# McKay correspondence and tautological bundles on ALE crepant resolutions

Anda Degeratu
University of Freiburg  Thomas Walpuski
Imperial College London

(30 July 2012)

###### Abstract

We establish a higher dimensional analogue of Kronheimer and Nakajima’s
geometric McKay correspondence [Kronheimer1990]*Appendix A. Along the way we
prove a rigidity result for certain Hermitian–Yang–Mills metrics on
tautological bundles on crepant resolutions of ${\bf C}^{3}/G$.

##  1 Introduction

In recent years, the term _McKay correspondence_ has come to describe any
relation between the group theory of a finite subgroup $G$ of
$\mathrm{SL}(n,{\bf C})$ and the topology of a crepant resolution of ${\bf
C}^{n}/G$. Its starting point is Du Val’s observation [DuVal1934] that for a
finite subgroup $G$ of $\mathrm{SL}(2,{\bf C})$ the exceptional divisor $E$ of
the unique crepant resolution $\widetilde{{\bf C}^{2}/G}$ of ${\bf C}^{2}/G$
is a tree of rational curves with self-intersection $-2$ and that these trees
are in one-to-one correspondence with the simply-laced Coxeter–Dynkin
diagrams. Brieskorn [Brieskorn1968] further observed that the the intersection
matrix of the irreducible components of $E$ is exactly the negative Cartan
matrix of the simple Lie algebra associated to the Coxeter–Dynkin tree. This
way a correspondence between the finite subgroups of $\mathrm{SL}(2,{\bf C})$
and the simply-laced Coxeter–Dynkin diagrams was established. This
correspondence was rediscovered from a combinatorial point of view by McKay
[McKay1980], who noticed that for finite subgroups of $\mathrm{SL}(2,{\bf C})$
there exists a bijection between the set ${\rm Irr}(G)$ of irreducible
representations and the set of vertices of the extended Coxeter–Dynkin
diagram. To show this, he relates the matrix
$A=(a_{\rho\sigma})_{\rho,\sigma\in{\rm Irr}(G)}$ corresponding to the
decomposition of the tensor products

$\displaystyle{\bf C}^{2}\otimes\rho=\bigoplus_{\sigma\in{\rm
Irr}(G)}a_{\rho\sigma}\sigma$

to the Cartan matrix $\tilde{C}$ of the extended Coxeter–Dynkin diagram via
$A=2I-\tilde{C}$. These observations together give a complete description of
the topology of $\widetilde{{\bf C}^{2}/G}$ in terms of $G$ and its embedding
in $\mathrm{SL}(2,{\bf C})$, which is now known under the name of the
(classical) _McKay correspondence_ : The irreducible components
$\Sigma_{\rho}$ of $E$ are labelled by the non-trivial irreducible
representations $\rho$ of $G$ and

(1.1) $\displaystyle\Sigma_{\rho}\cdot\Sigma_{\sigma}=-c_{\rho\sigma}.$

However, the role of the irreducible representations of $G$ in the above
description is still elusive. Hence, it is desirable to find a geometrical
interpretation for them.

The first geometric realization of this classical McKay correspondence was
given by Gonzalez-Sprinberg and Verdier [GonzalezSprinberg1983] at the level
of $K$–theory. They showed that the irreducible representation $\rho$ of
$G\subset\mathrm{SL}(2,{\bf C})$ give rise to holomorphic bundles
$\mathcal{R}_{\rho}$ on $\widetilde{{\bf C}^{2}/G}$, called _tautological
bundles_ , and these form a basis in $K$–theory. This way they obtained an
isomorphism between the representation ring of $G$, the $G$–equivariant
$K$–theory of ${\bf C}^{2}$ and the $K$–theory of the crepant resolution.
Moreover, they showed that the first Chern classes of the tautological bundles
form a basis in cohomology, dual to the basis given by the irreducible
components of the exceptional divisor. Using a concrete description of the
crepant resolution, Kapranov and Vasserot [Kapranov2000] lifted this
correspondence to the derived category and showed that it is given by a
Fourier–Mukai transform. Yet another interpretation was given by Kronheimer
and Nakajima [Kronheimer1990]. Using the index theorem, they obtained the
Cartan matrix as the multiplicative matrix of the first Chern classes of the
tautological bundles

(1.2)
$\displaystyle\int_{X}c_{1}(\mathcal{R}_{\rho})c_{1}(\mathcal{R}_{\sigma})=-(C^{-1})_{\rho\sigma},$

see [Kronheimer1990]*Theorem A.7.

The string theory insight of Dixon, Harvey, Vafa and Witten [Dixon1986]
suggested that such McKay correspondences between the topology of a crepant
resolution and the representation theory of the finite group should also hold
in higher dimension. They introduced the orbifold Euler number, a number
expressed in terms of the group $G$ and its action on ${\bf C}^{n}$, and
conjectured that it is equal to the Euler characteristic of any crepant
resolution of ${\bf C}^{n}/G$, whenever such a resolution exists. This has
sparked an enormous body of work in mathematics including among other results
the case-by-case proof of this conjecture for finite subgroups of
$\mathrm{SL}(3,{\bf C})$ of Ito [Ito1995], Markushevich [Markushevich1997] and
Roan [Roan1996], its refinement for Betti and Hodge numbers of Batyrev and
Dais [Batyrev1996], the notion of orbifold cohomology of Chen and Ruan
[Chen2004], and the new versions of the McKay correspondence of Ito and Reid
[Ito1994].

Recall that the existence and uniqueness of a crepant resolution depends
crucially on the dimension: In dimension two and three a crepant resolution
always exists; however, it is unique only in dimension two, as in dimension
three any flop gives another one. In higher dimensions crepant resolutions
need not exist at all. Moreover, above dimension two the link between finite
groups and Coxeter–Dynkin diagrams is missing. Hence, the insight gained from
the geometrical interpretation of the classical McKay correspondence is of key
importance to understanding its extensions to higher dimensions. In the case
of finite subgroups of $\mathrm{SL}(3,{\bf C})$, the $K$–theory interpretation
was carried out by Ito and Nakajima [Ito2000], while the derived category
equivalence was established by Bridgeland, King and Reid [Bridgeland2001] and
Craw and Ishii [Craw2004]. All these give additive information about the
cohomology of the crepant resolution and they are invariant under its choice.
In this work, we obtain the counterpart of Kronheimer and Nakajima’s result
and make the first steps towards understanding the multiplicative structure of
the cohomology of crepant resolutions of ${\bf C}^{3}/G$.

We consider the case of a finite subgroup $G$ of $\mathrm{SL}(3,{\bf C})$
which acts freely on ${\bf C}^{3}\setminus\\{0\\}$. This forces $G$ to be
cyclic of prime order. Let $X$ be a projective crepant resolution of ${\bf
C}^{3}/G$. From the work of Craw and Ishii, we know that $X$ is a moduli space
$\mathscr{M}_{\theta}$ of $G$–constelations (a sheaf theoretic generalisation
of the notion of $G$–orbit) which are stable with respect the parameter
$\theta\in\Theta_{\bf Q}$, see Definition 2.3. As such, $X$ naturally comes
with a collection of holomorphic bundles $\mathcal{R}_{\rho}$, one for each
representation $\rho$ of $G$. The bundles corresponding to the irreducible
representations of $G$ form a basis in $K$–theory, and hence their Chern
characters form a basis in $H^{*}(X,{\bf R})$. Using the natural action of $G$
on ${\bf C}^{3}$, one can define a matrix $C$, see (7.5), which is the higher
dimensional analogue of the Cartan matrix associated to the finite subgroups
of $\mathrm{SL}(2,{\bf C})$ from the matrix $A$.

######  Theorem 1.3.

Let $G$ be a finite subgroup of $\mathrm{SL}(3,{\bf C})$ which acts with an
isolated fixed point on ${\bf C}^{3}$. Let $X$ be a projective crepant
resolution of ${\bf C}^{3}/G$. Then

(1.4) $\displaystyle\int_{X}\widetilde{\rm
ch}(\mathcal{R}_{\rho})\wedge\widetilde{\rm
ch}(\mathcal{R}_{\sigma}^{*})=-\left(C^{-1}\right)_{\rho\sigma},$

for all nontrivial $\rho,\sigma\in{\rm Irr}(G)$. Here $\widetilde{\rm
ch}:={\rm ch}-\mathop{\mathrm{rk}}\nolimits$ is the reduced Chern character.

######  Remark 1.5.

This result has first appeared in a preprint of the first named author
[Degeratu2003], but it had a gap in its proof.

Formula (1.4) should be viewed as an analogue of Kronheimer’s and Nakajima’s
formula (1.2). It determines a part of the multiplicative structure in
cohomology of all the projective crepant resolutions of ${\bf C}^{3}/G$ that
only depends on the finite group $G$ and its embedding in $\mathrm{SL}(3,{\bf
C})$. Thus we have found a new McKay correspondence. More precisely, note that
the left-hand side of (1.4) can be written as

(1.6)
$\displaystyle\int_{X}c_{1}(\mathcal{R}_{\rho})^{2}c_{1}(\mathcal{R}_{\sigma})-c_{1}(\mathcal{R}_{\rho})c_{1}(\mathcal{R}_{\sigma})^{2}.$

Using the Chern-Weil theory,
$c_{1}(\mathcal{R}_{\rho})=\frac{1}{2\pi}F_{A_{\theta,\rho}}$ where
$F_{A_{\theta,\rho}}$ is the curvature of the natural $(1,1)$–connection
induced on $\mathcal{R}_{\rho}$. By Theorem 3.9, $F_{A_{\theta,\rho}}$ defines
a class in the weighted $L^{2}$–cohomology of $X$, which can be identified,
using a result of Hausel, Hunsicker and Mazzeo [Hausel2004]*Corollary 4, with
$H^{k}_{c}(X,{\bf C})$. Moreover, since on the crepant resolution $X$, we have
$H^{2}_{c}(X,{\bf C})\cong H^{1,1}(X,{\bf C})$, with the last generated by the
Poincaré duals of the exceptional divisors, the class corresponding to
$c_{1}(\mathcal{R}_{\rho})$ can be represented by compactly supported
$(1,1)$–forms. Hence, (1.4) describes the part of triple pairing

(1.7) $\displaystyle\int_{X}\mskip 0.5mu\colon\thinspace H^{1,1}(X,{\bf
C})\times H^{1,1}(X,{\bf C})\times H^{1,1}(X,{\bf C})\to{\bf C},$

mapping $(\alpha,\beta,\gamma)$ to $\int_{X}\alpha\wedge\beta\wedge\gamma$
where $\gamma=\alpha-\beta$ and
$\alpha,\beta\in\\{c_{1}(\mathcal{R}_{\rho})\mid\rho\in{\rm Irr}(G)\\}$. The
issues of completely determinig the multiplicative structure in cohomology as
well as of describing the entire part which is invariant under the choice of
the crepant resolution are still open and will be investigated in future work.

We prove Theorem 1.3 in same manner as Kronheimer and Nakajima by studying the
index formula for certain Dirac operators whose index we show to be zero. The
condition that the group $G$ acts with an isolated fixed point on ${\bf
C}^{3}$ ensures that any crepant resolution of the orbifold ${\bf C}^{3}/G$ is
an ALE space, in which case the index of the Dirac operator is given by the
Atiyah–Patodi–Singer (APS) index theorem [Atiyah1975a]. For all the other
finite subgroups of $\mathrm{SL}(3,{\bf C})$, the geometry of the crepant
resolution is that of a QALE manifold, as introduced by Joyce [Joyce2000]. The
generalization of the APS index theorem to QALE manifolds is work in progress
by the first named author [Degeratu2012a].

In the course of proving Theorem 1.3, we also gain more insight into the
geometry of the crepant resolutions $X=\mathscr{M}_{\theta}$. Since
$\mathscr{M}_{\theta}$ is constructed via geometric invariant theory (GIT), it
follows that the complex manifold $M_{\theta}$ underlying
$\mathscr{M}_{\theta}$ can be constructed as a Kähler quotient. This
construction, which we present in Section 3, is the generalisation to higher
dimension of Kronheimer’s construction of ALE gravitational instantons
[Kronheimer1989] and was first carried out by Sardo-Infirri
[SardoInfirri1996]. One of its consequences is that $M_{\theta}$ carries a
natural Kähler metric $g_{\theta}$. Our second main result establishes the
existence of special metrics on $M_{\theta}$ and its tautological bundles
$\mathcal{R}_{\rho}$.

######  Theorem 1.8.

Let $G$ be a finite subgroup of $\mathrm{SL}(3,{\bf C})$ acting with an
isolated fixed point on ${\bf C}^{3}$. Let $\theta\in\Theta_{{\bf Q}}$ be a
generic rational stability parameter. Then

  1. 1.

$M_{\theta}$ carries a ALE Ricci-flat Kähler metric $g_{\theta,\rm RF}$, which
is in the same Kähler class as $g_{\theta}$.

  2. 2.

for each $\rho\in{\rm Irr}(G)$, the holomorphic tautological bundle
$\mathcal{R}_{\rho}$ carries an infinitesimally rigid asymptotically flat
Hermitian–Yang–Mills metric.

The existence of the Ricci-flat Kähler metric on $M_{\theta}$ is a consequence
of Joyce’s proof of the Calabi conjecture for ALE crepant resolutions
[Joyce2000]*Section 8, while the existence of HYM metric is a consequence of
the properties of the Laplace operator on ALE manifolds. The difficulty lies
in proving the infinitesimal rigidity statement. In fact, the key ingredient
for proving both Theorem 1.3 and the rigidity of the HYM metric in Theorem 1.8
is the vanishing result in Lemma 5.1.

The results here are of interest in the context of higher dimensional gauge
theory. For example, one can use them to extend the second named author’s
construction of ${\rm G}_{2}$–instantons on generalized Kummer constructions
[Walpuski2011] to ${\rm G}_{2}$–manifolds arising from ${\rm G}_{2}$–orbifolds
with codimension $6$ singularities.

The paper is organised as follows. In Section 2 we briefly recall the
construction of crepant resolutions as moduli spaces of $G$–constelations,
introduce the Fourier–Mukai transform and collect the results of Bridgeland,
King and Reid [Bridgeland2001] and Craw and Ishii [Craw2004] that are relevant
for our work. In Section 3 we present the Kähler counterpart of the
construction of crepant resolutions and use it to describe its geometry. In
Section 4 we prove the existence part of Thereom 1.8, while in Section 5 we
prove the rigidity statement. Section 6 introduces the Dirac operator on
crepant resolutions and establishes its properties. We finish with the proof
of our main Theorem 1.3 in Section 7.

### Acknowledgements

The program of studying the geometry and topology of crepant resolutions using
methods from index theory was started by A.D. during her PhD work at MIT under
the supervision of Tom Mrowka, to whom she thanks for his support. A.D. would
also like to thank Tamás Hausel, Rafe Mazzeo and Mark Stern for useful
conversations about different aspects of this work.

Parts of this article are the outcome of work undertaken by T.W. for his PhD
thesis at Imperial College London, funded by the European Commission. T.W.
would like to thank his supervisor Simon Donaldson for his support.

##  2 Moduli spaces of $G$–constellations

Let $G$ be a finite subgroup of $\mathrm{SL}(3,{\bf C})$. We denote by ${\rm
Irr}(G)$ the set of irreducible representations, by $\mathrm{Rep}(G)$ the
representation ring, and by $R$ the regular representation of $G$.

######  Definition 2.1.

A _$G$ –sheaf_ on ${\bf C}^{3}$ is a coherent sheaf $\mathscr{F}$ together
with an action of $G$ which is equivariant with respect to the action of $G$
on ${\bf C}^{3}$. A _$G$ –constellation_ on ${\bf C}^{3}$ is a $G$–sheaf
$\mathscr{F}$ such that $H^{0}({\bf C}^{3},\mathscr{F})\cong R$ as
$G$–modules. An _isomorphism_ of $G$–constellations is an isomorphism of
sheaves intertwining the $G$–actions.

######  Remark 2.2.

The (set theoretic) support of a $G$–constellation is a $G$–orbit in ${\bf
C}^{3}$ and can thus be thought of as a point in ${\bf C}^{3}/G$. From this
point of view one can think of $G$–constellations as a sheaf-theoretic
generalisation of the notion of $G$–orbit.

######  Definition 2.3.

The set

$\displaystyle\Theta:=\left\\{\theta\in{\mathrm{Hom}}_{\bf
Z}(\mathrm{Rep}(G),{\bf Z}):\theta(R)=0\right\\}$

is called the _space of integral stability parameters_. The sets $\Theta_{\bf
Q}:=\Theta\otimes_{\bf Z}{\bf Q}$ and $\Theta_{\bf R}:=\Theta\otimes_{\bf
Z}{\bf Q}$ are called the _space of rational stability parameters_ and the
_space of real stability parameters_ , respectively. Given
$\theta\in\Theta_{\bf R}$, a $G$–constellation $\mathscr{F}$ is called
_$\theta$ –stable_ (resp.  _$\theta$ –semi-stable_) if each non-trivial proper
$G$–subsheaf $\mathscr{E}\subset\mathscr{F}$ satisfies
$\theta(H^{0}(\mathscr{E}))>0$ (resp. $\theta(H^{0}(\mathscr{E}))\geq 0$). A
real stability parameter $\theta\in\Theta_{\bf R}$ is called _generic_ , if
there is no non-trivial proper subrepresentation $S\subset R$ such that
$\theta(S)=0$.

The space of generic stability parameters is dense in $\Theta_{\bf R}$. If
$\theta\in\Theta_{{\bf R}}$ is generic, then every $\theta$–semi-stable
$G$–constellation is $\theta$–stable, c.f. [Craw2004]*Section 2.2.

######  Theorem 2.4 (Craw and Ishii [Craw2004]*Section 2.1).

If $\theta\in\Theta_{\bf Q}$ is generic, then there exists a smooth fine
moduli space $\mathscr{M}_{\theta}$ of $\theta$–stable $G$–constellations on
${\bf C}^{3}$. Moreover, associated to each representation $\rho$ of $G$ there
is a locally free sheaf $\mathscr{R}_{\rho}$ on $\mathscr{M}_{\theta}$. If
$\rho$ and $\sigma$ are two representations of $G$, then
$\mathscr{R}_{\rho\oplus\sigma}=\mathscr{R}_{\rho}\oplus\mathscr{R}_{\sigma}$.

###### Sketch of the Proof.

The construction of $\mathscr{M}_{\theta}$ uses GIT and is based on ideas of
King [King1994] and Sardo-Infirri [SardoInfirri1996].

A $G$–constellation on ${\bf C}^{3}$ is a $G$–equivariant
$\mathrm{Sym}^{\bullet}({\bf C}^{3})^{*}$–module structure on $R$, i.e., a
$G$–equivariant homomorphism $\mathrm{Sym}^{\bullet}({\bf
C}^{3})^{*}\to{\mathrm{End}}(R)$. Hence, to each point $B$ in

(2.5) $\displaystyle N:=\left\\{B\in\left({\mathrm{End}}(R)\otimes{\bf
C}^{3}\right)^{G}:[B\wedge B]=0\in{\mathrm{End}}(R)\otimes\Lambda^{2}{\bf
C}^{3}\right\\},$

we can associate the $G$–constellation defined by
$p\in\mathrm{Sym}^{\bullet}({\bf C}^{3})^{*}\mapsto p(B)\in{\mathrm{End}}(R)$.
In fact, every $G$–constellation arises this way. Furthermore, two points in
$N$ yield isomorphic $G$–constellations if and only if they are related by a
$G$–equivariant automorphism of $R$, i.e., an element of $\mathrm{GL}(R)^{G}$.
Since $R=\bigoplus_{\rho\in{\rm Irr}(G)}{\bf C}^{\dim\rho}\otimes\rho$,
Schur’s lemma gives

$\displaystyle\mathrm{GL}(R)^{G}=\prod_{\rho\in{\rm Irr}(G)}\mathrm{GL}({\bf
C}^{\dim\rho}).$

Since the diagonal ${\bf C}^{*}\subset\mathrm{GL}(R)^{G}$ acts trivially on
$N$, the action of $\mathrm{GL}(R)^{G}$ descends to an action of ${\bf
P}\mathrm{GL}(R)^{G}$. An integral stability parameter $\theta\in\Theta$ thus
determines a character $\chi_{\theta}\mskip 0.5mu\colon\thinspace{\bf
P}\mathrm{GL}(R)^{G}\to{\bf C}^{*}$ defined by

(2.6)
$\displaystyle\chi_{\theta}([g])=\chi_{\theta}([(g_{\rho})]):=\prod_{\rho\in{\rm
Irr}(G)}\mathop{\mathrm{missing}}{det}\nolimits(g_{\rho})^{\theta(\rho)}.$

King [King1994]*Proposition 3.1 proved that an element of $N$ is stable (resp.
semi-stable) in the sense of GIT with respect to $\chi_{\theta}$ if and only
if the corresponding $G$–constellation is $\theta$–stable (resp.
$\theta$–semi-stable). Let $N^{s}_{\theta}$ (resp. $N^{ss}_{\theta}$) be the
set GIT (semi-)stable points with respect to $\chi_{\theta}$ in $N$ and let

(2.7) $\displaystyle\mathscr{M}_{\theta}:=N^{s}_{\theta}/{\bf
P}\mathrm{GL}(R)^{G}$

be the corresponding GIT quotient. As schemes,
$\mathscr{M}_{k\theta}=\mathscr{M}_{\theta}$ for any $k\in{\bf N}$ and
therefore, the above construction extends to rational stability parameters
$\theta\in\Theta_{\bf Q}$ as well.

To see that $\mathscr{M}_{\theta}$ is indeed a fine moduli space, we construct
a _universal $G$–constellation_ $\mathscr{U}_{\theta}$ on
$\mathscr{M}_{\theta}\times{\bf C}^{3}$. For this, we identify

$\displaystyle{\bf P}\mathrm{GL}(R)^{G}\cong\prod_{\rho\in{\rm
Irr}_{0}(G)}\mathrm{GL}\left({\bf C}^{\dim\rho}\right),$

where ${\rm Irr}_{0}(G)$ is the set of _non-trivial_ irreducible
representations of $G$. In this way, ${\bf P}\mathrm{GL}(R)^{G}$ acts on $R$.
This makes $R\otimes\mathscr{O}_{N}$ into a ${\bf
P}\mathrm{GL}(R)^{G}$–equivariant sheaf on $N$. We denote its descend to
$\mathscr{M}_{\theta}$ by $\mathscr{R}$. Since the universal morphism
$R\otimes\mathscr{O}_{N}\to{\bf C}^{3}\otimes R\otimes\mathscr{O}_{N}$ is
${\bf P}\mathrm{GL}(R)^{G}$–equivariant, it descends to a universal morphism
$\mathscr{R}\to{\bf C}^{3}\otimes\mathscr{R}$ on $\mathscr{M}_{\theta}$. This
determines the universal $G$–constellation $\mathscr{U}_{\theta}$ on
$\mathscr{M}_{\theta}\times{\bf C}^{3}$. Concretely, $\mathscr{U}_{\theta}$ is
the sheaf obtained by pulling back $\mathscr{R}$ to
$\mathscr{M}_{\theta}\times{\bf C}^{3}$ with the action of $\mathscr{O}_{{\bf
C}^{3}}=\mathrm{Sym}^{\bullet}({\bf C}^{3})^{*}$ prescribed by the universal
morphism.

Let now $\rho\mskip 0.5mu\colon\thinspace G\to\mathrm{Aut}(R_{\rho})$ be a
representation of $G$. Since by the above identification ${\bf
P}\mathrm{GL}(R)^{G}$ acts on $R_{\rho}$, the construction that associates to
$R$ the sheaf $\mathscr{R}$ can be carried on for $R_{\rho}$, giving rise to
the sheaf $\mathscr{R}_{\rho}$. It is then clear that
$\mathscr{R}_{\rho\oplus\sigma}=\mathscr{R}_{\rho}\oplus\mathscr{R}_{\sigma}$.
∎

To obtain further insight into the spaces $\mathscr{M}_{\theta}$, it is
helpful to make use of the language of derived categories. We first recall the
_bounded derived category_ $D(\mathscr{A})$ associated with an abelian
category $\mathscr{A}$. For details we refer the reader to Bühler’s notes
[Buehler2007], as well as Thomas’ article [Thomas2001] and Huybrechts’ book
[Huybrechts2006], both of which underline the importance of derived categories
of coherent sheaves in algebraic geometry. Roughly speaking, $D(\mathscr{A})$
is obtained from the category of bounded chain complexes in $\mathscr{A}$ by
formally inverting quasi-isomorphisms. If $A,B\in\mathscr{A}$ are considered
as bounded chain complexes concentrated in degree zero, then
${\mathrm{Hom}}_{D(\mathscr{A})}(A,B)$ is a complex whose cohomology computes
$\mathrm{Ext}^{\bullet}(A,B)$, that is,

(2.8) $\displaystyle
H^{\bullet}({\mathrm{Hom}}_{D(\mathscr{A})}(A,B))=\mathrm{Ext}^{\bullet}(A,B).$

If $\mathscr{B}$ is another abelian category and $f\mskip
0.5mu\colon\thinspace\mathscr{A}\to\mathscr{B}$ is a left-exact functor, then
one associates to it a _right derived functor_ ${\bf R}f\mskip
0.5mu\colon\thinspace D(\mathscr{A})\to D(\mathscr{B})$. If $A\in\mathscr{A}$
is considered as a bounded chain complex concentrated in degree zero, then
${\bf R}f(A)$ is a complex which computes $R^{\bullet}f(A)$, that is,

(2.9) $\displaystyle H^{\bullet}({\bf R}f(A))=R^{\bullet}f(A).$

A similar construction holds for $g\mskip
0.5mu\colon\thinspace\mathscr{A}\to\mathscr{B}$ a right-exact functor,
producing a left derived functor ${\bf L}g.$ When working with derived
categories, it is customary to write $f$ and $g$ instead of ${\bf R}f$ and
${\bf L}g$. We follow this custom in this article.

An important example of such a derived category is $D({\rm Coh}(X))$, the
bounded derived category of coherent sheaves $D({\rm Coh}(X))$ over a scheme
$X$. If $X$ and $Y$ are two schemes and $K\in{\rm Coh}(X\times Y)$ is a
coherent sheaf, then the _Fourier–Mukai transform with kernel $K$_ is the
functor $\Phi_{K}\mskip 0.5mu\colon\thinspace D({\rm Coh}(X))\to D({\rm
Coh}(Y))$ defined by

$\displaystyle\Phi_{K}(-):=(p_{2})_{*}(p_{1}^{*}-\otimes K).$

Here $p_{1}^{*},(p_{2})_{*}$ and $\otimes$ are taken in the derived sense,
with $p_{1}$ and $p_{2}$ denoting the projections from $X\times Y$ to $X$ and
$Y$ respectively. If $f\mskip 0.5mu\colon\thinspace X\to Y$ is a morphism and
$\mathscr{O}_{\Gamma}$ denotes the structure sheaf of its graph $\Gamma\subset
X\times Y$, then $\Phi_{\mathscr{O}_{\Gamma}}$ is $f_{*}$.

In our context, we denote by $D(\mathscr{M}_{\theta})$ the bounded derived
category of coherent sheaves on $\mathscr{M}_{\theta}$ and by $D^{G}({\bf
C}^{3})$ the bounded derived category of $G$–equivariant coherent sheaves on
${\bf C}^{3}$, which is the same as the bounded derived category $D([{\bf
C}^{3}/G])$ of coherent sheaves on the stack $[{\bf C}^{3}/G]$. One of the key
ideas of Bridgeland, King and Reid [Bridgeland2001] is to introduce the
Fourier–Mukai transform $\Phi_{\theta}\mskip 0.5mu\colon\thinspace
D(\mathscr{M}_{\theta})\to D^{G}({\bf C}^{3})$ with kernel given by the
universal $G$–constelation $\mathscr{U}_{\theta}$

(2.10)
$\displaystyle\Phi_{\theta}(-)=q_{*}(p^{*}(-\otimes\rho_{0})\otimes\mathscr{U}_{\theta})$

to study crepant resolutions. Here $p\mskip
0.5mu\colon\thinspace\mathscr{M}_{\theta}\times{\bf
C}^{3}\to\mathscr{M}_{\theta}$ and $q\mskip
0.5mu\colon\thinspace\mathscr{M}_{\theta}\times{\bf C}^{3}\to{\bf C}^{3}$ are
the two projections. Using it we have the following descriptions of the
projective crepant resolutions of ${\bf C}^{3}/G$:

######  Theorem 2.11 (Craw and Ishii [Craw2004]*Proposition 2.2 and Theorem
2.5).

For each $\theta\in\Theta_{{\bf Q}}$, there exists a morphism
$\pi_{\theta}\mskip 0.5mu\colon\thinspace\mathscr{M}_{\theta}\to{\bf C}^{3}/G$
which associates to each isomorphism class of $G$–constellations its support.
If $\theta\in\Theta_{{\bf Q}}$ is generic, then $\pi_{\theta}$ is a projective
crepant resolution and the Fourier-Mukai transform $\Phi_{\theta}$ is an
equivalence of derived categories.

######  Remark 2.12.

Bridgeland, King and Reid [Bridgeland2001] first proved this result for
Nakamura’s $G$–Hilbert scheme. Craw and Ishii observed that the proof carries
over to the more general moduli spaces of $G$–constellations. The fact that
$\pi_{\theta}$ is a crepant resolution is essentially a consequence of
$\Phi_{\theta}$ being an equivalence of derived category, in which case there
exists a categorical criterion for a resolution to be crepant
[Bridgeland2001]*Lemma 3.1.

######  Theorem 2.13 (Craw and Ishii [Craw2004]*Theorem 1.1).

If $G$ is an abelian subgroup of $\mathrm{SL}(3,{\bf C})$, then every
projective crepant resolution of ${\bf C}^{3}/G$ is a moduli space of
$\theta$–stable $G$–constellations for some generic $\theta\in\Theta_{{\bf
Q}}$.

##  3 $\mathscr{M}_{\theta}$ via Kähler reduction

We now approach the previous discussion from the Kähler point of view. There
is no loss in assuming that the finite group $G\subset SL(3,{\bf C})$
preserves the standard Hermitian metric on ${\bf C}^{3}$, that is, $G\subset
SU(3)$. Moreover, we fix a $G$–invariant Hermitian metric on $R$. In this set-
up, $N$ defined in (2.5) is a cone in the Hermitian vector space
$({\mathrm{End}}(R)\otimes{\bf C}^{3})^{G}$. The space
$({\mathrm{End}}(R)\otimes{\bf C}^{3})^{G}$ is naturally a Kähler manifold
with Kähler form

$\displaystyle\omega(B,C):={\rm
Im}\sum_{\alpha=1}^{3}\mathop{\mathrm{tr}}\nolimits(B_{\alpha}C_{\alpha}^{*})=\sum_{\alpha=1}^{3}\frac{1}{2i}\mathop{\mathrm{tr}}\nolimits(B_{\alpha}C_{\alpha}^{*}-B_{\alpha}^{*}C_{\alpha}).$

Here we identify $B\in({\mathrm{End}}(R)\otimes{\bf C}^{3})^{G}$ with a triple
$(B_{1},B_{2},B_{3})$ of endomorphisms of $R$.

######  Proposition 3.1.

The action of ${{\bf P}{\rm U}}(R)^{G}$ on $({\mathrm{End}}(R)\otimes{\bf
C}^{3})^{G}$ by conjugation is Hamiltonian with moment map $\mu\mskip
0.5mu\colon\thinspace({\mathrm{End}}(R)\otimes{\bf
C}^{3})^{G}\to\left(\mathfrak{pu}(R)^{G}\right)^{*}$ given by

$\displaystyle\left<\mu(B),\xi\right>=\sum_{\alpha}\frac{1}{2i}\mathop{\mathrm{tr}}\nolimits(\xi[B_{\alpha},B_{\alpha}^{*}]).$

###### Proof.

It is enough to prove this for the action of $U(R)^{G}$. If
$\xi\in{\mathfrak{u}}(R)^{G}$, then the corresponding vector field $X_{\xi}$
on $({\mathrm{End}}(R)\otimes{\bf C}^{3})^{G}$ is given by
$X_{\xi}(B)=[\xi,B].$ Thus

$\displaystyle i(X_{\xi})\omega(\hat{B})$
$\displaystyle=\sum_{\alpha}\frac{1}{2i}\mathop{\mathrm{tr}}\nolimits\left([\xi,B_{\alpha}]\hat{B}_{\alpha}^{*}-[\xi,B_{\alpha}]^{*}\hat{B}_{\alpha}\right)$
$\displaystyle=\sum_{\alpha}\frac{1}{2i}\mathop{\mathrm{tr}}\nolimits\left(\xi([B_{\alpha},\hat{B}_{\alpha}^{*}]+[\hat{B}_{\alpha},B_{\alpha}^{*}])\right)=\left<{\rm
d}\mu(B)\hat{B},\xi\right>.$

∎

For $\theta\in\Theta_{\bf R}$, we define
$\zeta_{\theta}\in\left(\mathfrak{pu}(R)^{G}\right)^{*}$ by

(3.2) $\displaystyle\zeta_{\theta}(\xi):=-\sum_{\rho\in{\rm
Irr}(G)}i\theta(\rho)\mathop{\mathrm{tr}}\nolimits(\xi\cdot\pi_{\rho})$

for all $\xi\in\mathfrak{pu}(R)^{G}$. Here $\pi_{\rho}:R\to{\bf
C}^{\dim\rho}\otimes R_{\rho}$ is the projection onto the $\rho$-isotypical
component of the the regular representation, and $\xi\cdot\pi_{\rho}$ is
thought of as an element in ${\mathrm{End}}(R)$. Note that if $\theta$ is
integral, then $\zeta_{\theta}=-i{\rm
d}\chi_{\theta}\in\left(\mathfrak{pu}(R)^{G}\right)^{*}$, with $\chi_{\theta}$
the character associated to $\theta$ as defined in (2.6). Moreover, since the
centre of ${\mathfrak{u}}(R)^{G}$ is spanned by $\\{i\pi_{\rho}\mid\rho\in{\rm
Irr}(G)\\}$, we can identify $\Theta_{\bf R}$ with the centre of
$(\mathfrak{pu}(R)^{G})^{*}$ via $\theta\mapsto\zeta_{\theta}$.

With this identification, it follows that $\theta\in\Theta_{{\bf R}}$ is
generic if and only if for all proper subrepresentations $0\subsetneq
S\subsetneq R$ we have $\zeta_{\theta}(i\pi_{S})\neq 0$, with $\pi_{S}\mskip
0.5mu\colon\thinspace R\to S$ denoting the orthogonal projection onto $S$.

For each $\theta\in\Theta_{\bf R}$, we denote by

(3.3) $\displaystyle M_{\theta}:=\mu^{-1}(\zeta_{\theta})/{{\bf P}{\rm
U}}(R)^{G}$

corresponding Kähler quotient. Moreover, for each representation $\rho\mskip
0.5mu\colon\thinspace G\to\mathrm{GL}(R_{\rho})$ of $G$, ${{\bf P}{\rm
U}}(R)^{G}$ acts on $R_{\rho}$ and gives rise to the bundle

$\displaystyle\mathcal{R}_{\rho}=\mu^{-1}(\theta)\times_{{{\bf P}{\rm
U}}(R)^{G}}R_{\rho}$

on $M_{\theta}$. This bundle is holomorphic, since the holomorphic structure
on $\mu^{-1}(\theta)\times V$ is ${{\bf P}{\rm U}}(R)^{G}$–equivariant and
thus passes down to $\mathcal{R}_{\rho}$. We call $\mathcal{R}_{\rho}$ the
_tautological holomorphic bundle_ associated to the representation $\rho$ of
$G$.

We describe now the relation between the algebraic objects
$\mathscr{M}_{\theta}$ and $\mathscr{R}_{\rho}$ defined in Theorem 2.4 and the
holomorphic objects $M_{\theta}$ and $\mathcal{R}_{\rho}$ defined above.

######  Proposition 3.4.

If $B\in N\cap\mu^{-1}(\zeta_{\theta})$, then the $G$–sheaf $\mathscr{F}$
associated to $B$ is $\theta$–semi-stable. Therefore we have
$\mu^{-1}(\zeta_{\theta})\subset N^{ss}_{\theta}$ for all
$\theta\in\Theta_{\bf R}$.

###### Proof.

Let $\mathscr{E}$ be a non-trivial proper $G$–subsheaf of $\mathscr{F}$. Then
we can decompose the regular representation into two non-trivial proper
subrepresentations

$\displaystyle R=S\oplus T$

with $S:=H^{0}(\mathscr{E})$ and $T$ its orthogonal complement. Corresponding
to $\mathscr{E}$ there is a triple of matrices
$C\in{\mathrm{End}}(S)\otimes{\bf C}^{3}$. Moreover, since each component of
$B$ leaves $S$ invariant, we have

$\displaystyle B=\begin{pmatrix}C&D\\\ 0&E\end{pmatrix}.$

With this,

$\displaystyle\left<\mu(B),i\pi_{S}\right>=\frac{1}{2}\mathop{\mathrm{tr}}\nolimits_{S}\left([C,C^{*}]+DD^{*}\right)=\frac{1}{2}\mathop{\mathrm{tr}}\nolimits_{S}\left(DD^{*}\right)\geq
0.$

Since $\left<\mu(B),i\pi_{S}\right>=\theta(i\pi_{S})=\theta(S)$, it follows
that $\theta(H^{0}(\mathscr{E}))\geq 0$. ∎

King [King1994]*Theorem 6. 1 shows that the Kempf–Ness theorem holds in this
case: If $\theta\in\Theta$, then each ${\bf P}\mathrm{GL}(R)^{G}$–orbit which
is closed in $N^{ss}_{\theta}$ meets $\mu^{-1}(\zeta_{\theta})$ in precisely
one ${{\bf P}{\rm U}}(R)^{G}$–orbit, and meets no other orbit. From this we
have the following result:

######  Proposition 3.5.

Suppose $\theta\in\Theta_{\bf Q}$ is generic. Then the inclusion
$\mu^{-1}(\zeta_{\theta})\to N^{s}_{\theta}=N^{ss}_{\theta}$ induces a
biholomorphic map from $M_{\theta}$ to the analytification of
$\mathscr{M}_{\theta}$. This map identifies the holomorphic bundle
$\mathcal{R}_{\rho}$ with the analytification of the locally free sheaf
$\mathscr{R}_{\rho}$.

Now, for each $\theta\in\Theta_{R}$, let $g_{\theta}$ and $\omega_{\theta}$ be
the metric and the Kähler form on $M_{\theta}$ induced by the Kähler quotient
construction. We also have a natural ${{\bf P}{\rm U}}(R)^{G}$–connection
$A_{\theta}$ whose horizontal space is the orthogonal complement of the
tangent space to the orbit in $T_{B}(\mu^{-1}(\theta))$. To describe the
geometry of $M_{\theta}$ and the behaviour of $g_{\theta}$ and $A_{\theta}$,
we first need the following definitions:

######  Definition 3.6.

Let $G$ be a finite subgroup of ${\rm SU}(3)$ acting freely on ${\bf
C}^{3}\setminus\\{0\\}$. A noncompact Riemannian manifold $(X,g)$ of real
dimension $6$ is called an _ALE manifold asymptotic to ${\bf C}^{3}/G$ to
order $\tau>0$_, if there exists a compact subset $K\subset X$ and a
diffeomorphism $\pi\mskip 0.5mu\colon\thinspace\left({\bf C}^{3}\setminus
B_{R}\right)/G\to X\setminus K$ so that

$\displaystyle\partial_{k}(\pi^{*}g-g_{0})=O\left(r^{-\tau-k}\right)$

for all $0\leq k\leq 2$. Here $r:=|x|$ denotes the radius function on ${\bf
C}^{3}$ and $g_{0}$ denotes the standard metric on ${\bf C}^{3}$. The pair
$(X\setminus K,\pi)$ is called an _ALE end_ and the function $r$, extended
smoothly to $X$ so that $r\mskip 0.5mu\colon\thinspace X\to[1,\infty)$, is a
_radius function_ on $X$.

######  Definition 3.7.

Similarly, a connection $A$ on a complex vector bundle $E$ of rank $k$ on the
ALE manifold $(X,g)$ is called _asymptotically flat of order $\tau>0$_ if
there exists a flat connection $A_{0}$ on the ALE end of $X$ so that

$\displaystyle\nabla^{k}(A-A_{0})=O\left(r^{-\tau-k}\right)$

for $0\leq k\leq 1$.

######  Definition 3.8.

A Hermitian metric $h$ on a complex bundle $E$ on an ALE manifold $(X,g)$ is
called _asymptotically flat of order $\tau>0$_ if with respect a local
trivialization of $E$ on the ALE end of $X$ it satisfies

$\displaystyle\partial_{k}(h-h_{0})=O\left(r^{-\tau-k}\right)$

for all $0\leq k\leq 2$, where $h_{0}$ denotes the standard metric.

######  Theorem 3.9.

Let $G$ be a finite subgroup of $SU(3)$ acting feely on ${\bf
C}^{3}\setminus\\{0\\}$. Then the following holds:

  1. 1.

$M_{0}$ is isometric to the orbifold ${\bf C}^{3}/G$ with the induced orbifold
Kähler metric $g_{0}$, and the connection $A_{0}$ is flat.

  2. 2.

If $\theta\in\Theta_{\bf Q}$ generic, then $M_{\theta}$ is smooth and the
induced Kähler metric $g_{\theta}$ is ALE of order $4$.

  3. 3.

The ${{\bf P}{\rm U}}(R)^{G}$–connection $A_{\theta}$ is an $(1,1)$–connection
which is asymptotically flat of order $2$. In particular, its curvature decays
like $r^{-4}$.

In the case of finite subgroups of ${\rm SU}(2)$, the analogous theorem was
proven by Kronheimer [Kronheimer1989] and by Gocho and Nakajima [Gocho1992].
For the above theorem, the smoothness of the Kähler quotient $M_{\theta}$ for
$\theta\in\Theta_{{\bf Q}}$ generic follows from the identification with the
algebraic quotient $\mathscr{M}_{\theta}$ provided by Proposition 3.5 and the
result of Theorem 2.11. The first statement and the remaining of the second
were proved by Sardo-Infirri [SardoInfirri1996] by generalising Kronheimer’s
proof. The proof of the third statement is a direct generalization of Gocho
and Nakajima’s argument.

######  Remark 3.10.

It seems reasonable to expect that the second statement in Theorem 3.9 holds
for all generic $\theta\in\Theta_{{\bf R}}$. Because of the homogeneity of the
moment map, the statement follows for all $t\theta$ with
$\theta\in\Theta_{{\bf Q}}$ generic and $t$ a strictly positive real number.
On the other hand, when $\theta\in\Theta_{{\bf R}}$ generic, we can show that
${{\bf P}{\rm U}}(R)^{G}$ acts freely on $\mu^{-1}(\theta)$. However, to
conclude that $M_{\theta}$ is smooth, one still needs to show that
$\mu^{-1}(\theta)$ is contained in the smooth locus of $N$.

##  4 Ricci-flat metrics on $M_{\theta}$ and Hermitian–Yang–Mills metrics on
$\mathcal{R}_{\rho}$

In contrast to Kronheimer’s and Nakajima’s work [Kronheimer1989,
Kronheimer1990], the metric $g_{\theta}$ on $M_{\theta}$ obtained from the
Kähler reduction is not necessarily Ricci-flat and the connections $A_{\rho}$
on $\mathcal{R}_{\rho}$ are not necessarily Hermitian–Yang–Mills (HYM).
Indeed, Sardo-Infirri [SardoInfirri1996]*Example 7.1 showed that for the
finite subgroup $G={\bf Z}_{3}$ and an appropriate choice of $\theta$ generic,
the induced metric on $M_{\theta}=\mathcal{O}_{{\bf C}P^{2}}(-3)$ has non-
vanishing Ricci curvature. In this section, we show that $M_{\theta}$ does
admit a Ricci-flat Kähler metric and $\mathcal{R}_{\rho}$ carries an
asymptotically flat HYM connection, thus partially proving Theorem 1.8. The
existence of the Ricci-flat Kähler metric follows from the following result:

######  Theorem 4.1 (Joyce [Joyce2000]*Theorem 8.2.3).

Let $G$ be a finite subgroup of ${\rm SU}(n)$ acting freely on ${\bf
C}^{n}\setminus\\{0\\}$. Let $X$ be a smooth crepant resolution of ${\bf
C}^{n}/G$ with an ALE Kähler metric $g$ of order $\tau>n$. Then there exists a
unique Ricci-flat ALE Kähler metric $g_{\rm RF}$ in the Kähler class of $g$.
The metric $g_{\rm RF}$ is ALE of order $2n$.

######  Remark 4.2.

Joyce states this result only for ALE Kähler metrics of order $\tau=2n$;
however, his proof goes through for $\tau>n$.

######  Corollary 4.3.

Let $\theta\in\Theta_{{\bf Q}}$ generic and $(M_{\theta},g_{\theta})$ the
corresponding Kähler quotient. Then there exists an ALE Kähler Ricci-flat
metric $g_{\theta,{\rm RF}}$ of order $6$ on $M_{\theta}$ in the same Kähler
class as $g_{\theta}$.

######  Proposition 4.4.

Let $X$ be an ALE Kähler manifold and let $\mathcal{L}$ be a holomorphic line
bundle on $X$. If there is a Hermitian metric $h_{0}$ on $\mathcal{L}$ such
that

$\displaystyle\Lambda F_{h_{0}}=O\left(r^{-2-\varepsilon}\right)$

for some $\varepsilon>0$, then there exists an $\delta>0$ and a HYM metric $h$
on $\mathcal{L}$ asymptotic to $h_{0}$ to order $\delta$. Moreover, for any
$\gamma>0$, $h$ is the unique HYM metric on $\mathcal{L}$ asymptotic to
$h_{0}$ to order $\gamma$.

######  Corollary 4.5.

Let $\theta\in\Theta_{{\bf Q}}$ generic. Then for each $\rho\in{\rm Irr}(G)$
the tautological bundle $\mathcal{R}_{\rho}$ on $M_{\theta}$ carries an
asymptotically flat HYM metric with respect to $g_{\theta,{\rm RF}}$.

######  Remark 4.6.

Using some of the results derived in Section 5, one can show that the HYM
connection associated with $h$ in Proposition 4.4 is asymptotically flat of
order $5$. Thus the Hermitian metric $h$ is asymptotically flat of order $4$.

######  Remark 4.7.

Using heat flow methods, Bando [Bando1993] proved that every holomorphic
bundle $\mathcal{E}$ over an ALE Kähler manifold which admits a Hermitian
metric $h_{0}$ with $|F_{h_{0}}|=O\left(r^{-2-\varepsilon}\right)$ does in
fact carry a HYM metric.

The case of line bundles is much simpler and follows from the Laplace operator
being an isomorphism between certain weighted Sobolev spaces. For $k$ a
nonegative integer and $\delta\in{\bf R}$, we denote by $W^{k,2}_{\delta}(X)$
the completion of $C^{\infty}_{0}(X)$ with respect the norm

(4.8)
$\displaystyle\|f\|_{W^{k,2}_{\delta}}:=\sum_{j=0}^{k}\|r^{-\delta-m/2+j}\nabla^{j}f\|_{L^{2}}.$

Here $m$ denotes the real dimension of $X$. Let $\Delta_{\delta}\mskip
0.5mu\colon\thinspace W^{k+2,2}_{\delta}(X)\to W^{k,2}_{\delta-2}(X)$ denote
the corresponding completion of the Laplacian $\Delta$.

######  Proposition 4.9.

For $\delta\in(-m+2,0)$, the operator $\Delta_{\delta}$ is an isomorphism.

###### Proof.

The weighted Laplacian $\Delta_{\delta}\mskip 0.5mu\colon\thinspace
W^{k+2,2}_{\delta}(X)\to W^{k,2}_{\delta-2}(X)$ is a Fredholm operator if and
only if the weight parameter $\delta$ is not in its set of indicial roots at
infinity. This is a discrete set that does not intersect the interval
$(-m+2,0)$, see Bartnik [Bartnik1986]*Sections 1 and 2 for details. Moreover,
for $\delta<0$, the kernel of $\Delta_{\delta}$ is trivial by the maximum
principle. On the other hand, the cokernel of $\Delta_{\delta}$ is isomorphic
to the kernel of its formal adjoint $\Delta_{m-2-\delta}$. Therefore, for
$\delta\in(-m+2,0)$, $\Delta_{\delta}$ is an isomorphism. ∎

###### Proof of Proposition 4.4.

Any Hermitian metric on $\mathcal{L}$ is of the form $h=e^{f}h_{0}$, for $f\in
C^{\infty}(X)$. Then $F_{h}=F_{h_{0}}+\bar{\partial}\partial
f\in\Omega^{2}(X,i{\bf R})$ and thus

$\displaystyle i\Lambda F_{h}=i\Lambda F_{h_{0}}+\frac{1}{2}\Delta f.$

Since $\Lambda F_{h_{0}}\in W^{0,2}_{-2-\varepsilon}(X)$ for some
$\varepsilon>0$, by Proposition 4.9, there exists a unique $f$ such that
$\Delta f=-2i\Lambda F_{h_{0}}$ and $f=O\left(r^{-{\delta}}\right)$ for some
$\delta>0$. ∎

##  5 Rigidity of HYM metrics on the holomorphic tautological bundles

In this section we prove the rigidity statement in Theorem 1.8. This will be
an immediate consequence of Lemma 5.1, which is the main vanishing result of
this paper.

######  Lemma 5.1.

Let $\theta\in\Theta_{\bf Q}$ generic and let $M_{\theta}$ be equipped with an
ALE Kähler metric. Let $h$ be a Hermitian metric on $\mathcal{R}$ and let $A$
denote the associated Chern connection. Suppose that $A$ is asymptotically
flat of order $\tau>0$. Then the space

$\displaystyle\mathcal{H}^{1}_{A}:=\left\\{a\in\Omega^{0,1}(M_{\theta},\mathcal{E}nd(\mathcal{R})):\bar{\partial}_{A}a=\bar{\partial}_{A}^{*}a=0~{}\text{and}~{}\lim_{r\to\infty}\sup_{\partial
B_{r}}|a|=0\right\\}$

is trivial.

If the metric $h$ is HYM, then $\mathcal{H}^{1}_{A}$ is the space of
infinitesimal deformations. Hence, the HYM metrics on $\mathcal{R}_{\rho}$
constructed in the first part of Theorem 1.8 are infinitesimally rigid for all
$\rho\in{\rm Irr}(G)$, thus completing the proof of Theorem 1.8.

The strategy for proving Lemma 5.1 is as follows: We first reduce to a problem
in complex geometry, see Proposition 5.8. Using GAGA, we translate this into
an algebraic geometry problem, see (5.9), which we then solve using the
results of Bridgeland, King and Reid [Bridgeland2001] and Craw and Ishii
[Craw2004] discussed in Section 2.

It is a useful heuristic to think of bundles with decaying connections as
bundles on a compactification whose restrictions to the “divisor at infinity”
satisfy certain conditions, like being flat for example. Accordingly, we
compactify $M_{\theta}$ at infinity by gluing $M_{\theta}$ and $({\bf
P}^{3}\setminus\\{[0:0:0:1]\\})/G$ along
$M_{\theta}\setminus\pi_{\theta}^{-1}(0)=({\bf C}^{3}\setminus\\{0\\})/G$. The
resulting space $\bar{M}_{\theta}$ is not a complex manifold, but rather a
complex orbifold. We denote its divisor at infinity by $D$. This is a smooth
orbifold divisor, i.e., it lifts to a smooth divisors in covers of the
uniformising charts. The bundle $\mathcal{R}$ extends over $D$ to a bundle
$\bar{\mathcal{R}}$ on $\bar{M}_{\theta}$. The following result reduces the
proof of Lemma 5.1 to a problem in complex geometry.

######  Proposition 5.2.

$\mathcal{H}^{1}_{A}$ injects into
$H^{1}\left(\bar{M}_{\theta},\mathcal{E}nd\left(\bar{\mathcal{R}}\right)(-D)\right)$.

The proof of Proposition 5.2 requires two preparatory results.

######  Proposition 5.3.

Let $Z$ be a complex orbifold, $D$ be a smooth divisor in $Z$ and
$\mathcal{E}$ be a holomorphic bundle on $Z.$ Denote by $i\mskip
0.5mu\colon\thinspace D\hookrightarrow Z$ the inclusion of $D$ into $Z$. Then
the complex of sheaves $\left(\mathcal{A}^{\bullet},\bar{\partial}\right)$
defined by

$\displaystyle\mathcal{A}^{k}(U):=\left\\{\alpha\in\Omega^{0,k}(U,\mathcal{E}):i^{*}\alpha=0\right\\}$

for $U\subset Z$ open is an acyclic resolution of $\mathcal{E}(-D)$.

###### Proof.

Since the $i^{*}$ and $\bar{\partial}$ commute, $\mathcal{A}^{\bullet}$ forms
a complex. Moreover, it is clear that $\mathcal{E}(-D)$ is the kernel of
$\mathcal{A}^{0}\stackrel{{\scriptstyle\partial}}{{\to}}\mathcal{A}^{1}$.

The proof that $\mathcal{A}^{\bullet}$ is a resolution uses two ingredients:
the Grothendieck–Dolbeault Lemma and the fact that if $U$ is a sufficiently
small open set, then holomorphic sections on $D\cap U$ extend to $U$. We show
that these assertions hold also for orbifolds. Let $U$ be a small open set
which is covered by a uniformising chart $\tilde{U}/\Gamma$. Lifting
everything up to $\tilde{U}$, $\mathcal{E}$ corresponds to a
$\Gamma$–equivariant holomorphic bundle $\tilde{\mathcal{E}}$ and $D$ to a
$\Gamma$–equivariant smooth divisor $\tilde{D}$. If
$\alpha\in\Omega^{(0,k)}(U,\mathcal{E})$ satisfies $\bar{\partial}\alpha=0$,
then so does its lift
$\tilde{\alpha}\in\Omega^{(0,k)}(\tilde{U},\tilde{\mathcal{E}})^{\Gamma}$. If
$U$ (and thus $\tilde{U}$) is sufficiently small, then the usual
Grothendieck–Dolbeault Lemma yields
$\tilde{\beta}\in\Omega^{(0,k-1)}(\tilde{U},\tilde{\mathcal{E}})$ satisfying
$\bar{\partial}\tilde{\beta}=\tilde{\alpha}$. There is no loss in assuming
that $\tilde{\beta}$ is $\Gamma$–invariant and thus pushes down to the desired
primitive $\beta\in\Omega^{(0,k-1)}(U,\mathcal{E})$ of $\alpha$. We thus
obtain the Grothendieck–Dolbeault Lemma for orbifolds. Now, if $s$ is a
holomorphic section of $\mathcal{E}$ over $D\cap U$, we lift it to the
uniformising chart $\tilde{U}$, where, provided $U$ is sufficiently small, we
find a $\Gamma$-equivariant extension. We then push this extension down to
$U$. Hence, the proof of the second assertion in orbifold set-up.

Let now $U$ be a small open set of $Z$ and let
$\alpha\in\Omega^{0,k}(U,\mathcal{E})$ with $\bar{\partial}\alpha=0$. By the
Grothendieck–Dolbeault Lemma after possibly shrinking $U$, we can find
$\beta\in\Omega^{0,k}(U,\mathcal{E})$ satisfying $\bar{\partial}\beta=\alpha$.
If $k\geq 2$, we apply the Grothendieck-Dolbeault Lemma once more to obtain
$\gamma\in\Omega^{0,k-2}(U\cap D,\mathcal{E})$ such that
$\bar{\partial}\gamma=i^{*}\beta$. We extend $\gamma$ smoothly to all of $U$.
Then $\beta-\bar{\partial}\gamma\in\mathcal{A}^{k-1}(U)$ yields the desired
primitive of $\alpha$ on $U$. When $k=1$, we know that $\beta$ restricts to a
holomorphic section $\beta|_{D}$ of $\mathcal{E}|_{U\cap D}$, which can be
extended to a holomorphic section $\delta$ on $U$. Hence,
$\beta-\delta\in\mathcal{A}^{0}(U)$ is the desired primitive of $\alpha$.

Finally, $(\mathcal{A}^{\bullet},d)$ is an acyclic resolution of
$\mathcal{E}(-D)$, since the sheaves $\mathcal{A}^{\bullet}$ are
$C^{\infty}$–modules and therefore soft. ∎

######  Remark 5.4.

In the definition of $\mathcal{A}^{k}$ it is not strictly necessary to require
that $\alpha$ be smooth. In fact, a simple application of elliptic regularity
shows that it suffices that elements of $\mathcal{A}^{k}$ be in the Hölder
space $C^{n-k,\alpha}$, where $n$ denote the complex dimension of $Z$.

######  Proposition 5.5.

If $a\in\mathcal{H}^{1}_{A}$, then

(5.6) $\displaystyle\nabla^{k}a=O\left(r^{-5-k}\right)$

for all $k\geq 0$.

###### Proof.

First observe that using simple scaling considerations and standard elliptic
theory, (5.6) for $k>0$ follows from the case $k=0$.

It is rather straightforward to obtain $a=O\left(r^{-4}\right)$ using the
maximum principle. To obtain the stronger decay estimate it is customary to
make use of a refined Kato inequality, see, e.g., Bando, Kasue and Nakajima
[Bando1989]. The Kato inequality is a consequence of the following application
of the Cauchy-Schwarz inequality: $|\left<\nabla_{A}a,a\right>|\leq|\nabla a|\
|a|$. But, the equation $\bar{\partial}a=\bar{\partial}_{A}^{*}a=0$ imposes a
linear constraint on $\nabla_{A}a$, which is incompatible with equality in the
previous estimate unless $\nabla_{A}a=0$. Hence, there exists a constant
$\gamma<1$, such that $|{\rm d}|a||\leq\gamma|\nabla_{A}a|$ on the set
$U:=\\{x\in M_{\theta}\mskip 0.5mu\colon\thinspace a(x)\neq 0\\}$. A more
detailed analysis shows that $\gamma$ can be chosen to be $\sqrt{5/6}$. For a
systematic treatment of refined Kato inequalities we refer to the work of
Calderbank, Gauduchon and Herzlich [Calderbank2000].

We set $\gamma=\sqrt{5/6}$ and let $\sigma=2-1/\gamma^{2}=4/5$. Using the
refined Kato inequality for $a$, we have

$\displaystyle(2/\sigma)\Delta|a|^{\sigma}$
$\displaystyle=|a|^{\sigma-2}\left(\Delta|a|^{2}-2(\sigma-2)|{\rm
d}|a||^{2}\right)$
$\displaystyle\leq|a|^{\sigma-2}\left(\Delta|a|^{2}+2|\nabla_{A}a|^{2}\right)$
$\displaystyle=|a|^{\sigma-2}\left<a,\nabla_{A}^{*}\nabla_{A}a\right>.$

The Weitzenböck formula for $\nabla_{A}^{*}\nabla_{A}a$ gives

$\displaystyle(2/\sigma)\Delta|a|^{\sigma}$
$\displaystyle=|a|^{\sigma-2}\left(\left<\Delta_{\bar{\partial}}a,a\right>+\left<\\{R,a\\},a\right>+\left<\\{F_{A},a\\},a\right>\right),$

with $R$ the Riemannian curvature operator and $F_{A}$ the curvature of the
connection $A$. Since $\Delta_{\bar{\partial}}a=0$, and since by Theorem 3.9
the metric on $M_{\theta}$ is ALE of order $4$ and the curvature $F_{A}$
decays like $r^{-4}$, it follows that there exist positive constants
$c,\tau>0$ so that on $U$ we have

$\displaystyle(2/\sigma)\Delta|a|^{\sigma}\leq cr^{-2-\tau}|a|^{\sigma}.$

Set $f=|a|^{\sigma}$. We show that $f=O\left(r^{-4}\right)$, which is
equivalent to the desired decay estimate for $a$. Note that on $U$,

(5.7) $\displaystyle\Delta f\leq\frac{cf}{1+r^{2+\tau}}.$

Since $f$ is bounded, using [Joyce2000]*Theorem 8.3.6(a), we find
$g=O\left(r^{-\tau}\right)$, such that

$\displaystyle\Delta g=\begin{cases}(\Delta f)^{+}&\text{on}~{}U\\\
0&\text{on}~{}M_{\theta}\setminus U\end{cases}$

Here $(-)^{+}$ denotes taking the positive part. Then $f-g$ is a subharmonic
function on $M_{\theta}$ and must achieve its maximum at the boundary boundary
of $U$ or at infinity. Hence $f\leq g=O\left(r^{-\tau}\right)$. Then by (5.7),
$\Delta f=O\left(r^{-2-2\tau}\right)$ and the above procedure yields
$f=O\left(r^{-2\tau}\right)$. Reiterating this argument $k$–times gives
$f=O\left(r^{-k\tau}\right)$ for all $k<(n-2)/\tau$. For the biggest $k$ with
this property, we have $2+(k+1)\tau>n$. Then by [Joyce2000]*Theorem 8.3.6(b),
we can chose $g$ above such that $g=O\left(r^{-4}\right)$. Therefore,
$f=O\left(r^{-4}\right)$ as desired. ∎

###### Proof of Proposition 5.2.

Given $a\in\mathcal{H}^{1}_{A}$, we extend it to a $1$–form on
$\bar{M}_{\theta}$ vanishing along $D$. From Proposition 5.5 it follows that
$a$ vanishes to third order along $D$. Hence, $a$ is in $C^{2,\alpha}$ and we
can regard it as an element of $\mathcal{A}^{1}\left(\bar{M}_{\theta}\right)$.
Since $\bar{\partial}a=0$, by Proposition 5.3 it gives an element $[a]\in
H^{1}\left(\bar{M}_{\theta},\mathcal{E}nd\left(\bar{\mathcal{R}}\right)(-D)\right)$.
This defines a linear map $i\mskip 0.5mu\colon\thinspace\mathcal{H}^{1}_{A}\to
H^{1}\left(\bar{M}_{\theta},\mathcal{E}nd\left(\bar{\mathcal{R}}\right)(-D)\right)$.

We show now that $i$ is injective. For this, assume that there exists
$b\in\mathcal{A}^{0}\left(\bar{M}_{\theta}\right)$ so that
$a=\bar{\partial}b$. Since $b$ vanishes along $D$, its restriction to
$M_{\theta}$ decays like $r^{-1}$. Using this together with $a=O(r^{-5})$, we
can integrate by parts to obtain

$\displaystyle\|a\|_{L^{2}}^{2}=\int_{M_{\theta}}\left<a,\bar{\partial}b\right>\mathop{\mathrm{missing}}{dvol}\nolimits(g_{\theta})=\int_{M_{\theta}}\left<\bar{\partial}_{A}^{*}a,b\right>\mathop{\mathrm{missing}}{dvol}\nolimits(g_{\theta})=0.$

It follows that $a$ vanishes, and thus $i$ is injective. ∎

To prove Lemma 5.1 it now sufficies to establish the following result:

######  Proposition 5.8.

$H^{1}\left(\bar{M}_{\theta},\mathcal{E}nd\left(\bar{\mathcal{R}}\right)(-D)\right)=0$.

To prove this statement, we convert it into an algebraic geometry problem. In
the same way we compactified $M_{\theta}$, we can complete the scheme
$\mathscr{M}_{\theta}$ at infinity by attaching $D={\bf P}^{2}/G$. This yields
an algebraic stack $\bar{\mathfrak{M}}_{\theta}$. Moreover, $\mathscr{R}$
extends to a locally free sheaf $\bar{\mathscr{R}}$ on
$\bar{\mathfrak{M}}_{\theta}$. Using GAGA [Toen1999]*Théorème 5.10, we see
that Proposition 5.8 is equivalent to

(5.9) $\displaystyle
H^{1}\left(\bar{\mathfrak{M}}_{\theta},\mathscr{E}nd\left(\mathscr{R}\right)(-D)\right)=0.$

To show this, we use the following consequence of Theorem 2.11:

######  Proposition 5.10.

For $\theta\in\Theta_{\bf Q}$ generic,

$\displaystyle
H^{k}\left(\mathscr{M}_{\theta},\mathscr{R}_{\rho}^{*}\otimes\mathscr{R}_{\sigma}\right)=H^{k}\left({\bf
C}^{3},\mathscr{O}\otimes R_{\rho}^{*}\otimes R_{\sigma}\right)^{G},$

for all $\rho,\sigma\in{\rm Irr}(G)$. In particular, for $k>0$,

$\displaystyle
H^{k}(\mathscr{M}_{\theta},\mathscr{R}_{\rho}^{*}\otimes\mathscr{R}_{\sigma})=0.$

If $G$ acts freely on ${\bf C}^{3}\setminus\\{0\\}$, we have a commutative
diagram

where $i\mskip
0.5mu\colon\thinspace\mathscr{M}_{\theta}\setminus\pi_{\theta}^{-1}(0)\to\mathscr{M}_{\theta}$
and $j\mskip 0.5mu\colon\thinspace{\bf C}^{3}\setminus\\{0\\}\to{\bf C}^{3}$
are the inclusion maps.

###### Proof.

The first part is due to Craw and Ishii [Craw2004]*Lemma 5.4. Let us briefly
recall their proof: On the one hand, we have

$\displaystyle
H^{k}(\mathscr{M}_{\theta},\mathscr{R}_{\rho}^{*}\otimes\mathscr{R}_{\sigma})$
$\displaystyle=\mathrm{Ext}^{k}(\mathscr{O},\mathscr{R}_{\rho}^{*}\otimes\mathscr{R}_{\sigma})=\mathrm{Ext}^{k}(\mathscr{R}_{\rho},\mathscr{R}_{\sigma})$
$\displaystyle=H^{k}({\mathrm{Hom}}_{D(\mathscr{M}_{\theta})}(\mathscr{R}_{\rho},\mathscr{R}_{\sigma}))$

and

$\displaystyle H^{k}({\bf C}^{3},\mathscr{O}\otimes R_{\rho}^{*}\otimes
R_{\sigma})^{G}$ $\displaystyle=G\\!-\\!\mathrm{Ext}^{k}(\mathscr{O}\otimes
R_{\rho},\mathscr{O}\otimes R_{\sigma})$
$\displaystyle=H^{k}({\mathrm{Hom}}_{D^{G}({\bf C}^{3})}(\mathscr{O}\otimes
R_{\rho},\mathscr{O}\otimes R_{\sigma})).$

On the other hand, the inverse of the Fourier–Mukai transform $\Phi_{\theta}$
is given by

$\displaystyle\Phi_{\theta}^{-1}=\left(p_{*}\left(q^{*}(-)\otimes\mathscr{U}_{\theta}^{D}[3]\right)\right)^{G}=\left(-\otimes\bigoplus_{\rho}\mathscr{R}_{\rho}^{*}\otimes
R_{\rho}\right)^{G},$

see [Craw2004]*p. 267. Here $(-)^{D}$ denotes the derived dual. In particular,

(5.11) $\displaystyle\Phi_{\theta}^{-1}(\mathscr{O}\otimes
R_{\rho})=\mathscr{R}_{\rho}.$

Then, according to Theorem 2.11,
$H^{k}({\mathrm{Hom}}_{D(\mathscr{M}_{\theta})}(\mathscr{R}_{\rho},\mathscr{R}_{\sigma}))\cong
H^{k}({\mathrm{Hom}}_{D^{G}({\bf C}^{3})}(\mathscr{O}\otimes
R_{\rho},\mathscr{O}\otimes R_{\sigma}))$. Therefore,
$H^{k}(\mathscr{M}_{\theta},\mathscr{R}_{\rho}^{*}\otimes\mathscr{R}_{\sigma})\cong
H^{k}({\bf C}^{3},\mathscr{O}\otimes R_{\rho}^{*}\otimes R_{\sigma})^{G}$.

To prove the second part, we show that the following diagram commutes

Here $\Phi_{\mathscr{O}_{\Gamma}}$ is the Fourier–Mukai transform with kernel
$\mathscr{O}_{\Gamma}$, the structure sheaf of the graph of
$\pi_{\theta}\mskip
0.5mu\colon\thinspace\mathscr{M}_{\theta}\setminus\pi_{\theta}^{-1}(0)\to{\bf
C}^{3}\setminus\\{0\\}$. Note that under the identification $D^{G}({\bf
C}^{3}\setminus\\{0\\})=D\left(({\bf C}^{3}\setminus\\{0\\})/G\right)$,
$\Phi_{\mathscr{O}_{\Gamma}}$ becomes $\left(\pi_{\theta}\right)_{*}$.

Denote by $r$ and $s$ the projections from
$\mathscr{M}_{\theta}\setminus\pi_{\theta}^{-1}(0)\times{\bf
C}^{3}\setminus\\{0\\}$ to $\mathscr{M}_{\theta}\setminus\pi_{\theta}^{-1}(0)$
and ${\bf C}^{3}\setminus\\{0\\}$, respectively. Let $t\mskip
0.5mu\colon\thinspace\mathscr{M}_{\theta}\times{\bf
C}^{3}\setminus\\{0\\}\to{\bf C}^{3}\setminus\\{0\\}$ denote the projection
onto the second factor. The following diagram summarises the situation:

It follows, essentially from the definition of $\pi_{\theta}$, that

(5.12) $\displaystyle\left(\mathrm{id}_{\mathscr{M}_{\theta}}\times
j\right)^{*}\mathscr{U}_{\theta}=\left(i\times\mathrm{id}_{{\bf
C}^{3}\setminus\\{0\\}}\right)_{*}\mathscr{O}_{\Gamma}.$

Using (5.12) as well as the push-pull formula $(\mathscr{F}\otimes
f_{*}\mathscr{G})=f_{*}(f^{*}\mathscr{F}\otimes\mathscr{G})$, we have

$\displaystyle j^{*}\circ\Phi_{\theta}(-)$ $\displaystyle=j^{*}\circ
q_{*}\left(p^{*}(-\otimes\rho_{0})\otimes\mathscr{U}_{\theta}\right)$
$\displaystyle=t_{*}\left((\mathrm{id}_{\mathscr{M}_{\theta}}\times
j)^{*}p^{*}(-\otimes\rho_{0})\otimes(\mathrm{id}_{\mathscr{M}_{\theta}}\times
j)^{*}\mathscr{U}_{\theta}\right)$
$\displaystyle=t_{*}\left((\mathrm{id}_{\mathscr{M}_{\theta}}\times
j)^{*}p^{*}(-\otimes\rho_{0})\otimes(i\times\mathrm{id}_{{\bf
C}^{3}\setminus\\{0\\}})_{*}\mathscr{O}_{\Gamma}\right)$
$\displaystyle=t_{*}(i\times\mathrm{id}_{{\bf
C}^{3}\setminus\\{0\\}})_{*}\left((i\times
j)^{*}p^{*}(-\otimes\rho_{0})\otimes\mathscr{O}_{\Gamma}\right)$
$\displaystyle=s_{*}\left(r^{*}i^{*}(-\otimes\rho_{0})\otimes\mathscr{O}_{\Gamma}\right)$
$\displaystyle=\Phi_{\mathscr{O}_{\Gamma}}\circ i^{*}(-).$

This concludes the proof. ∎

Before we embark on the proof of (5.9), it is useful to recall some basic
properties of local cohomology, see, e.g., [Hartshorne1977]*Chapter III,
Exercise 2.3. Let $D$ be a closed subset of $X$ and let $\mathcal{E}$ be a
sheaf on $X$. Denote by $\Gamma_{D}(X,\mathscr{E})$ the subspace of
$\Gamma(X,\mathscr{E})$ consisting of sections whose support is contained in
$D$. The functor $\Gamma_{D}(X,-)$ is left-exact. Then
$H^{\bullet}_{D}(X,\mathscr{E}):=R^{\bullet}\Gamma_{D}(X,\mathscr{E})$ is
called the _local cohomology_ of $\mathcal{E}$ with respect to $D$. Local
cohomology is related to the usual cohomology of $\mathcal{E}$ by the
following long exact sequence

$\displaystyle\cdots\to H^{i}_{D}(X,\mathscr{E})\to H^{i}(X,\mathscr{E})\to
H^{i}(X\setminus D,\mathscr{E}|_{X\setminus
D})\stackrel{{\scriptstyle\delta}}{{\to}}H^{i+1}_{D}(X,\mathscr{E})\to\cdots$

Moreover, it satisfies _excision_ , that is, if $U$ is an open subset in $X$
containing $D$, then there is natural isomorphism

$\displaystyle H^{\bullet}_{D}(X,\mathcal{E})\cong
H^{\bullet}_{D}\left(U,\mathcal{E}|_{U}\right).$

###### Proof of Proposition 5.8.

We already reduced the proof of this to the proof of the vanishing (5.9).
Since by Proposition 5.10
$H^{1}(\mathscr{M}_{\theta},\mathscr{E}nd(\mathscr{R}))=0$, the long exact
sequence associated to the local cohomology yields

$\displaystyle
H^{0}(\mathscr{M}_{\theta},\mathscr{E}nd(\mathscr{R}))\stackrel{{\scriptstyle\delta}}{{\to}}H_{D}^{1}\left(\bar{\mathfrak{M}}_{\theta},\mathscr{E}nd\left(\bar{\mathscr{R}}\right)(-D)\right)\to
H^{1}\left(\bar{\mathfrak{M}}_{\theta},\mathscr{E}nd\left(\bar{\mathscr{R}}\right)(-D)\right)\to
0.$

We show that the first map in this sequence is an isomorphism. This gives the
desired vanishing,
$H^{1}(\bar{\mathfrak{M}}_{\theta},\mathscr{E}nd\left(\bar{\mathscr{R}}\right)(-D))=0$.

Let $H$ denote the hyperplane section in ${\bf P}^{3}$. By excision, we have

$\displaystyle
H_{D}^{1}\left(\bar{\mathfrak{M}}_{\theta},\mathscr{E}nd\left(\bar{\mathscr{R}}\right)(-D)\right)$
$\displaystyle\cong
H_{D}^{1}\left(\bar{\mathfrak{M}}_{\theta}\setminus\pi^{-1}(0),\mathscr{E}nd\left(\bar{\mathscr{R}}\right)(-D)\right)$
$\displaystyle=H_{H}^{1}\left([{\bf
P}^{3}\setminus\\{[0:0:0:1]\\}/G],\mathscr{O}(-1)\otimes{\mathrm{End}}(R)\right)$
$\displaystyle\cong H_{H}^{1}\left([{\bf
P}^{3}/G],\mathscr{O}(-1)\otimes{\mathrm{End}}(R)\right).$

Here and in the following we omit to make the appropriate restriction of
sheaves explicit, when confusion is unlikely to arise. Using the above, we
have the commutative diagram

We compose on the left with the commutative diagram in Proposition 5.10. Since
$\delta=\tilde{\delta}\circ i^{*}$, we obtain the commutative diagram

All the vertical arrows are isomorphisms. Moreover, by using the long exact
sequence associated to the local cohomology and

$\displaystyle H^{i}\left([{\bf
P}^{3}/G],\mathscr{O}(-1)\otimes{\mathrm{End}}(R)\right)=H^{i}\left({\bf
P}^{3},\mathscr{O}(-1)\otimes{\mathrm{End}}(R)\right)^{G}=0$

for $i=0$ and $1$, it follows that the bottom map is also an isomorphism.
Therefore the map $\delta$ must be an isomorphism. ∎

##  6 Dirac operators on $M_{\theta}$

Let $(X,g)$ be an ALE spin manifold asymptotic to ${\bf C}^{n}/G$ and let $E$
be a complex vector bundle over $X$ together with an asymptotically flat
connection $A$. Denote by $S^{\pm}$ be the spinor bundles on $X$ and by
$D^{\pm}_{E}$ the corresponding twisted Dirac operators. We denote by
$W^{k,2}_{\delta}(X,S^{\pm}\otimes E)$ the completions of the spaces of
compactly supported sections with respect the weighted Sobolev norm defined by
(4.8). Let $D^{\pm}_{E,\delta}\mskip 0.5mu\colon\thinspace
W^{k+1,2}_{\delta}(X,S^{\pm}\otimes E)\to W^{k,2}_{\delta-1}(Z,S^{\mp}\otimes
E)$ denote the corresponding completion of the Dirac operator $D^{\pm}_{E}$.

######  Theorem 6.1.

For $\delta\in(-2n-1,0)$, $D^{\pm}_{E,\delta}$ is Fredholm and its index is
given by

(6.2) $\displaystyle\mathop{\mathrm{index}}D^{+}_{E,\delta}=\int_{X}{\rm
ch}(E)\hat{A}(X)-\frac{\eta_{E}}{2}.$

Here $\eta_{E}(s):=\sum_{\lambda\neq 0}\mathrm{sign}(\lambda)|\lambda|^{-s}$
is the eta-function of the spectrum of the Dirac operator restricted to the
boundary at infinity $S^{2n-1}/G$ of the ALE manifold $X$, and $\eta_{E}(0)$
is the $eta$–invariant. Also $\hat{A}(X)$ is the Hirzebruch
$\hat{A}$–polynomial applied to the Pontrjagin forms $p_{i}(X)$ of the ALE
metric on $X$.

###### Proof.

The fact that $D^{\pm}_{E,\delta}$ is Fredholm is proved as in Proposition 4.9
by noting that the set of indicial roots does not intersect $(-2n-1,0)$. This
can be seen, for example, by realising that the indicial roots correspond to
the eigenvalues of the Dirac operator on $S^{2n-1}/G$ shifted by
$-\frac{2n-1}{2}$. The index formula follows from Atiyah–Patodi–Singer index
theorem [Atiyah1975a]. ∎

We consider the case when $X$ is a Calabi–Yau manifold and $E$ underlies a
holomorphic vector bundle $\mathcal{E}$ with a Hermitian metric $h$ whose
induced Chern connection is $A$. Then there exists a canonical spin structure
on $X$ with

$\displaystyle S^{+}=\Lambda^{0,\text{even}}T_{\bf C}^{*}X\quad\text{and}\quad
S^{-}=\Lambda^{0,\text{odd}}T_{\bf C}^{*}X.$

The corresponding twisted Dirac operator is

(6.3) $\displaystyle
D^{\pm}_{\mathcal{E}}=\sqrt{2}\left(\bar{\partial}+\bar{\partial}_{h}^{*}\right).$

Now, we take $X$ to be $M_{\theta}$ for $\theta\in\Theta_{\bf Q}$ generic and
$\mathcal{E}$ to be $\mathcal{R}$. Using Theorem 1.8, we equip $M_{\theta}$
with an ALE Calabi–Yau metric $g_{\theta,{\rm RF}}$ and $\mathcal{R}$ with an
HYM metric.

######  Proposition 6.4.

The operator $D^{+}_{\mathcal{E}nd(\mathcal{R}),\delta}$ is an isomorphism for
all $\delta\in(-5,0)$. In particular, we have

(6.5)
$\displaystyle\mathop{\mathrm{index}}D^{+}_{\mathcal{R}_{\rho}\otimes\mathcal{R}_{\sigma}^{*},\delta}=0,$

for all $\rho,\sigma\in{\rm Irr}(G)$.

###### Proof.

We prove that
$\mathop{\mathrm{index}}D^{+}_{\mathcal{E}nd(\mathcal{R}),\delta}=0$. Let
$\Omega$ be a nowhere vanishing holomorphic volume form on $M_{\theta}$, and
$*\mskip
0.5mu\colon\thinspace\Lambda^{p,q}T^{*}X\otimes\mathcal{E}\to\Lambda^{3-p,3-q}T^{*}X\otimes\mathcal{E}^{*}$
the Hodge–$*$–operator for some holomorphic bundle $\mathcal{E}$. Then, we
have an isomorphism of vector bundles $S^{+}\otimes\mathcal{E}\cong
S^{-}\otimes\mathcal{E}^{*}$, given by

$\displaystyle
S^{+}\otimes\mathcal{E}=\Lambda^{0,\text{even}}T^{*}M_{\theta}\otimes\mathcal{E}\stackrel{{\scriptstyle\Omega\wedge-}}{{\to}}\Lambda^{3,{\rm
even}}T^{*}M_{\theta}\otimes\mathcal{E}\stackrel{{\scriptstyle*}}{{\to}}\Lambda^{0,{\rm
odd}}T^{*}M_{\theta}\otimes\mathcal{E}^{*}=S^{-}\otimes\mathcal{E}^{*}.$

Similarly, $S^{-}\otimes\mathcal{E}\cong S^{+}\otimes\mathcal{E}^{*}$. Via
this identification $D^{+}_{\mathcal{E}}$ corresponds to
$D^{-}_{\mathcal{E}^{*}}$. Thus
$\mathop{\mathrm{index}}D^{+}_{\mathcal{E},\delta}=\mathop{\mathrm{index}}D^{-}_{\mathcal{E}^{*},\delta}=-\mathop{\mathrm{index}}D^{-}_{\mathcal{E}^{*},-5-\delta}$.
Hence, in the case when $\mathcal{E}=\mathcal{E}nd(\mathcal{R})$, taking
$\delta=-\frac{5}{2}$, it follows that
$\mathop{\mathrm{index}}D^{+}_{\mathcal{E}nd(\mathcal{R}),-\frac{5}{2}}=0$.
Since the index is constant for $\delta\in(-5,0)$, we must have
$\mathop{\mathrm{index}}D^{+}_{\mathcal{E}nd(\mathcal{R}),\delta}=0$ for all
$\delta\in(-5,0)$.

To complete the proof, we show that
$\mathop{\mathrm{coker}}D^{+}_{\mathcal{E}nd(\mathcal{R}),\delta}=0$ for
$\delta\in(-5,0)$. Since
$\mathop{\mathrm{coker}}D_{\mathcal{E}nd(\mathcal{R}),\delta}^{+}\cong\ker
D^{-}_{\mathcal{E}nd(\mathcal{R}),-5-\delta}$, it is enough to prove the last
is trivial. Let $(\phi_{1},\phi_{3})\in\ker
D^{-}_{\mathcal{E}nd(\mathcal{R})}$. Then we have
$\bar{\partial}_{h}^{*}\phi_{1}=0$ and
$\bar{\partial}\phi_{1}+\bar{\partial}_{h}^{*}\phi_{3}=0$. The second identity
gives

(6.6) $\displaystyle\bar{\partial}\bar{\partial}_{h}^{*}\phi_{3}=0.$

Arguing as in Proposition 5.5 shows that $\phi_{3}=O\left(r^{-4}\right)$ and
$\bar{\partial}^{*}_{h}\phi_{3}=O\left(r^{-5}\right)$. Hence, taking the
$L^{2}$–inner-product with $\phi_{3}$ in (6.6) shows that
$\bar{\partial}_{h}^{*}\phi_{3}=0$. Using the Kähler identity
$\partial_{A}=i[\Lambda,\bar{\partial}^{*}_{h}]$ where $\Lambda$ is the
adjoint of the wedge multiplication by the Kähler form, yields
$\partial_{A}\phi_{3}=0$. If we write $\phi_{3}=\bar{\Omega}\otimes s$ where
$\Omega$ is a holomorphic volume form on $X$ and $s$ is a smooth section of
$\mathcal{E}nd(\mathcal{R})$, then $\partial_{A}(\bar{\Omega}\otimes
s)=\bar{\Omega}\wedge\partial_{A}s=0$ and thus $\partial_{A}s=0$. Since the
metric on $\mathcal{R}$ is HYM, the Weitzenböck formula takes the form
$2\Delta_{\partial_{A}}=\nabla_{A}^{*}\nabla_{A}$. Therefore, it follows that
$s$ is parallel and must hence vanish, as it decays at infinity. This implies
that $\phi_{1}$ satisfies $\bar{\partial}\phi_{1}=0$ and
$\bar{\partial}^{*}\phi_{1}=0$. Then, by Lemma 5.1, $\phi_{1}$ must vanish
identically. ∎

##  7 Geometrical McKay correspondence

In this section we prove Theorem 1.3 and derive its consequences. The proof
uses the Atiyah–Patodi–Singer index theorem for ALE manifolds (6.2). In order
to apply it, we need to compute the eta-invariant term that appears in this
formula.

######  Proposition 7.1.

Let $G$ be a finite subgroup of $\mathrm{SL}(n,{\bf C})$ acting freely on
${\bf C}^{n}\setminus\\{0\\}$. Assume that $X$ is a smooth ALE spin manifold
assymptotic to ${\bf C}^{n}/G$ and let $(E,A)$ be a asymptotically flat bundle
on $X$ whose fiber at infinity is $E_{\infty}$. Then, the eta-invariant for
the Dirac operator $D_{E,\delta}$ on $X$ is given by

(7.2) $\displaystyle\eta_{E}=-\frac{2}{|G|}\sum_{\begin{subarray}{c}g\in G\\\
g\neq
e\end{subarray}}\frac{\chi_{E_{\infty}}(g)}{\sum_{i=0}^{n}(-1)^{i}\chi_{\Lambda^{i}{\bf
C}^{n}}(g)},$

provided $-2n+1<\delta<0$. In this formula $\chi_{E_{\infty}}$ denotes the
character of the representation corresponding to the action of $G$ on the
vector space $E_{\infty}$.

######  Remark 7.3.

Note that for any $g\in\mathrm{SL}(n,{\bf C})$,
$\sum_{i=0}^{n}(-1)^{i}\chi_{\Lambda^{i}{\bf
C}^{n}}(g)=\mathop{\mathrm{missing}}{det}\nolimits(\mathrm{id}-g)$. Since $G$
is chosen to act freely on ${\bf C}^{n}$,
$\mathop{\mathrm{missing}}{det}\nolimits(I_{n}-g)\neq 0$ for all $g\in
G\setminus\\{e\\}$, and thus all the denominators in formula (7.2) are non-
zero.

This proposition is a consequence of the Lefschetz fixed-point formula, in the
sense that $\eta_{E}$ is the contribution from the fixed locus under the
action of $G$ on ${\bf C}^{n}$. It can be also proved using the definition of
the eta-invariant as the analytic continuation at $0$ of the eta-series
corresponding to the spectrum of the Dirac operator on the boundary at
infinity of the orbifold ${\bf C}^{n}/G$. This last approach gives the
generalization of the above formula to the case of non-isolated singularities
[Degeratu2001].

Consider the virtual representation $\sum_{i=0}^{n}(-1)^{i}\Lambda^{i}{\bf
C}^{n}$ of $G$. For each $\rho\in{\rm Irr}(G)$ we have the decomposition into
irreducibles

(7.4) $\displaystyle\Lambda^{i}{\bf C}^{n}\otimes\rho=\sum_{\sigma\in{\rm
Irr}(G)}a^{(i)}_{\rho\sigma}\ \sigma.$

We define

(7.5) $\displaystyle
c_{\rho\sigma}:=\sum_{i=0}^{n}(-1)^{i}a_{\rho\sigma}^{(i)}.$

Let $\tilde{C}$ be the matrix with entries $c_{\rho\sigma}$ for
$\rho,\sigma\in{\rm Irr}(G)$, and let $C$ be the principal submatrix of
$\tilde{C}$ obtained by erasing the line and column corresponding to the
trivial representation.

When $G$ is a finite subgroup of $\mathrm{SL}(2,{\bf C})$, the matrix $C$ is
the Cartan matrix of the unique simple Lie algebra corresponding to $G$, while
$\tilde{C}$ is the extended version. This is the essence of the McKay
correspondence [McKay1980]. Note that for $n\geq 3$ this matrix is not the
Cartan matrix associated to a Lie algebra, nor is it a generalised Cartan
matrix as appearing in the context of Lie algebras.

When $n$ is even, the virtual representation
$\sum_{i=0}^{n}(-1)^{i}\Lambda^{i}{\bf C}^{n}$ is self-dual, while when $n$ is
odd, it is anti-self-dual. Hence $C$ is a symmetric matrix when $n$ is even,
and skew-symmetric when $n$ is odd. Moreover, note that when $G$ is abelian
(in particular, when $G$ acts freely on ${\bf C}^{n}\setminus\\{0\\}$), all
the irreducible representations are one-dimensional, and thus
$\sum_{\sigma\in{\rm Irr}(G)}a_{\rho\sigma}^{(i)}=\dim\,\Lambda^{i}{\bf
C}^{n}$. Combined with formula (7.5), this gives

(7.6) $\displaystyle\sum_{\sigma\in{\rm Irr}(G)}c_{\rho\sigma}=0$

for all $\rho\in{\rm Irr}(G)$.

Now we are ready to prove our main result Theorem 1.3. The Chern character we
are using is $\widetilde{\rm ch}:={\rm ch}-\mathop{\mathrm{rk}}\nolimits$, the
Chern character from the reduced topological $K$–theory
$\widetilde{K}(X)=\ker(\mathop{\mathrm{rk}}\nolimits\mskip
0.5mu\colon\thinspace K(X)\to{\bf Z})$ from $\bigoplus_{k\geq 1}H^{k}(X,{\bf
R})$.

###### Proof of Theorem 1.3.

Let $X$ be a projective crepant resolution of ${\bf C}^{3}/G$. By Theorem
2.13, $X$ is the moduli space $\mathscr{M}_{\theta}$ of $\theta$-stable
$G$–constelations for some $\theta\in\Theta_{{\bf Q}}$ generic. Using
Proposition 3.5, the analytification of $\mathscr{M}_{\theta}$ is the smooth
Kähler quotient $M_{\theta}$. We can thus assume that $X=M_{\theta}$.

Let ${\rm Irr}(G)$ be the set of irreducible representations of $G$,
$\rho_{0}$ be the trivial representation and ${\rm Irr}_{0}(G)$ be the set of
non-trivial irreducible representations. Note that since $G$ acts freely on
${\bf C}^{3}\setminus\\{0\\}$, the group $G$ must be abelian and, hence, all
its irreducible representations are one dimensional.

By Proposition 6.4, we know that
$\mathop{\mathrm{index}}D^{+}_{\mathcal{R}_{\rho}\otimes\mathcal{R}_{\sigma}^{*},\delta}=0$
for all weights $\delta\in(-5,0)$ and for all $\rho,\sigma\in{\rm Irr}(G)$.
Then (6.2) gives

(7.7) $\displaystyle\int_{M_{\theta}}{\rm
ch}(\mathcal{R}_{\rho}\otimes\mathcal{R}_{\sigma}^{*})\hat{A}(M_{\theta})=\frac{\eta_{R_{\rho}\otimes
R_{\sigma^{*}}}}{2}.$

Multiplying on the right by the matrix $\widetilde{C}$, we obtain

(7.8) $\displaystyle\sum_{\rho\in{\rm
Irr}(G)}c_{\tau\rho}\int_{M_{\theta}}{\rm
ch}(\mathcal{R}_{\rho}\otimes\mathcal{R}_{\sigma}^{*})\hat{A}(M_{\theta})=\sum_{\rho\in{\rm
Irr}(G)}c_{\tau\rho}\frac{\eta_{R_{\rho}\otimes R_{\sigma^{*}}}}{2}$

for all $\tau\in{\rm Irr}(G)$. The left-hand-side of (7.8) can be written as

$\sum_{\rho\in{\rm Irr}(G)}c_{\tau\rho}\int_{M_{\theta}}\tilde{\rm
ch}(\mathcal{R}_{\rho})\tilde{\rm ch}(\mathcal{R}_{\sigma}^{*})\\\
+\sum_{\rho\in{\rm Irr}(G)}c_{\tau\rho}\int_{M_{\theta}}{\rm
ch}(\mathcal{R}_{\rho})\hat{A}(M_{\theta})+\sum_{\rho\in{\rm
Irr}(G)}c_{\tau\rho}\int_{M_{\theta}}{\rm
ch}(\mathcal{R}_{\sigma}^{*})\hat{A}(M_{\theta})\\\ =\sum_{\rho\in{\rm
Irr}(G)}c_{\tau\rho}\int_{M_{\theta}}\tilde{\rm
ch}(\mathcal{R}_{\rho})\tilde{\rm
ch}(\mathcal{R}_{\sigma}^{*})+\sum_{\rho\in{\rm
Irr}(G)}c_{\tau\rho}\frac{\eta_{R_{\rho}}}{2}+\sum_{\rho\in{\rm
Irr}(G)}c_{\tau\rho}\frac{\eta_{R_{\sigma}^{*}}}{2}$

for all $\tau\in{\rm Irr}(G)$. Note that we have used the fact that, since
$M_{\theta}$ has complex dimension $3$,
$\hat{A}(M_{\theta})=1+\left(\hat{A}(M_{\theta})\right)_{4}$. Moreover, by
(7.6) the third term above vanishes. Therefore

(7.9) $\displaystyle\sum_{\rho\in{\rm
Irr}(G)}c_{\tau\rho}\int_{M_{\theta}}\tilde{\rm
ch}(\mathcal{R}_{\rho})\tilde{\rm
ch}(\mathcal{R}_{\sigma}^{*})=\sum_{\rho\in{\rm
Irr}(G)}c_{\tau\rho}\frac{\eta_{R_{\rho}\otimes
R_{\sigma^{*}}}}{2}-\sum_{\rho\in{\rm
Irr}(G)}c_{\tau\rho}\frac{\eta_{R_{\rho}}}{2}.$

Taking the characters of (7.4) and summing over $i$ with alternating signs, we
obtain

$\displaystyle\left(\chi_{{\bf C}^{3}}(g)-\chi_{\Lambda^{2}{\bf
C}^{3}}(g)\right)\chi_{\tau}(g)=-\sum_{\rho\in{\rm
Irr}(G)}c_{\tau\rho}\chi_{\rho}(g).$

This gives

$\displaystyle\chi_{\tau\otimes\sigma^{*}}(g)=-\sum_{\rho\in{\rm
Irr}(G)}c_{\tau\rho}\frac{\chi_{\rho\otimes\sigma^{*}}(g)}{\chi_{{\bf
C}^{3}}(g)-\chi_{\Lambda^{2}{\bf C}^{3}}(g)}$

for all $g\in G\setminus\\{e\\}$. Summing over all such $g$ and using formula
(7.2) for the eta-invariant, we obtain

(7.10)
$\displaystyle-2\left(\delta_{\tau\sigma}-\frac{1}{|G|}\right)=\sum_{\rho\in{\rm
Irr}(G)}c_{\tau\rho}\eta_{\mathcal{R}_{\rho}\otimes\mathcal{R}_{\sigma}^{*}}$

Hence, (7.9) yields

$\displaystyle\sum_{\rho\in{\rm
Irr}(G)}c_{\tau\rho}\int_{M_{\theta}}\tilde{\rm
ch}(\mathcal{R}_{\rho})\tilde{\rm
ch}(\mathcal{R}_{\sigma}^{*})=-\left(\delta_{\tau\sigma}-\frac{1}{|G|}\right)-\left(\delta_{\tau\rho_{0}}-\frac{1}{|G|}\right)$

for all $\tau,\sigma\in{\rm Irr}(G)$. In particular, for all $\tau\in{\rm
Irr}_{0}(G)$, this becomes

(7.11) $\displaystyle\sum_{\rho\in{\rm
Irr}(G)}c_{\tau\rho}\int_{M_{\theta}}\tilde{\rm
ch}(\mathcal{R}_{\rho})\tilde{\rm
ch}(\mathcal{R}_{\sigma}^{*})=-\delta_{\tau\sigma}.$

Since ${\rm ch}(\mathcal{R}_{\rho_{0}})-1=0$, it follows that the matrix
$C=(c_{\tau\rho})_{\tau,\rho\in{\rm Irr}_{0}(G)}$ is invertible and

$\displaystyle\int_{M_{\theta}}\tilde{\rm ch}(\mathcal{R}_{\rho})\tilde{\rm
ch}(\mathcal{R}_{\sigma}^{*})=-\left(C^{-1}\right)_{\rho\sigma},$

for all $\rho,\sigma\in{\rm Irr}_{0}(G)$, which is precisely (1.4). ∎

######  Remark 7.12.

Note that formula (7.11) gives that the matrix $C$ is invertible. In the case
of a finite subgroup of $\mathrm{SL}(2,{\bf C})$, the invertibility of $C$ was
a direct consequence of the McKay Correspondence, given that $C$ is the Cartan
matrix associated to a simply-laced Dynkin diagram [McKay1980].

## References


# Dynamic Simulation of Splashing Fluids

James F. O’Brien and Jessica K. Hodgins
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332-0280
[obrienj $|$<EMAIL_ADDRESS>

Abstract:  In this paper we describe a method for modeling the dynamic
behavior of splashing fluids. The model simulates the behavior of a fluid when
objects impact or float on its surface. The forces generated by the objects
create waves and splashes on the surface of the fluid. To demonstrate the
realism and limitations of the model, images from a computer-generated
animation are presented and compared with video frames of actual splashes
occuring under similar initial conditions.

From the proceedings of Computer Animation ’95, pages 198–205. Held April
19–21, 1995, in Geneva, Switzerland.

##  1 Introduction

The world is filled with natural phenomena that are remarkable in their form
and movements: a tree blowing in the wind, lightning arcing across the sky and
the simple elegance of ripples in a pool of water. We, along with others in
the computer graphics community, are attempting to emulate nature’s splendor
in our computer animations. Our approach to this problem is to add realism to
computer-generated images and animations through the use of models and
simulations that approximate the physical laws of nature.

While many natural phenomena appear simple, the true complexity of these
behaviors is considerable. For example, a rock falling into a pool of water
involves complex interactions between the rock and the water. The rock strikes
the pool’s surface, displacing water and creating a disturbance that travels
outward from the impact as it forms waves on the surface. Fluid that is
displaced with sufficient force breaks free of the pool and flies through the
air as spray. Cohesion between water molecules causes the formation of sheets
of water, and air forced beneath the surface generates foam and bubbles.

Computer animations have been produced by skilled animators manipulating
models of a computer-generated world using their intuition about the real or
imaginary physical laws of the animated universe. To provide the animator with
higher level tools for producing realistic computer animations of natural
phenomena, researchers have investigated the use of dynamically based
simulations. By modeling the physical effects of natural forces, the modeled
systems can display realistic behavior without explicit specification of the
movement for each object. Unfortunately, physically correct simulations of
complex natural phenomena are often computationally expensive, requiring many
hours of computation time to generate only a few seconds of motion.

In the following sections, we describe a method for modeling the dynamic
behavior of a fluid. The simulation method allows animation of impacts to the
surface of the fluid, splashes, and the waves that arise as a result of an
impact. The simulation also models the behavior and effects of objects
floating on the surface of a fluid. Because our goal is to provide a method
suitable for computer animation, we have compromised accuracy for reduced
computation time. Nonetheless, these methods model many of the visual effects
seen in the real world, as demonstrated by the images in Figures 1 and 7.

Figure 1:  Comparison between a video image of a physical splash and a
synthetic image generated using dynamic models of the fluid and the impacting
object.

##  2 Background

Fluids have been simulated using a number of different techniques. One of the
most accurate is to solve the 3D Navier-Stokes equations describing the fluid
system [4]. This approach requires dividing the space that the fluid occupies
into a lattice of cells and computing the behavior of the fluid as it moves
through the cells. Although this approach is well suited for scientific
simulations requiring a high level of detail, the lower bound of the
computational cost increases with the cube of the model’s resolution, making
it impractical to use this method for interactive computer animation.

Particle systems have been used to model waterfalls and other forms of falling
water [9]. The dynamics equations used in these simulations model particles
falling under the influence of gravity without modeling interactions among the
particles. Phenomena such as spray or loosely packed water droplets are easily
modeled by this technique. Systems that allow particle interactions have been
used to model fountains of fluid and amorphous solids [5]. While a large
volume could potentially be modeled with interacting particles, the number of
particles needed to fill a volume will grow with the cube of the resolution of
the model, and despite the simplified equations for the dynamics this method
can be computationally expensive for a large volume of particles.

Fournier and Reeves [2] and Peachey [7] implemented techniques that model the
movement of explicit waves over the surface of a fluid. They used functions
that describe the shape of a wave to displace a model representing the free
surface of the fluid. Spray from the top of the wave crests is modeled using a
particle system. Although these methods work well for modeling the propagation
of wave-trains, it is more difficult to apply them to simulations where no
explicit set of wave-fronts exist, and this approach does not specify how
wave-trains should be created as a result of an impact to the surface.

An efficient approach to the problem of simulating solid volumes of water is
to use a system that models a simplified subset of the fluid dynamics. Kass
and Miller [3] used this approach and represented the surface with a height
field, modeling the flow between adjacent columns of fluid. With this method,
surface artifacts, such as waves, do not need to be explicitly specified
because they arise naturally from the physical conditions occurring within the
system. The model can be used for large volumes of fluid without incurring
huge computational costs as the size of the height field is proportional to
the square, rather than the cube, of the resolution. Chen and Lobo implemented
a similar system that uses a simplification of the 3D Navier-Stokes Equations
to model fluid behavior in the presence of moving obstacles [1].

Although these systems provide a good set of tools for modeling the behavior
of waves and falling water, we would like to model the interaction of water
with other objects. Our method extends the approach described by Kass and
Miller to allow a wider range of behaviors such as the splashes caused by
impacts and the behavior of floating objects.

##  3 Simulation Model

Figure 2:  The fluid model is a three part system containing subsystems for
volume, surface and spray. These subsystems interact to model the dynamics of
the fluid body.

To simulate the surface behavior of a body of fluid, we use a three-part
system where each subsystem corresponds to a physical area of the fluid body:
the main volume, the free surface of the fluid, and disconnected components of
the fluid (spray). Each subsystem models the effects of one aspect of the
overall behavior of the fluid. Taken together, the subsystems, along with the
interfaces between them, form the basis for the model (Figure 2).

###  3.1 Volume Model

Figure 3:  Each vertical column is connected to its eight neighbors through a
set of directional horizontal pipes. The control points for the surface are
sampled between the columns so that the lines of the surface grid separate
adjacent columns.

To model the volume that makes up the main body of the fluid, we use a
formulation that divides the body into a rectilinear grid of connected
columns. The model assumes vertical isotropy within the system and all fluid
properties within a column are assumed to be constant. Flow between these
columns occurs through a set of virtual pipes that connect adjacent columns.
Pipes connect columns along the diagonals as well as along the axis-aligned
directions (Figure 3).

The equations to determine flow in the pipes are derived from the physical
laws for hydrostatic pressure. The static pressure of a column in the grid at
position $[i,j]$, $H_{ij}$, is

$H_{ij}=h_{ij}\rho g+p_{0}$ (1)

where $\rho$ is the density of the fluid, $g$ is the acceleration due to
gravity, $p_{0}$ is the atmospheric pressure in the system, and $h_{ij}$ is
the height of the column at $[i,j]$. The height of a column is related to the
volume of that column, $V_{ij}$, by

$h_{ij}={V_{ij}\over d_{x}d_{y}}$ (2)

where we approximate the cross-sectional column area with the product of the
nominal distance between mesh grid points in the $x$ and $y$ directions,
$d_{[x|y]}$. When used to determine the flow, equation (1) is an approximation
because it relies on the assumption that the fluid is not moving rapidly [6].

In addition to the pressure due to gravity, we model external forces to
simulate the impact of objects on the surface of the fluid. These forces are
described in Section 4, but for now we include an additional term, the
external pressure $E_{ij}$, and rewrite equation (1) for total pressure
$P_{ij}$:

$P_{ij}=h_{ij}\rho g+p_{0}+E_{ij}.$ (3)

As previously mentioned, each column is connected to its eight neighbors by a
set of virtual pipes. A flow velocity is maintained for each pipe in the
system and at each time step a force accelerates the flow of fluid in the
pipe. The force is determined by measuring the pressure differential, $\Delta
P_{ij\rightarrow kl}$, across the pipe between column $[i,j]$ and one of its
eight neighbors, $[k,l]\in\eta_{ij}$. Substituting into the elementary
equation $F=ma$, we can derive the acceleration of the fluid in a pipe,
$a_{ij\rightarrow kl}$:

$a_{ij\rightarrow kl}={{c(\Delta P_{ij\rightarrow kl})}\over{m}}$ (4)

where $c$ is the cross-sectional area of the pipe and $m$ is the mass of the
fluid in the pipe. Expressing the mass of the fluid in a pipe of length $l$,
as $m=\rho cl$, and assuming atmospheric pressure to be constant, equation (4)
can be rewritten:

$a_{ij\rightarrow kl}={\rho g(h_{ij}-h_{kl})+E_{ij}-E_{kl}\over\rho l}.$ (5)

Assuming that the acceleration is constant over some time period, $\Delta t$,
the flow in the pipe, $Q_{ij\rightarrow kl}$, is

$Q^{t+\Delta t}_{ij\rightarrow kl}=Q^{t}_{ij\rightarrow kl}+\Delta
t(ca_{ij\rightarrow kl})$ (6)

and the net volume change of a column during a time interval is

$\Delta V_{ij}=\Delta t\sum_{kl\in\eta_{ij}}\left[{Q^{t+\Delta
t}_{ij\rightarrow kl}+Q^{t}_{ij\rightarrow kl}\over 2}\right].$ (7)

In order to conserve volume within the system, some constraints must be added
to the system. Because the fluid is incompressible, the flow at one end of a
pipe must be equal in magnitude with opposite sign to the flow at the other
end:

$\forall ij:\forall kl\in\eta_{ij}:Q_{ij\rightarrow kl}=-Q_{kl\rightarrow
ij}.$ (8)

A negative volume does not make physical sense, and the following inequality
ensures that the volume in all columns remain positive:

$V^{t+\Delta t}_{ij}\geq 0\iff V^{t}_{ij}\geq-\Delta V^{t}_{ij}.$ (9)

To keep equation (9) true at the end of each integration step, the volume of
each column is tested. If a column has a negative volume, then all pipes that
are removing fluid from that column are scaled back. This procedure is
repeated until all columns in the grid have a positive volume.

The columns on the outer edges of the grid have pipes that lead out of the
area where columns are defined. Because no columns exist at the other end of
the virtual pipe, flow conditions must be specified to model boundary
conditions. To model a barrier, such as a wall, the flow is set to zero.
Phenomena such as fluid sources or sinks can be created by setting the flow to
a positive or negative constant.

###  3.2 Surface Model

The surface subsystem allows external objects to interact with the fluid
system. Objects that collide with or float on the surface exert forces on the
surface model. These forces are propagated as external pressure to the volume
subsystem. The vertical positioning of the surface elements is determined by
the volume of the columns.

The model for the surface subsystem is a rectilinear grid of control points
that define a mesh. The control points of the mesh map onto the column grid of
the volume model so that the control points are sampled between adjacent
columns (Figure 3). The vertical position of a grid point, $z_{ij}$, is
determined by averaging the height of the four columns surrounding the grid
point:

$z_{ij}={h_{i,j}+h_{i,j+1}+h_{i+1,j}+h_{i+1,j+1}\over 4}.$ (10)

Forces applied to each control point are reformulated as an external pressure,
$E_{ij}$, applied to the columns contacted by that control point:

$E_{ij}=-{f_{e}\over 4d_{x}d_{y}}$ (11)

where $f_{e}$ is the force applied to the control point. A downward force
(negative $z$ direction) will result in an increase in external pressure,
hence the negative sign. Grid points contact four columns and the external
pressure is evenly distributed across these columns.

###  3.3 Spray Model

To model droplets that are disconnected from the main body of the fluid, we
implemented a particle system using techniques similar to those described by
Reeves [8] and Sims [9]. Particles are created when the upward velocity of a
portion of the surface exceeds a threshold111The value for the threshold can
be adjusted to modify the appearance of the splash. Lower threshold values
will cause more particles to be generated over a larger area. The values used
to generate the images in this paper were between $2.0$ and $2.5m/s$.. Because
the vertical position of a column is determined by the volume of fluid in the
column, the column’s vertical velocity can be determined by the rate of change
in volume (the net flow into the column from its neighbors):

$\dot{h}_{ij}=\sum_{kl\in\eta_{ij}}{Q_{kl\rightarrow ij}\over d_{x}d_{y}}.$
(12)

The vertical velocity at the surface is

$\dot{z}_{ij}={\dot{h}_{i,j}+\dot{h}_{i,j+1}+\dot{h}_{i+1,j}+\dot{h}_{i+1,j+1}\over
4}$ (13)

and the horizontal velocity of the fluid at the surface is

$\dot{x}_{ij}={Q_{i,j\rightarrow i+1,j}+Q_{i,j+1\rightarrow i+1,j+1}\over 2}$
(14) $\dot{y}_{ij}={Q_{i,j\rightarrow i,j+1}+Q_{i+1,j\rightarrow i+1,j+1}\over
2}.$ (15)

Figure 4:  Initial velocity of ejected particles. The horizontal arrows at
bottom indicate direction of flow between columns, the thick vertical arrows
indicate upward velocity in columns and the thin arrows at top indicate the
velocity vectors for ejected particles.

When an area of the surface has an upward velocity greater than the threshold,
particles are distributed uniformly over that area and the initial velocities
for the particles are interpolated from the surface velocities (Figure 4).

To conserve the total volume in the system, the volume of each particle is
subtracted from the column from which it was created. Once created, the
particles fall under the influence of gravity and do not interact with each
other.

Particles are removed from the system when they fall back onto the surface,
and the volume of each particle is added to the column that absorbed it.
Particles that fly away from the fluid body are destroyed when they encounter
a ground plane or strike some other object.

##  4 External Objects

When an object collides with the fluid surface, we model the motion of the
fluid and of the object by computing the forces resulting from the collision
and applying them to the fluid surface and the object. The behavior of an
object with mass $m$, at a location $s_{0}$, moving at a velocity $\dot{s}$,
is described by:

$s=s_{0}+\dot{s}t+{1\over 2}\left(g+{f_{o}\over m}\Delta t^{2}\right)$ (16)

provided that the force, $f_{o}$, and the acceleration, $g$, remain constant
over the time interval.

The force acting on the fluid is equal in magnitude with opposite sign to the
force acting on the object ($f_{o}=-f_{e}$). If we assume that the object
remains in contact with the surface and does not bounce off, then the location
of the mass is determined by the location of the surface. The force acting at
the impact could be computed by solving the nonlinear system derived from
equation (16) and the fluid model system.

The force, $f_{o}$, is bound in the range $[0,f_{max}]$, where $f_{max}$ is
the force that would cause the mass to have a final position equal to the
initial position of the surface:

$f_{max}=m\left({2(z-s_{0}-\dot{s}t)\over t^{2}}-g\right).$ (17)

A search could be done to find the force that would minimize the difference
between the final position of the mass and the surface.

An alternative solution that exchanges accuracy for speed is to select $f_{o}$
in the range $[0,f_{max}]$ based on a heuristic measure of the expected
deceleration of the mass as it strikes the surface, and perform the
integration on the fluid system using the approximated value. Thus objects
with a small impact face would have lower values, and objects with a large
impact face would have larger values. We used this method to produce the
images for this paper.

##  5 Results and Conclusions

In developing our simulation model, we made several approximations to simplify
the calculations and to decrease the required computation time. These
simplifications introduce errors into the final solution. We justify this
approach by observing that computer animations do not generally require the
same degree of accuracy as a scientific experiment. Furthermore, animation is
often used to duplicate a known phenomenon, such as a ball falling into water,
rather than an unknown phenomena, such as the air flow within an experimental
jet engine. Because we are duplicating known phenomena, we can use recorded
footage or physical intuition to “tune” the simulation, thus allowing the
animator to create the desired effect while retaining many of the motion
characteristics exhibited by water in the real world. For example to improve
the correspondence shown in Figure 7 we adjusted the time during which the
falling object was in contact with the water’s surface. Other adjustable
parameters include the number of particles to be spawned by portions of the
surface that have exceeded the threshold, the volume of the ejected particles
and the amount of damping used during flow integration.

The images in Figure 5 were rendered from data generated using the methods
described above. These images show the splash that results when a physically
based simulation of a rigid body model of a human diver enters the water after
a 10 meter platform dive [10]. For the purposes of computing the intersection
of the diver with the pool’s surface, the human’s body was approximated by a
set of ellipsoids, each representing one of the diver’s body parts. The pool
covers an area of $16\times 16$ meters represented with a $241\times 241$
grid.

The system we have implemented runs at interactive speeds for simple problems
(about one-tenth real-time for a $11\times 11$ mesh), but the response time
degrades as the resolution of the simulation mesh becomes finer and as
multiple objects interact with the fluid surface. The images in Figure 6 show
three balls floating on a $2$ meter-square body of water. They were generated
using data from a simulation on a $61\times 61$ grid which took about 3
minutes actual time to complete $2$ seconds of simulation time. The simulation
for the images in Figure 5 took longer, a little over an hour for $20$ seconds
of motion, due to the higher resolution of the grid and the larger number of
objects. These timings were performed on an SGI $Indigo^{2}$ with an R4000
processor running at 100mHz. Simulation time might be improved by using a non-
uniform grid that would have a higher resolution in areas where impacts would
occur and lower resolution elsewhere.

The full spectrum of behaviors exhibited by fluids is very complex, and the
model we have described allows the relatively fast simulation of a subset of
these behaviors including waves, impacts, splashes, floating objects and other
behaviors that fit within the assumption of vertical isotropy. Phenomena that
occur primarily due to vertical effects within the fluid, such as turbulence
out of the horizontal plane, are beyond the scope of this model. Extending the
model to include additional horizontal effects or phenomena such as flowing
streams should be possible as would extensions to add foam or bubbles.

The visual appearance of the model could be improved by using a more
sophisticated spray model that took into account the cohesion between drops of
water such as the one described in [5]. Figure 7 shows a comparison between an
actual splash and a simulation where we attempted to match the initial
conditions of the actual splash. Although the simulated motion is similar to
that appearing in the captured video images, a clear difference can be seen in
the patterns formed by the ejected water droplets. The addition of a more
sophisticated spray model would allow the formation of sheets in the water,
improving the perceived realism of the splashes.

It is interesting to note that despite the obvious differences between the
images of real and simulated splashes, the resulting computer animation
appears realistic when viewed alone. An investigation as to what minimal set
of motion qualities are essential for believability might enable further
simplification and simulation at real-time speeds. Real-time execution would
allow the inclusion of a simulation such as this into an interactive virtual
reality system.

High resolution copies of the images in this paper and animated image
sequences are available through the World Wide Web at the URL

“http://www.cc.gatech.edu/gvu/animation/ Animation.html”.

## Acknowledgments

The authors would like to thank Wayne Wooten and Debbie Carlson for their help
in rendering images and Walter Patterson and Adrian Ferrier for their
suggestions on simulation methods.

This project was supported in part by NSF Grant No. IRI-9309189 and funding
from the Advanced Research Projects Agency.

## References

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  * [2] Fournier, A. and Reeves, W., “A Simple Model of Ocean Waves,” Proceedings of SIGGRAPH ‘86, in Computer Graphics, Vol. 20, No. 4, pp. 75-84, 1986.
  * [3] Kass, M., and Miller, G., “Rapid, Stable Fluid Dynamics for Computer Graphics,” Proceedings of SIGGRAPH ‘90, in Computer Graphics, Vol. 24, No. 4, pp. 49-57, 1990.
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  * [6] Munson, B., Young, D., and Okiishi, T., Fundamentals of Fluid Mechanics, John Wiley and Sons: Canada, 1990.
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  * [8] Reeves, W., “Particle Systems—A Technique for Modeling a Class of Fuzzy Objects,” Computer Graphics, Vol. 17, No. 3, pp. 359-376, 1983.
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  * [10] Wooten, W., and Hodgins, J., 1995. “Simulation of Human Diving,” Proceedings of Graphics Interface ’95, in press 1995.

Figure 5:  Image sequence from a computer-generated animation of a human diver
entering the water. Frames progress from top to bottom and from left to right
at intervals of 0.167 seconds. The motion of the diver was simulated using
rigid body dynamics and control laws for a variety of platform dives[10]. The
images were rendered with Photo-Realistic RenderMan on Silicon Graphics
workstations.

Figure 6:  Objects floating on a fluid surface. Three balls of equal density
but different sizes were dropped onto the fluid. After the balls settled onto
the surface, a fourth ball of higher density was dropped into the fluid. The
floating balls bob on the surface as the disturbance caused by the heavier
ball moves past them.

Figure 7: [See color plate.] A side-by-side comparison of video of an actual
splash and images from a computer simulation in which we have attempted to
match the initial conditions to those of the actual splash. The images proceed
top to bottom and left to right and are spaced at intervals of $0.233$
seconds. While the patterns formed in both image sequences are qualitatively
similar, substantial differences exist between the two. For example, the video
footage shows sheeting effects that are not present in the synthetic images.
The vertical column that appears in the center of the actual splash in frame
$3$ does not appear in the simulated splash until frame $4$. Finally, the
motion of the water in the video footage is more irregular.

A side-by-side comparison of video of an actual splash and images from a
computer simulation.