# Partial Resolutions of Affine Symplectic Singularities

Alberto San Miguel Malaney

###### Abstract.

We explore the relationship between the Poisson deformation theory, birational
geometry, and Springer theory of crepant partial resolutions of conical affine
symplectic singularities. We show that given any crepant partial resolution
$X^{\prime}$, the Poisson deformation functor of $X^{\prime}$ is
prorepresentable and unobstructed. Additionally, we define a version of the
Namikawa Weyl group $W_{X^{\prime}}$ for partial resolutions, such that
$W_{X^{\prime}}$ is a parabolic subgroup of the Namikawa Weyl group $W_{X}$
determined by the birational geometry of $X^{\prime}$, in particular by a face
of a relative movable cone. If $Y$ is a $\mathbb{Q}$-factorial terminalization
which covers $X^{\prime}$, we show there is a natural morphism from Poisson
deformations of $Y$ to those of $X^{\prime}$, and that this morphism is a
Galois covering with Galois group $W_{X^{\prime}}$, building on Namikawa’s
work in [NaPD2]. Finally, we put these partial resolutions and their universal
deformations into the context of McGerty and Nevins’ work in [MN], obtaining
some preliminary results concerning their Springer theory.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation
    2. 1.2 Outline of Results
    3. 1.3 Summary
    4. 1.4 Notation and Conventions
    5. 1.5 Acknowledgements
  2. 2 Lie Theory and Folding
  3. 3 Preliminaries and Background
  4. 4 Partial Symplectic Resolutions and Birational Geometry
  5. 5 Deformation Theory and Prorepresentability
    1. 5.1 Prorepresentability
    2. 5.2 Unobstructedness
    3. 5.3 Algebraization
  6. 6 Kleinian Singularities
    1. 6.1 Structure of Partial Resolutions
    2. 6.2 Identifying $W_{S^{\prime}}$
  7. 7 Main Theorem
  8. 8 Springer theory

##  1\. Introduction

In this paper we study the deformation theory, birational geometry, and
Springer theory of partial resolutions of affine symplectic singularities.

###  1.1. Motivation

The Springer resolution $\pi:\tilde{\mathcal{N}}\rightarrow\mathcal{N}$ of a
semisimple Lie algebra $\mathfrak{g}$ is a central object in geometric
representation theory. In particular, it is the unique symplectic resolution
of the nilpotent cone $\mathcal{N}\subset\mathfrak{g}$. More generally, for
every conjugacy class $\mathcal{P}$ of parabolic subalgebras of
$\mathfrak{g}$, we obtain the following partial resolution of $\mathcal{N}$:

$\tilde{\mathcal{N}}^{\mathcal{P}}:=\\{(x,\mathfrak{p})|x\in\mathfrak{p}\cap\mathcal{N},\mathfrak{p}\in\mathcal{P}\\}.$

If we let $\mathcal{P}$ be the set of Borel subalgebras we recover the
Springer resolution. Furthermore for any conjugacy class $\mathcal{P}$, each
Borel subalgebra $\mathfrak{b}\subset\mathfrak{g}$ is contained in a unique
element of $\mathcal{P}$, so we obtain birational and projective maps

$\tilde{\mathcal{N}}\xrightarrow{\eta}\tilde{\mathcal{N}}^{\mathcal{P}}\xrightarrow{\xi}\mathcal{N}.$

Let $W$ be the Weyl group of $\mathfrak{g}$. Then we can take the Weyl group
of the Levi subalgebra of any $\mathfrak{p}\in\mathcal{P}$ to obtain a
parabolic subalgebra $W_{\mathcal{P}}\subset W$ of the Weyl group. This gives
us a bijection between conjugacy classes of parabolic subalgebras of
$\mathfrak{g}$ and parabolic subgroups of $W$, both of which can also be
classified by subsets of the simple roots of $\mathfrak{g}$.

The relationship between the geometry of these partial resolutions and the
combinatorics of the Weyl group was studied extensively in [BM]. For example,
[BM, 2.8] states that for any $x\in\mathcal{N}$,

(1) $H^{*}(\xi^{-1}(x),\mathbb{Q})\cong
H^{*}(\pi^{-1}(x),\mathbb{Q})^{W_{\mathcal{P}}}.$

In proving results such as these it is essential that we can drop the
nilpotent condition to obtain a Grothendieck-Springer resolution as below,
where $\tilde{\eta}$ is generically a $W_{\mathcal{P}}$-cover and
$\tilde{\pi}:=\tilde{\xi}\circ\tilde{\eta}$ is generically a $W$-cover.

${{\tilde{\mathcal{N}}}}$${{\tilde{\mathcal{N}}^{\mathcal{P}}}}$${\mathcal{N}}$${{\tilde{\mathfrak{g}}}}$${{\tilde{\mathfrak{g}}^{\mathcal{P}}}}$${{\mathfrak{g}}}$$\scriptstyle{\tilde{\eta}}$$\scriptstyle{\tilde{\xi}}$$\scriptstyle{\eta}$$\scriptstyle{\xi}$

For a general conical affine symplectic singularity $X$, with a
$\mathbb{Q}$-factorial terminalization $\pi:Y\rightarrow X$, the
generalization of the Grothendieck-Springer resolution is the universal conic
deformation. These have been studied extensively by Namikawa, who in a series
of papers (primarily [NaFlops, NaPD1, NaPD2]) showed that the Poisson
deformation functors $\operatorname{PD}_{X}$ and $\operatorname{PD}_{Y}$ of
$X$ and $Y$ are prorepresentable and unobstructed, and that there is a natural
map

$\pi_{*}:\operatorname{PD}_{Y}\rightarrow\operatorname{PD}_{X}.$

In particular this implies that $X$ and $Y$ have universal formal deformations
over the formal neighborhood of zero in their tangent spaces. Using the
conical action on $X$, which also lifts to $Y$, these formal deformations can
be algebraicized to obtain universal conic deformations $\mathcal{X}$ and
$\mathcal{Y}$ (see [Lo, 2.6, 2.12]), which fit into the following diagram:

(2)
${Y}$${\mathcal{Y}}$${{\operatorname{PD}_{Y}(\mathbb{C}[\epsilon])}}$${X}$${\mathcal{X}}$${{\operatorname{PD}_{X}(\mathbb{C}[\epsilon])\cong\operatorname{PD}_{Y}(\mathbb{C}[\epsilon])/W_{X}}}$$\scriptstyle{\tilde{\pi}}$$\scriptstyle{\pi}$$\scriptstyle{\pi_{*}}$

In the above diagram $X$ and $Y$ embed in $\mathcal{X}$ and $\mathcal{Y}$ as
the central fibers, and $\pi_{*}$ is the quotient by $W_{X}$.

In the case where $\pi$ is a symplectic resolution, the viability of
symplectic Springer theory was further established by McGerty and Nevins in
[MN] using the above constructions. In particular [MN] employs a nearby cycles
construction over the universal conic deformation to show that the symplectic
Springer sheaf $\mathsf{Spr}:=\pi_{*}\mathbb{C}_{Y}$ is constructible with
respect to the stratification by symplectic leaves on $X$. Additionally,
letting $\mathsf{HC}:=\tilde{\pi}_{*}{\mathbb{C}_{\mathcal{Y}}}$ be the
symplectic Harish-Chandra sheaf, there is a restriction map

$\operatorname{End}(\mathsf{HC})\cong\mathbb{C}[W_{X}]\rightarrow\operatorname{End}(\mathsf{Spr}).$

which generalizes the classical isomorphism
$\mathbb{C}[W]\cong\operatorname{End}(\mathsf{Spr})$ in Springer theory. It
also produces an action of $W_{X}$ on the cohomology of the fibers of $\pi$.

###  1.2. Outline of Results

In this paper we introduce partial resolutions into the above story, treating
them as the appropriate analogue of parabolic subalgebras in the symplectic
setting. We introduce a natural definition of a symplectic partial resolution,
which we go on to show is equivalent to being crepant and to being covered by
a $\mathbb{Q}$-factorial terminalization. The main construction of the paper
is a generalization of the Namikawa Weyl group to crepant partial resolutions
$\rho:X^{\prime}\rightarrow X$ of conical affine symplectic singularities,
such that the Namikawa Weyl group $W_{X^{\prime}}$ of a partial resolution is
a parabolic subgroup of $W_{X}$.

In order to proceed we prove that the Poisson deformation functor
$\operatorname{PD}_{X^{\prime}}$ of $X^{\prime}$ is also prorepresentable and
unobstructed and that $X^{\prime}$ has a universal conic deformation over
$\operatorname{PD}_{X^{\prime}}(\mathbb{C}[\epsilon])$. Given a sequence of
maps

$Y\xrightarrow{\pi^{\prime}}X^{\prime}\xrightarrow{\rho}X$

where the composition $\pi:Y\rightarrow X$ is a $\mathbb{Q}$-factorial
terminalization of $X$, we also define and study the pushforward functors

$\operatorname{PD}_{Y}\xrightarrow{\pi^{\prime}_{*}}\operatorname{PD}_{X^{\prime}}\xrightarrow{\rho_{*}}\operatorname{PD}_{X}.$

This culminates in 7.3, the main theorem of the paper, which states that there
is a diagram

${Y}$${\mathcal{Y}}$${{\operatorname{PD}_{Y}(\mathbb{C}[\epsilon])}}$${{X^{\prime}}}$${{\mathcal{X}^{\prime}}}$${{\operatorname{PD}_{X^{\prime}}(\mathbb{C}[\epsilon])\cong\operatorname{PD}_{Y}(\mathbb{C}[\epsilon])/W_{X^{\prime}}}}$${X}$${\mathcal{X}}$${{\operatorname{PD}_{X}(\mathbb{C}[\epsilon])\cong\operatorname{PD}_{Y}(\mathbb{C}[\epsilon])/W_{X}}}$$\scriptstyle{i_{Y}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\tilde{\pi}^{\prime}}$$\scriptstyle{i_{X^{\prime}}}$$\scriptstyle{\pi^{\prime}_{*}}$$\scriptstyle{\rho}$$\scriptstyle{\tilde{\rho}}$$\scriptstyle{i_{X}}$$\scriptstyle{\rho_{*}}$

extending diagram (2) where $\pi^{\prime}_{*}$ is the quotient by
$W_{X^{\prime}}$. The diagram above can be seen as a justification for calling
$W_{X^{\prime}}$ the Namikawa Weyl group.

One key result and motivation for this paper is Namikawa’s result in [NaMori]
that a $\mathbb{Q}$-factorial terminalization $\pi:Y\rightarrow X$ of a
conical affine symplectic singularity is also a relative Mori Dream Space. We
explain that this implies that crepant partial resolutions of $X$ are
classified by faces of the Mori fan of $\overline{{\operatorname{Mov}(Y/X)}}$.
This provides a combinatorial description of the partial resolutions, and is
also the setting in which we first define the Namikawa Weyl group of a partial
resolution.

In particular to every simple generator $s$ of the Namikawa Weyl group
$W_{X}$, we assign a hyperplane $H_{s}\subset
H^{2}(Y^{\text{reg}},\mathbb{C})$, which bounds
$\overline{{\operatorname{Mov}(Y/X)}}$. Given a face $F$ of
$\overline{{\operatorname{Mov}(Y/X)}}$, we define the Namikawa Weyl group of
the corresponding partial resolution to be the parabolic subgroup of $W_{X}$
generated by all the simple generators $s$ such that $F\subset H_{s}$.

The nilpotent cone also has the special property that it has a unique
$\mathbb{Q}$-factorial terminalization. Whenever this is the case, each face
of the Mori fan will be given by intersecting
$\overline{{\operatorname{Mov}(Y/X)}}$ with a unique subset of the $H_{s}$
above. Thus we can view this as a way of expressing the bijection between
conjugacy classes of parabolic subalgebras, parabolic subgroups of the Weyl
group, and subsets of the simple roots. It also shows that these are in
natural bijection with crepant partial resolutions.

These bijections generalize to a bijection $[\rho:X^{\prime}\rightarrow
X]\mapsto W_{X^{\prime}}$ between crepant partial resolutions and parabolic
subgroups of $W_{X}$ anytime $X$ has a unique $\mathbb{Q}$-factorial
terminalization. As proved in 4.17, this map is not a bijection for every
conical affine symplectic singularity, but it is in general surjective. This
is because if there is more than one $\mathbb{Q}$-factorial terminalization,
then there will be faces in the interior of
$\overline{{\operatorname{Mov}(Y/X)}}$ which correspond to small contractions
which allow us to flop from one $\mathbb{Q}$-factorial terminalization to
another. As a corollary of this theorem we also prove in 4.18 that every
crepant partial resolution of a nilpotent cone $\mathcal{N}$ is of the form
$\tilde{\mathcal{N}}^{\mathcal{P}}\rightarrow\mathcal{N}$ described at the
beginning of the introduction.

In the final section of the paper we begin to study Springer theory of partial
resolutions as described above. We breifly highlight two instances of
interest. The first is a generalization of equation (1). There are simple
examples of conical affine symplectic singularities with trivial Namikawa Weyl
group but nontrivial partial resolutions, which shows the analogous equation
could not be true in general. However we show in 8.4 that the analogous
statement $H^{*}(\rho^{-1}(x),\mathbb{C})\cong
H^{*}(\pi^{-1}(x),\mathbb{C})^{W_{X^{\prime}}}$ is true as long as the
universal conic deformation $\mathcal{X}^{\prime}$ is smooth.

Secondly we introduce the algebra
$\operatorname{End}(\mathsf{Spr}^{\prime})\cong
H^{\text{BM}}_{top}(Z^{\prime})$ where
$\mathsf{Spr}^{\prime}:=\pi^{\prime}_{*}(\mathbb{C}_{Y})$ and
$Z^{\prime}:=Y\times_{X^{\prime}}Y$. As explained in diagram (47), this
algebra has a $W_{X^{\prime}}$-action that is compatible with the natural map
$\rho_{*}:\operatorname{End}(\mathsf{Spr}^{\prime})\rightarrow\operatorname{End}(\mathsf{Spr})$
and the inclusion
$\mathbb{C}[W_{X^{\prime}}]\hookrightarrow\mathbb{C}[W_{X}]$. Thus this
algebra provides a coarser invariant than $W_{X^{\prime}}$ which we can attach
to a partial resolution.

###  1.3. Summary

In Section 2 we review group actions on simply-laced Dynkin diagrams and fix a
formalism for the folded Dynkin diagrams we obtain by taking a quotient. We
note that unlike in some formalisms for folded Dynkin diagrams, we allow the
group action to permute nodes which have edges connecting them. For example we
will have $A_{4}/({\mathbb{Z}}/2{\mathbb{Z}})\cong C_{2}$. In Section 3 we
review some basics about symplectic singularities. In this section we recall
some facts about $\mathbb{Q}$-factorial terminalizations as well as the
definition of the Namikawa Weyl group.

In Section 4 we begin to study crepant partial resolutions of conical affine
symplectic singularities. In particular we define them and prove a few
equivalent characterizations of them. We review the main theorem of [NaMori],
which roughly states that if $\pi:Y\rightarrow X$ is a $\mathbb{Q}$-factorial
terminalization of a conical affine symplectic singularity, then it is also a
relative Mori Dream space. Using this we obtain a combinatorial description of
crepant partial resolutions by faces of the Mori fan of
$\overline{{\operatorname{Mov}(Y/X)}}$, and we use this description to define
the Namikawa Weyl group of a crepant partial resolution. In 4.17 we prove the
key result that the map from crepant partial resolutions to parabolic
subgroups of $W_{X}$ is surjective, and bijective if and only if $X$ has a
unique $\mathbb{Q}$-factorial terminalization.

In Section 5 we begin to study the deformation theory of crepant partial
resolutions. We use many of the techniques in [NaFlops] and [NaPD1] to show
that $\operatorname{PD}_{X^{\prime}}$ is prorepresentable and unobstructed for
any crepant partial resolution $\rho:X^{\prime}\rightarrow X$. Additionally we
define the pushforward functors
$\operatorname{PD}_{Y}\rightarrow\operatorname{PD}_{X^{\prime}}\rightarrow\operatorname{PD}_{X}$
on deformations, where $\pi:Y\rightarrow X$ is a $\mathbb{Q}$-factorial
terminalization which covers $\rho$. In Section 6 we continue by studying the
deformation theory of partial resolutions of Kleinian singularities. The main
strategy is that these partial resolutions have finitely many singular points,
with singularities which are also Kleinian and correspond to a subset of the
Dynkin diagram corresponding to the original Kleinian singularity. Using this
we are able to prove the main theorem for partial resolutions of Kleinian
singularities.

In Section 7 we adapt the argument in [NaPD2, 1.1] to prove the main theorem.
Given $Y\rightarrow X^{\prime}\rightarrow X$ as above, we proceed by looking
at the partial resolution and $\mathbb{Q}$-factorial terminalization locally
over a small neighborhood of a point in each codimension two symplectic leaf
of $X$. Thus we obtain full and partial resolutions of the Kleinian
singularity given by taking a transverse slice to the symplectic leaf. We are
able to use the results from the previous section to describe the pushforward
maps
$\operatorname{PD}_{Y}\rightarrow\operatorname{PD}_{X^{\prime}}\rightarrow\operatorname{PD}_{X}$
in the general case. Thus we obtain an explicit description of the universal
conic deformation of $\operatorname{PD}_{X^{\prime}}$, proving the main
theorem, 7.3.

Finally, in Section 8 we use the main theorem to study the symplectic Springer
theory of crepant partial resolutions, using many of the techniques from [MN].
We explain how many of the results in that paper which establish the viability
of symplectic Springer theory carry over for partial resolutions. Thus we are
able to define the partial symplectic Springer and Harish-Chandra sheaves, the
former of which is also constructible with respect to the stratification by
symplectic leaves. We end with a generalization of [BM, 2.8], showing that if
the universal conic deformation $\mathcal{X}^{\prime}$ is smooth, then
$H^{*}(\rho^{-1}(x),\mathbb{C})\cong
H^{*}(\pi^{-1}(x),\mathbb{C})^{W_{X^{\prime}}}$.

###  1.4. Notation and Conventions

All varieties and schemes will be assumed to be over $\mathbb{C}$. The results
of this paper primarily involve algebraic varieties, but if we use the word
“neighborhood” it will usually mean an open set in the complex analytic
topology. Germs will also be taken in the complex analytic topology. Finally,
$\epsilon$ will be assumed to square to zero, so that $\mathbb{C}[\epsilon]$
will always denote the dual numbers.

Throughout the paper $X$ will always be a conical affine symplectic
singularity, $\rho:X^{\prime}\rightarrow X$ will always be a crepant partial
resolution and $\pi:Y\rightarrow X$ will always be a $\mathbb{Q}$-factorial
terminalization of $X$, except in Section 8 where $\pi$ will always be a
symplectic resolution. If $\pi$ covers $\rho$ then $\pi^{\prime}:Y\rightarrow
X^{\prime}$ will be the map making $Y\rightarrow X^{\prime}\rightarrow X$
commute. Adding a prime symbol will denote the “partial” version. For example
the partial Springer and Harish-Chandra sheaves will be denoted
$\mathsf{Spr}^{\prime}$ and $\mathsf{HC}^{\prime}$. Additionally, whenever
there is a group action on a Dynkin diagram, we will denote the unfolded
Dynkin diagram and all of its data (such as the Cartan and Weyl group) with a
hat symbol. The folded Dynkin diagram and its data will have no hat. In
Section 6 there are no group actions so we will drop the hats entirely.

Finally we note that we call a map of normal varieties $f:Y\rightarrow X$ a
Galois cover if it is finite, surjective, and the degree of the map is
$|\textrm{Aut}(Y/X)|$. This last condition is equivalent to the extension
$[\mathbb{C}(Y),\mathbb{C}(X)]$ of function fields being Galois. In the
literature this is often called “generically Galois,” and we will refer to
$\textrm{Aut}(Y/X)$ as the Galois group. Given a Galois cover $f:Y\rightarrow
X$ and an open set $U\subset X$ in the complex analytic topology, we will also
refer to the map $f^{-1}(U)\rightarrow U$ as a Galois cover.

###  1.5. Acknowledgements

I want to thank David Ben-Zvi for all of his help and support, without which
this paper would not exist. I also want to thank Ivan Losev, Travis Schedler,
Gwyn Bellamy, Rok Gregoric, and Joakim Færgeman for helpful discussions.

##  2\. Lie Theory and Folding

Let $\hat{D}$ be a simply laced Dynkin diagram and let $\hat{\mathfrak{h}}$
and $\hat{W}$ be the corresponding Cartan subalgebra and Weyl group, with
$<,>$ the natural inner product on $\hat{\mathfrak{h}}$ given by the Killing
form. Furthermore let $I$ be the set of nodes of $\hat{D}$ and let
$\alpha_{i}\in\hat{\mathfrak{h}}$ and $s_{i}\in\hat{W}$ be the corresponding
simple root and simple reflection, for each $i\in I$. Let $G$ be a finite
group which acts on $\hat{D}$ by graph automorphism. We are going to use some
results from [St, Section 1] to obtain a folded Dynkin diagram $D:=\hat{D}/G$
with Cartan $\mathfrak{h}:=\hat{\mathfrak{h}}^{G}$ and Weyl group
$W:=\hat{W}^{G}$.

Let $J:=I/G$ be the set of orbits of the action of $G$ on $I$. For each $j\in
J$, let $\beta_{j}=\frac{1}{N}\sum_{i\in j}\alpha_{i}$, where $N$ is the
number of connected components of the subgraph of $\hat{D}$ given by the nodes
in $j$. Note that for the cases we are studying this is identical to the
$a_{j}$ in [St, p. 14]. Since the $\alpha_{i}$ form a basis for
$\hat{\mathfrak{h}}$, the permutation action of $G$ on the $\alpha_{i}$
induces an action of $G$ on $\hat{\mathfrak{h}}$, and
$\mathfrak{h}:=\hat{\mathfrak{h}}^{G}$ has a basis given by the $\beta_{j}$.
As explained in [NaPD1, 4.1], we can classify non-trivial free group actions
on simply laced Dynkin diagrams into the following classes:

${\mathbb{Z}}/2{\mathbb{Z}}\
\rotatebox[origin={c}]{-90.0}{$\circlearrowright$}\ A_{n}\text{ for }n\geq 2,\
{\mathbb{Z}}/2{\mathbb{Z}}\
\rotatebox[origin={c}]{-90.0}{$\circlearrowright$}\ D_{n}\text{ for }n\geq 3,\
{\mathbb{Z}}/3{\mathbb{Z}}\
\rotatebox[origin={c}]{-90.0}{$\circlearrowright$}\ D_{4},\
{\mathbb{Z}}/2{\mathbb{Z}}\
\rotatebox[origin={c}]{-90.0}{$\circlearrowright$}\ E_{6}.$

In all of the above cases, the matrix $[c_{jk}]_{j,k\in J}$ given by

(3)
$c_{jk}=\frac{2\langle\beta_{j},\beta_{k}\rangle}{\langle\beta_{j},\beta_{j}\rangle}.$

is the Cartan matrix of simple Lie algebra. We define $D=\hat{D}/G$ to be the
Dynkin diagram corresponding to this Lie algebra. In particular we have:

$\displaystyle A_{n}/({\mathbb{Z}}/2{\mathbb{Z}})\cong C_{\lceil
n/2\rceil}\text{ for }n\geq 2,\ D_{n}/({\mathbb{Z}}/2{\mathbb{Z}})\cong
B_{n-1}\text{ for }n\geq 3$ $\displaystyle
D_{4}/({\mathbb{Z}}/3{\mathbb{Z}})\cong G_{2},\
E_{6}/({\mathbb{Z}}/2{\mathbb{Z}})\cong F_{4}.$

######  Example 2.1.

We shall compute $A_{4}/({\mathbb{Z}}/2{\mathbb{Z}})\cong C_{2}$ explicitly.
We have $\beta_{1}=\frac{\alpha_{1}+\alpha_{4}}{2}$ and
$\beta_{2}=\alpha_{2}+\alpha_{3}$, since the middle two nodes of $A_{4}$ form
one connected component but the the outer two form two connected components.
Thus using the inner product from $A_{4}$ we obtain
$\langle\beta_{1},\beta_{1}\rangle=1,\langle\beta_{1},\beta_{2}\rangle=-1$ and
$\langle\beta_{2},\beta_{2}\rangle=2$. Using equation (3) above we obtain the
Cartan matrix for $C_{2}$.

Since the action of $G$ on $\hat{\mathfrak{h}}$ simply permutes the simple
roots, conjugation by any element of $G$ permutes the simple reflections
$s_{i}$. Thus $G$ normalizes $\hat{W}$, and we let $G$ act on $\hat{W}$ by
conjugation.

######  Theorem 2.2 ([NaPD2, 1.2]).

The subset of $\hat{W}$ which preserves $\mathfrak{h}$ is $W:=\hat{W}^{G}$.

Given $j\in J$, let $\hat{W}_{j}\subset\hat{W}$ be the parabolic subgroup
generated by $\alpha_{i}$ for all $i\in j$. Furthermore let $w^{0}_{j}$ be the
longest element of $\hat{W}_{j}$.

######  Theorem 2.3 ([St, 1.30, 1.32]).

For each $j\in J$, $w^{0}_{j}\in W$ and the restriction of $w^{0}_{j}$ to
$\mathfrak{h}$ is a simple reflection corresponding to $\beta_{j}$.
Additionally the $w^{0}_{j}$ generate $W$.

In particular 2.3 tells us that $W$ is simply the Weyl group corresponding to
the Dynkin diagram $D$. Finally, we want to compare the parabolic subgroups of
$W$ with those of $\hat{W}$. Thus let $J^{\prime}\subset J$ and let
$I^{\prime}\subset I$ be such that $i\in I^{\prime}$ if and only if $i\in j$
for some $j\in J^{\prime}$. We further let $\hat{W}^{\prime}$ be the parabolic
subgroup of $\hat{W}$ generated by $\alpha_{i}$ for each $i\in I^{\prime}$.

######  Theorem 2.4.

The intersection $\hat{W}^{\prime}\cap W$ is the parabolic subgroup
$W^{\prime}\subset W$ generated by $w^{0}_{j}$ for each $j\in J$.

###### Proof.

By definition $w^{0}_{j}$ is generated by the $s_{i}$ such that $i\in j$ so
$w^{0}_{j}\in\hat{W}^{\prime}$ for each $j\in J^{\prime}$. Furthermore by 2.3
we have $w^{0}_{j}\in W$ so $W^{\prime}\subset\hat{W}^{\prime}\cap W$.

We show the other inclusion by contradiction. Assume there is some
$w\in\hat{W}^{\prime}\cap W$ such that $w\notin W^{\prime}$. We have an
expansion $w=w^{0}_{j_{1}}...w^{0}_{j_{n}}$ for $j_{1},...,j_{n}\in J$ and
some $j_{l}$ is not in $J^{\prime}$. We can take $w$ to have minimal length
out of elements of $\hat{W}^{\prime}\cap W$ which are not in $W^{\prime}$.
Thus $j_{n}\notin J^{\prime}$, because otherwise multiplying by
$w^{0}_{j_{n}}$ on the right would create a shorter word in
$\hat{W}^{\prime}\cap W$ but not in $W^{\prime}$.

If we let $\Delta=\\{\beta_{j}\\}_{j\in J}$ and $\Phi=W\Delta$ be the simple
roots and the roots respectively, then by [St, 1.32],
$(\mathfrak{h},\Delta,\Phi,W|_{\mathfrak{h}})$ satisfies conditions 1.1-1.6 in
[St]. Clearly as a word in $W$, $l(ww^{0}_{j_{n}})=l(w)-1$ where $l$ is the
word length function. Thus by [St, 1.8], $ww^{0}_{j_{n}}$ makes one less
positive root of $\Phi$ negative than $w$.

Since $w\in\hat{W}^{\prime}$, [St, 1.15] tells us that as an element of
$\hat{W}$ it permutes all the positive roots of $\hat{\mathfrak{h}}$ whose
support is not contained in $I^{\prime}$. In particular it sends $\alpha_{i}$
to a positive root for each $i\in j_{n}$. Thus since $\beta_{j_{n}}$ is a
linear combination of these $\alpha_{i}$, it must map $\beta_{j_{n}}$ to the
positive cone of $\hat{\mathfrak{h}}$. Since the positive cone of
$\mathfrak{h}$ is contained in the positive cone of $\hat{\mathfrak{h}}$ and
$\beta_{j_{n}}\in\Phi$, $w$ must send $\beta_{j_{n}}$ to a positive root of
$\Phi$. By [St, 1.7], $w^{0}_{j_{n}}$ permutes all the positive roots of
$\Phi$ except for $\beta_{j_{n}}$, which it negates. Thus if
$\\{r_{1},...,r_{k}\\}\subset\Phi$ is the set of positive roots which $w$
sends to negative roots, the analogous set for $ww^{0}_{j_{n}}$ is
$\\{w^{0}_{j_{n}}(r_{1}),...,w^{0}_{j_{n}}(r_{k}),\beta_{j_{n}}\\}$. However
this contradicts the previous paragraph, so $\hat{W}^{\prime}\cap W\subset
W^{\prime}$ and we are done.

∎

For the remainder of the paper, whenever we have a Weyl group $W$ of a folded
Dynkin diagram $D$ as above, we define the $w^{0}_{j}$ to be the simple
generators of $W$ as a Coxeter group.

##  3\. Preliminaries and Background

######  Definition 3.1.

A symplectic singularity (or symplectic variety) is a normal algebraic variety
$X$ with a symplectic form $\omega$ on $X_{\text{reg}}$ such that for any
(equivalently, some) resolution of singularities $\pi:\tilde{X}\rightarrow X$,
$\pi^{*}(\omega)$ extends to a regular 2-form on $\tilde{X}$. If it extends to
a symplectic form then $\tilde{X}$ is called a symplectic resolution.

The symplectic form induces a Poisson bracket on
$\mathcal{O}(X_{\text{reg}})$, and since $X$ is normal, this bracket extends
to all of $X$, making $X$ a Poisson scheme. Throughout the paper we shall
study the universal Poisson deformations of affine symplectic singularities
and their (partial) symplectic resolutions. In order to ensure that universal
formal deformations algebraicize to deformations over affine spaces, we need
the following condition as well.

######  Definition 3.2.

A symplectic singularity $X$ is conical if it has an algebraic
$\mathbb{C}^{*}$-action such that $\mathcal{O}(X)$ has non-negative weights,
$\mathcal{O}(X)^{\mathbb{C}^{*}}=\mathbb{C}$, and $\omega_{X}$ is positively
weighted with respect to $\mathbb{C}^{*}$.

######  Remark 3.3.

The conditions that the $\mathbb{C}^{*}$-action has non-negative weights and
$\mathcal{O}(X)^{\mathbb{C}^{*}}=\mathbb{C}$ is equivalent to the
$\mathbb{C}^{*}$-action contracting $X$ to a unique cone point.

The following theorem is one of the most important facts about symplectic
singularities:

######  Theorem 3.4 ([Kal, 2.3]).

Every symplectic singularity has a finite stratification by symplectic leaves.

Throughout this paper we will be particularly interested in the codimension
two symplectic leaves. Thus we recall (see [NaNote, Cor. 1]) that a symplectic
singularity has terminal singularities if and only if it has no codimension
two symplectic leaves.

In general not every symplectic singularity has a symplectic resolution, but
they always have a $\mathbb{Q}$-factorial terminalization, which is defined
below:

######  Definition 3.5.

Given a symplectic singularity $X$, a $\mathbb{Q}$-factorial terminalization
(or simply $\mathbb{Q}$-terminalization) $\pi:Y\rightarrow X$ is a crepant
birational projective morphism such that $Y$ is a normal
$\mathbb{Q}$-factorial variety with terminal singularities.

We recall some essential facts about $\mathbb{Q}$-factorial terminalizations
below:

######  Theorem 3.6 ([[LoSRA, 2.1]).

Let $X$ be a symplectic singularity:

  1. (1)

$X$ has a $\mathbb{Q}$-factorial terminalization, $\pi:Y\rightarrow X$.

  2. (2)

$Y$ is a symplectic singularity and $\pi$ is a Poisson map.

  3. (3)

There is a $\mathbb{C}^{*}$-action on $Y$ such that $\pi$ is
$\mathbb{C}^{*}$-equivariant.

######  Remark 3.7.

The proof of part (1) is based primarily on [BCHM, 1.4.3]. Additionally, part
(3) was first proven in [NaFlops, A.7].

From now on let $X$ be a conical affine symplectic singularity, and
$\pi:Y\rightarrow X$ a $\mathbb{Q}$-factorial terminalization. Let $U\subset
X$ be the complement of the symplectic leaves of codimension four or greater.
This inherits the structure of a Poisson scheme from $X$, and since
$\mathbb{Q}$-factorial terminalizations do not have any codimension two
leaves, $\tilde{U}:=\pi^{-1}(U)$ is a symplectic resolution of $U$.

Given a codimension two symplectic leaf $\mathcal{L}_{i}\subset U$ and a point
$x_{i}\in\mathcal{L}_{i}$, we can take a transverse slice $S_{i}$ to $x_{i}$,
which will be a Kleinian singularity. Let $\hat{D}_{i}$ be the Dynkin diagram
corresponding to this singularity, with Cartan $\hat{\mathfrak{h}_{i}}$ and
Weyl group $\hat{W_{i}}$. Thus $\pi^{-1}(x_{i})$ is a union of projective
lines in the shape of $\hat{D}_{i}$, and we obtain a monodromy action of
$\pi_{1}(\mathcal{L}_{i},x_{i})$ on $\hat{D}_{i}$ by graph automorphism. This
induces an action of $\pi_{1}(\mathcal{L}_{i},x_{i})$ on
$\hat{\mathfrak{h}_{i}}$, and we define

$\mathfrak{h}_{i}:=\hat{\mathfrak{h}}_{i}^{\pi_{1}(\mathcal{L}_{i},x_{i})}\hskip
20.0pt\text{and}\hskip 20.0ptW_{i}=\\{w\in\hat{W_{i}}\ |\
w(\mathfrak{h}_{i})=\mathfrak{h}_{i}\\}.$

In particular, these are the Cartan and Weyl group of
$D_{i}:=\hat{D}_{i}/{\pi_{1}(\mathcal{L}_{i},x_{i})}$, as defined in Section
2.

######  Remark 3.8.

Although it is not always possible to take an algebraic transverse slice to
each codimension two symplectic leaf, we can always take either a formal slice
or a transverse slice in the complex analytic category.

######  Definition 3.9.

Let $\\{\mathcal{L}_{i}\\}_{i\in I}$ be the set of codimension two symplectic
leaves of $X$. Then the Namikawa Weyl group of $X$ is

$W_{X}:=\prod_{i\in I}W_{i}$

This group appears naturally when studying the Poisson deformations of $X$ and
$Y$, as explained in Section 1.

##  4\. Partial Symplectic Resolutions and Birational Geometry

######  Definition 4.1.

A map $\pi:Y\rightarrow X$ of normal varieties is a partial resolution if
$\pi$ is projective and birational and the restriction of $\pi$ to
$\pi^{-1}(X_{\textrm{reg}})$ is an isomorphism. Furthermore we say $\pi$ is a
symplectic partial resolution if the symplectic form
$\omega_{X_{\textrm{reg}}}$ extends to a symplectic form on
$Y_{\textrm{reg}}$.

######  Remark 4.2.

Traditionally a (partial) resolution of singularities is only required to be
proper, rather than projective. However for our purposes we will only be
interested in projective (partial) resolutions.

Before proceeding we recall that Zariski’s main theorem (see [Ha, III.11.4])
implies that for any partial resolution $\pi:Y\rightarrow X$, we have
$\pi_{*}\mathcal{O}_{Y}=\mathcal{O}_{X}$, i.e. $\pi$ is a contraction.

######  Theorem 4.3.

Any symplectic partial resolution $Y$ of a symplectic singularity $X$ is
itself a symplectic singularity.

###### Proof.

By definition $Y$ is normal and has a symplectic form
$\omega_{Y_{\textrm{reg}}}$ on its smooth locus which extends
$\omega_{X_{\textrm{reg}}}$. All that is left to check is that given a
resolution $\pi^{\prime}:Z\rightarrow Y$,
$\pi^{\prime-1}(\omega_{Y_{\textrm{reg}}})$ extends to a regular 2-form on
$Z$. Since $\pi^{\prime}\circ\pi$ is a resolution of $X$, and $X$ is a
symplectic singularity, $\omega_{X_{\textrm{reg}}}$ extends to a regular
2-form $\omega_{Z}$ on $Z$. Note that since $Z$ is smooth, the sheaf of
2-forms is a vector bundle, and thus restriction to a Zariski open set is
injective. Since we can treat $X_{\textrm{reg}}$ and $Y_{\textrm{reg}}$ as
Zariski open sets of $Z$, this means that the extensions
$\omega_{Y_{\textrm{reg}}}$ and $\omega_{Z}$ of $\omega_{X_{\textrm{reg}}}$
are unique. In particular $\omega_{Z}$ must extend
$\omega_{Y_{\textrm{reg}}}$.

∎

The following argument is identical for full resolutions (see [Fu, Prop. 1.1])
but we include it for completeness

######  Theorem 4.4.

A partial resolution is symplectic if and only if it is crepant.

###### Proof.

Let $X$ be a symplectic singularity with $\dim X=2n$ and let $\omega_{X}$ be
the symplectic form on $X_{\textrm{reg}}$. Then $\wedge^{n}\omega_{X}$
provides a trivialization of $K_{X}$ over $X_{\text{reg}}$ which extends to a
trivialization of $K_{X}$ by normality. Thus every symplectic singularity has
trivial canonical sheaf. In particular, given a partial symplectic resolution
$\pi:Y\rightarrow X$, we have $\pi^{*}(K_{X})=\pi^{*}0=0=K_{Y}$, so the map is
crepant.

Conversely, say $\pi:Y\rightarrow X$ is crepant. Since $X$ has symplectic
singularities, $\pi^{*}\omega_{X}$ extends to a regular 2-form $\omega_{Y}$ on
$Y_{\textrm{reg}}$. Additionally, $\wedge^{n}\omega_{X}$ extends to a non-
vanishing global section $f$ of $K_{X}\cong\mathcal{O}_{X}$. Thus $f$ pulls
back to a non-vanishing section of $K_{Y}\cong\mathcal{O}_{Y}$ which extends
$\pi^{*}(\wedge^{n}\omega_{X})=\wedge^{n}\pi^{*}\omega_{X}$ and thus
$\wedge^{n}\omega_{Y}$. Thus $\wedge^{n}\omega_{Y}$ is a non-vanishing section
of $K_{Y}|_{Y_{\text{reg}}}\cong\Omega^{n}_{Y_{\text{reg}}}$, which implies
that $\omega_{Y}$ is non-degenerate. Finally, since $\pi^{*}$ commutes with
differentials, $\operatorname{d}(\omega_{Y})$ extends
$\pi^{*}\operatorname{d}(\omega_{X})=0$ on $\pi^{-1}(X_{\text{reg}})$, which
extends uniquely to 0 on $Y_{\text{reg}}$. Thus $\omega_{Y}$ is closed and we
are done.

∎

The above theorem implies that if $\pi:Y\rightarrow X$ is a
$\mathbb{Q}$-factorial terminalization of a symplectic singularity $X$, then
$Y$ is a symplectic partial resolution of $X$. In fact they can be described
as the maximal elements of the poset of symplectic partial resolutions, as
described below:

######  Lemma 4.5.

A partial resolution $\rho:X^{\prime}\rightarrow X$ of a symplectic
singularity $X$ is symplectic if and only if it is covered by a
$\mathbb{Q}$-factorial terminalization of $X$.

###### Proof.

If $\rho:X^{\prime}\rightarrow X$ is symplectic then by 4.3 it has symplectic
singularities and thus by 3.6 there is some $\mathbb{Q}$-factorial
terminalization $\pi^{\prime}:Y\rightarrow X^{\prime}$. Thus
$\pi:=\rho\circ\pi^{\prime}$ is a $\mathbb{Q}$-factorial terminalization of
$X$ which covers $\rho$.

Conversely if $\pi^{\prime}:Y\rightarrow X^{\prime}$ is a
$\mathbb{Q}$-factorial terminalization which covers $\rho$, then the
symplectic form $\omega$ on $X_{\text{reg}}$ extends to $Y_{\text{reg}}$. In
particular since $\pi:=\rho\circ\pi^{\prime}$ and
$X^{\prime}_{\text{reg}}\subset Y_{\text{reg}}$, $\omega$ extends to
$X^{\prime}_{\text{reg}}$. Thus $\rho$ is symplectic.

∎

From now on we will only be interested in symplectic partial resolutions, so
we will simply called them partial resolutions and they will be assumed to be
symplectic.

Additionally, for the remainder of the paper let $X$ be a conical affine
symplectic singularity, and let $\rho:X^{\prime}\rightarrow X$ be a partial
resolution which is covered by a $\mathbb{Q}$-factorial terminalization
$\pi:Y\rightarrow X$. Since $X$ is affine we have
$H^{i}(X,\mathcal{O}_{X})=0,\forall i>0$. Furthermore since $X$, $X^{\prime}$,
and $Y$ all have rational singularities, we have quasi-isomorphisms
$\pi_{*}\mathcal{O}_{Y}\cong\mathcal{O}_{X}$ and
$\rho_{*}\mathcal{O}_{X^{\prime}}\cong\mathcal{O}_{X}$ in the derived
category, which implies
$H^{i}(Y,\mathcal{O}_{Y})=H^{i}(X^{\prime},\mathcal{O}_{X^{\prime}})=0,\forall
i>0$.

The following theorem will allow us pass back and forth between the settings
of birational geometry and Poisson deformation theory, as will become clearer
in the following section.

######  Theorem 4.6 ([LMM, 4.4.6]).

The first chern character map provides an isomorphism

$c_{1}:\operatorname{Pic}(Y^{\text{reg}})\otimes_{{\mathbb{Z}}}\mathbb{Q}\rightarrow
H^{2}(Y^{\text{reg}},\mathbb{Q}).$

######  Remark 4.7.

The analogue of the above theorem for the analytic Picard group can be deduced
via a standard computation using the exponential short exact sequence. However
the above statement applies to the algebraic Picard group, which requires an
argument using the GAGA principle.

Traditionally, the cones described in 4.8 below would be subsets of

$\operatorname{Pic}(Y/X)\otimes\mathbb{R}:=(\operatorname{Pic}(Y)/\pi^{*}(\operatorname{Pic}(X)))\otimes\mathbb{R}.$

However by [LMM, 7.1.1], the Picard group of every normal affine conical
scheme is trivial, and thus $\operatorname{Pic}(X)=0$. Additionally by [Ha,
6.5], $\operatorname{Cl}(Y)\cong\operatorname{Cl}(Y^{\text{reg}})$ where
$\operatorname{Cl}$ denotes the divisor class group, since $Y-Y^{\text{reg}}$
has codimension at least four. Finally since $Y$ and $Y^{\text{reg}}$ are
$\mathbb{Q}$-factorial, this implies
$\operatorname{Pic}(Y)\otimes\mathbb{Q}=\operatorname{Pic}(Y^{\text{reg}})\otimes\mathbb{Q}$.
So indeed we have

$\operatorname{Pic}(Y/X)\otimes\mathbb{R}\cong
H^{2}(Y^{\text{reg}},\mathbb{R}).$

As detailed in [NaPD2], $W_{X}$ acts on $H^{2}(Y^{\text{reg}},\mathbb{C})$ as
a finite reflection group, and the action is generically free. The following
celebrated theorem of Namikawa will allow us to classify its partial
resolutions.

######  Theorem 4.8 ([NaMori, Main Theorem, Remark 10]).

If $\pi:Y\rightarrow X$ is a $\mathbb{Q}$-factorial terminalization, then
there are finitely many hyperplanes $\\{H_{i}\\}\subset
H^{2}(Y^{\text{reg}},\mathbb{R})$ such that

1) The union $\mathcal{D}$ of the $H_{i}$ is invariant under the action of
$W_{X}$.

2) $\mathcal{D}$ is the locus over which the universal deformation map
$\mathcal{Y}\rightarrow\mathcal{X}$ is not an isomorphism, as described in
diagram (2).

3) For every $\mathbb{Q}$-factorial termination
$\pi^{\prime}:Y^{\prime}\rightarrow X$,
$\overline{{\operatorname{Amp}(\pi^{\prime})}}=\operatorname{Nef}(\pi^{\prime})$
is a unique chamber cut out by the hyperplanes.

4) The union of $\overline{{\operatorname{Amp}(\pi^{\prime})}}$ over every
$\mathbb{Q}$-factorial termination $\pi^{\prime}:Y^{\prime}\rightarrow X$ is
$\overline{{\operatorname{Mov}(Y/X)}}$.

In particular, $\pi:Y\rightarrow X$ is a relative Mori Dream Space in the
sense of [Ohta].

######  Remark 4.9.

As emphasized by Namikawa in [NaMori, p.2], a crepant resolution $Y\rightarrow
X$ is a relative Mori Dream space whenever $X$ has rational Gorenstein
singularities, but in the case of a conical affine symplectic singularity and
a $\mathbb{Q}$-factorial terminalization we have the added feature that the
Mori fan is given by full hyperplanes.

######  Corollary 4.10.

Partial resolutions of $X$ are in bijection with faces of the Mori fan of
$\overline{{\operatorname{Mov}(Y/X)}}$.

###### Proof.

As explained in [Ohta, 4.4], there is a bijection between faces in
$\overline{{\operatorname{Mov}(Y/X)}}$ and birational contractions
$Y\dashrightarrow X^{\prime}$ where $X^{\prime}$ is projective over $X$. Each
of these birational contractions is a contraction of some
$\mathbb{Q}$-factorial terminalization of $X$, and thus by Lemma 4.5 it is a
partial resolution of $X$. Conversely, given a partial resolution
$\rho:X^{\prime}\rightarrow X$ of $X$, it is covered by some
$\mathbb{Q}$-factorial terminalization, $\pi^{\prime}:Y^{\prime}\rightarrow
X^{\prime}$. This map is a contraction, so there is a face of
$\operatorname{Nef}(Y/X)$ which corresponds to $\rho:X^{\prime}\rightarrow X$.

∎

Because $\pi:Y\rightarrow X$ is a relative Mori Dream Space,
$\operatorname{Nef}(Y/X)$ is a closed polyhedral cone generated by semiample
divisors. The cone of effective curves $\operatorname{NE}_{1}(Y/X)$ is dual to
$\operatorname{Nef}(Y/X)$ and thus it is a polyhedral cone generated by
finitely many curves classes $[C_{k}]$. Thus every codimension one face $F$ of
$\operatorname{Nef}(Y/X)$ is of the form
$\operatorname{Nef}(Y/X)\cap[C_{k}]^{\perp}$ for a unique extremal ray,
$\langle[C_{k}]\rangle\subset\operatorname{NE}_{1}(Y/X)$. We can recover this
extremal ray as the kernel of
$\pi^{\prime}_{*}:\mathcal{N}_{1}(Y/X)\rightarrow\mathcal{N}_{1}(X^{\prime}/X)$,
where $\pi^{\prime}:Y\rightarrow X^{\prime}$ is the contraction corresponding
to $F$.

We follow [AW, Def 3.4] with the following definition

######  Definition 4.11.

Given a $\mathbb{Q}$-factorial terminalization $Y\rightarrow X$, the cone of
essential curves, $\operatorname{Ess}(Y/X)$ is the cone generated by curve
classes $[C_{i}]\in\mathcal{N}_{1}(Y/X)$ whose image intersects a codimension
two symplectic leaf of $X$.

As explained in Section 3, given any codimension two leaf
$\mathcal{L}_{i}\subset X$ and $x_{i}\in\mathcal{L}_{i}$, $\pi^{-1}(x_{i})$ is
given by a union of projective lines $C_{k}$ in the shape on the Dynkin
diagram $\hat{D}_{i}$ corresponding to the transverse slice to
$\mathcal{L}_{i}$. Since these curve classes are all contracted by $\pi$, they
also represent rays of $\operatorname{NE}_{1}(Y/X)$, and in particular they
generate $\operatorname{Ess}(Y/X)\subset\operatorname{NE}_{1}(Y/X)$.

Given a simple generator $s_{k}$ of $W_{X}$ corresponding to a node of
$D_{i}:=\hat{D}_{i}/\pi_{1}(\mathcal{L}_{i},x)$, we can take a projective line
$C_{k}$ which maps to some $x_{i}\in\mathcal{L}_{i}$ and which is in the orbit
of the monodromy action of $\pi_{1}(\mathcal{L}_{i},x_{i})$ which corresponds
to the same node of $D_{i}$ as $s_{k}$. It follows from [Wie, 4.4] that two of
the above projective lines $C_{k}$ and $C_{l}$ are in the same curve class if
they map to the same symplectic leaf and they are in the same orbit of the
$\pi_{1}(\mathcal{L}_{i},x)$-action. Thus $s_{k}\mapsto[C_{k}]$ is a well-
defined map

$\Phi:\\{\text{Simple generators of }W_{X}\\}\rightarrow\\{\text{Generators of
}\operatorname{Ess}(Y/X)\\}.$

######  Lemma 4.12.

The map $\Phi$ is a bijection.

###### Proof.

The map is surjective by construction because for every projective line
$C_{k}$ which maps to a codimension two leaf $\mathcal{L}_{i}$, we can take
the node of $D_{i}$ corresponding to its orbit under the monodromy action of
the fundamental group, and the corresponding simple generator $s_{k}$ will map
to $[C_{k}]$.

To prove injectivity, we recall that by [BPW, 2.17],
$\overline{{\operatorname{Mov}(Y/X)}}$ is a fundamental domain for the action
of $W_{X}$ on $H^{2}(Y^{\text{reg}},\mathbb{R})$. In particular this means
that the number of codimension one faces of
$\overline{{\operatorname{Mov}(Y/X)}}$ is equal to the number of simple
generators of $W_{X}$. By [AW, 3.5], $\operatorname{Ess}(Y/X)$ is dual to
$\overline{{\operatorname{Mov}(Y/X)}}$, and thus the number of generators of
$\operatorname{Ess}(Y/X)$ is also equal to the number of simple generators of
$W_{X}$. Thus $\Phi$ must be a bijection.

∎

######  Remark 4.13.

Let $U$ be the union of the open and codimension two leaves of $X$ and let
$\tilde{U}$ be $\pi^{-1}(U)$. Then the work in [Wie, Section 4] further shows
that if we only consider $\operatorname{NE}_{1}(\tilde{U}/U)$, then two
projective lines $C_{i}$ and $C_{j}$ as above are in the same curve class if
and only if they map to the same symplectic leaf and they are in the same
orbit of the monodromy action of $\pi_{1}(\mathcal{L},x)$. However without the
above proof it would be possible that two of these curves are not
algebraically equivalent in $\tilde{U}$ but are are algebraically equivalent
in $Y$.

Given a simple generator $s_{k}$ of the Namikawa Weyl group $W_{X}$, let
$H_{s_{k}}\subset H^{2}(Y^{\text{reg}},\mathbb{R})$ be the hyperplane
$\Phi(s_{k})^{\perp}$. Since $[\Phi(s_{k})]$ is a generator of
$\operatorname{Ess}(Y/X)$, $H_{s_{k}}$ bounds
$\overline{{\operatorname{Mov}(Y/X)}}$ and thus is one of the hyperplanes in
4.8. Additionally, if $\pi^{\prime}:Y\rightarrow X^{\prime}$ is a contraction
corresponding to a face $F\subset\operatorname{Nef}(Y/X)$, then by definition
$\pi^{\prime}$ contracts $[\Phi(s_{k})]$ if and only if $F\subset H_{s_{k}}$.

######  Definition 4.14.

Let $\rho:X^{\prime}\rightarrow X$ be the partial resolution of $X$
corresponding to a face $F\subset\overline{{\operatorname{Mov}(Y/X)}}$. Then
we define the Namikawa Weyl group $W_{X^{\prime}}$ of this partial resolution
to be the parabolic subgroup $W_{X^{\prime}}\subset W_{X}$ generated by all
the $s_{k}$ such that $F\subset H_{s_{k}}$.

######  Remark 4.15.

As far as we are aware, the Namikawa Weyl group has previously only been
defined for conical affine symplectic singularities, although the above
definition has similar motivation. We shall show later that it satisfies a
similar condition as the Namikawa Weyl group $W_{X}$ stemming from Poisson
deformations theory.

######  Theorem 4.16.

The above construction does not depend on a choice of $\mathbb{Q}$-factorial
terminalization.

###### Proof.

Any two $\mathbb{Q}$-factorial terminalizations are related by a finite
sequence of flops. In particular let $\pi_{1}:Y_{1}\rightarrow X$ and
$\pi_{2}:Y_{2}\rightarrow X$ be two $\mathbb{Q}$-factorial terminalizations
which are related by a flop $Y_{1}\rightarrow X^{\prime}\leftarrow Y_{2}$.
Then the contractions $Y_{1}\rightarrow X^{\prime}$ and $Y_{2}\rightarrow
X^{\prime}$ are both small, so they cannot contract any of the curve classes
which map to codimension two leaves in $X$, which would constitute a
divisorial contraction. In particular the above flop $Y_{1}\dashrightarrow
Y_{2}$ restricts to an isomorphism
$\pi_{1}^{-1}(U)\xrightarrow{\sim}\pi_{2}^{-1}(U)$ where $U\subset X$ is the
union of the open and codimension two leaves of $X$. In particular the
essential cones $\operatorname{Ess}(Y_{1}/X)$ and
$\operatorname{Ess}(Y_{2}/X)$ are canonically identified.

∎

######  Theorem 4.17.

The map

$\Psi:\\{\text{Partial resolutions of }X\\}\rightarrow\\{\text{Parabolic
subgroups of }W_{X}\\}$

given by $[\rho:X^{\prime}\rightarrow X]\mapsto W_{X^{\prime}}$ is surjective,
and it is bijective if and only if $X$ has a unique $\mathbb{Q}$-factorial
terminalization.

###### Proof.

Let $W^{\prime}\subset W_{X}$ be the parabolic subgroup of $W_{X}$ generated
by $s_{1},...,s_{l}$. Because the hyperplanes $H_{s}$ bound
$\overline{{\operatorname{Mov}(Y/X)}}$, they are in general position, and in
particular $H_{s_{1}}\cap...\cap H_{s_{l}}$ has codimension $l$. It also
bounds $\overline{{\operatorname{Mov}(Y/X)}}$, so we can pick some codimension
$l$ face $F^{\prime}$ of $\overline{{\operatorname{Mov}(Y/X)}}$ contained in
$H_{s_{1}}\cap...\cap H_{s_{l}}$. Since $F^{\prime}$ is codimension $l$, it
cannot be contained in any of the other hyperplanes $H_{s}$. Thus the partial
resolution corresponding to $F^{\prime}$ has Namikawa Weyl group $W^{\prime}$.

If $X$ has a unique $\mathbb{Q}$-factorial terminalization then the Mori fan
of $\overline{{\operatorname{Mov}(Y/X)}}$ is simply given by the faces which
bound $\overline{{\operatorname{Mov}(Y/X)}}=\operatorname{Nef}(Y/X)$. These
are all of the form $\operatorname{Nef}(Y/X)\cap H_{s_{1}}\cap...\cap
H_{s_{i}}$, for some unique subset $\\{s_{1},...,s_{i}\\}$ of the simple
roots. Thus $\Psi$ is a bijection. However if there is more than one
$\mathbb{Q}$-factorial terminalization, they both have trivial Namikawa Weyl
group, so $\Psi$ is not a bijection.

∎

######  Corollary 4.18.

Every partial resolution of the nilpotent cone $\mathcal{N}$ of a semisimple
Lie algebra $\mathfrak{g}$ is of the form
$\rho:\tilde{\mathcal{N}}^{\mathcal{P}}\rightarrow\mathcal{N}$ with

$\tilde{\mathcal{N}}^{\mathcal{P}}:=\\{(x,\mathfrak{p})|x\in\mathfrak{p}\cap\mathcal{N},\mathfrak{p}\in\mathcal{P}\\},\
\ \ \ \rho:(x,\mathfrak{p})\mapsto x$

for some conjugacy class ${\mathcal{P}}$ of parabolic subalgebras.

###### Proof.

Each of these partial resolutions is covered by the Springer resolution
$\mu:\tilde{\mathcal{N}}\rightarrow\mathcal{N}$ because every Borel subalgebra
is contained in a unique element of ${\mathcal{P}}$. Thus by Lemma 4.5 all of
these partial resolutions are symplectic. Additionally they are in bijection
with subsets of the simple roots of $\mathfrak{g}$, which are also in
bijection with parabolic subgroups of the Weyl group of $\mathfrak{g}$.
Furthermore the Weyl group of $\mathfrak{g}$ is the Namikawa Weyl group of
$\mathcal{N}$. Since the Springer resolution is the unique symplectic
resolution (or $\mathbb{Q}$-factorial terminalization) of $\mathcal{N}$, by
4.17, these must be all the partial resolutions.

∎

######  Example 4.19.

The following example is taken from the introduction of arxiv v7 of [NaBir].

Let $X$ be the closure of the subregular orbit of the nilpotent cone of
$\mathfrak{sl}_{4}$. Note that by [He, Rem. 1], $X$ is normal. Furthermore let
$\\{\alpha_{1},\alpha_{2},\alpha_{3}\\}$ be the simple roots of
$\mathfrak{sl}_{4}$ so that $\alpha_{2}$ corresponds to the middle node of the
Dynkin diagram. Then $X$ has three symplectic resolutions, given by the
Springer maps $T^{*}(G/P_{\alpha_{i}})\rightarrow X$ for $P_{\alpha_{1}}$,
$P_{\alpha_{2}}$, and $P_{\alpha_{3}}$ elements of the three conjugacy classes
of minimal parabolics of $SL_{4}$. The Mori fan of
$\overline{{\operatorname{Mov}(T^{*}(G/P_{\alpha_{i}})/X)}}$ is below.

The Namikawa Weyl group is ${\mathbb{Z}}/2{\mathbb{Z}}$ and it acts by
reflection across the bottom hyperplane. This hyperplane is also $H_{s}$, for
$s$ the unique simple generator of the Namikawa Weyl group. Thus the three
partial resolutions corresponding to faces which are contained in $H_{s}$ have
Namikawa Weyl group ${\mathbb{Z}}/2{\mathbb{Z}}$, while the other five have
trivial Namikawa Weyl group.

##  5\. Deformation Theory and Prorepresentability

In this section we study the Poisson deformation theory of partial resolutions
of conical affine symplectic singularities. Let $\textrm{Art}_{\mathbb{C}}$ be
the category of Artinian local rings with residue field $\mathbb{C}$. Given a
Poisson scheme (or complex analytic space) $X$, we define the Poisson
deformation 2-functor
$\mathcal{P}\mathcal{D}_{X}:\textrm{Art}_{\mathbb{C}}\rightarrow\text{Groupoids}$
where $\mathcal{P}\mathcal{D}_{X}(A)$ is the groupoid where objects are flat
Poisson schemes $X_{A}$ over $\operatorname{Spec}(A)$ such that
$X_{A}\times_{\operatorname{Spec}(A)}\operatorname{Spec}(k)\cong X$ and
morphisms are given by isomorphisms which restrict to the identity on $X$ over
$\operatorname{Spec}(k)$. Finally functors are given by base change.

If we take the underlying functor to Sets,
$PD_{X}:\textrm{Art}_{\mathbb{C}}\rightarrow\textrm{Sets}$ given by
$PD_{X}(A)=\textrm{Ob}(\mathcal{P}\mathcal{D}_{X}(A))/\cong$, we get the usual
Poisson deformation functor, studied extensively by Namikawa in [NaFlops,
NaPD1, NaPD2, NaMori]. In particular Namikawa proves in these papers that for
a $\mathbb{Q}$-factorial terminalization $\pi:Y\rightarrow X$ of a conical
affine symplectic singularity $X$, $\operatorname{PD}_{X}$ and
$\operatorname{PD}_{Y}$ are both prorepresentable and unobstructed. Throughout
this section we shall adapt many of the techniques and arguments in those
papers to partial resolutions.

###  5.1. Prorepresentability

The proof of 5.2 below is very similar to that of other deformations of
algebraic objects, and we assume it is known to experts. However we include it
for the benefit of the reader, since as far as we are aware it is absent from
other papers on symplectic singularities. Before the proof we recall the
following key lemma:

######  Lemma 5.1 ([Stacks, Lemma 0D2J]).

Assume we have the diagram of rings

${{B^{\prime}}}$${{A^{\prime}}}$${\ {B}}$${{A}}$

with $A^{\prime}\rightarrow A$ surjective and
$B^{\prime}=B\times_{A}A^{\prime}$. Additionally, for a ring $R$, let
$\textrm{Flat}_{R}$ be the full subcategory of $\operatorname{Mod}_{R}$
consisting of flat modules. Then there is an equivalence of categories

$\textrm{Flat}_{B^{\prime}}\rightarrow\textrm{Flat}_{B}\times_{\textrm{Flat}_{A}}\textrm{Flat}_{A^{\prime}}$

given by
$M\mapsto(M\otimes_{B^{\prime}}B)\times_{(M\otimes_{B^{\prime}}A)}(M\otimes_{B^{\prime}}A^{\prime})$.

######  Theorem 5.2.

$\mathcal{P}\mathcal{D}_{X}$ satisfies the Rim-Schlessinger conditions, as
defined in [Stacks, Definition 06J2].

###### Proof.

By [Stacks, Lemma 06J5], we have to show that given a diagram in
$\textrm{Art}_{\mathbb{C}}$ as in Lemma 5.1,
$\mathcal{P}\mathcal{D}_{X}(B^{\prime})\rightarrow\mathcal{P}\mathcal{D}_{X}(B)\times_{\mathcal{P}\mathcal{D}_{X}(A)}\mathcal{P}\mathcal{D}_{X}(A^{\prime})$
is an equivalence of categories. Note that since any
$A\in\textrm{Art}_{\mathbb{C}}$ is a thickening of $k$, any
$X_{A}\in\mathcal{P}\mathcal{D}_{X}(A)$ is a thickening of $X$ and thus has
the same underlying topological space. In particular we can think of $X_{A}$
as a sheaf of Poisson $A$-algebras on $X$. Thus given an object
$(X_{B},X_{A^{\prime}},\psi)\in\mathcal{P}\mathcal{D}_{X}(B)\times_{\mathcal{P}\mathcal{D}_{X}(A)}\mathcal{P}\mathcal{D}_{X}(A^{\prime})$,
we can explicitly construct an Poisson scheme
$X_{B^{\prime}}\in\mathcal{P}\mathcal{D}_{X}(B^{\prime})$ by setting

$X_{B^{\prime}}(U):=X_{B}(U)\times_{\psi,X_{A}(U)}X_{A^{\prime}}(U).$

Since all the data commutes with localization, this indeed forms a sheaf of
Poisson $B^{\prime}$-algebras on $X$ (in fact, the pushout
$X_{B}\sqcup_{\psi,X_{A}}X_{A^{\prime}}$), and thus an element of
$\mathcal{P}\mathcal{D}_{X}(B^{\prime})$. Furthermore by Lemma 5.1 the
resulting sheaf is flat. This shows that the functor is essentially
surjective.

To show that the functor is fully faithful, let
$X_{1},X_{2}\in\mathcal{P}\mathcal{D}_{X}(B^{\prime})$, and let $U$ be an open
set in $|X|=|X_{1}|=|X_{2}|$. If we consider $X_{1}(U)$ only as a flat
$B^{\prime}$-module, Lemma 5.1 states that it is a fiber product of flat
modules over $B$ and $A^{\prime}$. Furthermore its algebra and Poisson
structures are both defined by homomorphisms in $\textrm{Flat}_{B^{\prime}}$.
Since morphisms between fiber products are determined by their restrictions to
each component, this tells us that $X_{1}(U)$ and $X_{2}(U)$ are fiber
products as a Poisson $B$-algebras. Thus
$\operatorname{Hom}(X_{1}(U),X_{2}(U))$ is isomorphic to

$\operatorname{Hom}(X_{1}|_{B}(U),X_{2}|_{B}(U))\times_{\operatorname{Hom}(X_{1}|_{A}(U),X_{2}|_{A}(U))}\operatorname{Hom}(X_{1}|_{A^{\prime}}(U),X_{2}|_{A^{\prime}}(U))$

This bijection commutes with localization and thus holds at a level of Poisson
schemes, showing the map is fully faithful.

∎

We recall some constructions and theorems from [NaPD1]. In particular,
computations regarding the Poisson deformation functor $\operatorname{PD}_{X}$
are usually made using Poisson cohomology. If $X$ is smooth, and
$\mathcal{X}\in\operatorname{PD}_{X}(S)$ is a deformation of $X$ over
$T:=\operatorname{Spec}(S)$, $\text{HP}^{*}(\mathcal{X}/T)$ is given by the
cohomology of the Poisson-Lichnerowicz complex $(\wedge^{\geq
1}\Theta_{\mathcal{X}/T},\delta)$, where $\delta$ is defined in [NaFlops,
p.13] and $\Theta_{\mathcal{X}/T}$ is the sheaf of relative differentials. The
following key theorem allows us to compute Poisson cohomology using the de
Rham complex.

######  Theorem 5.3 ([NaFlops, Prop. 9]).

For a smooth symplectic variety ($\mathcal{X},\omega$) over $T$, the
symplectic form induces a quasi-isomorphism between $(\wedge^{\geq
1}\Theta_{\mathcal{X}/T},\delta)$ and the truncated de Rham complex
$(\Omega_{\mathcal{X}/T}^{\geq 1},d)$.

We define $PG_{\mathcal{X}/T}$ to be the sheaf which associates to any open
set $U\subset X$ the set of Poisson automorphisms of $\mathcal{X}|_{U}$ which
restrict to the identity over the central fiber of $T$. This sheaf is
essential to showing that $\operatorname{PD}_{X}$ is prorepresentable, and as
indicated above, it can be computed using Poisson cohomology. In particular,
let $P\Theta_{\mathcal{X}/T}$ be the first cohomology sheaf of $(\wedge^{\geq
1}\Theta_{\mathcal{X}/T},\delta)$, and let $P\Theta^{0}_{\mathcal{X}/T}$ be
the subset which restricts to zero over the central fiber.

######  Theorem 5.4 ([NaPD1, 2.2]).

For a smooth $T$-scheme $\mathcal{X}$, the sheaf $PG_{\mathcal{X}/T}$ is
isomorphic to $P\Theta^{0}_{\mathcal{X}/T}$.

For the remainder of this section we fix a partial resolution
$\rho:X^{\prime}\rightarrow X$ of a conical affine symplectic singularity $X$,
and let $\pi:Y\rightarrow X$ be a $\mathbb{Q}$-factorial terminalization which
covers $\rho$ via the map $\pi^{\prime}:Y\rightarrow X^{\prime}$. We are
interested in showing that the Poisson deformation functor
$\operatorname{PD}_{X^{\prime}}$ is prorepresentable. As a tool, we will also
be considering Zariski open subsets $V\subset X^{\prime}$ such that
$\rho^{-1}(X_{\text{reg}})\subset V$, but in all the cases we will be
interested in we will have
$\operatorname{PD}_{V}\cong\operatorname{PD}_{X^{\prime}}$.

The following lemma will be essential for proving that $X^{\prime}$ and $V$
satisfy the final condition in Schlessinger’s criterion.

######  Lemma 5.5.

Let $V\subset X^{\prime}$ be a Zariski open subset such that
$\rho^{-1}(X_{\text{reg}})\subset V$. Then
$H^{1}(V_{\text{reg}},\mathbb{C})=0$.

###### Proof.

Clearly the Hurewicz homomorphism
$\textrm{ab}:\pi_{1}(V_{\text{reg}})\rightarrow
H^{1}(V_{\text{reg}},{\mathbb{Z}})\cong\textrm{Ab}(\pi_{1}(V_{\text{reg}}))$
is surjective. Additionally, since $\rho$ is a partial resolution,
$X_{\text{reg}}$ is a Zariski open subset of $X^{\prime}_{\text{reg}}$ and
thus of $V_{\text{reg}}$. By [Ko, 2.10.1], this means that the map
$i_{*}:\pi_{1}(X_{\text{reg}})\rightarrow\pi_{1}(V_{\text{reg}})$ induced by
the inclusion is surjective. Thus $h:=\textrm{ab}\circ i_{*}$ is surjective as
well.

The profinite completion map is left adjoint to the forgetful map from
profinite groups to groups and thus it is a right exact functor. In particular
the induced map
$\hat{h}:\hat{\pi}_{1}(X_{\text{reg}})\rightarrow\hat{H}^{1}(V_{\text{reg}},{\mathbb{Z}})$
is surjective. By [NaPi1, 1.1] $\hat{\pi}_{1}(X_{\text{reg}})$ is finite, so
$\hat{H}^{1}(V_{\text{reg}},{\mathbb{Z}})$ is finite as well. Since
$H^{1}(V_{\text{reg}},{\mathbb{Z}})$ is abelian it must be finite because
otherwise it would have a ${\mathbb{Z}}$ summand and
$\hat{H}^{1}(V_{\text{reg}},{\mathbb{Z}})$ would have a $\hat{{\mathbb{Z}}}$
summand. Thus $H^{1}(V_{\text{reg}},\mathbb{C})=0$.

∎

In the following lemma we apply the reasoning of [NaPD1, 2.5] to partial
resolutions:

######  Lemma 5.6.

Let $V\subset X^{\prime}$ be a Zariski open subset such that
$\rho^{-1}(X_{\text{reg}})\subset V$. Then for every surjection
$S\rightarrow\overline{S}$ in $\textrm{Art}_{\mathbb{C}}$ and every
$\mathcal{V}\in\mathcal{P}\mathcal{D}_{V}(S)$ the induced map
$H^{0}(V,PG_{\mathcal{V}/T})\rightarrow
H^{0}(V,PG_{\overline{\mathcal{V}}/\overline{T}})$ is surjective, where
$\overline{\mathcal{V}}$ is the restriction of $\mathcal{X}$ to
$\overline{T}:=\operatorname{Spec}({\overline{S}})$.

###### Proof.

Every Poisson automorphism of $\mathcal{V}$ is determined by its restriction
to the smooth locus $\mathcal{U}$ over $T$, and similarly every Poisson
automorphism of $\overline{\mathcal{V}}$ is determined by the smooth locus
$\overline{\mathcal{U}}$ over $\overline{T}$. Note that $\mathcal{U}$ and
$\overline{{\mathcal{U}}}$ are deformations of the smooth locus $U$ of $V$.
Thus it is enough to show that the restriction map
$H^{0}(\mathcal{U},PG_{\mathcal{U}/T})\rightarrow
H^{0}(\overline{{\mathcal{U}}},PG_{\overline{\mathcal{U}}/\overline{T}})$ is
surjective. By 5.4, this is isomorphic to the restriction map
$P\Theta^{0}_{\mathcal{U}/T}\rightarrow
P\Theta^{0}_{\overline{\mathcal{U}}/\overline{T}}$. In particular this means
it is sufficient to show the map $H^{0}(U,P\Theta_{\mathcal{U}/T})\rightarrow
H^{0}(U,P\Theta_{\overline{\mathcal{U}}/\overline{T}})$ is surjective.

Since $\mathcal{U}$ and $\overline{\mathcal{U}}$ are smooth over $T$ and
$\overline{T}$ respectively, we can apply 5.3 and compute Poisson cohomology
as the cohomology of the truncated de Rham complex. In particular we have
$P\Theta_{\mathcal{U}/T}\cong H^{1}(U,\Omega_{\mathcal{U}/T}^{\geq 1})$, and
similarly for $\overline{\mathcal{U}}$. To compute this we use the following
distinguished trangle

(4) $0\rightarrow\Omega_{\mathcal{U}/T}^{\geq
1}\rightarrow\Omega_{\mathcal{U}/T}\rightarrow\mathcal{O}_{\mathcal{U}}\rightarrow
0.$

Note that by [NaFlops, p.18] $H^{i}(U,\Omega_{\mathcal{U}/S})\cong
H^{i}(U,S)$. This distinguished triangle is functorial in
$\textrm{Art}_{\mathbb{C}}$, so taking the long exact sequence in cohomology
for both $\mathcal{U}$ and $\overline{\mathcal{U}}$ we obtain the diagram

${{H^{0}(U,S)}}$${{H^{0}(\mathcal{U},\mathcal{O}_{\mathcal{U}})}}$${{\text{HP}^{1}(\mathcal{U}/T)}}$${{H^{1}(U,S)}}$${{H^{0}(U,S)}}$${{H^{0}(\overline{\mathcal{U}},\mathcal{O}_{\overline{\mathcal{U}}})}}$${{\text{HP}^{1}(\overline{\mathcal{U}}/\overline{T})}}$${{H^{1}(U,\overline{S})}}$

By Lemma 5.5, $H^{1}(U,S)=H^{1}(U,\mathbb{C})\otimes S=0$ and similarly
$H^{1}(U,\overline{S})=0$. Thus since
$H^{0}(\mathcal{U},\mathcal{O}_{\mathcal{U}})\rightarrow
H^{0}(\overline{\mathcal{U}},\mathcal{O}_{\overline{\mathcal{U}}})$ is
surjective,
$\text{HP}^{1}(\mathcal{U}/T)\rightarrow\text{HP}^{1}(\overline{\mathcal{U}}/\overline{T})$
is surjective as well. Using a spectral sequence argument we have
$\text{HP}^{1}(\mathcal{U}/T)\cong H^{0}(X,P\Theta_{\mathcal{U}/T})$ and
$\text{HP}^{1}(\overline{\mathcal{U}}/\overline{T})\cong
H^{0}(X,P\Theta_{\overline{\mathcal{U}}/\overline{T}})$.

∎

Given $A\in\textrm{Art}_{\mathbb{C}}$, we have a forgetful map
$\operatorname{PD}_{X}(A)\rightarrow\textrm{Def}_{X}(A)$ from Poisson
deformations to flat deformations. Note that Poisson and flat deformations
both form presheaves in the complex analytic topology. Following Namikawa, we
let $PT^{1}_{X}$ and $T^{1}_{X}$ be the sheafification of the presheaves of
first order Poisson and flat deformations respectively. Thus for any open set
$U\subset X$ there are canonical maps
$\operatorname{PD}_{U}(\mathbb{C}[\epsilon])\rightarrow PT^{1}_{X}(U)$ and
$\textrm{Def}_{U}(\mathbb{C}[\epsilon])\rightarrow T^{1}_{X}(U)$. Finally we
also have a forgetful map of sheaves $PT^{1}_{X}\rightarrow T^{1}_{X}$, and
for any $U\subset X$ all of the above maps fit into the commutative diagram:

${{\operatorname{PD}_{U}(\mathbb{C}[\epsilon])}}$${{\textrm{Def}_{U}(\mathbb{C}[\epsilon])}}$${{PT^{1}_{X}(U)}}$${{T^{1}_{X}(U)}}$

The kernel of the map composition
$\operatorname{PD}_{U}(\mathbb{C}[\epsilon])\rightarrow T^{1}_{X}(U)$ is the
set of first order Poisson deformations which are locally trivial as flat
deformations. We denote this $\operatorname{PD}_{lt,U}(\mathbb{C}[\epsilon])$.
Additionally let $\mathcal{H}$ be the subsheaf of $T^{1}_{X}$ given by the
image of $PT^{1}_{X}$. Thus we have the exact sequence

(5)
$0\rightarrow\operatorname{PD}_{lt,U}(\mathbb{C}[\epsilon])\rightarrow\operatorname{PD}_{U}(\mathbb{C}[\epsilon])\rightarrow\Gamma(U,\mathcal{H})$

######  Theorem 5.7.

Let $X$ a symplectic singularity such that every symplectic leaf has
codimension 0 or 2 and $H^{i}(X,\mathcal{O}_{X})=0$ for $i=1,2$. Then
$\operatorname{PD}_{lt,X}(\mathbb{C}[\epsilon])=H^{2}(X,\mathbb{C})$.

###### Proof.

The proof is identical [NaPD1, 1.11].

∎

This theorem has the following key corollary:

######  Corollary 5.8.

Let $X^{\prime}$ be a partial resolution of an affine symplectic singularity
and let $V\subset X^{\prime}$ be the union of the open stratum and the
codimension two strata. Then
$\operatorname{PD}_{lt,V}(\mathbb{C}[\epsilon])=H^{2}(V,\mathbb{C})$.

###### Proof.

We have that $H^{i}(X^{\prime},\mathcal{O}_{X^{\prime}})=0$ for all $i>0$.
Additionally since $X^{\prime}$ is Cohen-Macaulay and $X^{\prime}-V$ has
codimension four, by a depth argument we have $H^{i}(V,\mathcal{O}_{V})=0$ for
$i=1,2$. Thus by 5.7 we are done.

∎

Finally we are in a position to prove the main result of this subsection.

######  Theorem 5.9.

Given a partial resolution $\rho:X^{\prime}\rightarrow X$ of a conical affine
symplectic singularity $X$, $\operatorname{PD}_{X^{\prime}}$ is
prorepresentable.

###### Proof.

By [Stacks, Lemma 06I0] $\operatorname{PD}_{X^{\prime}}$ satisfies the first
two conditions of Schlessinger’s criterion ([Sch, 2.11]) as long as
$\mathcal{P}\mathcal{D}_{X^{\prime}}$ satisfies the Rim-Schlessinger
conditions, and $\mathcal{P}\mathcal{D}_{X^{\prime}}(k)$ is a point. We proved
that $\mathcal{P}\mathcal{D}_{X^{\prime}}$ satisfies the Rim-Schlessinger
conditions in 5.2. Additionally, by [Stacks, Lemma 06J8], the final condition
is equivalent to the condition that for every small surjection
$S\rightarrow\overline{S}$ in $\textrm{Art}_{\mathbb{C}}$, and every
$\mathcal{X}\in\mathcal{P}\mathcal{D}_{X}(S)$ the induced map
$\textrm{Aut}(\mathcal{X})\rightarrow\textrm{Aut}(\overline{\mathcal{X}})$ is
surjective, where $\overline{\mathcal{X}}$ is the restriction of $\mathcal{X}$
to $\overline{S}$. This is precisely the content of Lemma 5.6.

The only statement left to prove is that
$\operatorname{PD}_{X^{\prime}}(\mathbb{C}[\epsilon])$ is finite dimensional.
By [NaPD1, p. 12] we have that
$\operatorname{PD}_{X^{\prime}}(\mathbb{C}[\epsilon])\cong\operatorname{PD}_{V}(\mathbb{C}[\epsilon])$
where $V\subset X^{\prime}$ is the union of the smooth and codimension two
leaves of $X^{\prime}$. We apply diagram (5) to $V$. By [NaPD1, 1.5],
$\mathcal{H}$ is a finite dimensional local system on the singular locus of
$V$, so $\Gamma(V,\mathcal{H})$ is finite dimensional. By 5.8,
$\operatorname{PD}_{lt,V}(\mathbb{C}[\epsilon])$ is finite dimensional as
well. So by diagram (5),
$\operatorname{PD}_{X^{\prime}}(\mathbb{C}[\epsilon])$ is finite dimensional.

∎

Next we relate $\operatorname{PD}_{X^{\prime}}$ to the Poisson deformation
functors of $X$ and $Y$.

######  Lemma 5.10.

We have natural pushforward maps
$\operatorname{PD}_{Y}\xrightarrow{\pi^{\prime}_{*}}\operatorname{PD}_{X^{\prime}}\xrightarrow{\rho_{*}}\operatorname{PD}_{X}$.

###### Proof.

We construct the map $\pi^{\prime}_{*}$ as the construction of $\rho_{*}$ is
identical. Following [Wa], to obtain a pushforward of flat deformations it is
sufficient to show that
$\pi^{\prime}_{*}\mathcal{O}_{Y}\cong\mathcal{O}_{X^{\prime}}$ and
$R^{1}\pi^{\prime}_{*}\mathcal{O}_{Y}=0$. As explained in [Stacks, Lemma
0E3X], the map of flat deformations
$\overline{{\pi}}^{\prime}_{*}:\textrm{Def}_{Y}\rightarrow\textrm{Def}_{X^{\prime}}$
is given by taking $\mathcal{Y}\in\textrm{Def}_{Y}(A)$ as a sheaf of flat
$A$-algebras on $Y$ and pushing it forward to a sheaf of $A$-algebras on
$X^{\prime}$.

We take a resolution of singularities $\nu:Z\rightarrow Y$, which exists
because $Y$ has rational singularities. Note that $\pi^{\prime}\circ\nu$ is a
resolution of singularities of $X^{\prime}$. Furthermore since $Y$ and
$X^{\prime}$ have rational singularities $\nu_{*}\mathcal{O}_{Z}$ is quasi-
isomorphic to $\mathcal{O}_{Y}$ in the derived category and
$(\pi^{\prime}\circ\nu)_{*}(\mathcal{O}_{Z})\cong(\pi^{\prime}_{*}\circ\nu_{*})(\mathcal{O}_{Z})$
is also quasi-isomorphic to $\mathcal{O}_{X^{\prime}}$ in the derived
category. Thus $\pi^{\prime}_{*}\mathcal{O}_{Y}$ is quasi-isomorphic to
$\mathcal{O}_{X^{\prime}}$ which indeed implies that
$\pi^{\prime}_{*}\mathcal{O}_{Y}\cong\mathcal{O}_{X^{\prime}}$ and
$R^{1}\pi^{\prime}_{*}\mathcal{O}_{Y}=0$. So we have a pushforward map of flat
deformations
$\overline{{\pi}}^{\prime}_{*}:\textrm{Def}_{Y}\rightarrow\textrm{Def}_{X^{\prime}}$.

To extend this to a pushforward map
$\pi^{\prime}_{*}:\operatorname{PD}_{Y}\rightarrow\operatorname{PD}_{X^{\prime}}$
we observe that since $\pi^{\prime}$ is an isomorphism over
$X^{\prime}_{\text{reg}}$, a Poisson deformation of $Y$ restricts to a Poisson
deformation of $X^{\prime}_{\text{reg}}$. Let
$\mathcal{Y}\in\operatorname{PD}_{Y}(A)$, for some
$A\in\textrm{Art}_{\mathbb{C}}$. Since the above pushforward map of flat
deformations is the identity over $X^{\prime}_{\text{reg}}$, we can simply
extend this Poisson structure on
$\mathcal{Y}|_{X^{\prime}_{\text{reg}}}\cong\overline{{\pi}}^{\prime}_{*}(\mathcal{Y})|_{X^{\prime}_{\text{reg}}}$
to $\overline{{\pi}}^{\prime}_{*}(\mathcal{Y})$ by normality. ∎

######  Remark 5.11.

The above pushforward maps compose to standard pushforward map
$\pi_{*}:\operatorname{PD}_{Y}\rightarrow\operatorname{PD}_{X}$ studied by
Namikawa. This map (as well as the map $\rho_{*}$) can be described more
succintly by sending some $\mathcal{Y}\in\operatorname{PD}_{Y}(A)$ to
$\operatorname{Spec}\Gamma(Y,\mathcal{O}_{\mathcal{Y}})$, which is a
deformation of $X=\operatorname{Spec}\Gamma(Y,\mathcal{O}_{Y})$.

###  5.2. Unobstructedness

The primary goal of this section is to prove that
$\operatorname{PD}_{X^{\prime}}$ is unobstructed. In the course of the proof
we will need to consider $\tilde{V}:=\pi^{\prime-1}(V)$ where $V$ is the union
of the open and codimension two leaves of $X^{\prime}$. We will also be
considering the union $U$ of the open and codimension two leaves of $X$, as
well as $U^{\prime}:=\rho^{-1}(U)$ and $\tilde{U}:=\pi^{-1}(U)$. Throughout
this section we prove that all the functors in the diagram below are
unobstructed and that all the horizontal maps are equivalences. We note that
many of these functors were already proven to be unobstructed in [NaFlops,
Cor. 13] and [NaPD1, 5.1], and many of the arguments in this subsection for
the other spaces are adapted from these.

Considering all the spaces mentioned above, we obtain the following
commutative diagram:

(6)
${{\operatorname{PD}_{Y}}}$${{\operatorname{PD}_{Y_{\text{reg}}}}}$${{\operatorname{PD}_{\tilde{V}}}}$${{\operatorname{PD}_{\tilde{U}}}}$${{\operatorname{PD}_{X^{\prime}}}}$${{\operatorname{PD}_{V}}}$${{\operatorname{PD}_{U^{\prime}}}}$${{\operatorname{PD}_{X}}}$${{\operatorname{PD}_{U}}}$$\scriptstyle{r_{\tilde{V}}}$$\scriptstyle{(\pi^{\prime}|_{\tilde{V}})_{*}}$$\scriptstyle{r_{\tilde{U}}}$$\scriptstyle{(\pi^{\prime}|_{\tilde{U}})_{*}}$$\scriptstyle{r_{U^{\prime}}}$$\scriptstyle{(\rho|_{U^{\prime}})_{*}}$$\scriptstyle{r_{Y_{\text{reg}}}}$$\scriptstyle{\pi^{\prime}_{*}}$$\scriptstyle{\rho_{*}}$$\scriptstyle{r_{V}}$$\scriptstyle{r_{U}}$

######  Lemma 5.12.

All of the functors in diagram (6) are prorepresentable.

###### Proof.

This follows from Schlessinger’s criterion. 5.2 implies that the first two
conditions are satisfied. The third criterion is satisfied by applying the
short exact sequence (5) to each space. Finally the last criterion is implied
by Lemma 5.6, since all the spaces considered satisfy the hypotheses of that
lemma.

∎

######  Lemma 5.13.

The functors $\operatorname{PD}_{Y_{\text{reg}}}$,
$\operatorname{PD}_{\tilde{V}}$, and $\operatorname{PD}_{\tilde{U}}$ in the
top row of diagram (6) are all unobstructed, and the restriction maps between
them are equivalences.

###### Proof.

Because $X$, $X^{\prime}$, and $Y$ are all Cohen-Macaulay and $X-U$,
$X^{\prime}-V$, and $Y-Y^{\text{reg}}$ all have codimension 4, by a depth
argument we have
$H^{i}(U,\mathcal{O}_{U})=H^{i}(V,\mathcal{O}_{V})=H^{i}(Y^{\text{reg}},\mathcal{O}_{Y^{\text{reg}}})=0$
for $i=1,2$. Furthermore since $U$ and $V$ have rational singularities, we
have
$H^{i}(\tilde{U},\mathcal{O}_{\tilde{U}})=H^{i}(\tilde{V},\mathcal{O}_{\tilde{V}})=0$
for $i=1,2$. Since $\tilde{U}$ and $\tilde{V}$ are Zariski open subsets of
$Y^{\text{reg}}$, we can use the functoriality of the short exact sequence of
sheaves

$0\rightarrow\Omega_{Y^{\text{reg}}}^{\geq
1}\rightarrow\Omega_{Y^{\text{reg}}}\rightarrow\mathcal{O}_{Y^{\text{reg}}}\rightarrow
0.$

under restriction. Additionally we recall that since $Y^{\text{reg}}$ is
smooth, by [NaFlops, Prop 8],
$\operatorname{PD}_{Y^{\text{reg}}}(\mathbb{C}[\epsilon])\cong
H^{2}(Y^{\text{reg}},\Omega_{Y^{\text{reg}}}^{\geq 1})$ and the analogous
statements are true for $\tilde{U}$ and $\tilde{V}$. Thus taking long exact
sequences in cohomology, we have the diagram

${{H^{1}(Y^{\text{reg}},\mathcal{O}_{Y^{\text{reg}}})}}$${{\operatorname{PD}_{Y^{\text{reg}}}(\mathbb{C}[\epsilon])}}$${{H^{2}(Y^{\text{reg}},\mathbb{C})}}$${{H^{2}(Y^{\text{reg}},\mathcal{O}_{Y^{\text{reg}}})}}$${{H^{1}(\tilde{V},\mathcal{O}_{\tilde{V}})}}$${{\operatorname{PD}_{\tilde{V}}(\mathbb{C}[\epsilon])}}$${{H^{2}(\tilde{V},\mathbb{C})}}$${{H^{2}(\tilde{V},\mathcal{O}_{\tilde{V}})}}$${{H^{1}(\tilde{U},\mathcal{O}_{\tilde{U}})}}$${{\operatorname{PD}_{\tilde{U}}(\mathbb{C}[\epsilon])}}$${{H^{2}(\tilde{U},\mathbb{C})}}$${{H^{2}(\tilde{U},\mathcal{O}_{\tilde{U}})}}$

Since all the cohomology groups of the structure sheaves are zero, this
implies that $r_{\tilde{V}}(\mathbb{C}[\epsilon])$ and
$r_{\tilde{U}}(\mathbb{C}[\epsilon])$ is diagram (6) are given by restriction
in cohomology. Furthermore it follows from the proof of [Na1, Prop. 2] that
these maps are both isomorphisms.

Next we show these functors are unobstructed. We begin by briefly recalling
the $T^{1}$-lifting criterion, which can be proved using the argument in
[NaFlops, Cor. 13]. Let $S_{n}:=k[x]/(x^{n})$ and
$T_{n}:=\operatorname{Spec}(S_{n})$. For any deformation functor $D$ and
$X_{n}\in D(S_{n})$, we let $T^{1}(X_{n}/T_{n})$ be the set of elements of
$D(S_{n}[\epsilon])$ which restrict to $X_{n}$ over $T_{n}$. The
$T^{1}$-lifting criterion says that if $D$ is a prorepresentable functor, then
it is unobstructed if and only if the natural map

$T^{1}(X_{n}/T_{n})\rightarrow T^{1}(X_{n-1}/T_{n-1})$

induced by $S_{n}[\epsilon]\rightarrow S_{n-1}[\epsilon]$ is a surjection for
all positive $n$ and any (equivalently, some) $X_{n}\in D(S_{n})$. If $D$ is
the Poisson deformation functor $\operatorname{PD}_{Z}$ of some symplectic
singularity $Z$, we denote $T^{1}(X_{n}/T_{n})$ as
$\operatorname{PD}_{Z}(X_{n}/T_{n},T_{n}[\epsilon])$, following [NaFlops, p.
15].

We have shown that all of the functors in diagram (6) are prorepresentable, so
we will use the $T^{1}$-lifting criterion. We now consider the deformation
functors of the smooth symplectic manifolds $Y_{\text{reg}}$, $\tilde{V}$, and
$\tilde{U}$. We prove the $T^{1}$-lifting criterion for $\tilde{V}$ since the
other two proofs are identical. Applying [NaFlops, Prop. 8, Cor. 10] and the
fact that $H^{i}(\tilde{V},\mathcal{O}_{\tilde{V}})=0$ for $i=1,2$, we obtain
for any $\tilde{V}_{n}\in\operatorname{PD}_{\tilde{V}}(S_{n})$

$\operatorname{PD}_{\tilde{V}}(\tilde{V}_{n}/T_{n},T_{n}[\epsilon])\cong
HP^{2}(\tilde{V}_{n}/T_{n})\cong H^{2}(\tilde{V},S_{n})\cong
H^{2}(\tilde{V},\mathbb{C})\otimes S_{n}.$

Additionally, the above isomorphisms are functorial in
$\textrm{Art}_{\mathbb{C}}$. Thus since $S_{n}\rightarrow S_{n-1}$ is
surjective, $\operatorname{PD}_{\tilde{V}}$ is unobstructed, and the same
argument applies to the other two functors. Finally, the maps $r_{\tilde{V}}$
and $r_{\tilde{U}}$ of diagram (6) are maps between prorepresentable and
unobstructed deformations functors which induce isomorphisms on first order
deformations, so they are equivalences.

∎

######  Theorem 5.14.

All the functors in diagram (6) are unobstructed, and they all have tangent
spaces of the same dimension.

###### Proof.

We begin by observing that [NaPD1, Section 4] applies to any morphism
$Y\rightarrow X$ where $X$ is a symplectic singularity whose singularities are
all locally isomorphic to $(S,0)\times(\mathbb{C}^{2n-2},0)$, for $S$
Kleinian, and $Y$ is a symplectic resolution. In particular it applies to
$\pi^{\prime}|_{\tilde{V}}:\tilde{V}\rightarrow V$,
$\pi^{\prime}|_{\tilde{U}}:\tilde{U}\rightarrow U^{\prime}$, and
$\pi|_{\tilde{U}}:\tilde{U}\rightarrow U$.

The sheaf $R^{2}\pi^{\prime}_{*}\mathbb{C}_{\tilde{V}}$ is given by the
sheafification of the of the presheaf

$W\mapsto\operatorname{PD}_{\pi^{\prime-1}(W)}(\mathbb{C}[\epsilon]).$

Thus we have an exact sequence

(7)
$0\rightarrow\operatorname{PD}_{\tilde{V}}^{\pi^{\prime}}(\mathbb{C}[\epsilon])\rightarrow\operatorname{PD}_{\tilde{V}}(\mathbb{C}[\epsilon])\rightarrow\Gamma(V,R^{2}\pi^{\prime}_{*}\mathbb{C}_{\tilde{V}})$

where $\operatorname{PD}_{\tilde{V}}^{\pi^{\prime}}$ is the subfunctor of
$\operatorname{PD}_{\tilde{V}}$ given by deformations which are trivial on the
preimage of a sufficiently small analytic neighborhood of every point in $V$.
These types of subfunctors will be studied more extensively in the next two
sections. By the argument in [NaMori, Lem. 3], the map
$\operatorname{PD}_{\tilde{V}}^{\pi^{\prime}}(\mathbb{C}[\epsilon])\rightarrow\operatorname{PD}_{\tilde{V}}(\mathbb{C}[\epsilon])$
is given by the pullback map in cohomology $H^{2}(V,\mathbb{C})\rightarrow
H^{2}(\tilde{V},\mathbb{C})$. Furthermore, by the Leray spectral sequence the
above sequence (7) is short exact.

Next we apply the short exact sequence (5) to $V$, obtaining

(8)
$0\rightarrow\operatorname{PD}_{lt,V}(\mathbb{C}[\epsilon])\rightarrow\operatorname{PD}_{V}(\mathbb{C}[\epsilon])\rightarrow\Gamma(V,\mathcal{H}).$

For the next proof we want to show that the sequence above is also short
exact, in particular that the final map is surjectve. We can apply [NaPD1,
4.2] to $\tilde{V}\rightarrow V$ to obtain
$\dim\Gamma(V,\mathcal{H})=\dim\Gamma(V,R^{2}\pi^{\prime}_{*}\mathbb{C}_{\tilde{V}})$
and 5.7 to obtain $\dim\operatorname{PD}_{lt,V}(\mathbb{C}[\epsilon])=\dim
H^{2}(V,\mathbb{C})$. Thus we have

$\displaystyle\dim\operatorname{PD}_{V}(\mathbb{C}[\epsilon])$
$\displaystyle\leq\dim\operatorname{PD}_{lt,V}(\mathbb{C}[\epsilon])+\dim\Gamma(V,\mathcal{H})$
$\displaystyle=\dim
H^{2}(V,\mathbb{C})+\dim\Gamma(V,R^{2}\pi^{\prime}_{*}\mathbb{C}_{\tilde{V}})$
$\displaystyle=\dim
H^{2}(\tilde{V},\mathbb{C})=\dim\operatorname{PD}_{\tilde{V}}(\mathbb{C}[\epsilon])$

Since all of the singularities of $V$ are of the form
$(S,0)\times(\mathbb{C}^{2n-2},0)$ and
$\pi^{\prime}|_{\tilde{V}}:\tilde{V}\rightarrow V$ is a symplectic resolution,
the argument in part $(iii)$ of [NaPD1, 5.1] implies that the pushforward map

$(\pi^{\prime}|_{\tilde{V}})_{*}:\operatorname{Spec}(R_{\tilde{V}})\rightarrow\operatorname{Spec}(R_{V})$

has finite closed fiber, where $R_{\tilde{V}}$ and $R_{V}$ are the complete
local rings which prorepresent $\operatorname{PD}_{\tilde{V}}$ and
$\operatorname{PD}_{V}$ respectively. Letting $\mathfrak{m}_{\tilde{V}}$ and
$\mathfrak{m}_{V}$ be the maximal ideals of $R_{\tilde{V}}$ and $R_{V}$
respectively, this means that $\dim
R_{\tilde{V}}/\mathfrak{m}_{V}R_{\tilde{V}}=0$. Thus we have

$\dim R_{\tilde{V}}\leq\dim R_{V}+\dim
R_{\tilde{V}}/\mathfrak{m}_{V}R_{\tilde{V}}=\dim R_{V}.$

We also know $\dim
R_{V}\leq\dim_{\mathbb{C}}\mathfrak{m}_{V}/(\mathfrak{m}_{V})^{2}$ and $\dim
R_{\tilde{V}}=\dim_{\mathbb{C}}\mathfrak{m}_{\tilde{V}}/(\mathfrak{m}_{\tilde{V}})^{2}$
since $\operatorname{PD}_{\tilde{V}}$ is unobstructed and thus $R_{\tilde{V}}$
is regular. Combining the above equations we obtain

$\dim_{\mathbb{C}}\mathfrak{m}_{\tilde{V}}/(\mathfrak{m}_{\tilde{V}})^{2}\leq\dim
R_{V}\leq\dim_{\mathbb{C}}\mathfrak{m}_{V}/(\mathfrak{m}_{V})^{2}.$

However these two tangent spaces are given by
$\operatorname{PD}_{\tilde{V}}(\mathbb{C}[\epsilon])$ and
$\operatorname{PD}_{V}(\mathbb{C}[\epsilon])$ respectively. Since we proved
above that
$\dim\operatorname{PD}_{V}(\mathbb{C}[\epsilon])\leq\dim\operatorname{PD}_{\tilde{V}}(\mathbb{C}[\epsilon])$,
the inequalities above must be equalities. Since $\dim
R_{V}=\dim_{\mathbb{C}}\mathfrak{m}_{V}/(\mathfrak{m}_{V})^{2}$, $R_{V}$ is a
regular local ring and $\operatorname{PD}_{V}=\operatorname{Hom}(R_{V},-)$ is
unobstructed. This equality also implies that the sequence (8) is a short
exact sequence.

Note that the only features of the resolution
$\pi^{\prime}|_{\tilde{V}}:\tilde{V}\rightarrow V$ that we used were that all
of the singularities of $V$ are of the form
$(S,0)\times(\mathbb{C}^{2n-2},0)$, that
$\pi^{\prime}|_{\tilde{V}}:\tilde{V}\rightarrow V$ is a symplectic resolution,
and that $H^{i}(V,\mathcal{O}_{V})=0$ for $i=1,2$ (to apply 5.7). These all
apply to $\pi^{\prime}|_{\tilde{U}}:\tilde{U}\rightarrow U^{\prime}$, and
$\pi|_{\tilde{U}}:\tilde{U}\rightarrow U$ as well. In particular
$H^{i}(U,\mathcal{O}_{U})=H^{i}(U^{\prime},\mathcal{O}_{U^{\prime}})=H^{i}(\tilde{U},\mathcal{O}_{\tilde{U}})=0$
for $i=1,2$, since $U$ and $U^{\prime}$ have rational singularities. Thus the
same argument applies to show that $\operatorname{PD}_{U^{\prime}}$ and
$\operatorname{PD}_{U}$ are unobstructed and that the dimensions of the
prorepresenting rings are the same as that of $\operatorname{PD}_{\tilde{U}}$.

Finally, we only have left to prove that the Poisson deformation functors of
$Y$, $X^{\prime}$, and $X$ are unobstructed. For this we use the fact that
$Y-Y_{\text{reg}}$, $X^{\prime}-V$, and $X-U$ all have codimension at least
four. We will explain the details for $X^{\prime}$ as the other two cases are
identical. For any deformation
$X^{\prime}_{n}\in\operatorname{PD}_{X^{\prime}}(S_{n})$ we can restrict it to
a deformation $V_{n}\in\operatorname{PD}_{V}(S_{n})$. Applying the reasoning
of [NaFlops, Prop. 13], since $X^{\prime}-V$ has codimension at least four, we
have

$\operatorname{PD}_{X^{\prime}}(X^{\prime}_{n}/T_{n},T_{n}[\epsilon])\cong\operatorname{PD}_{V}(V_{n}/T_{n},T_{n}[\epsilon]).$

Thus $\operatorname{PD}_{X^{\prime}}$ satisfies the $T^{1}$-lifting criterion
if and only if $\operatorname{PD}_{V}$ does. Thus
$\operatorname{PD}_{X^{\prime}}$ is unobstructed. Finally since
$\operatorname{PD}_{X^{\prime}}$ and $\operatorname{PD}_{V}$ are both
unobstructed and $r_{V}$ induces an isomorphism on first order deformations it
is an equivalence.

As stated above the same argument shows that $r_{Y_{\text{reg}}}$ and $r_{U}$
are equivalences and in particular that $\operatorname{PD}_{Y}$ and
$\operatorname{PD}_{X}$ are unobstructed.

∎

In the above arguments we have shown that all of the horizontal maps in
diagram (6) are equivalences, except for $r_{U^{\prime}}$. We show that below:

######  Theorem 5.15.

The map $r_{U^{\prime}}$ is an equivalence.

###### Proof.

We have shown that $\operatorname{PD}_{V}$ and
$\operatorname{PD}_{U^{\prime}}$ are both prorepresentable and unobstructed,
so we only need to show that $r_{U^{\prime}}$ induces an isomorphism on first
order deformations. We first study $r_{\tilde{U}}$.

The construction of sequence (7) is functorial under pullback from the right
column to the left column in:

${{\tilde{U}}}$${{\tilde{V}}}$${{U^{\prime}}}$${V}$$\scriptstyle{i_{\tilde{U}}}$$\scriptstyle{\pi^{\prime}|_{\tilde{U}}}$$\scriptstyle{\pi^{\prime}|_{\tilde{V}}}$$\scriptstyle{i_{U}}$

Additionally, we showed in the previous argument that the sequences we obtain
are short exact. Thus we have the diagram:

(9)
${0}$${{H^{2}(V,\mathbb{C})}}$${{H^{2}(\tilde{V},\mathbb{C})}}$${{\Gamma(V,R^{2}\pi^{\prime}_{*}\mathbb{C}_{\tilde{V}})}}$${0}$${0}$${{H^{2}(U^{\prime},\mathbb{C})}}$${{H^{2}(\tilde{U},\mathbb{C})}}$${{\Gamma(U^{\prime},R^{2}\pi^{\prime}_{*}\mathbb{C}_{\tilde{V}})}}$${0}$$\scriptstyle{i_{\tilde{U}}^{*}}$$\scriptstyle{(\pi^{\prime}|_{\tilde{V}})^{*}}$$\scriptstyle{(\pi^{\prime}|_{\tilde{U}})^{*}}$$\scriptstyle{i_{U^{\prime}}^{*}}$

We know from the proof of Lemma 5.13 that $i_{\tilde{U}}^{*}$ is an
isomorphism. This implies that the right column of diagram (9) is surjective.
To prove this map is injective we begin by observing that
$\tilde{V}-\tilde{U}$ has codimension at least two because $\pi|_{\tilde{V}}$
is a symplectic resolution and thus must be semismall, and $X-U$ has
codimension at least four. This implies that the singular locus
$\Sigma_{V-U^{\prime}}$ of $V-U^{\prime}$ must be codimension at least three,
because otherwise its preimage would be a subset of $\tilde{V}-\tilde{U}$ of
codimension less that two. Finally this implies that $\Sigma_{V-U^{\prime}}$
has codimension at least one in the singular locus $\Sigma_{V}$ of $V$. In
particular this means that the singular locus $\Sigma_{U^{\prime}}$ of
$U^{\prime}$ intersects every connected component of $\Sigma_{V}$. Thus since
$R^{2}\pi^{\prime}_{*}\mathbb{C}_{\tilde{V}}$ is a local system on
$\Sigma_{V}$, the restriction map in the right column of diagram (9) is
injective. Thus it is an isomorphism, and by the commutativity of diagram (9),
$i_{U^{\prime}}^{*}$ is an isomorphism.

Finally we study the map
$\operatorname{PD}_{V}(\mathbb{C}[\epsilon])\rightarrow\operatorname{PD}_{U^{\prime}}(\mathbb{C}[\epsilon])$
induced by $r_{U^{\prime}}$ directly. We apply the short exact sequence (5) to
$U^{\prime}$ and $V$, noting that it is functorial under restriction.
Furthermore in the proof of 5.14 we saw that the sequences below are also
short exact. Thus we have the diagram:

(10)
${0}$${{\operatorname{PD}_{lt,V}(\mathbb{C}[\epsilon])}}$${{\operatorname{PD}_{V}(\mathbb{C}[\epsilon])}}$${{\Gamma(V,\mathcal{H})}}$${0}$${0}$${{\operatorname{PD}_{lt,U^{\prime}}(\mathbb{C}[\epsilon])}}$${{\operatorname{PD}_{U^{\prime}}(\mathbb{C}[\epsilon])}}$${{\Gamma(U^{\prime},\mathcal{H})}}$${0}$

We recall from the proof of 5.7 in [NaPD1, 1.11] that
$\operatorname{PD}_{lt,V}(\mathbb{C}[\epsilon])$ is given by
$H^{2}(V,\tilde{\Omega}_{V}^{\geq 1})$, where $(\tilde{\Omega}_{V}^{\geq
1},d)$ is the truncated de Rham complex of $V$ as a $V$-manifold. Like the de
Rham complex of a smooth variety, it sits in a distinguished triangle

$\tilde{\Omega}_{V}^{\geq
1}\rightarrow\tilde{\Omega}_{V}\rightarrow\mathcal{O}_{V}\rightarrow\tilde{\Omega}_{V}^{\geq
1}[1].$

This is also true for
$\operatorname{PD}_{lt,U^{\prime}}(\mathbb{C}[\epsilon])$, and furthermore the
above short exact sequence is functorial under restriction, so we obtain the
diagram

(11)
${{H^{1}(V,\mathcal{O}_{V})}}$${{\operatorname{PD}_{lt,V}(\mathbb{C}[\epsilon])}}$${{H^{2}(V,\mathbb{C})}}$${{H^{2}(V,\mathcal{O}_{V})}}$${{H^{1}(U^{\prime},\mathcal{O}_{U^{\prime}})}}$${{\operatorname{PD}_{lt,U^{\prime}}(\mathbb{C}[\epsilon])}}$${{H^{2}(U^{\prime},\mathbb{C})}}$${{H^{2}(U^{\prime},\mathcal{O}_{U^{\prime}})}}$

However we showed above that
$H^{i}(V,\mathcal{O}_{V})=H^{i}(U^{\prime},\mathcal{O}_{U^{\prime}})=0$ for
$i=1,2$. Thus by diagram (11) the restriction map
$\operatorname{PD}_{lt,V}(\mathbb{C}[\epsilon])\rightarrow\operatorname{PD}_{lt,U^{\prime}}(\mathbb{C}[\epsilon])$
is given by the restriction map in cohomology

$i_{U^{\prime}}^{*}:H^{2}(V,\mathbb{C})\rightarrow
H^{2}(U^{\prime},\mathbb{C})$

which we proved was an isomorphism.

Additionally applying [NaPD1, 4.2] to both $\tilde{V}\rightarrow V$ and
$\tilde{U}\rightarrow U^{\prime}$ and using the isomorphism in the right
column of diagram (9) we have

$\dim\Gamma(V,\mathcal{H})=\dim\Gamma(V,R^{2}\pi^{\prime}_{*}\mathbb{C}_{\tilde{V}})=\dim\Gamma(U^{\prime},R^{2}\pi^{\prime}_{*}\mathbb{C}_{\tilde{V}})=\dim\Gamma(U^{\prime},\mathcal{H}).$

Since $\mathcal{H}$ is also a local system on the singular locus $\Sigma_{V}$
of V, the argument from before that $\Sigma_{U^{\prime}}$ intersects every
connected component of $\Sigma_{V}$ implies that the restriction map
$\Gamma(V,\mathcal{H})\rightarrow\Gamma(U^{\prime},\mathcal{H})$ is injective,
and thus an isomorphism since both spaces have the same dimension. Thus by the
five lemma diagram (10) shows that

$r_{U^{\prime}}(\mathbb{C}[\epsilon]):\operatorname{PD}_{V}(\mathbb{C}[\epsilon])\rightarrow\operatorname{PD}_{U^{\prime}}(\mathbb{C}[\epsilon])$

is an isomorphism, so $r_{U^{\prime}}$ is an equivalence. ∎

######  Corollary 5.16.

The maps

$R_{X}\xrightarrow{\rho^{\sharp}}R_{X^{\prime}}\xrightarrow{\pi^{\prime\sharp}}R_{Y}$

induced by

$\operatorname{PD}_{Y}\xrightarrow{\pi^{\prime}_{*}}\operatorname{PD}_{X^{\prime}}\xrightarrow{\rho_{*}}\operatorname{PD}_{X}$

are injections.

###### Proof.

By 5.14, $R_{X}$, $R_{X^{\prime}}$, and $R_{Y}$ all have the same dimension.
Additionally, the argument in part $(iii)$ of [NaPD1, 5.1] shows that the maps
$R_{X}\rightarrow R_{X^{\prime}}\rightarrow R_{Y}$ all have closed finite
fiber. If the map $\rho^{\sharp}:R_{X}\rightarrow R_{X^{\prime}}$ induced by
$\rho_{*}$ was not injective then the map $R_{X}/I\rightarrow R_{X^{\prime}}$
is injective, where $I$ is the kernel of $\rho^{\sharp}$. This map has closed
finite fiber since $\rho^{\sharp}$ does, so we have

$\dim R_{X^{\prime}}\leq\dim R_{X}/I<\dim R_{X}.$

This is a contradiction so $\rho^{\sharp}$ is injective, and the same argument
shows the map $R_{X^{\prime}}\rightarrow R_{Y}$ is injective.

∎

###  5.3. Algebraization

Since $\operatorname{PD}_{X}$, $\operatorname{PD}_{X^{\prime}}$, and
$\operatorname{PD}_{Y}$ are all prorepresentable and unobstructed, $X$,
$X^{\prime}$ and $Y$ have compatible universal formal deformations
$\hat{\mathcal{X}}$, $\hat{\mathcal{X}^{\prime}}$, and $\hat{\mathcal{Y}}$
over the formal neighborhoods of zero in their respective tangent spaces. In
particular, the functors
$\operatorname{PD}_{Y}\rightarrow\operatorname{PD}_{X^{\prime}}\rightarrow\operatorname{PD}_{X}$
induce maps
$\hat{\mathcal{Y}}\rightarrow\hat{\mathcal{X}^{\prime}}\rightarrow\hat{\mathcal{X}}$.
Thus we have the following commutative diagram, where $\hat{V}$ denotes the
formal completion of a vector space $V$ at zero.

(12)
${Y}$${{\hat{\mathcal{Y}}}}$${{\widehat{\operatorname{PD}_{Y}(\mathbb{C}[\epsilon])}}}$${{X^{\prime}}}$${{\hat{\mathcal{X}^{\prime}}}}$${{\widehat{\operatorname{PD}_{X^{\prime}}(\mathbb{C}[\epsilon])}}}$${X}$${{\hat{\mathcal{X}}}}$${{\widehat{\operatorname{PD}_{X}(\mathbb{C}[\epsilon])}}}$$\scriptstyle{i_{Y}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\hat{\pi}^{\prime}}$$\scriptstyle{i_{X^{\prime}}}$$\scriptstyle{\rho}$$\scriptstyle{\hat{\rho}}$$\scriptstyle{i_{X}}$$\scriptstyle{\pi^{\prime}_{*}}$$\scriptstyle{\rho_{*}}$

We recall that $X$ has a $\mathbb{C}^{*}$-action satisfying 3.2, which lifts
to $Y$ and thus to $X^{\prime}$ by 3.6. Using the work in [NaFlops, Appendix
A], [NaPD1, 5.4], and [Lo, 2.6], we can use the $\mathbb{C}^{*}$ action to
algebraicize the universal formal deformations above to universal conic
deformations over the full tangent spaces. First we recall the definition of a
universal conic deformation, as described in [Lo, p. 7]. A similar
construction is also described in [MN, Section 2.3].

######  Definition 5.17.

Let $Z$ be a Poisson variety with a $\mathbb{C}^{*}$ action that rescales the
Poisson bracket with weight $-d$ for $d\in{\mathbb{Z}}_{>0}$. Then a graded
deformation of $Z$ over a positively graded ring $B$ is a pair $(Z_{B},i)$
where $Z_{B}$ is a flat Poisson scheme over $B$ equipped with a
$\mathbb{C}^{*}$-action and
$i:Z_{B}\times_{\operatorname{Spec}(B)}0\xrightarrow{\sim}Z$ is an isomorphism
of the central fiber of $Z_{B}$ with $Z$, satisfying:

  1. (1)

The $\mathbb{C}^{*}$-action on $Z_{B}$ transforms the relative Poisson bracket
with weight $-d$.

  2. (2)

The $d$-th power of the scaling action on $\operatorname{Spec}(B)$ lifts to
the $\mathbb{C}^{*}$ action on $Z_{B}$.

  3. (3)

$i$ intertwines the $\mathbb{C}^{*}$ actions on $Z_{B}$ and $Z$.

Furthermore a graded deformation $\mathcal{Z}\rightarrow V$ is a universal
conic deformation if given any graded deformation
$Z_{B}\rightarrow\operatorname{Spec}(B)$ of $Z$, there is a
$\mathbb{C}^{*}$-equivariant map $\operatorname{Spec}(B)\rightarrow V$ and an
isomorphism of graded deformations
$Z_{B}\xrightarrow{\sim}\operatorname{Spec}(B)\times_{V}\mathcal{Z}$.

######  Theorem 5.18.

Diagram (12) algebraicizes to the diagram of universal conic deformations

${Y}$${\mathcal{Y}}$${{\operatorname{PD}_{Y}(\mathbb{C}[\epsilon])}}$${{X^{\prime}}}$${{\mathcal{X}^{\prime}}}$${{\operatorname{PD}_{X^{\prime}}(\mathbb{C}[\epsilon])}}$${X}$${\mathcal{X}}$${{\operatorname{PD}_{X}(\mathbb{C}[\epsilon])}}$$\scriptstyle{i_{Y}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\tilde{\pi}^{\prime}}$$\scriptstyle{i_{X^{\prime}}}$$\scriptstyle{\rho}$$\scriptstyle{\tilde{\rho}}$$\scriptstyle{i_{X}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g}$

###### Proof.

Let
$A:=\operatorname{lim}\Gamma(\hat{\mathcal{X}}_{n},\mathcal{O}_{\hat{\mathcal{X}}_{n}})$
as a topological ring so that $\hat{\mathcal{X}}\cong\operatorname{Spf}(A)$.
Similarly let
$A^{\prime}:=\operatorname{lim}\Gamma(\hat{\mathcal{X}}^{\prime}_{n},\mathcal{O}_{\hat{\mathcal{X}}^{\prime}_{n}})$
and
$B:=\operatorname{lim}\Gamma(\hat{\mathcal{Y}}_{n},\mathcal{O}_{\hat{\mathcal{Y}}_{n}})$
so that we have natural maps
$\hat{\mathcal{X}^{\prime}}\rightarrow\operatorname{Spf}(A^{\prime})$ and
$\hat{\mathcal{Y}}\rightarrow\operatorname{Spf}(B)$.

We can extend diagram (12) to the following diagram of formal spaces.

${{\hat{\mathcal{Y}}}}$${{\hat{\mathcal{X}}^{\prime}_{B}:=\hat{\mathcal{X}^{\prime}}\times_{\operatorname{Spf}(A^{\prime})}\operatorname{Spf}(B)}}$${{\operatorname{Spf}(B)}}$${{\widehat{\operatorname{PD}_{Y}(\mathbb{C}[\epsilon])}\cong\operatorname{Spf}(R_{Y})}}$${{\hat{\mathcal{X}^{\prime}}}}$${{\operatorname{Spf}(A^{\prime})}}$${{\widehat{\operatorname{PD}_{X^{\prime}}(\mathbb{C}[\epsilon])}\cong\operatorname{Spf}(R_{X^{\prime}})}}$${{\hat{\mathcal{X}}\cong\operatorname{Spf}(A)}}$${{\widehat{\operatorname{PD}_{X}(\mathbb{C}[\epsilon])}\cong\operatorname{Spf}(R_{X})}}$$\scriptstyle{\pi^{\prime}_{*}}$$\scriptstyle{\rho_{*}}$$\scriptstyle{\hat{\rho}}$$\scriptstyle{\hat{\rho}_{A^{\prime}}}$$\scriptstyle{\hat{\rho}_{B}}$$\scriptstyle{\hat{\pi}^{\prime}_{B}}$$\scriptstyle{\hat{\pi}^{\prime}}$

We also define $\hat{\pi}_{B}:=\hat{\rho}_{B}\circ\hat{\pi}^{\prime}_{B}$. By
[EGAIII, 5.4.5], we can algebraicize all of the maps above which are
projective maps to the formal spectrum of a ring. Thus we can algebraicize the
maps
$\hat{\rho}_{A^{\prime}}:\hat{\mathcal{X}^{\prime}}\rightarrow\operatorname{Spf}(A^{\prime})$
and
$\hat{\rho}_{B}:\hat{\mathcal{X}}^{\prime}_{B}\rightarrow\operatorname{Spf}(B)$
as well as the composition
$\hat{\pi}_{B}:\hat{\mathcal{Y}}\rightarrow\operatorname{Spf}(B)$. Let the
algebraizations of these maps be
$\mathcal{X}^{\prime}_{A^{\prime}}\rightarrow\operatorname{Spec}(A^{\prime})$,
$\mathcal{X}^{\prime}_{B}:=\mathcal{X}^{\prime}_{A^{\prime}}\times_{\operatorname{Spec}(A^{\prime})}\operatorname{Spec}(B)\rightarrow\operatorname{Spec}(B)$,
and $\mathcal{Y}_{B}\rightarrow\operatorname{Spec}(B)$ respectively. By
[EGAIII, 5.4.1] the map

$\operatorname{Hom}_{\operatorname{Spec}(B)}(\mathcal{Y}_{B},\mathcal{X}^{\prime}_{B})\rightarrow\operatorname{Hom}_{\operatorname{Spf}(B)}(\hat{\mathcal{Y}},\hat{\mathcal{X}}^{\prime}_{B})$

given by mapping a morphism to its formal completion is a bijection. Thus we
can algebraicize the map
$\hat{\mathcal{Y}}\rightarrow\hat{\mathcal{X}}^{\prime}_{B}$ to a map
$\mathcal{Y}_{B}\rightarrow\mathcal{X}^{\prime}_{B}$. Finally, base changing
the map $\operatorname{Spec}(B)\rightarrow\operatorname{Spec}(A)$ clearly
algebraicizes
$\hat{\mathcal{X}}^{\prime}_{B}\rightarrow\hat{\mathcal{X}}^{\prime}$ to a map
$\mathcal{X}^{\prime}_{B}\rightarrow\mathcal{X}^{\prime}_{A^{\prime}}$. Thus
we obtain the following diagram of schemes, where we also define
$\pi_{B}:=\pi^{\prime}_{B}\circ\rho_{B}$.

(13)
${{\mathcal{Y}_{B}}}$${{\mathcal{X}^{\prime}_{B}:=\mathcal{X}^{\prime}_{A^{\prime}}\times_{\operatorname{Spec}(A^{\prime})}\operatorname{Spec}(B)}}$${{\operatorname{Spec}(B)}}$${{\operatorname{Spec}(R_{Y})}}$${{\mathcal{X}^{\prime}_{A^{\prime}}}}$${{\operatorname{Spec}(A^{\prime})}}$${{\operatorname{Spec}(R_{X^{\prime}})}}$${{\operatorname{Spec}(A)}}$${{\operatorname{Spec}(R_{X})}}$$\scriptstyle{\rho_{A^{\prime}}}$$\scriptstyle{\rho_{B}}$$\scriptstyle{\pi^{\prime}_{B}}$

The next step is to use the tools from [NaFlops, Appendix A] to further
algebraicize the above diagram to a new diagram where we have replaced the
rings with the subrings generated by $\mathbb{C}^{*}$-eigenvalues. We recall
from [NaFlops, p. 32] that $R_{Y}\cong\mathbb{C}[[t_{i},...,t_{m}]]$, where
each of the $t_{i}$ are $\mathbb{C}^{*}$-eigenvalues of the same weight. Thus
we can apply [NaFlops, A.2] to $R_{Y}$, as well as $R_{X}$ and
$R_{X^{\prime}}$ since they are subrings of $R_{Y}$. In particular since
$R_{Y}$ is the ring of functions of the formal neighborhood of zero in
$\operatorname{PD}_{Y}(\mathbb{C}[\epsilon])$, [NaFlops, A.2] lets us recover
$\operatorname{PD}_{Y}(\mathbb{C}[\epsilon])$ as the spectrum of the subring
of $R_{Y}$ generated by $\mathbb{C}^{*}$-eigenvalues. Applying the same
argument to $R_{X}$ and $R_{X^{\prime}}$ we algebraicize the right column of
diagram (13) to a sequence of maps

$\operatorname{PD}_{Y}(\mathbb{C}[\epsilon])\rightarrow\operatorname{PD}_{X^{\prime}}(\mathbb{C}[\epsilon])\rightarrow\operatorname{PD}_{X}(\mathbb{C}[\epsilon])$

which are also finite and surjective by [NaFlops, A.4].

Similarly, we can algebraicize the second column of diagram (13) on the right
by replacing $B$, $A^{\prime}$, and $A$ by their subrings generated by
$\mathbb{C}^{*}$-eigenvalues, which we will denote $T$, $S^{\prime}$, and $S$
respectively. Since the maps $\rho_{A^{\prime}}$, $\rho_{B}$, and $\pi_{B}$
are projective maps to affine schemes, we can apply the argument of [NaFlops,
A.5] to algebraicize them. In order to algebraicize the map
$\mathcal{Y}_{B}\rightarrow\mathcal{X}^{\prime}_{B}$, what we do is we show
that since it is projective, $\mathcal{Y}_{B}$ and $\mathcal{X}^{\prime}_{B}$
have compatible line bundles. This allows us to carry out the procedure in
[NaFlops, A.5] simultaneously for both spaces. We explain this in more detail
below.

The map $\mathcal{X}^{\prime}_{B}\rightarrow\operatorname{Spec}(B)$ is
projective, and thus it factors as

$\mathcal{X}^{\prime}_{B}\hookrightarrow\mathbb{P}_{B}^{n}\cong\operatorname{Spec}(B)\times\mathbb{P}^{n}\rightarrow\operatorname{Spec}(B).$

Since $\mathcal{Y}_{B}\rightarrow\mathcal{X}^{\prime}_{B}$ is also projective,
the composition
$\mathcal{Y}_{B}\rightarrow\mathcal{X}^{\prime}_{B}\hookrightarrow\mathbb{P}_{B}^{n}$
is projective, and thus we have a diagram

(14)
${{\mathcal{Y}_{B}}}$${{\mathbb{P}_{\mathbb{P}_{B}^{n}}^{m}\cong\times\mathbb{P}_{B}^{n}\times_{B}\mathbb{P}_{B}^{m}}}$${{\mathbb{P}_{B}^{(n+1)(m+1)-1}}}$${{\mathcal{X}^{\prime}_{B}}}$${{\mathbb{P}_{B}^{n}\cong\operatorname{Spec}(B)\times\mathbb{P}^{n}}}$$\scriptstyle{\pi^{\prime}_{B}}$$\scriptstyle{i_{\mathcal{Y}_{B}}}$$\scriptstyle{i_{\mathcal{X}^{\prime}_{B}}}$$\scriptstyle{\gamma}$$\scriptstyle{s}$

where the final map is given by the Segre embedding. By [Stacks, Lemma 01WD]
the Segre embedding $s$ is induced by the line bundle
$\mathcal{L}:=\mathcal{O}_{\mathbb{P}_{B}^{n}}(1)\boxtimes\mathcal{O}_{\mathbb{P}_{B}^{m}}(1)$
and thus $s\circ i_{\mathcal{Y}}$ is induced by the line bundle
$\mathcal{L}_{\mathcal{Y}_{B}}:=i_{\mathcal{Y}_{B}}^{*}(\mathcal{L})$. At the
same time the embedding $i_{\mathcal{X}^{\prime}_{B}}$ is induced by the line
bundle
$\mathcal{L}_{\mathcal{X}^{\prime}_{B}}:=i_{\mathcal{X}^{\prime}_{B}}^{*}(\mathcal{O}_{\mathbb{P}_{B}^{n}}(1))$.
The natural map
$\mathcal{O}_{\mathbb{P}_{B}^{m}}\rightarrow\mathcal{O}_{\mathbb{P}_{B}^{m}}(1)$
induces a natural map

$\gamma^{*}(\mathcal{O}_{\mathbb{P}_{B}^{n}}(1))\cong\mathcal{O}_{\mathbb{P}_{B}^{n}}(1)\boxtimes\mathcal{O}_{\mathbb{P}_{B}^{m}}\rightarrow\mathcal{L}\cong\mathcal{O}_{\mathbb{P}_{B}^{n}}(1)\boxtimes\mathcal{O}_{\mathbb{P}_{B}^{m}}(1).$

Pulling back along $i_{\mathcal{Y}_{B}}$ we obtain a map

$(i_{\mathcal{Y}_{B}}^{*}\circ\gamma^{*})(\mathcal{O}_{\mathbb{P}_{B}^{n}}(1))\cong(\pi^{\prime*}_{B}\circ
i_{\mathcal{X}^{\prime}_{B}}^{*})(\mathcal{O}_{\mathbb{P}_{B}^{n}}(1))\cong\pi^{\prime*}_{B}\mathcal{L}_{\mathcal{X}^{\prime}_{B}}\rightarrow
i_{\mathcal{Y}_{B}}^{*}\mathcal{L}\cong\mathcal{L}_{\mathcal{Y}_{B}}.$

By adjunction we obtain a natural map
$\mathcal{L}_{\mathcal{X}^{\prime}_{B}}\rightarrow\pi^{\prime}_{B*}\mathcal{L}_{\mathcal{Y}_{B}}$,
which we can push forward along $\rho_{B}$ to obtain a map
$\rho_{B*}\mathcal{L}_{\mathcal{X}^{\prime}_{B}}\rightarrow\pi_{B*}\mathcal{L}_{\mathcal{Y}_{B}}$.
This in turn gives us a map of graded $B$-algebras

(15) $S^{\mathcal{X}^{\prime}_{B}}:=\bigoplus_{i\geq
0}\Gamma(\mathcal{X}^{\prime}_{B},\mathcal{L}_{\mathcal{X}^{\prime}_{B}}^{\otimes
i})\rightarrow S^{\mathcal{Y}_{B}}:=\bigoplus_{i\geq
0}\Gamma(\mathcal{Y}_{B},\mathcal{L}_{\mathcal{Y}_{B}}^{\otimes i}).$

These are precisely the algebras used in the proof of [NaPD1, A.5] to
algebraicize the maps
$\pi_{B}:\mathcal{Y}_{B}\rightarrow\operatorname{Spec}(B)$ and
$\rho_{B}:\mathcal{X}^{\prime}_{B}\rightarrow\operatorname{Spec}(B)$. Thus we
can use the map (15) to carry out the procedure functorially and algebraicize
the maps
$\mathcal{Y}_{B}\rightarrow\mathcal{X}^{\prime}_{B}\rightarrow\operatorname{Spec}(B)$
to a sequence of projective and birational maps

$\mathcal{Y}\rightarrow\mathcal{X}^{\prime}_{T}\rightarrow\operatorname{Spec}(T).$

Combining these algebraizations we obtain a final diagram

(16)
${\mathcal{Y}}$${{\mathcal{X}^{\prime}_{T}:=\mathcal{X}^{\prime}\times_{\operatorname{Spec}(S^{\prime})}\operatorname{Spec}(T)}}$${{\operatorname{Spec}(T)}}$${{\operatorname{PD}_{Y}(\mathbb{C}[\epsilon])}}$${{\mathcal{X}^{\prime}}}$${{\operatorname{Spec}(S^{\prime})}}$${{\operatorname{PD}_{X^{\prime}}(\mathbb{C}[\epsilon])}}$${{\mathcal{X}\cong\operatorname{Spec}(S)}}$${{\operatorname{PD}_{X}(\mathbb{C}[\epsilon])}}$

where the map $\mathcal{X}^{\prime}_{T}\rightarrow\mathcal{X}^{\prime}$ is
defined by base change and the diagonal maps are simply defined by
composition. Finally the method in [NaFlops, Lemma 22] extends to give
$\mathcal{Y}$, $\mathcal{X}^{\prime}$, and $\mathcal{X}$ relative Poisson
structures over $\operatorname{PD}_{X^{\prime}}(\mathbb{C}[\epsilon])$,
$\operatorname{PD}_{X^{\prime}}(\mathbb{C}[\epsilon])$, and
$\operatorname{PD}_{X}(\mathbb{C}[\epsilon])$ respectively. In particular, it
is clear that $B$ has a Poisson structure over $R_{Y}$, and this induces a
Poisson structure of $T$ over
$\mathbb{C}[\operatorname{PD}_{Y}(\mathbb{C}[\epsilon])]$. Similarly, we
obtain Poisson structures of $\operatorname{Spec}(S^{\prime})$ over
$\operatorname{PD}_{X^{\prime}}(\mathbb{C}[\epsilon])$ and
$\mathcal{X}\cong\operatorname{Spec}(S)$ over
$\operatorname{PD}_{X}(\mathbb{C}[\epsilon])$. Because the maps
$\mathcal{Y}\rightarrow\mathcal{X}^{\prime}_{T}\rightarrow\operatorname{Spec}(T)$
and $\mathcal{X}^{\prime}\rightarrow\operatorname{Spec}(S^{\prime})$ are
birational we can can use the relative Poisson structures on
$\operatorname{Spec}(T)$ and $\operatorname{Spec}(S^{\prime})$ to induce
relative Poisson structures on $\mathcal{Y}$, $\mathcal{X}^{\prime}_{T}$, and
$\mathcal{X}^{\prime}$.

Restricting to the right column and the diagonal sequence in diagram (16) we
have the desired algebraization. All that is left is to note that it is shown
in [Lo, 2.6] and [Lo, 2.12] that
$\mathcal{Y}\rightarrow\operatorname{PD}_{Y}(\mathbb{C}[\epsilon])$ and
$\mathcal{X}\rightarrow\operatorname{PD}_{X}(\mathbb{C}[\epsilon])$ are
universal conic deformations, and the argument in [Lo, 2.6] carries over to
show that
$\mathcal{X}^{\prime}\rightarrow\operatorname{PD}_{X^{\prime}}(\mathbb{C}[\epsilon])$
is a universal conic deformation as well.

∎

##  6\. Kleinian Singularities

###  6.1. Structure of Partial Resolutions

Let $\pi:\tilde{S}\rightarrow S$ be the unique crepant resolution of a
Kleinian singularity $S$ corresponding to the Lie algebra $\mathfrak{g}$ with
Cartan subalgebra $\mathfrak{h}$ and Weyl group $W$. Recall that by [NaPD1,
3.1] we have universal deformations of $S$ and $\tilde{S}$, obtained from the
Grothendieck-Springer resolution of $\mathfrak{g}$:

(17)
${{\tilde{S}}}$${{\tilde{\mathcal{S}}}}$${{\operatorname{PD}_{\tilde{S}}(\mathbb{C}[\epsilon])\cong
H^{2}(\tilde{S},\mathbb{C})\cong\mathfrak{h}}}$${S}$${\mathcal{S}}$${{\operatorname{PD}_{S}(\mathbb{C}[\epsilon])\cong\mathfrak{h}/W}}$

where $\mathfrak{h}\rightarrow\mathfrak{h}/W$ is the standard quotient map. In
particular, $\mathcal{S}$ can be constructed explicitly by taking a Slodowy
slice to the subregular nilpotent orbit of $\mathfrak{g}$ and giving it the
relative Poisson structure over $\mathfrak{h}/W$ induced from the Kostant-
Kirillov 2-form. Furthermore we can construct $\tilde{\mathcal{S}}$ by taking
the preimage of $\mathcal{S}$ under the Grothendieck-Springer map
$\tilde{\mathfrak{g}}\rightarrow\mathfrak{g}$. We note that both of these
deformations inherit a graded structure from the Grothendieck-Springer
resolution and thus by [NaPD1, 3.1] they are universal conic deformations.

Additionally, the exceptional fiber $\pi^{-1}(0)$ of $\pi$ is a union of
projective lines in the shape of the Dynkin diagram $D_{S}$ corresponding to
$S$. This gives us a series of isomorphisms

${{\operatorname{Pic}(\tilde{S})}}$${{H^{2}(\tilde{S},\mathbb{C})}}$${{\mathfrak{h}}}$$\scriptstyle{c_{1}}$

If we let $[C_{1}],...,[C_{k}]$ be the exceptional curves corresponding to the
simple roots $\alpha_{1},....,\alpha_{k}$ respectively, then the above
isomorphisms are given by

${{[C_{i}]}}$${{[C_{i}]}}$${{\alpha_{i}}}$

It is proven in [NaPD1, 3.1] that the universal deformation of the germ
$(S,0)$ of $S$ at zero is the same as the universal deformation of $S$.
Similarly, [NaPD1, 3.1] proves that if we let $F:=\pi^{-1}(0)$ then the
universal deformation of $(\tilde{S},F)$ is given by the universal deformation
of $\tilde{S}$ above. In particular this means that we have the following
commutative diagram, where the horizontal maps are given by restriction.

(18)
${{\operatorname{PD}_{\tilde{S}}}}$${{\operatorname{PD}_{(\tilde{S},F)}}}$${{\operatorname{PD}_{S}}}$${{\operatorname{PD}_{(S,0)}}}$$\scriptstyle{\sim}$$\scriptstyle{\pi_{*}}$$\scriptstyle{(\pi|_{(\tilde{S},F)})_{*}}$$\scriptstyle{\sim}$

The above diagram shows that all the data of the Poisson deformations is
determined locally near the singularity. This is useful for the study of
partial resolutions of Kleinian singularities because they all have finitely
many singular points and locally around each singular point they are
isomorphic to Kleinian singularities corresponding to subsets of the Dynkin
diagram of the original Kleinian singularity. We now study these in more
detail.

Let $\rho:S^{\prime}\rightarrow S$ be a partial resolution of $S$. Since
$\tilde{S}$ is the unique crepant resolution of $S$, we have a map
$\pi^{\prime}:\tilde{S}\rightarrow S^{\prime}$. Let $D_{S^{\prime}}\subset
D_{S}$ be the subset of the Dynkin diagram corresponding to the projective
lines contracted by $\pi^{\prime}$.

By [Nak, p. 16-18] the singular locus $\Sigma$ of $S^{\prime}$ is a set of
points in bijection with the connected components $D_{i}$ of $D_{S^{\prime}}$.
Furthermore, locally the singularity near a point $x_{i}\in\Sigma$
corresponding to a component $D_{i}$ is a Kleinian singularity $S_{i}$ of the
same type. For the rest of the section let $\mathfrak{h}_{i}$ and $W_{i}$ be
the Cartan and Weyl group associated to this Kleinian singularity and let
$T^{\prime}_{i}$ denote the germ $(S^{\prime},x_{i})$. Thus we have a
restriction map

(19)
$\prod_{x_{i}\in\Sigma}r_{T^{\prime}_{i}}:\operatorname{PD}_{S^{\prime}}\rightarrow\operatorname{PD}_{(S^{\prime},\Sigma)}=\prod_{x_{i}\in\Sigma}\operatorname{PD}_{T^{\prime}_{i}}\cong\prod_{x_{i}\in\Sigma}\operatorname{PD}_{(S_{i},0)}.$

Additionally if we let $\tilde{F_{i}}:=\pi^{\prime-1}(x_{i})$ and let
$\tilde{T}_{i}$ be the germ $(\tilde{S},\tilde{F_{i}})$, we obtain the diagram

(20)
$\prod_{x_{i}\in\Sigma}r_{\tilde{T}_{i}}:\operatorname{PD}_{\tilde{S}}\rightarrow\operatorname{PD}_{(\tilde{S},\pi^{\prime-1}(\Sigma))}=\prod_{x_{i}\in\Sigma}\operatorname{PD}_{\tilde{T}_{i}}\cong\prod_{x_{i}\in\Sigma}\operatorname{PD}_{(S_{i},F_{i})}$

where $F_{i}$ is the exceptional fiber of the crepant resolution of $S_{i}$.

The kernel of the restriction functor in diagram (19) is the subfunctor
$\operatorname{PD}_{lt,S^{\prime}}$ of $\operatorname{PD}_{S^{\prime}}$ given
by locally trivial deformations while the kernel of restriction functor in
diagram (20) is given by the subfunctor
$\operatorname{PD}_{\tilde{S}}^{\pi^{\prime}}$ of
$\operatorname{PD}_{\tilde{S}}$. This is defined in [NaMori, p. 7], and we
reproduce the definition below:

######  Definition 6.1.

Given a sequence of partial resolutions
$X^{\prime}\xrightarrow{\mu}X^{\prime\prime}\rightarrow X$ of a conical affine
symplectic singularity $X$, the subfunctor
$\operatorname{PD}_{X^{\prime}}^{\mu}\subset\operatorname{PD}_{X^{\prime}}$ is
given by letting $\operatorname{PD}_{X^{\prime}}^{\mu}(A)$ be the set of
Poisson deformations over $A$ which are trivial on the preimage of a
sufficiently small analytic neighborhood of every $x\in X^{\prime\prime}$, for
any $A\in\textrm{Art}_{\mathbb{C}}$.

In particular [NaMori, p. 7] also tells us the following about this functor.

######  Theorem 6.2 ([NaMori, p. 7]).

Using the notation of 6.1, if $X^{\prime}$ is a symplectic resolution of $X$,
then the functor $\operatorname{PD}_{X^{\prime}}^{\mu}$ is prorepresentable
and unobstructed, with tangent space
$\operatorname{PD}_{X^{\prime}}^{\mu}(\mathbb{C}[\epsilon])\cong
H^{2}(X^{\prime\prime},\mathbb{C})$.

Assembling the above information we obtain the following diagram:

(21)
${{\operatorname{PD}_{\tilde{S}}^{\pi^{\prime}}}}$${{\operatorname{PD}_{\tilde{S}}}}$${{\displaystyle\prod_{x_{i}\in\Sigma}\operatorname{PD}_{\tilde{T}_{i}}}}$${{\operatorname{PD}_{lt,S^{\prime}}}}$${{\operatorname{PD}_{S^{\prime}}}}$${{\displaystyle\prod_{x_{i}\in\Sigma}\operatorname{PD}_{T^{\prime}_{i}}}}$$\scriptstyle{\prod
r_{\tilde{T}_{i}}}$$\scriptstyle{\pi_{*}^{\prime}}$$\scriptstyle{\prod(\pi^{\prime}|_{\tilde{T}_{i}})_{*}}$$\scriptstyle{\prod
r_{T^{\prime}_{i}}}$$\scriptstyle{\pi_{*,lt}^{\prime}}$

Recall that for every $A\in\textrm{Art}_{\mathbb{C}}$, each
$\tilde{\mathcal{S}}\in\operatorname{PD}_{\tilde{S}}^{\pi^{\prime}}(A)$ is
trivial on the preimage of some neighborhood of each $x_{i}$. Thus
$\pi_{*}^{\prime}(A)(\tilde{\mathcal{S}})$ is trivial on some neighborhood of
each $x_{i}$ so it is an element of $\operatorname{PD}_{lt,S^{\prime}}(A)$.
Thus for every $A\in\textrm{Art}_{\mathbb{C}}$, $\pi_{*}^{\prime}(A)$ maps
$\operatorname{PD}_{\tilde{S}}^{\pi^{\prime}}(A)$ to
$\operatorname{PD}_{lt,S^{\prime}}(A)$, inducing the map $\pi_{*,lt}^{\prime}$
in the above diagram. We will prove below that this map induces an isomorphism
on first order deformations.

######  Lemma 6.3.

The map $\pi_{*,lt}^{\prime}(\mathbb{C}[\epsilon])$ is an isomorphism.

###### Proof.

For each $x_{i}\in\Sigma$, let $U_{i}$ be a sufficiently small neighborhood of
$x_{i}$ in $S^{\prime}$ and $\tilde{U}_{i}:=\pi^{\prime-1}(U_{i})$.

An element
$\mathcal{S}^{\prime}\in\operatorname{PD}_{lt,S^{\prime}}(\mathbb{C}[\epsilon])$
can be given by the data of locally trivial Poisson deformations
$\mathcal{U}_{i}\in\operatorname{PD}_{lt,U_{i}}(\mathbb{C}[\epsilon])$ and
$\mathcal{S}^{\prime}_{0}\in\operatorname{PD}_{lt,S^{\prime}-\Sigma}(\mathbb{C}[\epsilon])\cong\operatorname{PD}_{S^{\prime}-\Sigma}(\mathbb{C}[\epsilon])$
along with an identification $\Phi_{i}$ of each
$\mathcal{U}_{i}|_{U_{i}-x_{i}}$ with
$\mathcal{S}^{\prime}_{0}|_{U_{i}-x_{i}}$. However the identifications
$\Phi_{i}$ are only defined up to automorphism of $\mathcal{U}_{i}$ and
$\mathcal{S}^{\prime}_{0}$. In particular by normality, every automorphism of
$\mathcal{U}_{i}|_{U_{i}-x_{i}}$ extends to an automorphism of
$\mathcal{U}_{i}$. This means that $\Phi_{i}$ is the trivial gluing datum.

Since every locally trivial deformation of $U_{i}$ is trivial, this is means
that $\mathcal{S}^{\prime}$ is determined by $\mathcal{S}^{\prime}_{0}$, and
in particular an element of
$\operatorname{PD}_{lt,S^{\prime}}(\mathbb{C}[\epsilon])$ is given by the data
of an element of $\operatorname{PD}_{S^{\prime}-\Sigma}(\mathbb{C}[\epsilon])$
which restricts to the trivial deformation on each $U_{i}-x_{i}$.

Similarly, and element
$\tilde{\mathcal{S}}\in\operatorname{PD}_{\tilde{S}}^{\pi^{\prime}}(\mathbb{C}[\epsilon])$
is given by elements
$\tilde{\mathcal{U}_{i}}\in\operatorname{PD}_{\tilde{U}_{i}}^{\pi^{\prime}}(\mathbb{C}[\epsilon])$
and
$\tilde{\mathcal{S}}_{0}\in\operatorname{PD}_{\tilde{S}-\pi^{\prime-1}(\Sigma)}^{\pi^{\prime}}(\mathbb{C}[\epsilon])\cong\operatorname{PD}_{\tilde{S}-\pi^{\prime-1}(\Sigma)}(\mathbb{C}[\epsilon])$
along with an identification $\tilde{\Phi}_{i}$ of each
$\tilde{\mathcal{U}}_{i}|_{\tilde{U}_{i}-\pi^{\prime-1}(x_{i})}$ with
$\tilde{\mathcal{S}}_{0}|_{\tilde{U_{i}}-\pi^{\prime-1}(x_{i})}$. Replacing
each $U_{i}$ with a smaller set if necessary, each $\tilde{\mathcal{U}_{i}}$
must be trivial by the definition of
$\operatorname{PD}_{\tilde{S}}^{\pi^{\prime}}(\mathbb{C}[\epsilon])$. Thus an
element
$\tilde{\mathcal{S}}\in\operatorname{PD}_{\tilde{S}}^{\pi^{\prime}}(\mathbb{C}[\epsilon])$
is equivalent to the data of a deformation
$\tilde{\mathcal{S}}_{0}\in\operatorname{PD}_{\tilde{S}-\pi^{\prime-1}(\Sigma)}(\mathbb{C}[\epsilon])$
along with a trivialization $\tilde{\Phi}_{i}$ over each
$\tilde{U}_{i}^{0}:=\tilde{U}_{i}-\pi^{\prime-1}(x_{i})$. As above, we wish to
show that each $\tilde{\Phi}_{i}$ extends to $\tilde{U}_{i}$.

Let $T:=\operatorname{Spec}(\mathbb{C}[\epsilon])$. By 5.4, recall that the
sheaf of Poisson automorphisms of $\tilde{\mathcal{S}}$ is given by the the
subsheaf of the first cohomology sheaf $P\Theta_{\tilde{S}/T}$ of the Poisson-
Lichnerowicz complex $(\wedge^{\geq 1}\Theta_{\tilde{\mathcal{S}}/T},\delta)$
which vanishes over the central fiber. Since $\tilde{S}$ is smooth, 5.3 tells
us that the symplectic form gives us a quasi-isomorphism $(\wedge^{\geq
1}\Theta_{\tilde{\mathcal{S}}/T},\delta)\cong(\Omega_{\tilde{\mathcal{S}}/T}^{\geq
1},d)$. Thus we can apply the short exact sequence (4). Taking long exact
sequences in cohomology associated to sections over $\tilde{U}_{i}$ and
$\tilde{U}_{i}^{0}$ we obtain the diagram

(22)
${{H^{0}(\tilde{U}_{i},\mathbb{C}[\epsilon])}}$${{H^{0}(\tilde{U}_{i},\mathcal{O}_{\tilde{\mathcal{S}}})}}$${{H^{0}(\tilde{U}_{i},\Omega_{\tilde{\mathcal{S}}/T}^{\geq
1})}}$${{H^{1}(\tilde{U}_{i},\mathbb{C}[\epsilon])}}$${{H^{0}(\tilde{U}_{i}^{0},\mathbb{C}[\epsilon])}}$${{H^{0}(\tilde{U}_{i}^{0},\mathcal{O}_{\tilde{\mathcal{S}}})}}$${{H^{0}(\tilde{U}_{i}^{0},\Omega_{\tilde{\mathcal{S}}/T}^{\geq
1})}}$${{H^{1}(\tilde{U}_{i}^{0},\mathbb{C}[\epsilon])}}$

Both $H^{1}(\tilde{U}_{i},\mathbb{C}[\epsilon])$ and
$H^{1}(\tilde{U}_{i}^{0},\mathbb{C}[\epsilon])$ are clearly trivial, as
$\tilde{U}_{i}$ is homotopic to a bouquet of projective curves and
$\tilde{U}_{i}^{0}$ is isomorphic to a quotient of $\mathbb{C}^{2}$ by a
finite group.

Next we recall the construction of the pushforward of deformations ([Wa],
[Stacks, Lemma 0E3X]). In particular the underlying flat deformation of
$\pi^{\prime}_{*}(\tilde{\mathcal{S}})$ is given by considering
$\mathcal{O}_{\tilde{\mathcal{S}}}$ as a sheaf of
$\mathbb{C}[\epsilon]$-algebras on $\tilde{S}$ and taking the usual
pushforward under $\pi^{\prime}$ to get a sheaf of
$\mathbb{C}[\epsilon]$-algebras on $S^{\prime}$. Thus
$\pi^{\prime}_{*}\mathcal{O}_{\tilde{\mathcal{S}}}\cong\mathcal{O}_{\mathcal{S}^{\prime}}$
as sheaves on $\tilde{S}$ and $S^{\prime}$ respectively. In particular this
means that if a function $f\in
H^{0}(\tilde{U}_{i}^{0},\mathcal{O}_{\tilde{\mathcal{S}}})$ extends uniquely
to a function on $U_{i}$, it must also extend uniquely to a function on
$\tilde{U}_{i}$. As explained before, all of these functions extend uniquely
to $U_{i}$ by normality, thus the restriction map in diagram (22)
$H^{0}(\tilde{U}_{i},\mathcal{O}_{\tilde{\mathcal{S}}})\rightarrow
H^{0}(\tilde{U}_{i}^{0},\mathcal{O}_{\tilde{\mathcal{S}}})$ is an isomorphism.
Finally the map $H^{0}(\tilde{U}_{i},\mathbb{C}[\epsilon])\rightarrow
H^{0}(\tilde{U}_{i}^{0},\mathbb{C}[\epsilon])$ is clearly an isomorphism since
both $\tilde{U}_{i}$ and $\tilde{U}_{i}^{0}$ have a single connected
component. Thus by the commutativity of the diagram, the restriction map
$H^{0}(\tilde{U}_{i},\Omega_{\mathcal{S}/T}^{\geq 1})\rightarrow
H^{0}(\tilde{U}_{i}^{0},\Omega_{\mathcal{S}/T}^{\geq 1})$ is an isomorphism as
well. Furthermore by a spectral sequence argument we have isomorphisms
$H^{0}(\tilde{U}_{i},\Omega_{\mathcal{S}/T}^{\geq 1})\cong
H^{0}(\tilde{U}_{i},P\Theta_{\tilde{S}/T})$ and
$H^{0}(\tilde{U}_{i}^{0},\Omega_{\mathcal{S}/T}^{\geq 1})\cong
H^{0}(\tilde{U}_{i}^{0},P\Theta_{\tilde{S}/T})$. Thus indeed every
automorphism of $\tilde{U}_{i}^{0}$ extends uniquely to an automorphism of
$\tilde{U}_{i}$ and the identifications $\tilde{\Phi_{i}}$ are trivial.

Recapping, this means every
$\tilde{\mathcal{S}}\in\operatorname{PD}_{\tilde{S}}^{\pi^{\prime}}(\mathbb{C}[\epsilon])$
is given uniquely by its restriction $\tilde{\mathcal{S}}_{0}$ to
$\tilde{S}-\pi^{\prime-1}(\Sigma)$. This means that elements of
$\operatorname{PD}_{\tilde{S}}^{\pi^{\prime}}(\mathbb{C}[\epsilon])$ are given
by the data of a deformation
$\tilde{\mathcal{S}}_{0}\in\operatorname{PD}_{\tilde{S}-\pi^{\prime-1}(\Sigma)}(\mathbb{C}[\epsilon])$
whose restriction to each $\tilde{U}_{i}^{0}$ is trivializable. Since
$\pi^{\prime}$ maps $\tilde{S}-\pi^{\prime-1}(\Sigma)$ isomorphically onto
$S^{\prime}-\Sigma$, this means that
$\pi_{*,lt}^{\prime}(\mathbb{C}[\epsilon])$ is injective. By 6.2 and 5.7 we
have
$\operatorname{PD}_{\tilde{S}}^{\pi^{\prime}}(\mathbb{C}[\epsilon])\cong\operatorname{PD}_{lt,S^{\prime}}(\mathbb{C}[\epsilon])\cong
H^{2}(S^{\prime},\mathbb{C})$, so $\pi_{*,lt}^{\prime}(\mathbb{C}[\epsilon])$
is an isomorphism.

∎

We are going to make a slight detour and prove a generalization of Lemma 6.3
for a general conical affine symplectic singularity. We have split up Lemma
6.3 and Lemma 6.4 for the convenience of the reader, since the proof for
Kleinian singularities is simpler.

######  Lemma 6.4.

Let $Y\xrightarrow{\pi^{\prime}}X^{\prime}\xrightarrow{\rho}X^{\prime\prime}$
be a sequence of partial resolutions of a conical affine symplectic
singularity, with $\pi:=\rho\circ\pi^{\prime}$. Furthermore let $U\subset
X^{\prime\prime}$ be the union of the open and codimension two symplectic
leaves and let $U^{\prime}:=\rho^{-1}(U)$ and $\tilde{U}:=\pi^{-1}(U)$. Then
the pushforward maps

$\operatorname{PD}_{\tilde{U}}^{\pi}(\mathbb{C}[\epsilon])\xrightarrow{\pi^{\prime}_{*}(\mathbb{C}[\epsilon])}\operatorname{PD}_{U^{\prime}}^{\rho}(\mathbb{C}[\epsilon])\xrightarrow{\rho_{*}(\mathbb{C}[\epsilon])}\operatorname{PD}_{lt,U}(\mathbb{C}[\epsilon])$

are both isomorphisms.

###### Proof.

We cover the singular locus $\Sigma$ of $U$ by sufficiently small analytic
open sets $U_{j}$ for $j\in[1,n]$, such that they are all isomorphic to open
subsets of $S\times\mathbb{C}^{2n-2}$ for a Kleinian singularity $S$. This is
possible by [NaPD1, 1.3], and Namikawa calls these sets admissible.
Additionally let $U_{0}:=U-\Sigma$ be the smooth locus of $U$ and let
$U^{\prime}_{j}:=\rho^{-1}(U_{j})$ for $j\in[0,n]$. We want to show that every
element of $\operatorname{PD}_{U^{\prime}}^{\rho}(\mathbb{C}[\epsilon])$ is
determined by its restriction to $U^{\prime}_{0}$, and thus show that

$\rho_{*}(\mathbb{C}[\epsilon]):\operatorname{PD}_{U^{\prime}}^{\rho}(\mathbb{C}[\epsilon])\rightarrow\operatorname{PD}_{lt,U}(\mathbb{C}[\epsilon])$

is an injection.

An element of
$\mathcal{U}^{\prime}\in\operatorname{PD}_{U^{\prime}}(\mathbb{C}[\epsilon])$
is given by the data of a first order Poisson deformation
$\mathcal{U}^{\prime}_{j}$ of each $U^{\prime}_{j}$, and compatible
identifications on intersections. Let $\Phi_{jk}$ be the identification
between $\mathcal{U}^{\prime}_{j}$ and $\mathcal{U}^{\prime}_{k}$ on
$U^{\prime}_{j}\cap U^{\prime}_{k}$ for $j<k\in[0,n]$. These identifications
must be compatible on triple intersections, and furthermore they are only
defined up to global automorphism of $\mathcal{U}^{\prime}_{j}$, for
$j\in[0,n]$. In particular the gluing data is best described as the middle
Čech cohomology group of the following chain complex:

(23)
$\bigoplus_{j\in[0,n]}H^{0}(U^{\prime}_{j},PG_{\mathcal{U}^{\prime}/T})\xrightarrow{r_{1}}\bigoplus_{j<k\in[0,n]}H^{0}(U^{\prime}_{j}\cap
U^{\prime}_{k},PG_{\mathcal{U}^{\prime}/T})\xrightarrow{r_{2}}\bigoplus_{j<k<l\in[0,n]}H^{0}(U^{\prime}_{j}\cap
U^{\prime}_{k}\cap U^{\prime}_{l},PG_{\mathcal{U}^{\prime}/T})$

We shall briefly explain why every element of $H^{0}(U^{\prime}_{0}\cap
U^{\prime}_{k},PG_{\mathcal{U}^{\prime}/T})$ extends uniquely to an element of
$H^{0}(U^{\prime}_{k},PG_{\mathcal{U}^{\prime}/T})$, for $k\in[1,n]$. Every
element of $H^{0}(U^{\prime\textrm{sm}}_{k},PG_{\mathcal{U}^{\prime}/T})$
extends uniquely to an element of
$H^{0}(U^{\prime}_{k},PG_{\mathcal{U}^{\prime}/T})$ by normality, where
$U^{\prime\textrm{sm}}_{k}$ is the smooth locus of $U^{\prime}_{k}$. Then
since $U^{\prime\textrm{sm}}_{k}$ and $U^{\prime}_{0}\cap
U^{\prime}_{k}\subset U^{\prime\textrm{sm}}_{k}$ are both smooth we can apply
the argument from Lemma 6.3 using (22) to show that every automorphism of
$U^{\prime}_{0}\cap U^{\prime}_{k}$ extends uniquely to
$U^{\prime\textrm{sm}}_{k}$. Similarly, every element of
$H^{0}(U^{\prime}_{0}\cap U^{\prime}_{k}\cap
U^{\prime}_{l},PG_{\mathcal{U}^{\prime}/T})$ extends uniquely to an element of
$H^{0}(U^{\prime}_{k}\cap U^{\prime}_{l},PG_{\mathcal{U}^{\prime}/T})$, for
$k<l\in[1,n]$. This last claim implies that $r_{2}$ in diagram (23) surjects
onto the component with $j=0$. Since $\dim H^{0}(U^{\prime}_{0}\cap
U^{\prime}_{k}\cap U^{\prime}_{l},PG_{\mathcal{U}^{\prime}/T})=\dim
H^{0}(U^{\prime}_{k}\cap U^{\prime}_{l},PG_{\mathcal{U}^{\prime}/T})$ for all
$k<l\in[1,n]$, this means that

$\dim\ker(r_{2})\leq\sum_{k\in[1,n]}\dim H^{0}(U^{\prime}_{0}\cap
U^{\prime}_{k},PG_{\mathcal{U}^{\prime}/T}).$

However, since every element of $H^{0}(U^{\prime}_{0}\cap
U^{\prime}_{k},PG_{\mathcal{U}^{\prime}/T})$ extends uniquely to an element of
$H^{0}(U^{\prime}_{k},PG_{\mathcal{U}^{\prime}/T})$, for $k\in[1,n]$

$\dim\operatorname{Im}(r_{1})\geq\sum_{k\in[1,n]}\dim
H^{0}(U^{\prime}_{k},PG_{\mathcal{U}^{\prime}/T})=\sum_{k\in[1,n]}\dim
H^{0}(U^{\prime}_{0}\cap U^{\prime}_{k},PG_{\mathcal{U}^{\prime}/T}).$

Thus we have $\ker(r_{2})/\operatorname{Im}(r_{1})=0$ and the gluing data is
always trivial.

Now if
$\mathcal{U}^{\prime}\in\operatorname{PD}_{U^{\prime}}^{\rho}(\mathbb{C}[\epsilon])$
then by definition each $\mathcal{U}^{\prime}_{j}$ must be trivial for
$j\in[1,n]$. Thus an element of
$\operatorname{PD}_{U^{\prime}}^{\rho}(\mathbb{C}[\epsilon])$ is equivalent to
the data of a Poisson deformation $\mathcal{U}^{\prime}_{0}$ of
$U^{\prime}_{0}$ which is trivializable on each of the $U^{\prime}_{0}\cap
U^{\prime}_{j}$, for $j\in[1,n]$. In particular since $\rho$ maps
$U^{\prime}_{0}$ homeomorphically onto $U_{0}$ this tells us that the map
$\rho_{*}:\operatorname{PD}_{U^{\prime}}^{\rho}(\mathbb{C}[\epsilon])\rightarrow\operatorname{PD}_{lt,U}(\mathbb{C}[\epsilon])$
is indeed an injection. Clearly the same argument applies to
$\pi_{*}:\operatorname{PD}_{\tilde{U}}^{\pi}(\mathbb{C}[\epsilon])\rightarrow\operatorname{PD}_{lt,U}(\mathbb{C}[\epsilon])$,
so we obtain a series of injections

$\operatorname{PD}_{\tilde{U}}^{\pi}(\mathbb{C}[\epsilon])\xrightarrow{\pi^{\prime}_{*}}\operatorname{PD}_{U^{\prime}}^{\rho}(\mathbb{C}[\epsilon])\xrightarrow{\rho_{*}}\operatorname{PD}_{lt,U}(\mathbb{C}[\epsilon])$

Applying 5.7 and the argument in [NaMori, p. 7] for 6.2 we have

$\operatorname{PD}_{\tilde{U}}^{\pi}(\mathbb{C}[\epsilon])\cong\operatorname{PD}_{lt,U}(\mathbb{C}[\epsilon])\cong
H^{2}(U,\mathbb{C}),$

so the above injections must both be isomorphisms. ∎

######  Corollary 6.5.

Using the notation from Lemma 6.4, all of the functors in the sequence

$\operatorname{PD}_{\tilde{U}}^{\pi}\xrightarrow{\pi^{\prime}_{*}}\operatorname{PD}_{U^{\prime}}^{\rho}\xrightarrow{\rho_{*}}\operatorname{PD}_{lt,U}$

are prorepresentable and unobstructed and the maps between them are
equivalences.

###### Proof.

The prorepresentability of these functors follows from the work in Section
5.1. Additionally the argument for 6.2 in [NaMori, p. 7] proves that
$\operatorname{PD}_{\tilde{U}}^{\pi}$ is unobstructed. Finally, Lemma 6.4
shows that the maps $\pi^{\prime}_{*}$ and $\rho_{*}$ induce isomorphisms on
first order deformations, and this is only possible if all of the
prorepresenting rings are regular.

∎

Next we are going to study the rows of diagram (21), and in particular the
following result will show that they induce short exact sequences on first
order deformations.

######  Lemma 6.6.

The maps $\prod r_{\tilde{T}_{i}}$ and $\prod r_{T^{\prime}_{i}}$ in diagram
(21) are both surjective.

###### Proof.

We begin with $\prod r_{T^{\prime}_{i}}$. Considering the bottom row of
diagram (21) and taking first order deformations we obtain the exact sequence

$0\rightarrow\operatorname{PD}_{lt,S^{\prime}}(\mathbb{C}[\epsilon])\rightarrow\operatorname{PD}_{S^{\prime}}(\mathbb{C}[\epsilon])\rightarrow\prod_{x_{i}\in\Sigma}\operatorname{PD}_{T^{\prime}_{i}}(\mathbb{C}[\epsilon]).$

By 5.14 we have $\dim\operatorname{PD}_{S^{\prime}}(\mathbb{C}[\epsilon])=\dim
H^{2}(\tilde{S},\mathbb{C})$. Furthermore by 5.8 we have
$\dim\operatorname{PD}_{lt,S^{\prime}}(\mathbb{C}[\epsilon])=\dim
H^{2}(S^{\prime},\mathbb{C})$. Finally by (19), (18), and (17), the dimension
of $\operatorname{PD}_{T^{\prime}_{i}}(\mathbb{C}[\epsilon])$ is equal to the
number of projective lines contracted to $x_{i}$ by $\pi^{\prime}$. The sum of
these dimensions over $x_{i}\in\Sigma$ is the total number of projective lines
contracted by $\pi^{\prime}$, which is precisely $\dim
H^{2}(\tilde{S},\mathbb{C})-\dim H^{2}(S^{\prime},\mathbb{C})$. Thus the above
sequence must be short exact.

Moving on to $\prod r_{\tilde{T}_{i}}$, we consider the top row of diagram
(21) and take first order deformations to obtain

$0\rightarrow\operatorname{PD}_{\tilde{S}}^{\pi^{\prime}}(\mathbb{C}[\epsilon])\rightarrow\operatorname{PD}_{\tilde{S}}(\mathbb{C}[\epsilon])\rightarrow\prod\operatorname{PD}_{\tilde{T}_{i}}(\mathbb{C}[\epsilon]).$

Similarly to above, we have
$\dim\operatorname{PD}_{\tilde{S}}(\mathbb{C}[\epsilon])=\dim
H^{2}(\tilde{S},\mathbb{C})$ by diagram (17) and
$\dim\operatorname{PD}_{\tilde{S}}^{\pi^{\prime}}(\mathbb{C}[\epsilon])=\dim
H^{2}(S^{\prime},\mathbb{C})$ by 6.2. Finally we can also apply (19), (18),
and (17) to show that
$\sum_{x_{i}\in\Sigma}\dim\operatorname{PD}_{\tilde{T}_{i}}(\mathbb{C}[\epsilon])$
is the number of projective line contracted by $\pi^{\prime}$, which is $\dim
H^{2}(\tilde{S},\mathbb{C})-\dim H^{2}(S^{\prime},\mathbb{C})$. Thus the above
sequence is also short exact.

Since all of the functors in (21) are prorepresentable and unobstructed, the
fact that $\prod r_{\tilde{T}_{i}}$ and $\prod r_{T^{\prime}_{i}}$ induce
surjections on first order deformation implies that the functors are
surjective when evaluated on any Artinian ring.

∎

In the next section we are going to need an analogue of diagram (18) and
[NaPD1, 3.1] for partial resolutions of Kleinian singularities, and we now
have all the tools to prove it below.

######  Lemma 6.7.

Let
$\mathcal{S}^{\prime}\rightarrow\operatorname{PD}_{S^{\prime}}(\mathbb{C}[\epsilon])$
be the universal conic deformation of $S^{\prime}$. Then it is also a
universal deformation of the germ $T^{\prime}:=(S^{\prime},\rho^{-1}(0))$.
Additionally
$\mathcal{S}^{\prime}\times\mathbb{C}^{2n-2}\rightarrow\operatorname{PD}_{S^{\prime}}(\mathbb{C}[\epsilon])$
is a universal deformation of the germ
$V^{\prime}:=T^{\prime}\times(\mathbb{C}^{2n-2},0)$.

###### Proof.

First we note that we can apply the arguments in Section 5.1 in the complex
analytic topology to show that $\operatorname{PD}_{T^{\prime}}$ and
$\operatorname{PD}_{V^{\prime}}$ are prorepresentable.

We first prove the lemma for $\operatorname{PD}_{T^{\prime}}$. Clearly
$\mathcal{S}^{\prime}$ is a deformation of the germ $T^{\prime}$ and the
induced map

(24)