is divided into two
components. The first component is the total time to compute the embeddings of
all slices, and the second one is the latency per image in its runtime.

In the test environment, the embedding computation for a $352\times 352\times
240$ MRI image takes 162.9 seconds. For the same test dataset, the latency of
an end-to-end process within the same segmentation cycle, including timestamp
logging, is 0.612 seconds (measured at event 60 in the “Stable Phase”, shown
in Figure 4).

##  4 Discussion and Conclusion

The integration of 3D Slicer and SAM enables researchers to conduct
segmentation on medical images using a state-of-the-art foundation model. We
have validated the capability of this model for segmenting images with a
latency of 0.6 seconds. Our integration enables segmentation through prompts
that can be automatically propagated to subsequent slices to streamline the
segmentation process. This functionality could be particularly useful in
large-scale image annotation when processing time is critical.

In the future, the performance of SAM for segmentation of medical images can
be enhanced by using medical image datasets and building upon the SA-1B
generic model. With the SAM’s support for text prompts (Kirillov et al.,
2023), future work may also include using text inputs as commands to perform
segmentation tasks in 3D Slicer. Additionally, specialized medical AIs such as
MONAI (Cardoso et al., 2022) may use SAM as an initialization tool to improve
their inference and training efficiency.

As shown in Figure 4, we have noted that the initialization phase, especially
at the SAM Server end, is not stable. The instability is evaluated by
comparing the time latency of two consecutive events at the initialization and
stable phase. The time latency of different tasks has a significant variation
at the initialization phase but the instability typically disappeared in 5
seconds after initialization. Further work is needed to optimize the latency
of the events in the segmentation cycles and the process of generating the
image masks.

Overall, our results show that SAMM has a small latency when used as a prompt-
based, semi-automatic segmentation method. The integration with 3D Slicer
allows researchers to validate how SAM can be applied for segmentation of
medical images. Moreover, the validation process can be significantly
streamlined with the help of an open-source platform with numerous off-the-
shelf tools.

## References

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  * Sinha and Dolz [2021] Ashish Sinha and Jose Dolz.  Multi-scale self-guided attention for medical image segmentation.  _IEEE Journal of Biomedical and Health Informatics_ , 25(1):121–130, 2021.  doi:10.1109/JBHI.2020.2986926.
  * Wang et al. [2020] Eric Ke Wang, Chien-Ming Chen, Mohammad Mehedi Hassan, and Ahmad Almogren.  A deep learning based medical image segmentation technique in internet-of-medical-things domain.  _Future Generation Computer Systems_ , 108:135–144, 2020\.  ISSN 0167-739X.  doi:10.1016/j.future.2020.02.054.
  * Gao et al. [2023] Cong Gao, Benjamin D. Killeen, Yicheng Hu, Robert B. Grupp, Russell H. Taylor, Mehran Armand, and Mathias Unberath.  Synthetic data accelerates the development of generalizable learning-based algorithms for X-ray image analysis.  _Nat Mach Intell_ , 5(3):294–308, March 2023\.  ISSN 2522-5839.  doi:10.1038/s42256-023-00629-1.
  * Fedorov et al. [2012] Andriy Fedorov, Reinhard Beichel, Jayashree Kalpathy-Cramer, Julien Finet, Jean-Christophe Fillion-Robin, Sonia Pujol, Christian Bauer, Dominique Jennings, Fiona Fennessy, Milan Sonka, John Buatti, Stephen Aylward, James V. Miller, Steve Pieper, and Ron Kikinis.  3d slicer as an image computing platform for the quantitative imaging network.  volume 30, pages 1323–1341, 2012.  doi:10.1016/j.mri.2012.05.001.
  * Hintjens [2013] Pieter Hintjens.  _ZeroMQ: Messaging for Many Applications_.  O’Reilly Media, Inc., 2013.
  * Harris et al. [2020] Charles R. Harris, K. Jarrod Millman, and Stéfan J. van der Walt et al.  Array programming with NumPy.  _Nature_ , 585(7825):357–362, September 2020\.  doi:10.1038/s41586-020-2649-2.
  * Cardoso et al. [2022] M Jorge Cardoso, Wenqi Li, Richard Brown, Nic Ma, Eric Kerfoot, Yiheng Wang, Benjamin Murrey, Andriy Myronenko, Can Zhao, Dong Yang, et al.  Monai: An open-source framework for deep learning in healthcare.  _arXiv preprint arXiv:2211.02701_ , 2022.


Normal form of bimeromorphically contractible holomorphic Lagrangian
submanifolds

Ekaterina Amerik111Both authors acknowledge support of HSE University basic
research program; also partially supported by ANR (France) project FANOHK
Misha Verbitsky222Partially supported by FAPERJ E-26/202.912/2018 and CNPq -
Process 310952/2021-2.
Keywords: hyperkähler manifold, Lagrangian submanifold, normal form,
holomorphically symplectic manifold 2010 Mathematics Subject Classification:
53C26

Abstract

Let $M$ be a holomorphically symplectic complex manifold, not necessarily
compact or quasiprojective, and $X\subset M$ a compact Lagrangian submanifold.
We construct a deformation to the normal cone, showing that a neighbourhood of
$X$ can be deformed to its neighbourhood in $T^{*}X$. This is used to study
Lagrangian submanifolds which can be bimeromorphically contracted to a point.
We prove that such submanifolds are biholomorphic to ${\mathbb{C}}P^{n}$, and
show that a certain neighbourhood of $X$ is symplectically biholomorphic to a
neighbourhood of the zero section of its cotangent bundle. This gives a
holomorphic version of the Weinstein’s normal neighbourhood theorem.

###### Contents

  1. 1 Introduction
    1. 1.1 Holomorphically symplectic Moser’s lemma and the deformation to the normal cone
    2. 1.2 Holomorphic Weinstein theorem
  2. 2 Contractible Lagrangian submanifolds
    1. 2.1 Vo Van Tan’s characterization of contractible subvarieties
    2. 2.2 Contractible subvarieties are isotropic
    3. 2.3 An MMP-style proof of 2
    4. 2.4 Lagrangian contractible submanifolds are uniruled
    5. 2.5 Uniruled Lagrangian submanifolds
  3. 3 Moser’s lemma and normal form of Lagrangian submanifolds
    1. 3.1 Moser’s lemma for holomorphic symplectic and C-symplectic structures
    2. 3.2 Moser’s lemma for a neighbourhood of a holomorphic Lagrangian submanifold
  4. 4 Deformation to the normal cone of a Lagrangian submanifold
    1. 4.1 Deformation to the normal cone: an introduction
    2. 4.2 Deformation to the normal cone in holomorphic symplectic category

##  1 Introduction

###  1.1 Holomorphically symplectic Moser’s lemma and the deformation to the
normal cone

A smooth complex manifold $M$ is holomorphically symplectic if it carries a
closed holomorphic $2$-form $\Omega$, non-degenerate at each point. Clearly,
in this case $M$ is of even dimension $2n$ and the canonical bundle of $M$ is
trivial. A subvariety $X\subset M$ is called Lagrangian if the restriction of
$\Omega$ to $X$ is zero and $\dim(X)=n$, in other words, $X$ is isotropic of
maximal possible dimension.

It is not hard to see that the holomorphically symplectic form defines the
complex structure in a unique way (3.1). This gives a way to describe the
holomorphically symplectic structures symplectically, without referring to the
complex structure. We define a C-symplectic structure (3.1) on a real manifold
of dimension $4n$ as a closed complex-valued form $\Omega$ such that
$\Omega^{n+1}=0$ and $\Omega^{n}\wedge\overline{\Omega}^{n}$ is non-
degenerate. Treated in this fashion, the holomorphic symplectic structure
gains the features we could expect from the symplectic form. In [SV], a
C-symplectic version of Moser’s lemma was proven. Just like in case of Moser’s
lemma, it was shown that a smooth family $\Omega_{t}$ of cohomologous
C-symplectic structures is trivialized by an appropriate flow of
diffeomorphisms. However, a significant caveat applies: Moser’s lemma fails
unless one requires the vanishing of the cohomology group $H^{1}({\cal
O}_{M,I_{t}})$, where $I_{t}$ the family of complex structures on $M$ induced
by the C-symplectic forms $\Omega_{t}$, and ${\cal O}_{M,I_{t}}$ is the sheaf
of holomorphic functions on $M$ with the complex structure $I_{t}$.

Weinstein normal neighbourhood theorem is one of the most immediate and
important applications of Moser’s lemma. This theorem, stated in the
traditional symplectic context, identifies a tubular neighbourhood of a
Lagrangian submanifold with its neighbourhood in its cotangent space, equipped
with the standard symplectic structure.

This paper is an attempt to apply this reasoning to C-symplectic manifolds.
However, the condition $H^{1}({\cal O}_{M,I_{t}})=0$ is quite restrictive. It
is not always true, and can otherwise be hard to establish, that $H^{1}({\cal
O}_{M,I_{t}})=0$ for a neighbourhood of a holomorphic Lagrangian submanifold.
One exception is when the holomorphic Lagrangian submanifold $S\subset M$ is
bimeromorphically contractible, that is, there exists a holomorphic,
bimeromorphic map
$M\stackrel{{\scriptstyle\pi}}{{{\>\longrightarrow\>}}}M_{0}$, bijective on
$M\backslash S$, which contracts $S$ to a point $s$. As follows from Grauert-
Riemenschneider theorem, in this situation the preimage $V:=\pi^{-1}(U)\subset
M$ of a Stein neighbourhood $U$ of $s\in M_{0}$ satisfies $H^{1}({\cal
O}_{V})=0$ (2.2).

To establish the existence of a normal form, we prove that the neighbourhood
$V\supset S$ can be deformed to a neighbourhood of $S$ in its cotangent
bundle. For complex manifolds this construction, called “deformation to the
normal cone”, is quite standard. In 4.2, we show that the deformation to the
normal cone can be performed in the holomorphically symplectic category. We
construct a smooth family of holomorphic symplectic manifolds connecting a
neighbourhood of a holomorphic Lagrangian submanifold $S\subset M$ to its
neighbourhood in $T^{*}S$.

The content of this paper is split between Sections 3 and 4, where the
applications of C-symplectic Moser’s lemma are discussed and the deformation
to the normal cone is constructed, and Section 2, where we show that any
bimeromorphically contractible Lagrangian submanifold is biholomorphic to the
complex projective space.

###  1.2 Holomorphic Weinstein theorem

Recently, Lagrangian subvarieties have attracted considerable attention in
algebraic geometry of holomorphic symplectic manifolds. One of the reasons is
their relation to the birational geometry: Lagrangian subvarieties appear as
exceptional loci of birational contractions of certain type. These are well-
known to be isomorphic to the projective space whenever smooth (a brief sketch
of proof is given in the beginning of Section 2). However, the general, non-
algebraic and non-compact case, apparently has not been studied up to now. In
this situation, the exceptional subvariety of a contraction is defined as
follows.

Definition 1.1: Let $X\subset Y$ be a complex subvariety in a complex manifold
$Y$. We say that $X$ can be bimeromorphically contracted if there exists a
proper morphism of complex varieties $Y{\>\longrightarrow\>}Y_{0}$ mapping $X$
to a point, which is an isomorphism outside of $X$.

Our main result is an analogue of Weinstein tubular neighbourhood theorem for
bimeromorphically contractible Lagrangian subvarieties, as follows.

Theorem 1.2:  Let $(M,I,\Omega)$ be a holomorphically symplectic manifold (not
necessarily compact or quasi-projective), and $E\subset(M,I)$ a compact
holomorphic Lagrangian submanifold. Assume that $E$ can be bimeromorphically
contracted to a point. Then $E$ is isomorphic to ${\mathbb{C}}P^{n}$.
Moreover, $E$ has a neighbourhood which is biholomorphically symplectomorphic
to a neighbourhood of the zero section in $T^{*}{\mathbb{C}}P^{n}$.

The first statement shall be proved in the next section, and the proof of the
second part shall be given in Subsection 4.2, after recalling Moser’s isotopy
lemma and its holomorphic version in Section 3.

Remark 1.3: Weinstein tubular neighbourhood theorem fails when $E$ is a fiber
of a holomorphic Lagrangian fibration on a hyperkähler manifold (say, on an
elliptic K3 surface). Indeed, the normal bundle $NE$ is trivial, but the
elliptic curve in the elliptic family varies, hence its neighbourhood cannot
be isomorphic to a neighbourhood of the zero section in the total space of
$T^{*}E=E\times{\mathbb{C}}$.

##  2 Contractible Lagrangian submanifolds

The main result of this section is the following theorem.

Theorem 2.1:  Let $X$ be a bimeromorphically contractible compact Lagrangian
submanifold of a holomorphic symplectic manifold $M$ of dimension $2n$, not
necessarily compact or quasiprojective. Then $X$ is isomorphic to
${\mathbb{C}}P^{n}$.

This section is devoted to the proof, in fact we shall provide two versions of
the argument. These versions have a common beginning which proves that $X$ is
Moishezon. Then one may proceed either by adaptation of log-MMP111This part
has been added after the first version of this paper has been written, and is
suggested by the subsequent discussions with C. Shramov and Yu. Prokhorov.
following [Kaw], [N], or by using the positivity properties of the cotangent
bundle restricted to a movable curve (see [CP], [CPT]).

To begin with, we recall that in the projective setting, this result is well-
known, see e.g. [CMS], chapter 8. The main steps of the proof in the
projective case (that is, assuming the contraction morphism $f:M\to M_{0}$
projective, as well as $M$ and $M_{0}$) are briefly sketched as follows.

  * •

Step 1: The contraction morphism is a log-extremal contraction with respect to
a certain boundary divisor $\Delta$ on $M$. More precisely, one takes $H$
ample om $M_{0}$, then $f^{*}H$ is big, and writes $f^{*}H=A+E$ where $A$ is
ample on $M$ and $E$ effective on $M$. Then $(M,\varepsilon E)$ is a klt-pair
for a small rational number $\varepsilon$, and the contraction is log-extremal
for this pair, since $E$ is $f$-negative. By the log Minimal Model theory, the
contraction locus $X$ is uniruled, i. e. covered by rational curves (see e.g.
[Kaw]).

  * •

Step 2: According to Z. Ran [Ra], Corollary 5.1, any rational curve in a
holomorphic symplectic manifold $M$ (not necessarily algebraic or compact)
deforms in a family of dimension at least $2n-2$.

  * •

Step 3: If $C\subset X$ is a rational curve, it is contracted by $f$, so that
any deformation of $C$ inside $M$ must be contracted as well and therefore
lies in $X$. Hence any rational curve in $X$ deforms in a family of dimension
at least $2n-2$. In particular this is true for a covering family of minimal
rational curves222I.e. such that all of its members passing through a general
point are irreducible. Informally, $X$ should not covered by rational curves
splitting off some curves in this family.. Now the results by S. Kebekus [Ke]
imply that this is possible only when $X\cong{\mathbb{P}}^{n}$.333See [CMS],
Theorem 4.2. Since [CMS] is known to be not always accurate, we prefer to rely
on [Ke], who proves in Proposition 3.1 that the base $H_{x}$ of a family of
minimal rational curves passing through the general point $x$ and covering the
variety $X$ is ${\mathbb{P}}^{n-1}$, and in the following subsection 3.2
computes that the total space $U_{x}$ of this family is ${\mathbb{P}}({\cal
O}_{{\mathbb{P}}^{n-1}}\oplus{\cal O}_{{\mathbb{P}}^{n-1}}(-1))$, so that the
evaluation map contracts the exceptional section onto ${\mathbb{P}}^{n}$.

In this section we give a proof of 2 which does not require any projectivity
assumption. We start with a contraction criterion by Ancona and Vo Van Tan
[AnVo], and obtain in particular that $X$ is Moishezon (i.e. bimeromorphic to
projective). Then the results by Campana–Peternell–Toma and Campana–Paun [CP]
(formulated in the projective context in [CP], but remarked by the authors to
be of bimeromorphic nature) imply uniruledness. Applying Step 2, we obtain a
$2n-2$-dimensional family of minimal rational curves on $X$. It remains to
remark that Kebekus’ proof is also valid when $X$ is Moishezon.

Alternatively (we thank Yuri Prokhorov and Costya Shramov for many indications
on this argument), once we know that $X$ is Moishezon we can remark that
either $X$ has rational curves, or $X$ is projective (or both) ([Sh]). Then in
the first case we repeat Step 2 and Step 3, and in the second case we make a
small adjustment to reduce to the projective case.

Remark 2.2:  As soon as we know that $X\cong{\mathbb{P}}^{n}$, we can apply
Grauert’s cohomological criterion ([G], §4, Satz 7, Corollar) to show that a
neighbourhood of $X$ in $M$ is biholomorphic to a neighbourhood of the zero
section in the cotangent bundle of ${\mathbb{P}}^{n}$: see [CMS], Example 8.1
and Lemma 8.2.

Later on in this paper we also prove the Weinstein normal form theorem for the
contractible Lagrangian submanifold (Subsection 4.2), i.e., that there exists
a biholomorphism taking the symplectic form on $M$ into the standard
symplectic form on the cotangent bundle.

###  2.1 Vo Van Tan’s characterization of contractible subvarieties

Here we recall the main results about the bimeromorphically contractible
subvarieties, following [G] [Vo], [AnVo]. All varieties and analytic spaces
are assumed to be connected.

Definition 2.3: A complex analytic space $X$ is called 1-convex if it admits a
proper holomorphic, bimeromorphic map $p:X\to Y$ to a Stein space $Y$, and
$p_{*}{\cal O}_{X}={\cal O}_{Y}$.

Remark 2.4: By Remmert Reduction theorem ([Rem]), a complex variety is
holomorphically convex if and only if it admits a proper, holomorphic map with
connected fibers to a Stein variety. The 1-convexity is a stronger property
which requires this map to be bimeromorphic.

Definition 2.5: Given a coherent sheaf $A$ on $X$, denote by $L(A)$ the
complex analytic space obtained as the relative spectrum of the sheaf of rings
$\bigoplus\operatorname{Sym}^{k}(A)$ over $X$.

Remark 2.6: When $A$ is a vector bundle, $L(A)$ is the total space of the dual
bundle $A^{*}$.

Definition 2.7: A coherent sheaf $A$ on a compact complex analytic space $X$
is called ample if for any coherent sheaf ${\cal F}$ on $X$ there exists $k>0$
such that $\operatorname{Sym}^{k}(A)\otimes{\cal F}$ is globally generated. It
is called cohomologically positive if there exists $k>0$ such that
$H^{i}(X,\operatorname{Sym}^{k}(A)\otimes{\cal F})=0$ for all $i>0$, and
weakly positive if the zero section of $L(A)$ admits a 1-convex neighbourhood.

Claim 2.8: Suppose that $A$ is a torsion-free coherent sheaf on $X$. Then $A$
is weakly positive if and only if the zero section $X\subset L(A)$ is
bimeromorphically contractible.

Proof. Step 1: The group ${\mathbb{C}}^{*}$ acts on $L(A)$ by taking any
compact subvariety which is not contained in the zero section to a non-compact
family of compact subvarieties. Since 1-convexity implies that all positive-
dimensional compact subvarieties of $L(A)$ are contained in a compact subset,
this is impossible, and any bimeromorphically contractible subset of $L(A)$ is
contained in the zero section.

Step 2: The 1-convexity means that $L(A)$ is equipped with a bimeromorphic
contraction $f$ to a Stein variety. Since the zero section $X\subset L(A)$ is
compact, and a Stein variety has no compact complex subvarieties, the image
$f(X)$ is zero-dimensional, hence it is contracted. As follows from Step 2,
nothing else is contracted.

Weak positivity has been introduced by Grauert in [G], who proved that for
line bundles on compact complex manifolds weak positivity is equivalent to
positivity, i.e. ampleness, and that a compact complex space carrying a weakly
positive vector bundle is projective algebraic ([G], §3, Satz 1 and 2). These
results were subsequently extended by Vo Van Tan, [Vo], as follows.

Theorem 2.9: ([Vo, Theorem 1]) Let $A$ be a coherent sheaf on a compact
irreducible complex space $X$. Then the three conditions defined above
(ampleness, weak positivity, cohomological positivity) are equivalent.

Theorem 2.10: ([Vo, Corollary 6]) A compact irreducible complex space carries
a torsion-free weakly positive coherent sheaf if and only if it is Moishezon.

Remark 2.11:  Grauert proved in [G, §3, Satz 8], that a compact submanifold
$X$ in an analytic space $M$ with weakly positive conormal bundle is
bimeromorphically contractible, but observed that the converse is not true,
that is, the conormal bundle of a contractible submanifold does not have to be
weakly positive. He asked whether the contractibility of $X$ implies that the
conormal sheaf of some analytic space structure on $X$ is weakly positive.
This question has been answered affirmatively by Vo Van Tan and Ancona.

More precisely, there is the following contractibility criterion:

Theorem 2.12: ([Vo, Corollary 7], [AnVo, Corollary 3]) Let $M$ be a complex
analytic space and $X\subset M$ a compact subvariety. Then $X$ is
bimeromorphically contractible if and only if there exists an ideal sheaf
${\cal J}\subset{\cal O}_{M}$ such that $\operatorname{Supp}({\cal
O}_{M}/{\cal J})=X$ and the conormal sheaf, which is defined as ${\cal
J}/{\cal J}^{2}$, is ample on $X$.

Remark 2.13: To say that $X$ carries an ample coherent torsion-free sheaf is
equivalent to saying that there is an ample coherent sheaf with sheaf-
theoretic support equal to $X$.

Remark 2.14:  In particular, it follows that a bimeromorphically contractible
subvariety is Moishezon.

###  2.2 Contractible subvarieties are isotropic

Recall the following classical theorem of complex geometry by Grauert and
Riemenschneider [GR], Satz 2.4 (see also [T], Corollary I).

Theorem 2.15:  Let $f:\;U{\>\longrightarrow\>}V$ be a bimeromorphic
contraction of complex spaces with $U$ smooth. Then $R^{i}f_{*}(K_{U})=0$ for
$i>0$, where $K_{U}$ is the canonical bundle of $U$.

Corollary 2.16:  Let $X\subset M$ be a bimeromorphically contractible
subvariety of a manifold with trivial canonical bundle. Then any open
neighbourhood of $X$ in $M$ contains a tubular neighbourhood $U\supset X$ such
that $H^{i}({\cal O}_{U})=0$ for any $i>0$.

Proof: Let $f:\;M{\>\longrightarrow\>}M_{0}$ be the bimeromorphic contraction,
mapping $X$ to a point $x\in M_{0}$. Consider a Stein neighbourhood $V\ni x$,
and let $U:=f^{-1}(V)$. By assumption, $K_{M}={\cal O}_{M}$. Grauert-
Riemenschneider theorem (2.2) implies that $R^{i}f_{*}({\cal O}_{U})=0$. The
Grothendieck spectral sequence with $E_{2}$-table $H^{j}(R^{i}f_{*}({\cal
O}_{U}))$ converges to $H^{i+j}({\cal O}_{U})$, giving $H^{k}({\cal
O}_{U})=H^{k}(f_{*}{\cal O}_{U})=H^{k}({\cal O}_{V})=0$ because $V$ is Stein.

The fact that a contractible subvariety is Moishezon, together with Grauert-
Riemenschneider theorem, yields the following result which is well-known in
the projective case (see e. g. [Ka]):

Proposition 2.17: Let $M$ be a holomorphic symplectic manifold and $X$ a
contractible subvariety. Then $X$ is isotropic, that is, the restriction of
the holomorphic symplectic form $\Omega$ to $X$ is zero.

Proof: We may assume that $M$ is a neighbourhood of $X$ as above, so that
$H^{2}({\cal O}_{M})=0$. Let $h:X^{\prime}\to X\subset M$ be a resolution of
singularities of $X$ with $X^{\prime}$ projective (it exists because $X$ is
Moishezon, 2.1). The Dolbeault cohomology class of the $(2,0)$-form
$h^{*}\overline{\Omega}$ is zero, because the Dolbeault cohomology class of
$\overline{\Omega}$ itself is zero on $M$. Since $X^{\prime}$ is smooth
projective, by Hodge theory $h^{*}{\Omega}=0$, so that $\Omega$ must be zero
on $X$.

###  2.3 An MMP-style proof of 2

Let $f:M\to M_{0}$ be our bimeromorphic contraction. Without loss of
generality, we replace $M_{0}$ by a small Stein neighbourhood of $x=f(X)$. In
particular, we may assume that $M$ retracts on $X$. We have established that
$X$ is Moishezon. It is known, see [P] in dimension 3, [Sh], [VP] in arbitrary
dimension, that all Moishezon varieties either contain rational curves, or are
projective, or both.

Case 1: $X$ contains a rational curve. Since $X$ is isotropic, we know that
$\dim(X)\leqslant n$. By [Ra], any rational curve in $M$ deforms in a family
of dimension at least $2n-2$. Since $M_{0}$ is Stein, all rational curves in
$M$ are contained in the contraction locus $X$. We claim they cover $X$ and
the dimension of $X$ is $n$. Indeed without loss of generality we may consider
a family of “minimal”, or “generically unsplit” curves: this means that all
curves through a general point $z$ of the subvariety $Z\subset X$ which they
cover, are irreducible. Then by bend-and-break there is only a finite number
of minimal curves through $z$ and any other point of $Z$. Dimension count
gives $\dim(Z)\geqslant n$ and $Z=X$. One proceeds to the proof of 2.5 to
finish the argument.

Case 2: $X$ is projective.

Claim 2.18: Let $M$ be a complex manifold with trivial canonical bundle
admitting a holomorphic birational contraction
$f:\;M{\>\longrightarrow\>}M_{0}$, contracting a submanifold $X\subset M$ to a
point $x$. If $X$ is projective, then some ample line bundle $L$ on $X$
extends, together with its sections, to a line bundle $L^{\prime}$ on
$f^{-1}(U)$, where $U$ is a suitable neighbourhood of $x$.

Proof: We choose $U$ in such a way that it is Stein and $X$ is a deformation
retract of $f^{-1}(U)$. To simplify the notation, we may assume that
$U=M_{0}$, and $f^{-1}(U)=M$. Consider the exponential exact sequence
$0{\>\longrightarrow\>}{\mathbb{Z}}_{M}\stackrel{{\scriptstyle
2\pi\sqrt{-1}\>}}{{{\>\longrightarrow\>}}}{\cal
O}_{M}{\>\longrightarrow\>}{\cal O}_{M}^{*}{\>\longrightarrow\>}0$. From this
exact sequence (and a similar sequence on $X$) we obtain the following
diagram, with all rows exact.

$\begin{CD}H^{1}(M,{\cal O}_{M})@>{}>{}>H^{1}(M,{\cal
O}_{M}^{*})@>{}>{}>H^{2}(M,{\mathbb{Z}})@>{}>{}>H^{2}(M,{\cal O}_{M})\\\
@V{}V{}V@V{}V{}V@V{}V{}V@V{}V{}V\\\ H^{1}(X,{\cal O}_{X})@>{}>{}>H^{1}(X,{\cal
O}_{X}^{*})@>{}>{}>H^{2}(X,{\mathbb{Z}})@>{}>{}>H^{2}(X,{\cal O}_{X})\end{CD}$

The vertical arrow
$H^{2}(M,{\mathbb{Z}}){\>\longrightarrow\>}H^{2}(X,{\mathbb{Z}})$ in this
diagram is an isomorphism, and $H^{2}(M,{\cal O}_{M})=0$ by Grauert-
Riemenschneider (2.2). This implies that any $\eta\in H^{2}(X,{\mathbb{Z}})$
which belongs to the image of $H^{1}(X,{\cal O}_{X}^{*})$ can be lifted to an
element of $H^{2}(M,{\mathbb{Z}})$ which belongs to the image of
$H^{1}(M,{\cal O}_{M}^{*})$. Therefore, the restriction map is surjective on
the set of connected components of $H^{1}(X,{\cal O}_{X}^{*})$. Since
ampleness is a numerical condition, a connected component of
$\operatorname{Pic}(X)$ containing an ample line bundle consists of ample line
bundles.

Therefore some ample $L$ on $X$ extends to a line bundle $L^{\prime}$ on $M$.
The proof of [N], Proposition 1.4, shows that for a sufficiently large $k$,
the restriction map $(L^{\prime})^{\otimes k}\to L^{\otimes k}$ is surjective
on the global sections.

Hence the contraction $f$ is projective. Now we obtain the uniruledness by the
proof of [Kaw], Theorem 2. Indeed, this theorem is a consequence of [Kaw],
Lemma 2, where the crucial step is the vanishing of higher direct images from
[N], Theorem 3.6, valid in the complex-analytic context. Once the uniruledness
is established, we again proceed to 2.5.

Remark 2.19:  As noted in the proof of [N], Theorem 3.6, we are in fact
dealing with a version of log-MMP here. Namely, to apply log-MMP one needs a
boundary divisor negative on $X$. In the projective setting, it is given by
decomposing the big divisor $f^{*}H$, $H$ ample, in the sum of ample and
effective. In the complex-analytic setting one remarks that
$(L^{\prime})^{-1}$ is effective, indeed $f_{*}((L^{\prime})^{-1})$ has
sections because $M_{0}$ is Stein. This gives the boundary divisor negative on
$X$.

In the rest of this section, we give another proof of 2, where the
uniruledness of $X$ is obtained directly from Ancona-Vo Van Tan criterion with
the help of [CP].

###  2.4 Lagrangian contractible submanifolds are uniruled

Here we apply Vo Van Tan’s results to a bimeromorphically contractible
Lagrangian submanifold $X$ and prove that it is uniruled (that is, covered by
rational curves).

Lemma 2.20:  i) A quotient of an ample sheaf is ample;

ii) A tensor power of an ample sheaf is again ample;

iii) If $\operatorname{Supp}({\cal A})=X$, the restriction of an ample sheaf
${\cal A}$ to a curve on $X$, which is not contained in the singular set of
${\cal A}$, is of strictly positive degree.

Proof: The first assertion is true because the $k$-th symmetric power of a
sheaf surjects onto the $k$-th symmetric power of its quotient, the second is
proved using the representation theory, exactly as in the vector bundle case
treated in [H], and the third results from the fact that a globally generated
sheaf of positive rank on a curve has strictly positive degree, unless it is
trivial.

Lemma 2.21:  Let $X\subset M$ be a smooth submanifold, and ${\cal
J}\subset{\cal O}_{M}$ an ideal sheaf defining an analytic subspace of $M$
with support $X$ (that is, $\operatorname{Supp}({\cal O}_{M}/{\cal J})=X$).
Denote by $J_{X}$ the ideal of $X$. Then there exists a non-zero map from
${\cal J}/{\cal J}^{2}$ to
$\operatorname{Sym}^{r}(J_{X}/J_{X}^{2})=J_{X}^{r}/J^{r+1}_{X}$ for some $r$.

Proof: We have ${\cal J}\subset J_{X}$. Choose $r$ such that ${\cal J}\subset
J_{X}^{r}$ and ${\cal J}\subsetneq J_{X}^{r+1}$. Then the natural map ${\cal
J}{\>\longrightarrow\>}J_{X}^{r}/J^{r+1}_{X}$ is non-zero. However, since
${\cal J}\subset J_{X}$, this map vanishes on ${\cal J}^{2}$, defining a non-
zero sheaf morphism ${\cal J}/{\cal
J}^{2}{\>\longrightarrow\>}J_{X}^{r}/J^{r+1}_{X}$.

Corollary 2.22: If $X$ is a contractible Lagrangian submanifold in a
holomorphic symplectic $M$, then for some $r$, $\operatorname{Sym}^{r}TM$
contains an ample subsheaf.

Proof: For such an $X$, the holomorphic symplectic form provides an
isomorphism between the conormal bundle and the tangent bundle. Take ${\cal
J}$ as in 2.1 and apply the above lemma.

In the sequel, we will need the following theorem, implicit in the work of
Campana and Paun [CP] and the preceding work of Campana, Peternell and Toma
[CPT]. Define a movable curve $C$ on $X$ as a member of a dominating family of
irreducible curves.

Theorem 2.23:  Let $X$ be a Moishezon manifold. Assume that for some $r>0$,
$\operatorname{Sym}^{r}TX$ has a subsheaf ${\cal E}$ such that $deg({\cal
E}|_{C})>0$ for a movable curve $C$. Then $X$ is uniruled.

Proof: Let $\alpha$ be the cohomology class of $C$, then, in the terminology
of [CP], $\mu_{\alpha,max}({\operatorname{Sym}^{r}TX})>0$. Then also
$\mu_{\alpha,max}(TX)>0$ ([CP], Theorem 2.9), and the maximal destabilizing
sheaf ${\cal F}$ of $TX$ has positive $\alpha$-slope (together with all its
quotients). If $X$ was projective, we would conclude immediately that $X$ is
uniruled, by [CP], Theorem 1.1. Otherwise, we let $\pi:X^{\prime}\to X$ be a
projective modification. The class $\pi^{*}\alpha$ is a movable class on
$X^{\prime}$. Indeed, by [BDPP] a movable class $\beta$ on a projective
manifold is characterized by $D\beta\geqslant 0$ for all effective divisor
classes $D$. Setting $\beta=\pi^{*}\alpha$, we get that this condition is
satisfied by the projection formula: indeed $\alpha\pi_{*}D=0$ when $D$ is
contracted by $\pi$, and is non-negative otherwise. Now in the same way as in
Lemma 2.12 of [CP], the subsheaf ${\cal F}^{\prime}\subset TX^{\prime}$
defined by ${\cal F}^{\prime}=\pi^{*}{\cal F}\cap TX^{\prime}$ is of positive
slope with respect to the movable class $\pi^{*}\alpha$, and we conclude by
the main results of [CP] that $X^{\prime}$, and hence also $X$, is uniruled.

Corollary 2.24: A contractible Lagrangian submanifold of a holomorphic
symplectic manifold is uniruled.

###  2.5 Uniruled Lagrangian submanifolds

Following [CMS, HY, Ke], we use the uniruledness (2.4) to prove the following
theorem.

Theorem 2.25:  Let $X\subset M$ be a bimeromorphically contractible Lagrangian
submanifold of a holomorphically symplectic $2n$-dimensional manifold $M$.
Then $X$ is biholomorphic to ${\mathbb{C}}P^{n}$.

Proof: By [Ra], Corollary 5.1, any rational curve $C$ on $M$ deforms in a
family of dimension at least $2n-2$. When $C\subset X$, any deformation of $C$
also lies in $X$, because $X$ is contractible. We conclude that any family of
rational curves in $X$, $\dim(X)=n$, has dimension at least $2n-2$. Such
families exist because $X$ is uniruled.

We apply this to a complete family of minimal rational curves covering $X$.
“Minimal” in this context means that all curves from the family passing
through a sufficiently general point $x$ of $X$ are irreducible. It is easy to
see by noetherian induction that such minimal rational curves exist. We obtain
a family of irreducible curves through $x$ of dimension at least $n-1$. These
curves must cover $X$, by bend-and-break (yielding that there are only
finitely many curves connecting $x$ to another point $y$) and by compacity of
families of cycles on Moishezon manifolds. Now we are in the situation studied
by Kebekus for projective manifolds, see [Ke]. The arguments carry over
verbatim to the Moishezon case and prove that $X$ is isomorphic to the
projective space.

##  3 Moser’s lemma and normal form of Lagrangian submanifolds

###  3.1 Moser’s lemma for holomorphic symplectic and C-symplectic structures

The complex structure on a holomorphically symplectic manifold is uniquely
determined by the holomorphic symplectic form. This was the starting point of
[SV]. In [SV] the authors obtain a version of holomorphically symplectic
Moser’s lemma which accounts for the deformations of complex structure as well
as for the deformations of the holomorphic symplectic form.

Moser’s lemma is the following fundamental result of symplectic geometry.

Theorem 3.1: Let $\omega_{t}$, $t\in[0,1]$ be a smooth family of symplectic
structures on a compact manifold $M$. Assume that the cohomology class
$[\omega_{t}]\in H^{2}(M)$ is constant in $t$. Then there exists a smooth
family $\Psi_{t}\in\operatorname{\sf Diff}_{0}(M)$ of diffeomorphisms such
that $\Psi_{t}^{*}\omega_{0}=\omega_{t}$.

Proof: [McDS].

Remark 3.2: The proof proceeds by constructing a vector field and integrating
it to a flow of diffeomorphisms. It is for integration that one needs
compacity. In particular, one can obtain a similar statement replacing a
compact manifold $M$ by a neighbourhood of a compact submanifold $X\subset M$.

For a holomorphically symplectic version of this lemma, we recall the notion
of C-symplectic structures. A C-symplectic structure is a holomorphically
symplectic form understood abstractly, that is, without fixing a complex
structure (which is uniquely determined by the holomorphically symplectic form
nevertheless).

Definition 3.3:  ([BDV]) Let $M$ be a smooth $4n$-dimensional manifold. A
closed complex-valued form $\Omega$ on $M$ is called C-symplectic if
$\Omega^{n+1}=0$ and $\Omega^{n}\wedge\overline{\Omega}^{n}$ is a non-
degenerate volume form.

Theorem 3.4:  Let $\Omega\in\Lambda^{2}(M,{\mathbb{C}})$ be a C-symplectic
form, and $T^{0,1}_{\Omega}(M):=\ker\Omega$, where

$\ker\Omega:=\\{v\in TM\otimes{\mathbb{C}}\ \ |\ \ \Omega\hskip
2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip 2.0ptv=0\\}.$

Then
$T^{0,1}_{\Omega}(M)\oplus\overline{T^{0,1}_{\Omega}(M)}=TM\otimes_{\mathbb{R}}{\mathbb{C}}$,
hence the sub-bundle $T^{0,1}_{\Omega}(M)$ defines an almost complex structure
$I_{\Omega}$ on $M$. Moreover, $I_{\Omega}$ is integrable, and $\Omega$ is
holomorphically symplectic on $(M,I_{\Omega})$.

Proof: [BDV, Proposition 2.12], [Ve, Theorem 3.5].

Now we can state the C-symplectic version of Moser’s lemma.

Theorem 3.5: Let $(M,I_{t},\Omega_{t})$, $t\in[0,1]$ be a family of
C-symplectic forms on a compact manifold, with corresponding complex
structures $I_{t}$. Assume that the cohomology class $[\Omega_{t}]\in
H^{2}(M,{\mathbb{C}})$ is constant, and $H^{0,1}(M,I_{t})=0$, where
$H^{0,1}(M,I_{t})=H^{1}(M,{\cal O}_{(M,I_{t})})$ is the first cohomology of
the sheaf of holomorphic functions. Then there exists a smooth family of
diffeomorphisms $V_{t}\in\operatorname{\sf Diff}_{0}(M)$, such that
$V_{t}^{*}\Omega_{0}=\Omega_{t}$.

Proof: [SV, Theorem 2.5 (2)].

For non-compact manifolds, a slightly more cumbersome version of Moser’s lemma
can be stated.

Theorem 3.6:  Let $\pi\colon{\cal X}\to\Delta$ be a smooth family of
holomorphic symplectic manifolds (not necessarily compact) over the unit disc,
locally trivial as a family of $C^{\infty}$ manifolds. Denote by ${\cal
X}_{t}=\pi^{-1}(t)$ its fiber, and let $\Omega_{t}\in H^{0}({\cal
X}_{t},\Omega^{2}_{{\cal X}_{t}})$ be its holomorphic symplectic form,
smoothly depending on $t$. Using the $C^{\infty}$ trivialization to identify
cohomology groups of the fibres, assume that the cohomology class of
$\Omega_{t}$ does not depend on $t\in\Delta$, and $H^{1}({\cal X}_{t},{\cal
O}_{{\cal X}_{t}})=0$. Let $K\subset{\cal X}_{t_{0}}$ be a compact subset.
Then there exists an open neighbourhood $U\subset\Delta$ of $t_{0}\in\Delta$,
and an open subset ${\widetilde{U}}\subset\pi^{-1}(U)$, with
$K\subset{\widetilde{U}}$, with the following property. The set
${\widetilde{U}}$ is locally trivially fibered over $U$, with all fibres
${\widetilde{U}}\cap\pi^{-1}(t)$, $t\in U$ isomorphic as holomorphically
symplectic manifolds.

Proof: [SV, Theorem 2.5 (1)].

We will use an equivalent version of this result.

Theorem 3.7:  Let $\pi\colon{\cal X}\to\Delta$ be a smooth family of
holomorphic symplectic manifolds (not necessarily compact) over the unit disc,
trivial as a family of $C^{\infty}$ manifolds. Denote by ${\cal
X}_{t}=\pi^{-1}(t)$ its fiber, and let $\Omega_{t}\in H^{0}({\cal
X}_{t},\Omega^{2}_{{\cal X}_{t}})$ be its holomorphic symplectic form,
smoothly depending on $t$. Using the $C^{\infty}$ trivialization to identify
cohomology groups of the fibres, assume that the cohomology class of
$\Omega_{t}$ does not depend on $t\in\Delta$, and $H^{1}({\cal X}_{t},{\cal
O}_{{\cal X}_{t}})=0$. Let $K_{t}\subset{\cal X}_{t}$ be a locally trivial
family of compact subsets, and ${\cal K}\subset{\cal X}$ their union. Then
there exists an open neighbourhood $U\subset\Delta$ of $t_{0}\in\Delta$, and
an open subset ${\widetilde{U}}\subset\pi^{-1}(U)$, with
$K_{t}\subset{\widetilde{U}}$, with the following property. The set
${\widetilde{U}}$ is locally trivially fibered over $U$, with all fibres
${\widetilde{U}}\cap\pi^{-1}(t)$, $t\in U$ isomorphic as holomorphically
symplectic manifolds.

Proof: Chose $U\subset\Delta$ and $\widetilde{U}\subset{\cal X}$ as in 3.1.
Denote by $\Delta_{\lambda}$ the disc of radius $\lambda$ centered in 0. We
need to show that ${\cal K}\cap\pi^{-1}(\Delta_{\lambda})\subset\widetilde{U}$
for $\lambda$ sifficiently small. Introduce a Riemannian metric on ${\cal X}$.
Since $\widetilde{U}\subset{\cal X}$ is open, it contains an
$\varepsilon$-neighbourhood of $K_{0}$, for a sufficiently small
$\varepsilon>0$. Choose $\lambda$ in such a way that $K_{t}$ belongs to an
$\varepsilon$-neighbourhood $K_{0}(\varepsilon)$ of $K_{0}$ for all
$t\in\Delta_{\lambda}$; this is possible to do, because the Hausdorff distance
$d_{H}(K_{t},K_{0})$ converges to 0 as $t$ converges to 0. For such $\lambda$,
clearly, one has

${\cal K}\cap\pi^{-1}(\Delta_{\lambda})\subset
K_{0}(\varepsilon)\subset\widetilde{U}.$

###  3.2 Moser’s lemma for a neighbourhood of a holomorphic Lagrangian
submanifold

We will apply 3.1 when $K$ is bimeromorphically contractible Lagrangian
submanifolds. We need the following version of Moser’s lemma.

Lemma 3.8:  Let $(M,I_{t},\Omega_{t})$, $t\in[0,1]$ be a smooth family of
C-symplectic manifolds (not necessarily compact), with all $\Omega_{t}$ exact,
and $E_{t}\subset(M,I_{t})$ be a family of compact holomorphic Lagrangian
subvarieties. Assume that $H^{0,1}(M,I_{t})=0$. Then $E_{t}$ have open
neighbourhoods $U_{t}$ in $M$ such that $(U_{t},I_{t},\Omega_{t},E_{t})$ is
trivialized by a flow of diffeomorphisms.

Proof: Follows immediately from 3.1.

In the situation described by 2.2, any deformation of a neighbourhood of a
holomorphic Lagrangian subvariety is locally trivial by 3.2, because the
relevant cohomology groups vanish. This brings

Corollary 3.9:  Let $(M,I,\Omega)$ be a holomorphically symplectic manifold
(not necessarily compact), and $E\subset(M,I)$ a compact holomorphic
Lagrangian subvariety biholomorphic to ${\mathbb{C}}P^{n}$. Assume that a
neighbourhood of $E$ can be smoothly deformed to a neighbourhood of the zero
section in $T^{*}E$ as a C-symplectic manifold. Then $E$ has a neighbourhood
which is isomorphic to a neighbourhood of $E$ in $T^{*}E$ as a holomorphically
symplectic manifold.

Proof: We apply 3.2 to the deformation $(U_{t},I_{t},\Omega_{t},E_{t})$
connecting a neighbourhood $U_{0}$ of $E$ in $M$ to a neighbourhood $U_{1}$ of
$E$ in $T^{*}E$. Note that since $E\cong{\mathbb{P}}^{n}$, all $E_{t}$ are
isomorphic to ${\mathbb{P}}^{n}$ as well. Therefore they are all
bimeromorphically contractible inside $U_{t}$: indeed $N^{*}E_{t}\cong TE_{t}$
is isomorphic to the cotangent bundle of ${\mathbb{P}}^{n}$ and hence ample,
and we conclude by 2.1. Hence $H^{i}({\cal O}_{U_{t}}),i>0,$ of a tubular
neighbourhood of $E_{t}$ in $(M,I_{t})$ vanish by 2.2, and 3.2 applies.

Remark 3.10: The exactness condition on the symplectic forms is satisfied
automatically in a sufficiently small neighbourhood $U_{t}$ of $E_{t}$,
because $U_{t}$ retracts on $E_{t}$.

##  4 Deformation to the normal cone of a Lagrangian submanifold

To apply 3.2, we need to deform a neighbourhood of a contractible Lagrangian
submanifold to its neighbourhood in its cotangent space. This construction is
well known in algebraic geometry under the name “deformation to the normal
cone”; however, we have to make sure it works in the holomorphic symplectic
category.

###  4.1 Deformation to the normal cone: an introduction

Let $X\subset M$ be a complex submanifold in a complex manifold. Deformation
to the normal cone is a holomorphic deformation of a neighbourhood of
$X\subset M$ over the disk such that its central fiber is the total space of
the normal bundle $NX$, and the rest of the fibers are $M$. It is obtained as
follows.

Let $X\subset M$ be a complex subvariety. Consider a product
$M_{1}:=M\times\Delta$ of $M$ with the disk $\Delta$, and let
$\widetilde{M}_{1}$ be the blow-up of $M_{1}$ in $X\times\\{0\\}$. Denote by
$\widetilde{\pi}_{1}:\;\widetilde{\cal X}{\>\longrightarrow\>}\Delta$ the
blow-down composed with the projection. The preimage
$\widetilde{\pi}_{1}^{-1}(0)$ is a union of two irreducible components, the
proper preimage of $M\times\\{0\\}$, denoted $D_{1}$, and the blow-up divisor,
denoted $D_{2}$.

Definition 4.1: The deformation to the normal cone is the complement
$\widetilde{M}:=\widetilde{M_{1}}\backslash D_{1}$. considered as a fibration
over the disk $\Delta$.

Clearly, the central fiber of the natural projection
$\widetilde{M}{\>\longrightarrow\>}\Delta$ is $D_{2}\backslash(D_{1}\cap
D_{2})$.

Claim 4.2: When $X$ is smooth, and $M$ is a tubular neighbourhood of $X$ in
$M$, the complement $D_{2}\backslash(D_{1}\cap D_{2})$ is naturally isomorphic
to $NX$, and the “deformation to the normal cone” family
$\widetilde{\pi}:\;\widetilde{M}{\>\longrightarrow\>}\Delta$ is locally
trivial in the smooth category.

Proof. Step 1: The blow-up divisor
$E={\mathbb{P}}N_{M_{1}}X={\mathbb{P}}(NX\oplus{\cal O}_{X})$, and its
intersection with $D_{1}$ is the set of all $l\in{\mathbb{P}}(NX\oplus{\cal
O}_{X})$ tangent to $M\times\\{0\\}$. We identify this intersection with
${\mathbb{P}}(NX)$. This gives an isomorphism $D_{2}\backslash(D_{1}\cap
D_{2})={\mathbb{P}}(NX\oplus{\cal
O}_{X})\backslash{\mathbb{P}}(NX)=\operatorname{Tot}(NX)$.

Step 2: Now, the tubular neighbourhood of $X\subset M$ is diffeomorphic to
$\operatorname{Tot}(NX)$, hence all fibers of
$\widetilde{\pi}:\;\widetilde{M}{\>\longrightarrow\>}\Delta$ are
diffeomorphic.

###  4.2 Deformation to the normal cone in holomorphic symplectic category

The following theorem, together with 3.2, implies the holomorphic version of
the Weinstein normal neighbourhood theorem (1.2).

Theorem 4.3:  Let $(M,\Omega)$ be a holomorphically symplectic manifold, and
$X\subset M$ a holomorphically Lagrangian submanifold. Then there exists a
smooth, holomorphic deformation of a neighbourhood of $X$ in $M$ over the disk
$\Delta$, such that its central fiber is biholomorphic to a neighbourhood of
$X$ in $T^{*}X$, the rest of the fibers are biholomorphic to a neighbourhood
of $X$ in $M$, and the holomorphic symplectic form on $T^{*}X$ can be smoothly
extended to the holomorphic symplectic form on the rest of the fibers.

Remark 4.4: Note that this deformation in complex analytic category is already
constructed: it is the “deformation to the normal cone” family. However, to
apply the C-symplectic Moser lemma, we need to have a smooth family of
holomorphically symplectic forms on its fibers.

Proof of 4.2. Step 1: Let
$\widetilde{M}\stackrel{{\scriptstyle\widetilde{\pi}}}{{{\>\longrightarrow\>}}}\Delta$
be the deformation to the normal cone family, and $t$ the coordinate on
$\Delta$. Locally in $X$ we can write $X$ by a system of holomorphic equations
$q_{1}=q_{2}=...=q_{n}=0$, and the holomorphically symplectic form as
$\Omega=\sum_{i=1}^{n}dp_{i}\wedge dq_{i}$. We are going to prove that
$t^{-1}\Omega$ is extended to a non-degenerate form on the central fiber of
$\widetilde{\pi}$.

Since this extension is unique, if would suffice to prove that $t^{-1}\Omega$
can be extended to the central fiber locally in $X$, and to check that it is
non-degenerate.

Step 2: The coordinates on the central fiber of the deformation to the normal
cone family
$\widetilde{M}\stackrel{{\scriptstyle\widetilde{\pi}}}{{{\>\longrightarrow\>}}}\Delta$
are given by $p_{1},...,p_{n},\widetilde{q}_{1},...,\widetilde{q}_{n}$.
Trivializing the neighbourhood of $x\times\Delta\in X\times\Delta$ along
$\Delta$ in the usual way, we write $\widetilde{q}_{i}=t^{-1}q_{i}$: this is
the standard way to write coordinates on the blow-up.

Locally, we can always find a $\theta\in\Omega^{1}M$ such that
$d\theta=\Omega$ and $\theta{\left|{}_{{\phantom{|}\\!\\!}_{X}}\right.}=0$.
For example, we can take $\theta=\sum_{i}q_{i}dp_{i}$ in the above
coordinates. Writing $\theta$ in the coordinates $\widetilde{q}_{i},p_{i}$, we
get $\theta=t\sum_{i}\widetilde{q}_{i}dp_{i}$. Then
$d(t^{-1}\theta)=\sum_{i}d\widetilde{q}_{i}\wedge dp_{i}$ is a holomorphic
form on $\widetilde{M}$. Restricted to the fibers $\pi^{-1}(u)$ of the
projection
$\widetilde{M}\stackrel{{\scriptstyle\widetilde{\pi}}}{{{\>\longrightarrow\>}}}\Delta$,
this form is equal to $t^{-1}\Omega+d(t^{-1})\wedge\theta$; the second term
restricted to $\pi^{-1}(u)$ vanishes, hence the restrictions of
$d(t^{-1}\theta)$ extend $t^{-1}\Omega$ holomorphically to $\widetilde{M}$.

Step 3: It remains to show that $d(t^{-1}\theta)$ is non-degenerate on the
central fiber of
$\widetilde{M}\stackrel{{\scriptstyle\widetilde{\pi}}}{{{\>\longrightarrow\>}}}\Delta$.
Writing $\Omega=\sum_{i=1}^{n}dp_{i}\wedge dq_{i}$ as above and passing to the
coordinates $q_{i}=t\widetilde{q}_{i}$, we obtain
$\Omega=\sum_{i=1}^{n}tdp_{i}\wedge
d\widetilde{q}_{i}+\sum_{i=1}^{n}dp_{i}\wedge\widetilde{q}_{i}dt$. Since the
last term vanishes on the fibers, the form
$t^{-1}\Omega=\sum_{i=1}^{n}dp_{i}\wedge d\widetilde{q}_{i}$ is smooth, non-
degenerate on the central fiber, and equal to $t^{-1}\Omega$ on the general
fibers. This proves 4.2.

Acknowledgements: We are grateful to Frédéric Campana, Andreas Höring, Dmitry
Kaledin, Yuri Prokhorov, Costya Shramov and Andrey Soldatenkov for their
insightful discussions, and to Arnaud Beauville for his valuable email
communication.

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Ekaterina Amerik
Laboratory of Algebraic Geometry,
National Research University HSE,
Department of Mathematics, 6 Usacheva Str., Moscow, Russia,
<EMAIL_ADDRESS>also:
Université Paris-11,
Laboratoire de Mathématiques,
Campus d’Orsay, Bâtiment 425, 91405 Orsay, France

Misha Verbitsky
Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Estrada Dona Castorina, 110
Jardim Botânico, CEP 22460-320
Rio de Janeiro, RJ - Brasil
<EMAIL_ADDRESS>
also:
Laboratory of Algebraic Geometry,
National Research University HSE,
Department of Mathematics, 6 Usacheva Str.,
Moscow, Russia.