\newcommand{\R}{{\Bbb R}}
\newcommand{\C}{{\Bbb C}}
\newcommand{\M}{{\cal M}}

\chapter*{Preface}

Hyperk\"ahler manifolds have appeared at first within the framework
of differential geometry as an example of Riemannian manifolds with
holonomy of a special restricted type. However, they have soon
exhibited such diverse and unexpected links with various branches of
mathematics that now hyperk\"ahler geometry by itself forms a
separate research subject. Among the traditional areas fused within
this new subject are differential and algebraic geometry of complex
manifolds, holomorphic symplectic geometry, geometric representation
theory, Hodge theory and many others. The most recent addition to
the list is the link between hyperk\"ahler geometry and theoretical
physics: it turns out that hyperk\"ahler manifolds play a critical
part in the modern version of the string theory, which is in itself
the basis for the future unified field theory and quantum gravity.

Perhaps because the structure of a hyperk\"ahler manifold is so
rich, such manifolds are quite rare and hard to construct. Thus
every new example or a class of examples of hyperk\"ahler manifolds
is of considerable interest. The main goal of this book is to
describe two recent developments in this area, one dealing with
compact hyperk\"ahler manifolds, the other - with a rather general
class of non-compact ones. In order to make the presentation as
self-contained as possible, we have included much preliminary
material and gave an exposition of most of the basic facts of the
theory. We believe that this makes it possible to read the book
without any prior knowledge of hyperk\"ahler geometry and to use it
as an introduction to the subject. On the other hand, it is our hope
that the new examples of hyperk\"ahler manifolds constructed here
would be of interest to a specialist in the field.

For the detailed description of the new results proved in the book
the reader is referred to the introductions to the individual
chapters. In this general introduction we restrict ourselves to
giving a brief historical overview of the theory of hyperk\"ahler
manifolds and indicating the place of our results in the general
framework of hyperk\"ahler geometry.

\section*{Historical overview}
Recall that one can define a K\"ahler manifold as a Riemannian
manifold $M$ equipped with an almost complex structure parallel with
respect to the Levi-Civita connection. It is well-known that such an
almost complex structure is automatically integrable, thus every
K\"ahler $M$ is a complex manifold. Moreover, the Riemannian metric
and the complex structure together define a non-degenerate closed
$2$-form $\omega$ on $M$, thus making $M$ a symplectic manifold.

The notion of a hyperk\"ahler manifold is obtained from this
definition by replacing the field of complex numbers with the
algebra of quaternions. A hyperk\"ahler manifold is by definition a
Riemannian manifold $M$ equipped with two anticommuting almost
complex structures parallel with respect to the Levi-Civita
connection. These two almost complex structures generate an action
of the quaternion algebra in the tangent bundle to $M$, which is
also parallel. Every quaternion $h$ with $h^2=-1$ defines an almost
complex struture on $M$. This almost complex structure is parallel,
hence integrable. Thus every hyperk\"ahler manifold is canonically
complex, and in many different ways. 

It is convenient to fix once and for all a quaternion $I$ with
$I^2=-1$ and to consider a hyperk\"ahler manifold $M$ as complex by
means of the corresponding complex structure. It is canonically
K\"ahler. Moreover, one can combine the other complex structures on
$M$ with the Riemannian metric and obtain, apart from the K\"ahler
$2$-form $\omega$, a canonical closed non-degenerate holomorphic
$2$-form $\Omega$ on $M$. Thus every hyperk\"ahler manifold carries
canonical K\"ahler and holomorphically symplectic structures.

A Riemannian manifold of dimension $4n$ is hyperk\"ahler if and only
if its holonomy group is contained in the symplectic group
$Sp(n)$. As such, hyperk\"ahler manifolds first appeared in the
classification of all possible holonomy groups given by
M. Berger\index{names}{Berger, M.} \cite{Ber}. The term
``hyperk\"ahler manifold'' was introduced by E,
Calabi\index{names}{Calabi, E.} in his paper \cite{Cal}, where he
also constructed several non-trivial examples of hyperk\"ahler
metrics. All of Calabi's examples were non-compact. In fact, all
these manifolds were total spaces of cotangent bundles to K\"ahler
manifolds.

At the time of the original paper of Calabi\index{names}{Calabi,
E.}'s, it seemed that hyperk\"ahler manifolds are a rather unusual
phenomenon, not unlike, for example, sporadic simple
groups. However, starting with the beginning of the eighties, there
has been a wave of discoveries in the area, and we now know a lot of
examples of hyperk\"ahler metrics which occur ``in the
nature''. These examples split naturally into two groups, depending
on whether the underlying complex manifold is compact.

A powerful tool for constructing compact hyperk\"ahler manifolds is
the famous Calabi-Yau\index{names}{Yau, S.-T.}\index{names}{Calabi,
E.} Theorem, which provides a Ricci-flat K\"ahler metric on every
compact manifold of K\"ahler type with trivial canonial bundle. Its
usefulness for the hyperk\"ahler geometry lies in the fact that
every hyperk\"ahler manifold is canonically holomorphically
symplectic. The converse statement is far from being true: a
holomorphically symplectic K\"ahler manifold does not have to be
hyperk\"ahler. However, the converse is true if we require in
addition that the hololomorphic symplectic form is parallel with
respect to the Levi-Civita connection. It is easy to see that every
K\"ahler manifold equipped with a parallel holomorphic symplectic
form is hyperk\"ahler.

In general it is very hard to check whether a holomorphic symplectic
form on a compact K\"ahler manifold is parallel. However, there
exists a theorem of S. Bochner\index{names}{Bochner, S.}'s
\cite{Boch} which claims that this is always the case when the
K\"ahler metric is Ricci-flat.  Since the canonical bundle of a
holomorphically symplectic manifold is trivial, indeed, trivialized
by the power of the symplectic from, the Calabi-Yau Theorem shows
that every compact holomorphically symplectic manifold of K\"ahler
type carries a Ricci-flat K\"ahler metric. This metric must be
hyperk\"ahler by the Bochner Theorem. Thus every compact
holomorphically symplectic manifold of K\"ahler type is
hyperk\"ahler.

Well-known examples of compact holomorphically symplectic manifolds
of dimension $2$ are abelian complex surfaces and K3 surfaces. In
higher dimensions non-trivial examples of such manifolds have been
given by A. Beauville\index{names}{Beauville, A.} \cite{Beau},
extending earlier results of A. Fujiki\index{names}{Fujiki, A.}
\cite{Fu}. Beauville's examples are the Hilbert schemes of points on
an abelian or a K3 surface. All these manifolds are of K\"ahler
type, hence hyperk\"ahler.

Given a compact hyperk\"ahler manifold $M$, one can consider moduli
spaces $\M$ of stable holomorphic vector bundles on $M$ with fixed
Chern classes. When $M$ is $4$-dimensional, hence either an abelian
surface or a K3 surface, the moduli space $\M$ is known to be smooth
and hyperk\"ahler (see \cite{Kob} for an excellent exposition of
these results). When the Chern classes are such that $\M$ is
compact, we obtain in this way a new compact hyperk\"ahler
manifold. This situation was studied in detail by
S. Mukai\index{names}{Mukai, S.} in \cite{Muk1}, \cite{Muk2}. It is
now known that some of the compact moduli spaces of stable bundles
on a K3 surface are deformationally equivalent to hyperk\"ahler
manifolds of the type constructed by
Beauville\index{names}{Beauville, A.}. Conjecturally all these
moduli spaces lie in the Beauville's deformation class. We refer the
reader to \cite{Huy} for an overview of this subject.

These results were partially generalized to higher dimensions in
\cite{Ver}. The moduli space of stable bundles on a
higher-dimensional hyperk\"ahler manifold is no longer automatically
smooth. However, it is a singular hyperk\"ahler variety in the sense
of \cite{Ver2}. This implies, in particular, that it is
hyperk\"ahler near every smooth point. Moreover, a singular
hyperk\"ahler veriety can be canonically desingularized to a smooth
hyperk\"ahler manifold.

An outstanding problem in the theory of compact hyperk\"ahler
manifolds is to find an example of such a manifold which would be
simply connected and not equivalent deformationally to a product of
the ones constructed by Beauville. Several important results
(\cite{OGra}) on this subject have appeared recently, but the
problem is still not completely closed. The first chapter of the
present book describes a different approach to this subject. The
results of Mukai\index{names}{Mukai, S.} and Verbitsky are extended
and strengthened in a way that conjecturally leads to the hoped-for
examples of compact hyperk\"ahler manifolds belonging to a new
deformation class.

\hfill

Both the Calabi-Yau\index{names}{Calabi, E.}\index{names}{Yau,
S.-T.} Theorem and the Bochner\index{names}{Bochner, S.} Theorem are
results of a global nature and cannot be used to construct {\em
non-compact} hyperk\"ahler manifolds. Two general methods are known
which can be used for this purpose. The first one uses the link
between the theory of hyperk\"ahler manifolds and the holomorphic
geometry provided by the notion of the twistor space. The twistor
space construction, introduced by R. Penrose\index{names}{Penrose,
R.}, has a long and glorious history. The reader is referred to
\cite{HiKLR} for a detailed exposition of this subject. Here we only
mention that the twistor space $X$ for a $4n$-dimensional
hyperk\"ahler manifold $M$ is a holomorphic manifold of complex
dimension $2n+2$ canonically associated to $M$, and that the
holomorphic structure on the twistor space $X$ embodies most of the
differential-geometric properties of the hyperk\"ahler manifold $M$.

There exists a theorem which allows one to reconstruct a
hyperk\"ahler manifold $M$ from its twistor space $X$ equipped with
some additional structures. Therefore, if it is impossible to
construct $M$ explicitly, one can construct $X$ instead. In this way
one can, for example, construct an infinite-dimensional family of
hyperk\"ahler metrics on the vector space $\C^{2n}$ (see
\cite{HiKLR}).

Another general method of constructing non-compact hyperk\"ahler
manifolds is the famous hyperk\"ahler reduction technique introduced
by Hitchin\index{names}{Hitchin, N. J.} \cite{HitchLefschetz},
Hitchin et al. \cite{HiKLR}. It is this technique that led to recent
discoveries of hyperk\"ahler structures on many interesting
manifolds. Fortunately, the hyperk\"ahler reduction is well-covered
in the literature (see, for example, \cite{HitchBour}). Therefore we
will only list the most important examples of hyperk\"ahler
manifolds obtained by this method.
\begin{itemize}
\item One of the original examples of hyperk\"ahler manifolds given
by Calabi was the total space of the cotangent bundle to a complex
projective space. This manifold admits an elementary construction
via the hyperk\"ahler reduction. 

\item Let $\Gamma \in SL(2,\C)$ be a finite subgroup. The quotient
$\C^2/\Gamma$ has an isolated singularity at $0$ which can be blown
up to a non-singular complex manifold. This manifold can be equipped
with a hyperk\"ahler metric by means of the hyperk\"ahler
reduction. The resulting metric is asymptotically locally Euclidean
(ALE) at infinity. It was discovered by P.B. Kronheimer
\index{names}{Kronheimer, P. B.} \cite{Kron3} and generalized by
Kronheimer and H. Nakajima \index{names}{Nakajima, H.}
\cite{KN}. Nakajima \cite{Naka} has recently generalized this example
even further and obtained a whole family of hyperk\"ahler manifolds
called the quiver varieties.

\item Let $G$ be a compact Lie group, and let $LG$ be the
(infinite-dimensional) Lie group of maps from the unit cricle $S^1$
to $G$. A hyperk\"ahler metric on the quotient $LG/G$ has been
constructed by S. Donaldson\index{names}{Donaldson, S.} \cite{Don}.

\item Let $S$ be a compact complex curve, and let $\M$ be the moduli
space of bundles on $S$ equipped with a flat
connection. Hitchin\index{names}{Hitchin, N. J.}  \cite{HitchCurves}
has constructed a hyperk\"ahler structure on the space $\M$. This
construction has been recently generalized by
C. Simpson\index{names}{Simpson, C.} \cite{Simp} to the case when
$S$ is an arbitrary projective complex manifold.
\end{itemize}

A related group of examples is obtained by considering the solutions
to a system of ordinary differential equations called the Nahm
equations. These equations first appeared in the work of
Schmid\index{names}{Schmid, W.} \cite{Schmid} on the variations of
Hodge structures. They have been used by
Kronheimer\index{names}{Kronheimer, P. B.}
\cite{Kron1},\cite{Kron2} to obtain a hyperk\"ahler metric on an
orbit in the coadjoint representstion of an arbitrary semisimple
complex ie group $G$. More recently Kronheimer's method was used in
papers \cite{Biq},\cite{DS} to obtain new examples of hyperk\"ahler
manifolds. This method is not directly related to the hyperk\"ahler
reduction but shares many features with it. Some of the metrics
obtained by reduction can be also constructed via the Nahm
equations, and vice versa.

Unfortunately, while hyperk\"ahler reduction is a generous source of
new hy\-per\-k\"ah\-ler metrics, it can only be pushed so far. One
of the problems which seems to lie outside of the scope of this
approach is that of constructing a hyperk\"ahler metric on the total
space of the cotangent bundle to a non-homogeneous K\"ahler
manifold. The second chapter of the present book describes such a
construction. This construction is local and works for an arbitrary
K\"ahler manifold. The methods used are necessarily different from
the ones already exploited in the literature and consist of explicit
but cumbersome application of the deformation theory in the spirit
of Kodaira and Spencer\index{names}{Kodaira,
K.}\index{names}{Spencer, D.} \cite{Kod}.

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