identified in a neighbourhood of $X$.

Let $M$ be a hyperkähler manifold, $\operatorname{Tw}(M)$ its twistor space,
and $\pi:\;\operatorname{Tw}(M){\>\longrightarrow\>}{\mathbb{C}}P^{1}$ the
twistor projection. Let $l\subset\operatorname{Tw}(M)$ be a rational curve,
such that the restriction of $\pi$ to $l$ is an identity. Such a curve gives a
section of $\pi$, and vice versa, every section of $\pi$ corresponds to such a
curve. The set of sections of the projection $\pi$ is called the space of
twistor lines, denoted by $\operatorname{Lin}$, or $\operatorname{Lin}(M)$.
This space is equipped with complex structure, by Douady ([Do]).

Let $m\in M$ be a point. Consider a twistor line
${s_{m}}:\;I{\>\longrightarrow\>}(I\times m)\in{\mathbb{C}}P^{1}\times
M=\operatorname{Tw}$. Then $s_{m}$ is called a horisontal twistor line. The
space of horisontal twistor lines is a real analytic subvariety in
$\operatorname{Lin}$, denoted by $\operatorname{Hor}$, or
$\operatorname{Hor}(M)$. Clearly, the set $\operatorname{Hor}$ is naturally
identified with $M$.

Proposition 6.4:  (Hitchin, Karlhede, Lindström, Roček) Let $M$ be a
hyperkähler manifold, $\operatorname{Tw}(M)$ its twistor space, $I$,
$J\in{\mathbb{C}}P^{1}$ induced complex structures, and $\operatorname{Lin}$
the space of twistor lines. The complex manifolds $(M,I)$ and $(M,J)$ are
naturally embedded to $\operatorname{Tw}(M)$:

$(M,I)=\pi^{-1}(I),\ \ (M,J)=\pi^{-1}(J).$

Consider a point $s\in\operatorname{Lin}$,
$s:\;{\mathbb{C}}P^{1}{\>\longrightarrow\>}\operatorname{Tw}(M)$. Let

$ev_{I,J}:\;\operatorname{Lin}(M){\>\longrightarrow\>}(M,I)\times(M,J)$

be the map defined by $ev_{I,J}(s)=(s(I),s(J))$. Assume that $I\neq J$. Then
there exists a neighbourhood $U$ of
$\operatorname{Hor}\subset\operatorname{Lin}$, such that the restriction of
$ev_{I,J}$ to $U$ is an open embedding.

Proof: [HKLR], [V-d3].

Consider the anticomplex involution $i$ of ${\mathbb{C}}P^{1}\cong S^{2}$
which corresponds to the central symmetry of $S^{2}$. Let
$\iota:\;\operatorname{Tw}{\>\longrightarrow\>}\operatorname{Tw}$ be the
corresponding involution of the twistor space
$\operatorname{Tw}(M)={\mathbb{C}}P^{1}\times M$,
$(x,m){\>\longrightarrow\>}(i(x),m)$. It is clear that $\iota$ maps
holomorphic subvarieties of $\operatorname{Tw}(M)$ to holomorphic
subvarieties. Therefore, $\iota$ acts on $\operatorname{Lin}$ as an
anticomplex involution. For $J=-I$, we obtain a local identification of
$\operatorname{Lin}$ in a neighbourhood of $\operatorname{Hor}$ with
$(M,I)\times(M,-I)$, that is, with $(M,I)$ times its complex conjugate.
Therefore, the space of twistor lines is a complexification of $(M,I)$. The
natural anticomplex involution of 6.1 coincides with $\iota$. This gives an
identification of $\operatorname{Hor}$ and the real analytic manifold
underlying $(M,I)$.

We shall explain how to construct the natural ${\mathbb{C}}^{*}$-action on a
local ring of a hyperkähler manifold, using the machinery of twistor lines.

Fix a point $x_{0}\in M$ and induced complex structures $I$, $J$, such that
$I\neq\pm J$. Let $V_{1}$, $V_{2}$ be neighbourhoods of
$s_{x_{0}}\in\operatorname{Lin}$, and $U_{1}$, $U_{2}$ be neighbourhoods of
$(x_{0},x_{0})$ in $(M,I)\times(M,-I)$, $(M,J)\times(M,-J)$, such that the
evaluation maps $ev_{I,-I}$, $ev_{J,-J}$ induce isomorphisms

$ev_{I,-I}:\;V_{1}\tilde{\>\longrightarrow\>}U_{1},\ \ \ \
ev_{J,-J}:\;V_{2}\tilde{\>\longrightarrow\>}U_{2}.$

Let $B$ be an open neighbourhood of $x_{0}\in M$, such that
$(B,I)\times(B,-I)\subset U_{1}$ and $(B,I)\times(B,-I)\subset U_{2}$. Denote
by $V_{I}\subset V_{1}$ be the preimage of $(B,I)\times(B,-I)$ under
$ev_{I,-I}$, and by $V_{J}\subset V_{2}$ be the preimage of
$(B,J)\times(B,-J)$ under $ev_{J,-J}$. Let
$p_{I}:\;V_{I}{\>\longrightarrow\>}(B,I)$ be the evaluation,
$s{\>\longrightarrow\>}s(I)$, and $e_{I}:\;(B,I){\>\longrightarrow\>}V_{1}$
the map associating to $x\in B$ the unique twistor line passing through
$(x,x_{0})\subset(B,I)\times(B,-I)$. In the same fashion, we define $e_{J}$
and $p_{J}$. We are interested in the composition

$\Psi_{I,J}:=e_{I}\circ p_{J}\circ e_{J}\circ
p_{I}:\;(B_{0},I){\>\longrightarrow\>}(B,I)$

which is defined in a smaller neighbourhood $B_{0}\subset B$ of $x_{0}\in M$.

The following proposition is the focal point of this Subsection: we explain
the map $\Psi_{I,J}$ of [V-d2], [V-d3] is geometric terms (in [V-d2], [V-d3]
this map was defined algebraically).

Proposition 6.5:  Consider the map
$\Psi_{I,J}:\;(B_{0},I){\>\longrightarrow\>}(B,I)$ defined above. By
definition, $\Psi_{I,J}$ preserves the point $x_{0}\in B_{0}\subset B$. Let
$d\Psi_{I,J}$ be the differential of $\Psi_{I,J}$ acting on the tangent space
$T_{x_{0}}B_{0}$. Assume that $I\neq\pm J$. Then $d\Psi_{I,J}$ is a
multiplication by a scalar $\lambda\in{\mathbb{C}}$, $0<|\lambda|<1$.

Proof: The map $\Psi_{I,J}$ was defined in [V-d2], [V-d3] using the
identifications between the real analytic varieties underlying $(M,I)$ and
$(M,J)$. We proved that $\Psi_{I,J}$ defined this way acts on $T_{x_{0}}B_{0}$
as a multiplication by the scalar $\lambda\in{\mathbb{C}}$, $0<|\lambda|<1$.
It remains to show that the map $\Psi_{I,J}$ defined in [V-d2], [V-d3]
coincides with $\Psi_{I,J}$ defined above.

Consider the natural identification

$(B,I)\times(B,-I)\sim(B,J)\times(B,-J),$

which is defined in a neighbourhood $B_{\mathbb{C}}$ of $(x_{0},x_{0})$. There
is a natural projection $a_{I}:\;B_{\mathbb{C}}{\>\longrightarrow\>}(M,I)$.
Consider the embedding $b_{I}:\;(B,I){\>\longrightarrow\>}B_{\mathbb{C}}$,
$x{\>\longrightarrow\>}(x,x_{0})$, defined in a neighbourhood of
$x_{0}\in(B,I)$. In a similar way we define $a_{J}$, $b_{J}$. In [V-d2],
[V-d3] we defined $\Psi_{I,J}$ as a composition $b_{I}\circ a_{J}\circ
b_{J}\circ a_{I}$. Earlier in this Subsection, we described a local
identification of $(B,I)\times(B,-I)$ and $\operatorname{Lin}(B)$. Clearly,
under this identification, the maps $a_{I}$, $b_{I}$ correspond to $p_{I}$,
$e_{I}$. Therefore, the definition of $\Psi_{I,J}$ given in this paper is
equivalent to the definition given in [V-d2], [V-d3].

###  6.2 The automorphism $\Psi_{I,J}$ acting on hyperholomorphic
sheaves

In this section, we prove that hyperholomorphic sheaves are equivariant with
respect to the map $\Psi_{I,J}$, considered as an automorphism of the local
ring ${\cal O}_{x_{0}}(M,I)$.

Theorem 6.6:  Let $M$ be a hyperkähler manifold, not necessarily compact,
$x_{0}\in M$ a point, $I$ an induced complex structure and $F$ a reflexive
sheaf over $(M,I)$ equipped with a hyperholomorphic connection. Let $J\neq\pm
I$ be another induced complex structure, and $B_{0}$, $B$ the neighbourhoods
of $x_{0}\in M$ for which the map $\Psi_{I,J}:\;B_{0}{\>\longrightarrow\>}B$
was defined in 6.1. Assume that $\Psi_{I,J}:\;B_{0}{\>\longrightarrow\>}B$ is
an isomorphism. Then there exists a canonical functorial isomorphism of
coherent sheaves

$\Psi_{I,J}^{F}:\;F{\left|{}_{{\phantom{|}\\!\\!}_{B_{0}}}\right.}{\>\longrightarrow\>}\Psi_{I,J}^{*}(F{\left|{}_{{\phantom{|}\\!\\!}_{B}}\right.}).$

Proof: Return to the notation introduced in Subsection 6.1. Let $W:=V_{I}\cap
V_{J}$. By definition of $V_{I}$, $V_{J}$, the evaluation maps produce open
embeddings

$ev_{I,-I}:\;\operatorname{Lin}(W)\hookrightarrow(W,I)\times(W,-I),$

and

$ev_{J,-J}:\;\operatorname{Lin}(W)\hookrightarrow(W,J)\times(W,-J),$

Let $S\subset W$ be the singular set of
$F{\left|{}_{{\phantom{|}\\!\\!}_{W}}\right.}$,
$\operatorname{Tw}(S)\subset\operatorname{Tw}(W)$ the corresponding embedding,
and $L_{0}\subset\operatorname{Lin}(W)$ be the set of all lines
$l\in\operatorname{Lin}(W)$ which do not intersect $\operatorname{Tw}(S)$.
Consider the maps

$p_{I}:\;L_{0}\hookrightarrow(W,I)\backslash S$

and

$p_{J}:\;L_{0}\hookrightarrow(W,J)\backslash S$

obtained by restricting the evaluation map
$p_{I}:\;\operatorname{Lin}(M){\>\longrightarrow\>}(M,I)$ to
$L_{0}\subset\operatorname{Lin}(M)$. Since $F$ is equipped with a
hyperholomorphic connection, the vector bundle
$F{\left|{}_{{\phantom{|}\\!\\!}_{(M,J)\backslash S}}\right.}$ has a natural
holomorphic structure. Let
$\underline{F}_{1}:=p_{I}^{*}\left(F{\left|{}_{{\phantom{|}\\!\\!}_{(M,I)\backslash
S}}\right.}\right)$ and
$\underline{F}_{2}:=p_{J}^{*}\left(F{\left|{}_{{\phantom{|}\\!\\!}_{(M,J)\backslash
S}}\right.}\right)$ be the corresponding pullback sheaves over $L_{0}$, and
$F_{1}$, $F_{2}$ the sheaves on $\operatorname{Lin}(W)$ obtained as direct
images of $\underline{F}_{1}$, $\underline{F}_{2}$ under the open embedding
$L_{0}\hookrightarrow\operatorname{Lin}(W)$.

Lemma 6.7:  Under these assumptions, the sheaves $F_{1}$, $F_{2}$ are coherent
reflexive sheaves. Moreover, there exists a natural isomorphism of coherent
sheaves $\Psi_{1,2}:\;F_{1}{\>\longrightarrow\>}F_{2}$.

Proof: The complex codimension of the singular set $S$ in $(M,I)$ is at least
3, because $F$ is reflexive ([OSS], Ch. II, 1.1.10). Since $S$ is trianalytic
(3.3), this codimension is even. Thus,
$\operatorname{codim}_{\mathbb{C}}(S,(M,I))\geqslant 4$. Therefore,

$\operatorname{codim}_{\mathbb{C}}(\operatorname{Tw}(S),\operatorname{Tw}(M))\geqslant
4.$

Consider the set $L_{S}$ of all twistor lines $l\in\operatorname{Lin}(W)$
passing through $\operatorname{Tw}(S)$. For generic points
$x,y\in\operatorname{Tw}(W)$, there exists a unique line
$l\in\operatorname{Lin}(W)$ passing through $x,y$. Therefore,

$\operatorname{codim}_{\mathbb{C}}(L_{S},\operatorname{Lin}(W))=\operatorname{codim}_{\mathbb{C}}(\operatorname{Tw}(S),\operatorname{Tw}(M))-1\geqslant
3.$

By definition, $L_{0}:=\operatorname{Lin}(W)\backslash L_{S}$. Since $F_{1}$,
$F_{2}$ are direct images of bundles $\underline{F}_{1}$, $\underline{F}_{2}$
over a subvariety $L_{S}$ of codimension 3, these sheaves are coherent and
reflexive ([OSS], Ch. II, 1.1.12; see also 9.1). To show that they are
naturally isomorphic it remains to construct an isomorphism between
$\underline{F}_{1}$ and $\underline{F}_{2}$.

Let ${\cal F}$ be a coherent sheaf on $\operatorname{Tw}(W)$ obtained from
$F{\left|{}_{{\phantom{|}\\!\\!}_{W}}\right.}$ as in the proof of 3.3. The
singular set of ${\cal F}$ is
$\operatorname{Tw}(S)\subset\operatorname{Tw}(W)$. Therefore, the restriction
${\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{\operatorname{Tw}(W)\backslash\operatorname{Tw}(S)}}\right.}$
is a holomorphic vector bundle. For all horisontal twistor lines
$l_{x}\subset\operatorname{Tw}(W)\backslash\operatorname{Tw}(S)$, the
restriction ${\cal F}{\left|{}_{{\phantom{|}\\!\\!}_{l_{x}}}\right.}$ is
clearly a trivial vector bundle over $l_{x}\cong{\mathbb{C}}P^{1}$. A small
deformation of a trivial vector bundle is again trivial. Shrinking $W$ if
necessary, we may assume that for all lines $l\in L_{0}$, the restriction of
${\cal F}$ to $l\cong{\mathbb{C}}P^{1}$ is a trivial vector bundle.

The isomorphism
$\underline{\Psi}_{1,2}:\;\underline{F}_{1}{\>\longrightarrow\>}\underline{F}_{2}$
is constructed as follows. Let $l\in L_{0}$ be a twistor line. The restriction
${\cal F}{\left|{}_{{\phantom{|}\\!\\!}_{l}}\right.}$ is trivial. Consider $l$
as a map $l:\;{\mathbb{C}}P^{1}{\>\longrightarrow\>}\operatorname{Tw}(M)$. We
identify ${\mathbb{C}}P^{1}$ with the set of induced complex structures on
$M$. By definition, the fiber of $F_{1}$ in $l$ is naturally identified with
the space ${\cal F}{\left|{}_{{\phantom{|}\\!\\!}_{l(I)}}\right.}$, and the
fiber of $F_{2}$ in $l$ is identifies with ${\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{l(J)}}\right.}$. Since ${\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{l}}\right.}$ is trivial, the fibers of the
bundle ${\cal F}{\left|{}_{{\phantom{|}\\!\\!}_{l}}\right.}$ are naturally
identified. This provides a vector bundle isomorphism
$\underline{\Psi}_{1,2}:\;\underline{F}_{1}{\>\longrightarrow\>}\underline{F}_{2}$
mapping $\underline{F}_{1}{\left|{}_{{\phantom{|}\\!\\!}_{l}}\right.}={\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{l(I)}}\right.}$ to
$\underline{F}_{2}{\left|{}_{{\phantom{|}\\!\\!}_{l}}\right.}={\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{l(J)}}\right.}$. It remains to show that
this isomorphism is compatible with the holomorphic structure.

Since the bundle ${\cal F}{\left|{}_{{\phantom{|}\\!\\!}_{l}}\right.}$ is
trivial, we have an identification

${\cal F}{\left|{}_{{\phantom{|}\\!\\!}_{l(I)}}\right.}\cong{\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{l(J)}}\right.}=\Gamma({\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{l}}\right.}),$

where $\Gamma({\cal F}{\left|{}_{{\phantom{|}\\!\\!}_{l}}\right.})$ is the
space of global sections of ${\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{l}}\right.}$. Thus,
$F_{i}{\left|{}_{{\phantom{|}\\!\\!}_{l}}\right.}=\Gamma({\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{l}}\right.})$, and this identification is
clearly holomorphic. This proves 6.2.

We return to the proof of 6.2. Denote by $F_{J}$ the restriction of ${\cal F}$
to $(M,J)=\pi^{-1}(J)\subset\operatorname{Tw}(M)$. The map $\Psi_{I,J}$ was
defined as a composition $e_{I}\circ p_{J}\circ e_{J}\circ p_{I}$. The sheaf
$p_{I}^{*}F$ is by definition isomorphic to $F_{1}$, and $p_{J}^{*}F_{J}$ to
$F_{2}$. On the other hand, clearly, $e_{J}^{*}F_{2}=F_{J}$. Therefore,
$(p_{J}\circ e_{J})^{*}F_{2}\cong F_{2}$. Using the isomorphism $F_{1}\cong
F_{2}$, we obtain $(p_{J}\circ e_{J})^{*}F_{1}\cong F_{1}$. To sum it up, we
have the following isomorphisms:

$\begin{split}p_{I}^{*}F&\cong F_{1},\\\ (p_{J}\circ e_{J})^{*}F_{1}&\cong
F_{1},\\\ e_{I}^{*}F_{1}&\cong F.\end{split}$

A composition of these isomorphisms gives an isomorphism

$\Psi_{I,J}^{F}:\;F{\left|{}_{{\phantom{|}\\!\\!}_{B_{0}}}\right.}{\>\longrightarrow\>}\Psi_{I,J}^{*}(F{\left|{}_{{\phantom{|}\\!\\!}_{B}}\right.}).$

This proves 6.2.

###  6.3 A ${\mathbb{C}}^{*}$-action on a local ring of a hyperkähler manifold

Let $M$ be a hyperkähler manifold, non necessarily compact, $x\in M$ a point
and $I$, $J$ induced complex structures, $I\neq J$. Consider the complete
local ring ${\cal O}_{x,I}:=\hat{\cal O}_{x}(M,I)$. Throughout this section we
consider the map $\Psi_{I,J}$ (6.1) as an automorphism of the ring ${\cal
O}_{x,I}$. Let $\mathfrak{m}$ be the maximal ideal of ${\cal O}_{x,I}$, and
$\mathfrak{m}/\mathfrak{m}^{2}$ the Zariski cotangent space of $(M,I)$ in $x$.

By 6.1, $\Psi_{I,J}$ acts on $\mathfrak{m}/\mathfrak{m}^{2}$ as a
multiplication by a number $\lambda\in{\mathbb{C}}$, $0<|\lambda|<1$. (6.1)

Let $V_{\lambda^{n}}$ be the eigenspace corresponding to the eigenvalue
$\lambda^{n}$,

$V_{\lambda^{n}}:=\\{v\in{\cal O}_{x,I}\ \ |\ \
\Psi_{I,J}(v)=\lambda^{n}v\\}.$

Clearly, $\oplus V_{\lambda^{i}}$ is a graded subring in ${\cal O}_{x,I}$. In
[V-d2], (see also [V-d3]) we proved that the ring $\oplus V_{\lambda^{i}}$ is
dense in ${\cal O}_{x,I}$ with respect to the adic topology. Therefore, the
ring ${\cal O}_{x,I}$ is identified with the adic completion of $\oplus
V_{\lambda^{i}}$.

Consider at action of ${\mathbb{C}}^{*}$ on $\oplus V_{\lambda^{i}}$, with
$z\in{\mathbb{C}}^{*}$ acting on $V_{\lambda^{i}}$ as a multiplication by
$z^{i}$. This ${\mathbb{C}}^{*}$-action is clearly continuous, with respect to
the adic topology. Therefore, it can be extended to

${\cal O}_{x,I}=\widehat{\oplus V_{\lambda^{i}}}.$

Definition 6.8:  Let $M$, $I$, $J$, $x$, ${\cal O}_{x,I}$ be as in the
beginning of this Subsection. Consider the ${\mathbb{C}}^{*}$-action

$\Psi_{I,J}(z):\;{\cal O}_{x,I}{\>\longrightarrow\>}{\cal O}_{x,I}$

constructed as above. Then $\Psi_{I,J}(z)$ is called the canonical
${\mathbb{C}}^{*}$-action associated with $M$, $I$, $J$, $x$.

In the above notation, consider a reflexive sheaf $F$ on $(M,I)$ equipped with
a hyperholomorphic connection. Denote the germ of $F$ at $x$ by $F_{x}$,
$F_{x}:=F\otimes_{{\cal O}_{(M,I)}}{\cal O}_{x,I}$. From 6.2, we obtain an
isomorphism $F_{x}\cong\Psi^{*}_{I,J}F_{x}$. This isomorphism can be
interpreted as an automorphism

$\Psi^{F}_{I,J}:\;F_{x}{\>\longrightarrow\>}F_{x}$

satisfying

$\Psi^{F}_{I,J}(\alpha v)=\Psi_{I,J}(\alpha)v,$ (6.2)

for all $\alpha\in{\cal O}_{x,I}$, $v\in F_{x}$.

By (6.2), the automorphism $\Psi^{F}_{I,J}$ respects the filtration

$F_{x}\supset\mathfrak{m}F_{x}\supset\mathfrak{m}^{2}F_{x}\supset...$

Thus, it makes sense to speak of $\Psi^{F}_{I,J}$-action on
$\mathfrak{m}^{i}F_{x}/\mathfrak{m}^{i+1}F_{x}$.

Lemma 6.9:  The automorphism $\Psi^{F}_{I,J}$ acts on
$\mathfrak{m}^{i}F_{x}/\mathfrak{m}^{i+1}F_{x}$ as a multiplication by
$\lambda^{i}$, where $\lambda\in{\mathbb{C}}$ is the number considered in
(6.1).

Proof: By (6.2), it suffices to prove 6.3 for $i=0$. In other words, we have
to show that $\Psi^{F}_{I,J}$ acts as identity on $F_{x}/\mathfrak{m}F_{x}$.
We reduced 6.3 to the following claim.

Claim 6.10:  In the above assumptions, the automorphism $\Psi^{F}_{I,J}$ acts
as identity on $F_{x}/\mathfrak{m}F_{x}$.

Proof: In the course of defining the map $\Psi^{F}_{I,J}$, we identified the
space $\operatorname{Lin}(M)$ with a complexification of $(M,I)$, and defined
the maps

$p_{I}:\;\operatorname{Lin}(M){\>\longrightarrow\>}(M,I),\ \
p_{J}:\;\operatorname{Lin}(M){\>\longrightarrow\>}(M,I)$

(these maps are smooth, in a neighbourhood of
$\operatorname{Hor}\subset\operatorname{Lin}(M)$, by 6.1), and

$e_{I}:\;(B,I){\>\longrightarrow\>}\operatorname{Lin}(M),\ \
e_{J}:\;(B,J){\>\longrightarrow\>}\operatorname{Lin}(M)$

(these maps are locally closed embeddings). Consider $(M,J)$ as a subvariety
of $\operatorname{Tw}(M)$, $(M,J)=\pi^{-1}(J)$. Let ${\cal F}$ be the lift of
$F$ to $\operatorname{Tw}(M)$ (see the proof of 3.3 for details). Denote the
completion of ${\cal O}_{x}(M,J)$ by ${\cal O}_{x,J}$. Let $F_{J}$ denote the
${\cal O}_{x,J}$-module $\left({\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{(M,J)}}\right.}\right)\otimes_{{\cal
O}_{(M,J)}}\hat{\cal O}_{x,J}$. Consider the horisontal twistor line
$l_{x}\in\operatorname{Lin}(M)$. Let $\operatorname{Lin}_{x}(M)$ be the
spectre of the completion ${\cal O}_{x,\operatorname{Lin}}$ of the local ring
of holomorphic functions on $\operatorname{Lin}(M)$ in $l_{x}$. The maps
$p_{I}$, $p_{J}$, $e_{I}$, $e_{J}$ can be considered as maps of corresponding
formal manifolds:

$\begin{split}p_{I}:\;\operatorname{Lin}_{x}(M)&{\>\longrightarrow\>}\operatorname{Spec}({\cal
O}_{x,I}),\\\
p_{J}:\;\operatorname{Lin}_{x}(M)&{\>\longrightarrow\>}\operatorname{Spec}({\cal
O}_{x,J}),\\\ e_{I}:\;\operatorname{Spec}({\cal
O}_{x,I})&{\>\longrightarrow\>}\operatorname{Lin}_{x}(M),\\\
e_{J}:\;\operatorname{Spec}({\cal
O}_{x,J})&{\>\longrightarrow\>}\operatorname{Lin}_{x}(M),\end{split}$

As in Subsection 6.2, we consider the ${\cal
O}_{x,\operatorname{Lin}}$-modules $F_{1}:=p_{I}^{*}F_{x}$ and
$F_{2}:=p_{J}^{*}F_{J}$. By 6.2, there exists a natural isomorphism
$\Psi_{1,2}:\;F_{1}{\>\longrightarrow\>}F_{2}$.

Let $\mathfrak{m}_{l_{x}}$ be the maximal ideal of ${\cal
O}_{x,\operatorname{Lin}}$. Since the morphism $p_{I}$ is smooth, the space
$F_{1}/\mathfrak{m}_{l_{x}}F_{1}$ is naturally isomorphic to
$F_{x}/\mathfrak{m}F_{x}$. Similarly, the space
$F_{2}/\mathfrak{m}_{l_{x}}F_{2}$ is isomorphic to
$F_{J}/\mathfrak{m}_{J}F_{J}$, where $\mathfrak{m}_{J}$ is the maximal ideal
of ${\cal O}_{x,J}$. We have a chain of isomorphisms

$\begin{split}F_{x}/\mathfrak{m}F_{x}&\stackrel{{\scriptstyle
p_{I}^{*}}}{{{\>\longrightarrow\>}}}F_{1}/\mathfrak{m}_{l_{x}}F_{1}\stackrel{{\scriptstyle\Psi_{1,2}}}{{{\>\longrightarrow\>}}}F_{2}/\mathfrak{m}_{l_{x}}F_{2}\\\
&\stackrel{{\scriptstyle
e_{J}^{*}}}{{{\>\longrightarrow\>}}}F_{J}/\mathfrak{m}_{J}F_{J}\stackrel{{\scriptstyle
p_{J}^{*}}}{{{\>\longrightarrow\>}}}F_{2}/\mathfrak{m}_{l_{x}}F_{2}\\\
&\stackrel{{\scriptstyle\Psi_{1,2}^{-1}}}{{{\>\longrightarrow\>}}}F_{1}/\mathfrak{m}_{l_{x}}F_{1}\stackrel{{\scriptstyle
e_{I}^{*}}}{{{\>\longrightarrow\>}}}F_{x}/\mathfrak{m}F_{x}.\end{split}$ (6.3)

By definition, for any $f\in F_{x}/\mathfrak{m}F_{x}$, the value of
$\Psi^{F}_{I,J}(f)$ is given by the composide map of (6.3) applied to $f$. The
composition

$F_{2}/\mathfrak{m}_{l_{x}}F_{2}\stackrel{{\scriptstyle
e_{J}^{*}}}{{{\>\longrightarrow\>}}}F_{J}/\mathfrak{m}_{J}F_{J}\stackrel{{\scriptstyle
p_{J}^{*}}}{{{\>\longrightarrow\>}}}F_{2}/\mathfrak{m}_{l_{x}}F_{2}$ (6.4)

is identity, because the spaces $F_{2}/\mathfrak{m}_{l_{x}}F_{2}$ and
$F_{J}/\mathfrak{m}_{J}F_{J}$ are canonically identified, and this
identification can be performed via $e_{J}^{*}$ or $p_{J}^{*}$. Thus, the map
(6.3) is a composition

$F_{x}/\mathfrak{m}F_{x}\stackrel{{\scriptstyle
p_{I}^{*}}}{{{\>\longrightarrow\>}}}F_{1}/\mathfrak{m}_{l_{x}}F_{1}\stackrel{{\scriptstyle\Psi_{1,2}}}{{{\>\longrightarrow\>}}}F_{2}/\mathfrak{m}_{l_{x}}F_{2}\stackrel{{\scriptstyle\Psi_{1,2}^{-1}}}{{{\>\longrightarrow\>}}}F_{1}/\mathfrak{m}_{l_{x}}F_{1}\stackrel{{\scriptstyle
e_{I}^{*}}}{{{\>\longrightarrow\>}}}F_{x}/\mathfrak{m}F_{x}.$

This map is clearly equivalent to a composition

$F_{x}/\mathfrak{m}F_{x}\stackrel{{\scriptstyle
p_{I}^{*}}}{{{\>\longrightarrow\>}}}F_{1}/\mathfrak{m}_{l_{x}}F_{1}\stackrel{{\scriptstyle
e_{I}^{*}}}{{{\>\longrightarrow\>}}}F_{x}/\mathfrak{m}F_{x},$

which is identity according to the same reasoning which proved that (6.4) is
identity. We proved 6.3 and 6.3.

Consider the $\lambda^{n}$-eigenspaces $F_{\lambda^{n}}$ of $F_{x}$. Consider
the $\oplus V_{\lambda^{n}}$-submodule $\oplus F_{\lambda^{n}}\subset F_{x}$,
where $\oplus V_{\lambda^{n}}\subset{\cal O}_{x,I}$ is the ring defined in
Subsection 6.3. From 6.3 and (6.1) it follows that $\oplus F_{\lambda^{n}}$ is
dense in $F_{x}$, with respect to the adic topology on $F_{x}$. For
$z\in{\mathbb{C}}^{*}$, let $\Psi_{I,J}^{F}(z):\;\oplus
F_{\lambda^{n}}{\>\longrightarrow\>}\oplus F_{\lambda^{n}}$ act on
$F_{\lambda^{n}}$ as a multiplication by $z^{n}$. As in 6.3, we extend
$\Psi_{I,J}^{F}(z)$ to $F_{x}=\widehat{\oplus F_{\lambda^{n}}}$. This
automorphism makes $F_{x}$ into a ${\mathbb{C}}^{*}$-equivariant module over
${\cal O}_{x,I}$

Definition 6.11:  The constructed above ${\mathbb{C}}^{*}$-equivariant
structure on $F_{x}$ is called the canonical ${\mathbb{C}}^{*}$-equivariant
structure on $F_{x}$ associated with $J$.

###  6.4 Desingularization of ${\mathbb{C}}^{*}$-equivariant sheaves

Let $M$ be a hyperkähler manifold, $I$ an induced complex structure and $F$ a
reflexive sheaf with isolated singularities over $(M,I)$, equipped with a
hyperholomorphic connection. We have shown that the sheaf $F$ admits a
${\mathbb{C}}^{*}$-equivariant structure compatible with the canonical
${\mathbb{C}}^{*}$-action on the local ring of $(M,I)$. Therefore, 6 is
implied by the following proposition.

Proposition 6.12:  Let $B$ be a complex manifold, $x\in B$ a point. Assume
that there is an action $\Psi(z)$ of ${\mathbb{C}}^{*}$ on $B$ which fixes $x$
and acts on $T_{x}B$ be dilatations. Let $F$ be a reflexive coherent sheaf on
$B$, which is locally trivial outside of $x$. Assume that the germ $F_{x}$ of
$F$ in $x$ is equipped with a ${\mathbb{C}}^{*}$-equivariant structure,
compatible with $\Psi(z)$. Let $\widetilde{B}$ be a blow-up of $B$ in $x$, and
$\pi:\;\widetilde{B}{\>\longrightarrow\>}B$ the standard projection. Then the
pullback sheaf $\widetilde{F}:=\pi^{*}F$ is locally trivial on
$\widetilde{B}$.

Proof: Let $C:=\pi^{-1}(x)$ be the singular locus of $\pi$. The sheaf $F$ is
locally trivial outside of $x$. Let $d$ be the rank of
$F{\left|{}_{{\phantom{|}\\!\\!}_{B\backslash x}}\right.}$. To prove that
$\widetilde{F}$ is locally trivial, we need to show that for all points
$y\in\widetilde{B}$, the fiber
$\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{y}}\right.}$ is $d$-dimensional.
Therefore, to prove 6.4 it suffices to show that
$\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}$ is a vector bundle of
dimension $d$.

The variety $C$ is naturally identified with the projectivization
${\mathbb{P}}T_{x}B$ of the tangent space $T_{x}B$. The total space of
$T_{x}B$ is equipped with a natural action of ${\mathbb{C}}^{*}$, acting by
dilatations. Clearly, coherent sheaves on ${\mathbb{P}}T_{x}B$ are in one-to-
one correspondence with ${\mathbb{C}}^{*}$-equivariant coherent sheaves on
$T_{x}B$. Consider a local isomorphism
$\varphi:\;T_{x}B{\>\longrightarrow\>}B$ which is compatible with
${\mathbb{C}}^{*}$-action, maps $0\in T_{x}B$ to $x$ and acts as identity on
the tangent space $T_{0}(T_{x}B)=T_{x}B$. The sheaf $\varphi^{*}F$ is
${\mathbb{C}}^{*}$-equivariant. Clearly, the corresponding sheaf on
${\mathbb{P}}T_{x}B$ is canonically isomorphic with
$\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}$. Let $l\in T_{x}B$ be a
line passing through $0$, and $l\backslash 0$ its complement to $0$. Denote
the corresponding point of ${\mathbb{P}}T_{x}B$ by $y$. The restriction
$\varphi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{l\backslash 0}}\right.}$ is a
${\mathbb{C}}^{*}$-equivariant vector bundle. The
${\mathbb{C}}^{*}$-equivariant structure identifies all the fibers of the
bundle $\varphi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{l\backslash 0}}\right.}$.
Let $F_{l}$ be one of these fibers. Clearly, the fiber of
$\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}$ in $y$ is canonically
isomorphic to $F_{l}$. Therefore, the fiber of
$\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}$ in $y$ is
$d$-dimensional. We proved that $\pi^{*}F$ is a bundle.

##  7 Twistor transform and quaternionic-Kähler geometry

This Section is a compilation of results known from the literature. Subsection
7.1 is based on [KV] and the results of Subsection 7.2 are implicit in [KV].
Subsection 7.3 is based on [Sal], [N1] and [N2], and Subsection 7.4 is a
recapitulation of the results of A. Swann ([Sw]).

###  7.1 Direct and inverse twistor transform

In this Subsection, we recall the definition and the main properties of the
direct and inverse twistor transform for bundles over hyperkähler manifolds
([KV]).

The following definition is a non-Hermitian analogue of the notion of a
hyperholomorphic connection.

Definition 7.1: Let $M$ be a hyperkähler manifold, not necessarily compact,
and $(B,\nabla)$ be a vector bundle with a connection over $M$, not
necessarily Hermitian. Assume that the curvature of $\nabla$ is contained in
the space $\Lambda^{2}_{inv}(M,\operatorname{End}(B))$ of $SU(2)$-invariant
2-forms with coefficients in $\operatorname{End}(B)$. Then $(B,\nabla)$ is
called an autodual bundle, and $\nabla$ an autodual connection.

Let $\operatorname{Tw}(M)$ be the twistor space of $M$, equipped with the
standard maps
$\pi:\;\operatorname{Tw}(M){\>\longrightarrow\>}{\mathbb{C}}P^{1}$,
$\sigma:\;\operatorname{Tw}(M){\>\longrightarrow\>}M$.

We introduce the direct and inverse twistor transforms which relate autodual
bundles on the hyperkähler manifold $M$ and holomorphic bundles on its twistor
space $\operatorname{Tw}(M)$.

Let $B$ be a complex vector bundle on $M$ equipped with a connection $\nabla$.
The pullback $\sigma^{*}B$ of $B$ to $\operatorname{Tw}(M)$ is equipped with a
pullback connection $\sigma^{*}\nabla$.

Lemma 7.2:  ([KV], Lemma 5.1) The connection $\nabla$ is autodual if and only
if the connection $\sigma^{*}\nabla$ has curvature of Hodge type $(1,1)$.

Proof: Follows from 2.1.

In assumptions of 7.1, consider the $(0,1)$-part $(\sigma^{*}\nabla)^{0,1}$ of
the connection $\sigma^{*}\nabla$. Since $\sigma^{*}\nabla$ has curvature of
Hodge type $(1,1)$, we have

$\left((\sigma^{*}\nabla)^{0,1}\right)^{2}=0,$

and by 2.4, this connection is integrable. Consider $(\sigma^{*}\nabla)^{0,1}$
as a holomorphic structure operator on $\sigma^{*}B$.

Let ${\cal A}$ be the category of autodual bundles on $M$, and ${\cal C}$ the
category of holomorphic vector bundles on $\operatorname{Tw}(M)$. We have
constructed a functor

$(\sigma^{*}\bullet)^{0,1}:\;{\cal A}{\>\longrightarrow\>}{\cal C},$

$\nabla{\>\longrightarrow\>}(\sigma^{*}\nabla)^{0,1}$. Let
$s\in\operatorname{Hor}\subset\operatorname{Tw}(M)$ be a horisontal twistor
line (Subsection 6.1). For any $(B,\nabla)\in{\cal A}$, consider corresponding
holomorphic vector bundle $(\sigma^{*}B,(\sigma^{*}\nabla)^{0,1})$. The
restriction of $(\sigma^{*}B,(\sigma^{*}\nabla)^{0,1})$ to
$s\cong{\mathbb{C}}P^{1}$ is a trivial vector bundle. A converse statement is
also true. Denote by ${\cal C}_{0}$ the category of holomorphic vector bundles
$C$ on $\operatorname{Tw}(M)$, such that the restriction of $C$ to any
horisontal twistor line is trivial.

Theorem 7.3:  Consider the functor

$(\sigma^{*}\bullet)^{0,1}:\;{\cal A}{\>\longrightarrow\>}{\cal C}_{0}$

constructed above. Then it is an equivalence of categories.

Proof: [KV], Theorem 5.12.

Definition 7.4: Let $M$ be a hyperkähler manifold, $\operatorname{Tw}(M)$ its
twistor space and ${\cal F}$ a holomorphic vector bundle. We say that ${\cal
F}$ is compatible with twistor transform if the restriction of $C$ to any
horisontal twistor line $s\in\operatorname{Tw}(M)$ is a trivial bundle on
$s\cong{\mathbb{C}}P^{1}$.

Recall that a connection $\nabla$ in a vector bundle over a complex manifold
is called $(1,1)$-connection if its curvature is of Hodge type $(1,1)$.

Remark 7.5:  Let ${\cal F}$ be a holomorphic bundle over
$\operatorname{Tw}(M)$ which is compatible with twistor transform. Then ${\cal
F}$ is equipped with a natural $(1,1)$-connection $\nabla_{{\cal
F}}=\sigma^{*}\nabla$, where $(B,\nabla)$ is the corresponding autodual bundle
over $M$. The connection $\nabla_{{\cal F}}$ is not, generally speaking,
Hermitian, or compatible with a Hermitian structure.

###  7.2 Twistor transform and Hermitian structures on vector bundles

Results of this Subsection were implicit in [KV], but in this presentation,
they are new.

Let $M$ be a hyperkähler manifold, not necessarily compact, and
$\operatorname{Tw}(M)$ its twistor space. In Subsection 7.1, we have shown
that certain holomorphic vector bundles over $\operatorname{Tw}(M)$ admit a
canonical (1,1)-connection $\nabla{{}_{\cal F}}$ (7.1). This connection can be
non-Hermitian. Here we study the Hermitian structures on $({\cal
F},\nabla{{}_{\cal F}})$ in terms of holomorphic properties of ${\cal F}$.

Definition 7.6: Let $F$ be a real analytic complex vector bundle over a real
analytic manifold $X_{\mathbb{R}}$, and $h:\;F\times
F{\>\longrightarrow\>}{\mathbb{C}}$ a ${\cal O}_{X_{\mathbb{R}}}$-linear
pairing on $F$. Then $h$ is called semilinear if for all $\alpha\in{\cal
O}_{X_{\mathbb{R}}}\otimes_{\mathbb{R}}{\mathbb{C}}$, we have

$h(\alpha x,y)=\alpha\cdot h(x,y),\text{\ \ and\ \ }h(x,\alpha
y)=\overline{\alpha}\cdot h(x,y).$

For $X$ a complex manifold and $F$ a holomorphic vector bundle, by a
semilinear pairing on $F$ we understand a semilinear pairing on the underlying
real analytic bundle. Clearly, a real analytic Hermitian metric is always
semilinear.

Let $X$ be a complex manifold, $I:\;TX{\>\longrightarrow\>}TX$ the complex
structure operator, and $i:\;X{\>\longrightarrow\>}X$ a real analytic map. We
say that $X$ is anticomplex if the induced morphism of tangent spaces
satisfies $i\circ I=-I\circ I$. For a complex vector bundle $F$ on $X$,
consider the complex adjoint vector bundle $\overline{F}$, which coincides
with $F$ as a real vector bundle, with ${\mathbb{C}}$-action which is
conjugate to that defined on $F$. Clearly, for every holomorphic vector bundle
$F$, and any anticomplex map $i:\;X{\>\longrightarrow\>}X$, the bundle
$i^{*}\overline{F}$ is equipped with a natural holomorphic structure.

Let $M$ be a hyperkähler manifold, and $\operatorname{Tw}(M)$ its twistor
space. Recall that $\operatorname{Tw}(M)={\mathbb{C}}P^{1}\times M$ is
equipped with a canonical anticomplex involution $\iota$, which acts as
identity on $M$ and as central symmetry $I{\>\longrightarrow\>}-I$ on
${\mathbb{C}}P^{1}=S^{2}$. For any holomorphic bundle ${\cal F}$ on
$\operatorname{Tw}(M)$, consider the corresponding holomorphic bundle
$\iota^{*}\overline{{\cal F}}$.

Let $M$ be a hyperkähler manifold, $I$ an induced complex structure, $F$ a
vector bundle over $M$, equipped with an autodual connection $\nabla$, and
${\cal F}$ the corresponding holomorphic vector bundle over
$\operatorname{Tw}(M)$, equipped with a canonical connection $\nabla_{{\cal
F}}$. As usually, we identify $(M,I)$ and the fiber $\pi^{-1}(I)$ of the
twistor projection
$\pi:\;\operatorname{Tw}(M){\>\longrightarrow\>}{\mathbb{C}}P^{1}$. Let
$\nabla_{{\cal F}}=\nabla^{1,0}_{I}+\nabla^{0,1}_{I}$ be the Hodge
decomposition of $\nabla$ with respect to $I$.

Clearly, the operator $\nabla^{1,0}_{I}$ can be considered as a holomorphic
structure operator on $F$, considered as a complex vector bundle over
$(M,-I)$. (7.1)

Then the holomorphic structure operator on ${\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{(M,I)}}\right.}$ is equal to
$\nabla^{0,1}_{I}$, and the holomorphic structure operator on ${\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{(M,-I)}}\right.}$ is equal to
$\nabla^{1,0}_{I}$.

Assume that the bundle $({\cal F},\nabla_{{\cal F}})$ is equipped with a non-
degenerate semilinear pairing $h$ which is compatible with the connection.
Consider the natural connection $\nabla_{{\cal F}^{*}}$ on the dual bundle to
${\cal F}$, and its Hodge decomposition (with respect to $I$)

$\nabla_{{\cal F}^{*}}=\nabla^{1,0}_{{\cal F}^{*}}+\nabla^{0,1}_{{\cal
F}^{*}}.$

Clearly, the pairing $h$ gives a $C^{\infty}$-isomorphism of ${\cal F}$ and
the complex conjugate of its dual bundle, denoted as $\overline{{\cal
F}}^{*}$. Since $h$ is semilinear and compatible with the connection, it maps
the holomorphic structure operator $\nabla^{0,1}_{I}$ to the complex conjugate
of $\nabla^{1,0}_{{\cal F}^{*}}$. On the other hand, the operator
$\nabla^{1,0}_{{\cal F}^{*}}$ is a holomorphic structure operator in ${\cal
F}^{*}{\left|{}_{{\phantom{|}\\!\\!}_{(M,-I)}}\right.}$, as (7.1) claims. We
obtain that the map $h$ can be considered as an isomorphism of holomorphic
vector bundles

$h:\;{\cal F}{\>\longrightarrow\>}(\iota^{*}\overline{{\cal F}})^{*}.$

This correspondence should be thought of as a (direct) twistor transform for
bundles with a semilinear pairing.

Proposition 7.7:  (direct and inverse twistor transform for bundles with
semilinear pairing) Let $M$ be a hyperkähler manifold, and ${\cal C}_{sl}$ the
category of autodual bundles over $M$ equipped with a non-degenerate
semilinear pairing. Consider the category ${\cal C}_{hol,sl}$ of holomorphic
vector bundles ${\cal F}$ on $\operatorname{Tw}(M)$, compatible with twistor
transform and equipped with an isomorphism

${\mathfrak{h}}:\;{\cal F}{\>\longrightarrow\>}(\iota^{*}\overline{{\cal
F}})^{*}.$

Let ${\cal T}:\;{\cal C}_{sl}{\>\longrightarrow\>}{\cal C}_{hol,sl}$ be the
functor constructed above. Then ${\cal T}$ is an isomorphism of categories.

Proof: Given a pair ${\cal F},{\mathfrak{h}}:\;{\cal
F}{\>\longrightarrow\>}(\iota^{*}\overline{{\cal F}})^{*}$, we need to
construct a non-degenerate semilinear pairing $h$ on ${\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{(M,I)}}\right.}$, compatible with a
connection. Since ${\cal F}$ is compatible with twistor transform, it is a
pullback of a bundle $(F,\nabla)$ on $M$. This identifies the real analytic
bundles ${\cal F}{\left|{}_{{\phantom{|}\\!\\!}_{(M,I^{\prime})}}\right.}$,
for all induced complex structures $I^{\prime}$. Taking $I^{\prime}=\pm I$, we
obtain an identification of the $C^{\infty}$-bundles ${\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{(M,I)}}\right.}$, ${\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{(M,-I)}}\right.}$. Thus, $\mathfrak{h}$ can
be considered as an isomorphism of $F={\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{(M,I)}}\right.}$ and $(\overline{{\cal
F}})^{*}{\left|{}_{{\phantom{|}\\!\\!}_{(M,I)}}\right.}$. This allows one to
consider $\mathfrak{h}$ as a semilinear form $h$ on $F$. We need only to show
that $h$ is compatible with the connection $\nabla$. Since $\nabla_{{\cal F}}$
is an invariant of holomorphic structure, the map ${\mathfrak{h}}:\;{\cal
F}{\>\longrightarrow\>}(\iota^{*}\overline{{\cal F}})^{*}$ is compatible with
the connection $\nabla_{{\cal F}}$. Thus, the obtained above form $h$ is
compatible with the connection $\nabla_{{\cal
F}}{\left|{}_{{\phantom{|}\\!\\!}_{(M,I)}}\right.}=\nabla$. This proves 7.2.

###  7.3  $B_{2}$-bundles on quaternionic-Kähler manifolds

Definition 7.8:  ([Sal], [Bes]) Let $M$ be a Riemannian manifold. Consider a
bundle of algebras $\operatorname{End}(TM)$, where $TM$ is the tangent bundle
to $M$. Assume that $\operatorname{End}(TM)$ contains a 4-dimensional bundle
of subalgebras $W\subset\operatorname{End}(TM)$, with fibers isomorphic to a
quaternion algebra ${\mathbb{H}}$. Assume, moreover, that $W$ is closed under
the transposition map
$\bot:\;\operatorname{End}(TM){\>\longrightarrow\>}\operatorname{End}(TM)$ and
is preserved by the Levi-Civita connection. Then $M$ is called quaternionic-
Kähler.

Example 7.9: Consider the quaternionic projective space

${\mathbb{H}}P^{n}=({\mathbb{H}}^{n}\backslash 0)/{\mathbb{H}}^{*}.$

It is easy to see that ${\mathbb{H}}P^{n}$ is a quaternionic-Kähler manifold.
For more examples of quaternionic-Kähler manifolds, see [Bes].

A quaternionic-Kähler manifold is Einstein ([Bes]), i. e. its Ricci tensor is
proportional to the metric: $Ric(M)=c\cdot g$, with $c\in{\mathbb{R}}$. When
$c=0$, the manifold $M$ is hyperkähler, and its restricted holonomy group is
$Sp(n)$; otherwise, the restricted holonomy is $Sp(n)\cdot Sp(1)$. The number
$c$ is called the scalar curvature of $M$. Further on, we shall use the term
quaternionic-Kähler manifold for manifolds with non-zero scalar curvature.

The quaternionic projective space ${\mathbb{H}}P^{n}$ has positive scalar
curvature.

The quaternionic projective space is the only example of quaternionic-Kähler
manifold which we need, in the course of this paper. However, the formalism of
quaternionic-Kähler manifolds is very beautiful and significantly simplifies
the arguments, so we state the definitions and results for a general
quaternionic-Kähler manifold whenever possible.

Let $M$ be a quaternionic-Kähler manifold, and
$W\subset\operatorname{End}(TM)$ the corresponding 4-dimensional bundle. For
$x\in M$, consider the set ${\cal R}_{x}\subset
W{\left|{}_{{\phantom{|}\\!\\!}_{x}}\right.}$, consisting of all $I\in
W{\left|{}_{{\phantom{|}\\!\\!}_{x}}\right.}$ satisfying $I^{2}=-1$. Consider
${\cal R}_{x}$ as a Riemannian submanifold of the total space of
$W{\left|{}_{{\phantom{|}\\!\\!}_{x}}\right.}$. Clearly, ${\cal R}_{x}$ is
isomorphic to a 2-dimensional sphere. Let ${\cal R}=\cup_{x}{\cal R}_{x}$ be
the corresponding spherical fibration over $M$, and $\operatorname{Tw}(M)$ its
total space. The manifold $\operatorname{Tw}(M)$ is equipped with an almost
complex structure, which is defined in the same way as the almost complex
structure for the twistor space of a hyperkähler manifold. This almost complex
structure is known to be integrable (see [Sal]).

Definition 7.10:  ([Sal], [Bes]) Let $M$ be a quaternionic-Kähler manifold.
Consider the complex manifold $\operatorname{Tw}(M)$ constructed above. Then
$\operatorname{Tw}(M)$ is called the twistor space of $M$.

Note that (unlike in the hyperkähler case) the space $\operatorname{Tw}(M)$ is
Kähler. For quaternionic-Kähler manifolds with positive scalar curvature, the
anticanonical bundle of $\operatorname{Tw}(M)$ is ample, so
$\operatorname{Tw}(M)$ is a Fano manifold.

Quaternionic-Kähler analogue of a twistor transform was studied by T. Nitta in
a serie of papers ([N1], [N2] etc.) It turns out that the picture given in
[KV] for Kähler manifolds is very similar to that observed by T. Nitta.

A role of $SU(2)$-invariant 2-forms is played by so-called $B_{2}$-forms.

Definition 7.11: Let $SO(TM)\subset\operatorname{End}(TM)$ be a group bundle
of all orthogonal automorphisms of $TM$, and $G_{M}:=W\cap SO(TM)$. Clearly,
the fibers of $G_{M}$ are isomorphic to $SU(2)$. Consider the action of
$G_{M}$ on the bundle of 2-forms $\Lambda^{2}(M)$. Let
$\Lambda^{2}_{inv}(M)\subset\Lambda^{2}(M)$ be the bundle of
$G_{M}$-invariants. The bundle $\Lambda^{2}_{inv}(M)$ is called the bundle of
$B_{2}$-forms. In a similar fashion we define $B_{2}$-forms with coefficients
in a bundle.

Definition 7.12:  In the above assumptions, let $(B,\nabla)$ be a bundle with
connection over $M$. The bundle $B$ is called a $B_{2}$-bundle, and $\nabla$
is called a $B_{2}$-connection, if its curvature is a $B_{2}$-form.

Consider the natural projection
$\sigma:\;\operatorname{Tw}(M){\>\longrightarrow\>}M$. The proof of the
following claim is completely analogous to the proof of 2.1 and 7.1.

Claim 7.13:

(i)

Let $\omega$ be a $2$-form on $M$. The pullback $\sigma^{*}\omega$ is of type
$(1,1)$ on $\operatorname{Tw}(M)$ if and only if $\omega$ is a $B_{2}$-form on
$M$.

(ii)

Let $B$ be a complex vector bundle on $M$ equipped with a connection $\nabla$,
not necessarily Hermitian. The pullback $\sigma^{*}B$ of $B$ to
$\operatorname{Tw}(M)$ is equipped with a pullback connection
$\sigma^{*}\nabla$. Then, $\nabla$ is a $B_{2}$-connection if and only if
$\sigma^{*}\nabla$ has curvature of Hodge type $(1,1)$.

There exists an analogue of direct and inverse twistor transform as well.

Theorem 7.14:  For any $B_{2}$-connection $(B,\nabla)$, consider the
corresponding holomorphic vector bundle

$(\sigma^{*}B,(\sigma^{*}\nabla)^{0,1}).$

The restriction of $(\sigma^{*}B,(\sigma^{*}\nabla)^{0,1})$ to a line
$\sigma^{-1}(m)\cong{\mathbb{C}}P^{1}$ is a trivial vector bundle, for any
point $m\in M$. Denote by ${\cal C}_{0}$ the category of holomorphic vector
bundles $C$ on $\operatorname{Tw}(M)$, such that the restriction of $C$ to
$\sigma^{-1}(m)$ is trivial, for all $m\in M$, and by ${\cal A}$ the category
of $B_{2}$-bundles (not necessarily Hermitian). Consider the functor

$(\sigma^{*}\bullet)^{0,1}:\;{\cal A}{\>\longrightarrow\>}{\cal C}_{0}$

constructed above. Then it is an equivalence of categories.

Proof: It is easy to modify the proof of the direct and inverse twistor
transform theorem from [KV] to work in quaternionic-Kähler situation.

We will not use 7.3, except for its consequence, which was proven in [N1].

Corollary 7.15:  Consider the functor

$(\sigma^{*}\bullet)^{0,1}:\;{\cal A}{\>\longrightarrow\>}{\cal C}_{0}$

constructed in 7.3. Then $(\sigma^{*}\bullet)^{0,1}$ gives an injection
$\kappa$ from the set of equivalence classes of Hermitian $B_{2}$-connections
over $M$ to the set of equivalence classes of holomorphic connections over
$\operatorname{Tw}(M)$.

Let $M$ be a quaternionic-Kähler manifold. The space $\operatorname{Tw}(M)$
has a natural Kähler metric $g$, such that the standard map
$\sigma:\;\operatorname{Tw}(M){\>\longrightarrow\>}M$ is a Riemannian
submersion, and the restriction of $g$ to the fibers $\sigma^{-1}(m)$ of
$\sigma$ is a metric of constant curvature on
$\sigma^{-1}(m)={\mathbb{C}}P^{1}$ ([Sal], [Bes]).

Example 7.16:  In the case $M={\mathbb{H}}P^{n}$, we have
$\operatorname{Tw}(M)={\mathbb{C}}P^{2n+1}$, and the Kähler metric $g$ is
proportional to the Fubini-Study metric on ${\mathbb{C}}P^{2n+1}$.

Theorem 7.17:  (T. Nitta) Let $M$ be a quaternionic-Kähler manifold of
positive scalar curvature, $\operatorname{Tw}(M)$ its twistor space, equipped
with a natural Kähler structure, and $B$ a Hermitian $B_{2}$-bundle on $M$.
Consider the pullback $\sigma^{*}B$, equipped with a Hermitian connection.
Then $\sigma^{*}B$ is a Yang-Mills bundle on $\operatorname{Tw}(M)$, and $\deg
c_{1}(\sigma^{*}B)=0$.

Proof: [N2].

Let $\kappa$ be the map considered in 7.3. Assume that $M$ is a compact
manifold. In [N2], T. Nitta defined the moduli space of Hermitian
$B_{2}$-bundles. By Uhlenbeck-Yau theorem, Yang-Mills bundles are polystable.
Then the map $\kappa$ provides an embedding from the moduli of non-
decomposable Hermitian $B_{2}$-bundles to the moduli ${\cal M}$ of stable
bundles on $\operatorname{Tw}(M)$. The image of $\kappa$ is a totally real
subvariety in ${\cal M}$ ([N2]).

###  7.4 Hyperkähler manifolds with special ${\mathbb{H}}^{*}$-action and
quaternionic-Kähler manifolds of positive scalar curvature

Further on, we shall need the following definition.

Definition 7.18: An almost hypercomplex manifold is a smooth manifold $M$ with
an action of quaternion algebra in its tangent bundle For each
$L\in\mathbb{H}$, $L^{2}=-1$, $L$ gives an almost complex structure on $M$.
The manifold $M$ is caled hypercomplex if the almost complex structure $L$ is
integrable, for all possible choices $L\in\mathbb{H}$.

The twistor space for a hypercomplex manifold is defined in the same way as
for hyperkähler manifolds. It is also a complex manifold ([K]). The formalism
of direct and inverse twistor transform can be repeated for hypercomplex
manifolds verbatim.

Let ${\mathbb{H}}^{*}$ be the group of non-zero quaternions. Consider an
embedding $SU(2)\hookrightarrow{\mathbb{H}^{*}}$. Clearly, every quaternion
$h\in{\mathbb{H}}^{*}$ can be uniquely represented as $h=|h|\cdot g_{h}$,
where $g_{h}\in SU(2)\subset{\mathbb{H}}^{*}$. This gives a natural
decomposition ${\mathbb{H}}^{*}=SU(2)\times{\mathbb{R}}^{>0}$. Recall that
$SU(2)$ acts naturally on the set of induced complex structures on a
hyperkähler manifold.

Definition 7.19:  Let $M$ be a hyperkähler manifold equipped with a free
smooth action $\rho$ of the group
${\mathbb{H}}^{*}=SU(2)\times{\mathbb{R}}^{>0}$. The action $\rho$ is called
special if the following conditions hold.

(i)

The subgroup $SU(2)\subset{\mathbb{H}}^{*}$ acts on $M$ by isometries.

(ii)

For $\lambda\in{\mathbb{R}}^{>0}$, the corresponding action
$\rho(\lambda):\;M{\>\longrightarrow\>}M$ is compatible with the
hyperholomorphic structure (which is a fancy way of saying that
$\rho(\lambda)$ is holomorphic with respect to any of induced complex
structures).

(iii)

Consider the smooth ${\mathbb{H}^{*}}$-action
$\rho_{e}:\;{\mathbb{H}^{*}}\times\operatorname{End}(TM){\>\longrightarrow\>}\operatorname{End}(TM)$
induced on $\operatorname{End}(TM)$ by $\rho$. For any $x\in M$ and any
induced complex structure $I$, the restriction
$I{\left|{}_{{\phantom{|}\\!\\!}_{x}}\right.}$ can be considered as a point in
the total space of $\operatorname{End}(TM)$. Then, for all induced complex
structures $I$, all $g\in SU(2)\subset{\mathbb{H}^{*}}$, and all $x\in M$, the
map $\rho_{e}(g)$ maps $I{\left|{}_{{\phantom{|}\\!\\!}_{x}}\right.}$ to
$g(I){\left|{}_{{\phantom{|}\\!\\!}_{\rho_{e}(g)(x)}}\right.}$.

Speaking informally, this can be stated as “${\mathbb{H}}^{*}$-action
interchanges the induced complex structures”.

(iv)

Consider the automorphism of $S^{2}T^{*}M$ induced by $\rho(\lambda)$, where
$\lambda\in{\mathbb{R}}^{>0}$. Then $\rho(\lambda)$ maps the Riemannian metric
tensor $s\in S^{2}T^{*}M$ to $\lambda^{2}s$.

Example 7.20: Consider the flat hyperkähler manifold $M_{\rm
fl}={\mathbb{H}}^{n}\backslash 0$. There is a natural action of
${\mathbb{H}^{*}}$ on ${\mathbb{H}}^{n}\backslash 0$. This gives a special
action of ${\mathbb{H}^{*}}$ on $M_{\rm fl}$.

The case of a flat manifold $M_{\rm fl}={\mathbb{H}}^{n}\backslash 0$ is the
only case where we apply the results of this section. However, the general
statements are just as difficult to prove, and much easier to comprehend.

Definition 7.21:  Let $M$ be a hyperkähler manifold with a special action
$\rho$ of ${\mathbb{H}^{*}}$. Assume that $\rho(-1)$ acts non-trivially on
$M$. Then $M/\rho(\pm 1)$ is also a hyperkähler manifold with a special action
of ${\mathbb{H}^{*}}$. We say that the manifolds $(M,\rho)$ and $(M/\rho(\pm
1),\rho)$ are hyperkähler manifolds with special action of ${\mathbb{H}^{*}}$
which are special equivalent. Denote by $H_{sp}$ the category of hyperkähler
manifolds with a special action of ${\mathbb{H}^{*}}$ defined up to special
equivalence.

A. Swann ([Sw]) developed an equivalence between the category of quaternionic-
Kähler manifolds of positive scalar curvature and the category $H_{sp}$. The
purpose of this Subsection is to give an exposition of Swann’s formalism.

Let $Q$ be a quaternionic-Kähler manifold. The restricted holonomy group of
$Q$ is $Sp(n)\cdot Sp(1)$, that is, $(Sp(n)\times Sp(1))/\\{\pm 1\\}$.
Consider the principal bundle ${\cal G}$ with the fiber $Sp(1)/\\{\pm 1\\}$,
corresponding to the subgroup

$Sp(1)/\\{\pm 1\\}\subset(Sp(n)\times Sp(1))/\\{\pm 1\\}.$

of the holonomy. There is a natural $Sp(1)/\\{\pm 1\\}$-action on the space
${\mathbb{H}^{*}}/\\{\pm 1\\}$. Let

${\cal U}(Q):={\cal G}\times_{Sp(1)/\\{\pm 1\\}}{\mathbb{H}^{*}}/\\{\pm 1\\}.$

Clearly, ${\cal U}(Q)$ is fibered over $Q$, with fibers which are isomorphic
to
${\mathbb{H}^{*}}/\\{\pm 1\\}$. We are going to show that the manifold ${\cal
U}(Q)$ is equipped with a natural hypercomplex structure.

There is a natural smooth decomposition ${\cal U}(Q)\cong{\cal
G}\times{\mathbb{R}}^{>0}$ which comes from the isomorphism
${\mathbb{H}^{*}}\cong Sp(1)\times{\mathbb{R}}^{>0}$.

Consider the standard 4-dimensional bundle $W$ on $Q$. Let $x\in Q$ be a
point. The fiber $W{\left|{}_{{\phantom{|}\\!\\!}_{q}}\right.}$ is isomorphic
to $\mathbb{H}$, in a non-canonical way. The choices of isomorphism
$W{\left|{}_{{\phantom{|}\\!\\!}_{q}}\right.}\cong\mathbb{H}$ are called
quaternion frames in $q$. The set of quaternion frames gives a fibration over
$Q$, with a fiber $\operatorname{Aut}({\mathbb{H}})\cong Sp(1)/\\{\pm 1\\}$.
Clearly, this fibration coincides with the principal bundle ${\cal G}$
constructed above. Since ${\cal U}(Q)\cong{\cal G}\times{\mathbb{R}}^{>0}$, a
choice of $u\in{\cal U}(Q){\left|{}_{{\phantom{|}\\!\\!}_{q}}\right.}$
determines an isomorphism
$W{\left|{}_{{\phantom{|}\\!\\!}_{q}}\right.}\cong\mathbb{H}$.

Let $(q,u)$ be the point of ${\cal U}(Q)$, with $q\in Q$, $u\in{\cal
U}(Q){\left|{}_{{\phantom{|}\\!\\!}_{q}}\right.}$. The natural connection in
${\cal U}(Q)$ gives a decomposition

$T_{(q,u)}U(Q)=T_{u}\bigg{(}{\cal
U}(Q){\left|{}_{{\phantom{|}\\!\\!}_{q}}\right.}\bigg{)}\oplus T_{q}Q.$

The space ${\cal
U}(Q){\left|{}_{{\phantom{|}\\!\\!}_{q}}\right.}\cong{\mathbb{H}^{*}}/\\{\pm
1\\}$ is equipped with a natural hypercomplex structure. This gives a
quaternion action on $T_{u}\bigg{(}{\cal
U}(Q){\left|{}_{{\phantom{|}\\!\\!}_{q}}\right.}\bigg{)}$ The choice of
$u\in{\cal U}(Q){\left|{}_{{\phantom{|}\\!\\!}_{q}}\right.}$ determines a
quaternion action on $T_{q}Q$, as we have seen above. We obtain that the total
space of ${\cal U}(Q)$ is an almost hypercomplex manifold.

Proposition 7.22:  (A. Swann) Let $Q$ be a quaternionic-Kähler manifold.
Consider the manifold ${\cal U}(Q)$ constructed as above, and equipped with a
quaternion algebra action in its tangent space. Then ${\cal U}(Q)$ is a
hypercomplex manifold.

Proof: Clearly, the manifold ${\cal U}(Q)$ is equipped with a
${\mathbb{H}}^{*}$-action, which is related with the almost hypercomplex
structure as prescribed by 7.4 (ii)-(iii). Pick an induced complex structure
$I\in{\mathbb{H}}$. This gives an algebra embedding
${\mathbb{C}}{\>\longrightarrow\>}{\mathbb{H}}$. Consider the corresponding
${\mathbb{C}}^{*}$-action $\rho_{I}$ on an almost complex manifold $({\cal
U}(Q),I)$. This ${\mathbb{C}}^{*}$-action is compatible with the almost
complex structure. The quotient ${\cal U}(Q)/\rho(I)$ is an almost complex
manifold, which is naturally isomorphic to the twistor space
$\operatorname{Tw}(Q)$. Let $L^{*}$ be a complex vector bundle of all
$(1,0)$-vectors $v\in T(\operatorname{Tw}(Q))$ tangent to the fibers of the
standard projection $\sigma:\;\operatorname{Tw}(Q){\>\longrightarrow\>}Q$, and
$L$ be the dual vector bundle. Denote by $Tot_{\neq 0}(L)$ the complement
$\operatorname{Tot}(L)\backslash N$, where
$N=\operatorname{Tw}(Q)\subset\operatorname{Tot}(L)$ is the zero section of
$L$. Using the natural connection in $L$, we obtain an almost complex
structure on $\operatorname{Tot}(L)$.

Consider the natural projection $\varphi:\;Tot_{\neq
0}(L){\>\longrightarrow\>}Q$. The fibers $\varphi^{-1}(q)$ of $\varphi$ are
identified with the space of non-zero vectors in the total space of the
cotangent bundle $T^{*}\sigma^{-1}(q)\cong T^{*}({\mathbb{C}}P^{1})$. This
space is naturally isomorphic to

${\cal
G}{\left|{}_{{\phantom{|}\\!\\!}_{q}}\right.}\times{\mathbb{R}}^{>0}={\cal
U}(Q){\left|{}_{{\phantom{|}\\!\\!}_{q}}\right.}\cong{\mathbb{H}}^{*}/\\{\pm
1\\}.$

This gives a canonical isomorphism of almost complex manifolds

$({\cal U}(Q),I){\>\longrightarrow\>}Tot_{\neq 0}(L).$

Therefore, to prove that $({\cal U}(Q),I)$ is a complex manifold, it suffices
to show that the natural almost complex structure on $Tot_{\neq
0}(L)\subset\operatorname{Tot}(L)$ is integrable. Consider the natural
connection $\nabla_{L}$ on $L$. To prove that $\operatorname{Tot}(L)$ is a
complex manifold, it suffices to show that $\nabla_{L}$ is a holomorphic
connection. The bunlde $L$ is known under the name of holomorphic contact
bundle, and it is known to be holomorphic ([Sal], [Bes]).

Remark 7.23: The result of 7.4 is well known. We have given its proof because
we shall need the natural identification $Tot_{\neq 0}(L)\cong{\cal U}(Q)$
further on in this paper.

Theorem 7.24:  Let $Q$ be a quaternionic-Kähler manifold of positive scalar
curvature, and ${\cal U}(Q)$ the hypercomplex manifold constructed above. Then
${\cal U}(Q)$ admits a unique (up to a scaling) hyperkähler metric compatible
with the hypercomplex structure.

Proof: [Sw].

Consider the action of ${\mathbb{H}}^{*}$ on ${\cal U}(M)$ defined in the
proof of 7.4. This action satisfies the conditions (ii) and (iii) of 7.4. The
conditions (i) and (iv) of 7.4 are easy to check (see [Sw] for details). This
gives a functor from the category ${\cal C}$ of quaternionic-Kähler manifolds
of positive scalar curvature to the category $H_{sp}$ of 7.4.

Theorem 7.25:  The functor $Q{\>\longrightarrow\>}{\cal U}(Q)$ from ${\cal C}$
to $H_{sp}$ is an equivalence of categories.

Proof: [Sw].

The inverse functor from $H_{sp}$ to $C$ is constructed by taking a quotient
of $M$ by the action of ${\mathbb{H}}^{*}$. Using the technique of
quaternionic-Kähler reduction anf hyperkähler potentials ([Sw]), one can equip
the quotient $M/{\mathbb{H}}^{*}$ with a natural quaternionic-Kähler
structure.

##  8 ${\mathbb{C}}^{*}$-equivariant twistor spaces

In Section 7, we gave an exposition of the twistor transform, $B_{2}$-bundles
and Swann’s formalism. In the present Section, we give a synthesis of these
theories, obtaining a construction with should be thought of as Swann’s
formalism for vector bundles.

Consider the equivalence of categories $Q{\>\longrightarrow\>}{\cal U}(Q)$
constructed in 7.4 (we call this equivalence “Swann’s formalism”). We show
that $B_{2}$-bundles on $Q$ are in functorial bijective correspondence with
${\mathbb{C}}^{*}$-equivariant holomorphic bundles on $\operatorname{Tw}({\cal
U}(Q))$ (8.1).

In Subsection 8.4, this equivalence is applied to the vector bundle
$\pi^{*}(F)$ of 6. We use it to construct a canonical Yang-Mills connection on
$\pi^{*}(F){\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}$, where $C$ is a
special fiber of $\pi:\;\widetilde{M}{\>\longrightarrow\>}(M,I)$ (see 6 for
details and notation). This implies that the holomorphic bundle
$\pi^{*}(F){\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}$ is polystable (8.5).

###  8.1 $B_{2}$-bundles on quaternionic-Kähler manifolds and
${\mathbb{C}}^{*}$-equivariant holomorphic bundles over twistor spaces

For the duration of this Subsection, we fix a hyperkähler manifold $M$,
equipped with a special ${\mathbb{H}}^{*}$-action $\rho$, and the
corresponding quaternionic-Kähler manifold $Q=M/{\mathbb{H}}^{*}$. Denote the
natural quotient map by $\varphi:\;M{\>\longrightarrow\>}Q$.

Lemma 8.1:  Let $\omega$ be a 2-form over $Q$, and $\varphi^{*}\omega$ its
pullback to $M$. Then the following conditions are equivalent

(i)

$\omega$ is a $B_{2}$-form

(ii)

$\varphi^{*}\omega$ is of Hodge type $(1,1)$ with respect to some induced
complex structure $I$ on $M$

(iii)

$\varphi^{*}\omega$ is $SU(2)$-invariant.

Proof: Let $I$ be an induced complex structure on $M$. As we have shown in the
proof of 7.4, the complex manifold $(M,I)$ is idenified with an open subset of
the total space $\operatorname{Tot}(L)$ of a holomorphic line bundle $L$ over
$\operatorname{Tw}(Q)$. The map $\varphi$ is represented as a composition of
the projections
$h:\;\operatorname{Tot}(L){\>\longrightarrow\>}\operatorname{Tw}(Q)$ and
$\sigma_{Q}:\;\operatorname{Tw}(Q){\>\longrightarrow\>}Q$. Since the map $h$
is smooth and holomorphic, the form $\varphi^{*}\omega$ is of Hodge type
$(1,1)$ if and only if $\sigma_{Q}^{*}\omega$ is of type $(1,1)$. By 7.3 (i),
this happens if and only if $\omega$ is a $B_{2}$-form. This proves an
equivalence (i) $\Leftrightarrow$ (ii). Since the choice of $I$ is arbitrary,
the pullback $\varphi^{*}\omega$ of a $B_{2}$-form is of Hodge type $(1,1)$
with respect to all induced complex structures. By 2.1, this proves the
implication (i) $\Rightarrow$ (iii). The implication (iii) $\Rightarrow$ (ii)
is clear.

Proposition 8.2: Let $(B,\nabla)$ be a complex vector bundle with connection
over $Q$, and $(\varphi^{*}B,\varphi^{*}\nabla)$ its pullback to $M$. Then the
following conditions are equivalent

(i)

$(B,\nabla)$ is a $B_{2}$-form

(ii)

The curvature of $(\varphi^{*}B,\varphi^{*}\nabla)$ is of Hodge type (1,1)
with respect to some induced complex structure $I$ on $M$

(iii)

The bundle $(\varphi^{*}B,\varphi^{*}\nabla)$ is autodual

Proof: Follows from 8.1 applied to $\omega=\nabla^{2}$.

For any point $I\in{\mathbb{C}}P^{1}$, consider the corresponding algebra
embedding ${\mathbb{C}}\stackrel{{\scriptstyle
c_{I}}}{{\hookrightarrow}}{\mathbb{H}}$. Let $\rho_{I}$ be the action of
${\mathbb{C}}^{*}$ on $(M,I)$ obtained as a restriction of $\rho$ to
$c_{I}({\mathbb{C}}^{*})\subset{\mathbb{H}}^{*}$. Clearly from 7.4 (ii),
$\rho_{I}$ acts on $(M,I)$ by holomorphic automorphisms.

Consider $\operatorname{Tw}(M)$ as a union

$\operatorname{Tw}(M)=\bigcup_{I\in{\mathbb{C}}P^{1}}\pi^{-1}(I),\ \
\pi^{-1}(I)=(M,I)$

Gluing $\rho(I)$ together, we obtain a smooth ${\mathbb{C}}^{*}$-action
$\rho_{\mathbb{C}}$ on $\operatorname{Tw}(M)$.

Claim 8.3:  Consider the action
$\rho_{\mathbb{C}}:\;{\mathbb{C}}^{*}\times\operatorname{Tw}(M){\>\longrightarrow\>}\operatorname{Tw}(M)$
constructed above. Then $\rho_{\mathbb{C}}$ is holomorphic.

Proof: It is obvious from construction that $\rho_{\mathbb{C}}$ is compatible
with the complex structure on $\operatorname{Tw}(M)$.

Example 8.4: Let $M={\mathbb{H}}^{n}\backslash 0$. Since
$\operatorname{Tw}({\mathbb{H}}^{n})$ is canonically isomorphic to a total
space of the bundle ${\cal O}(1)^{n}$ over ${\mathbb{C}}P^{1}$, the twistor
space $\operatorname{Tw}(M)$ is $\operatorname{Tot}({\cal O}(1)^{n})$ without
zero section. The group ${\mathbb{C}}^{*}$ acts on $\operatorname{Tot}({\cal
O}(1)^{n})$ by dilatation, and the restriction of this action to
$\operatorname{Tw}(M)$ coincides with $\rho_{\mathbb{C}}$.

Consider the map $\sigma:\;\operatorname{Tw}(M){\>\longrightarrow\>}M$. Let
$(B,\nabla)$ be a $B_{2}$-bundle over $Q$. Since the bundle
$(\varphi^{*}B,\varphi^{*}\nabla)$ is autodual, the curvature of
$\sigma^{*}\varphi^{*}\nabla$ has type $(1,1)$. Let
$(\sigma^{*}\varphi^{*}B,(\sigma^{*}\varphi^{*}\nabla)^{0,1})$ be the
holomorphic bundle obtained from $(\varphi^{*}B,\varphi^{*}\nabla)$ by twistor
transform. Clearly, this bundle is ${\mathbb{C}}^{*}$-equivariant, with
respect to the natural ${\mathbb{C}}^{*}$-action on $\operatorname{Tw}(M)$. It
turns out that any ${\mathbb{C}}^{*}$-equivariant bundle ${\cal F}$ on
$\operatorname{Tw}(M)$ can be obtained this way, assuming that ${\cal F}$ is
compatible with twistor transform.

Theorem 8.5:  In the above assumptions, let ${\cal C}_{B_{2}}$ be the category
of of $B_{2}$-bundles on $Q$, and ${\cal
C}_{\operatorname{Tw},{\mathbb{C}}^{*}}$ the category of
${\mathbb{C}}^{*}$-equivariant holomorphic bundles on $\operatorname{Tw}(M)$
which are compatible with the twistor transform. Consider the functor

$(\sigma^{*}\varphi^{*})^{0,1}:{\cal C}_{B_{2}}{\>\longrightarrow\>}{\cal
C}_{\operatorname{Tw},{\mathbb{C}}^{*}},$

$(B,\nabla){\>\longrightarrow\>}(\sigma^{*}\varphi^{*}B,(\sigma^{*}\varphi^{*}\nabla)^{0,1})$,
constructed above. Then $(\sigma^{*}\varphi^{*})^{0,1}$ establishes an
equivalence of categories.

We prove 8.1 in Subsection 8.3.

Remark 8.6: Let $Q$ be an arbitrary quaternionic-Kähler manifold, and $M={\cal
U}(Q)$ the corresponding fibration. Then $M$ is hypercomplex, and its twistor
space is equipped with a natural holomorphic action of ${\mathbb{C}}^{*}$.
This gives necessary ingredients needed to state 8.1 for $Q$ with negative
scalar curvature. The proof which we give for $Q$ with positive scalar
curvature will in fact work for all quaternionic-Kähler manifolds.

Question 8.7: What happens with this construction when $Q$ is a hyperkähler
manifold?

In this paper, we need 8.1 only in the case $Q={\mathbb{H}}P^{n}$,
$M={\mathbb{H}}^{n}\backslash 0$, but the general proof is just as difficult.

###  8.2 ${\mathbb{C}}^{*}$-equivariant bundles and twistor transform

Let $M$ be a hyperkähler manifold, and $\operatorname{Tw}(M)$ its twistor
space. Recall that $\operatorname{Tw}(M)={\mathbb{C}}P^{1}\times M$ is
equipped with a canonical anticomplex involution $\iota$, which acts as
identity on $M$ and as central symmetry $I{\>\longrightarrow\>}-I$ on
${\mathbb{C}}P^{1}=S^{2}$.

Proposition 8.8:  Let $M$ be a hyperkähler manifold, and
$\operatorname{Tw}(M)$ its twistor space. Assume that $\operatorname{Tw}(M)$
is equipped with a free holomorphic action
$\rho(z):\;\operatorname{Tw}(M){\>\longrightarrow\>}\operatorname{Tw}(M)$ of
${\mathbb{C}}^{*}$, acting along the fibers of
$\pi:\;\operatorname{Tw}(M){\>\longrightarrow\>}{\mathbb{C}}P^{1}$. Assume,
moreover, that $\iota\circ\rho(z)=\rho(\overline{z})\circ\iota$, where $\iota$
is the natural anticomplex involution of $\operatorname{Tw}(M)$.111These
assumptions are automatically satisfied when $M$ is equipped with a special
${\mathbb{H}}^{*}$-action, and $\rho(z)$ is the corresponding
${\mathbb{C}}^{*}$-action on $\operatorname{Tw}(M)$. Let ${\cal F}$ be a
${\mathbb{C}}^{*}$-equivariant holomorphic vector bundle on
$\operatorname{Tw}(M)$. Assume that ${\cal F}$ is compatible with the twistor
transform. Let $\nabla_{{\cal F}}$ be the natural connection on ${\cal F}$
(7.1). Then $\nabla_{{\cal F}}$ is flat along the leaves of $\rho$.

Proof: First of all, let us recall the construction of the natural connection
$\nabla_{{\cal F}}$. Let ${\cal F}$ be an arbitrary bundle compatible with the
twistor transform. We construct $\nabla_{{\cal F}}$ in terms of the
isomorphism $\Psi_{1,2}$ defined in 6.2.

Consider an induced complex structure $I$. Let $F_{I}$ be the restriction of
${\cal F}$ to $(M,I)=\pi^{-1}(I)\subset\operatorname{Tw}(M)$. Consider the
evaluation map

$p_{I}:\;\operatorname{Lin}(M){\>\longrightarrow\>}(M,I)$

(Subsection 6.1). In a similar way we define the holomorphic vector bundle
$F_{-I}$ on $(M,-I)$ and the map
$p_{-I}:\;\operatorname{Lin}(M){\>\longrightarrow\>}(M,-I)$. Denote by
$F_{1}$, $F_{-1}$ the sheaves $p_{I}^{*}(F_{I})$, $p_{-I}^{*}(F_{-I})$. In
6.2, we constructed an isomorphism
$\Psi_{1,-1}:\;F_{1}{\>\longrightarrow\>}F_{-1}$.

Let us identify $\operatorname{Lin}(M)$ with $(M,I)\times(M,I)$ (this
idenitification is naturally defined in a neighbourhood of
$\operatorname{Hor}\subset\operatorname{Lin}(M)$ – see 6.1). Then the maps
$p_{I}$, $p_{-I}$ became projections to the relevant components. Let

$\begin{split}\overline{\partial}:\;F_{1}&{\>\longrightarrow\>}F_{1}\otimes
p_{I}^{*}\Omega^{1}(M,-I),\\\
\partial:\;F_{-1}&{\>\longrightarrow\>}F_{-1}\otimes
p_{-I}^{*}\Omega^{1}(M,I),\end{split}$

be the sheaf maps obtained as pullbacks of de Rham differentials (the tensor
product is taken in the category of coherent sheaves over
$\operatorname{Lin}(M)$). Twisting $\partial$ by an isomorphism
$\Psi_{1,-1}:\;F_{1}{\>\longrightarrow\>}F_{-1}$, we obtain a map

$\partial^{\Psi}:\;F_{1}{\>\longrightarrow\>}F_{1}\otimes
p_{I}^{*}\Omega^{1}(M,I).$

Adding $\overline{\partial}$ and $\partial^{\Psi}$, we obtain

$\nabla:\;F_{1}{\>\longrightarrow\>}F_{1}\otimes\bigg{(}p_{I}^{*}\Omega^{1}(M,I)\oplus
p_{I}^{*}\Omega^{1}(M,-I)\bigg{)}.$

Clearly, $\nabla$ satisfies the Leibniz rule. Moreover, the sheaf
$p_{I}^{*}\Omega^{1}(M,I)\oplus p_{I}^{*}\Omega^{1}(M,-I)$ is naturally
isomorphic to the sheaf of differentials over

$\operatorname{Lin}(M)=(M,I)\times(M,-I).$

Therefore, $\nabla$ can be considered as a connection in $F_{1}$, or as a real
analytic connection in a real analytic complex vector bundle underlying
$F_{I}$. From the definition of $\nabla_{{\cal F}}$ ([KV]), it is clear that
$\nabla_{{\cal F}}{\left|{}_{{\phantom{|}\\!\\!}_{(M,I)}}\right.}$ equals
$\nabla$.

Return to the proof of 8.2. Consider a ${\mathbb{C}}^{*}$-action $\rho_{I}(z)$
on $(M,I)$, $(M,-I)$ induced from the natural embeddings
$(M,I)\hookrightarrow\operatorname{Tw}(M)$,
$(M,-I)\hookrightarrow\operatorname{Tw}(M)$. Then $F_{I}$ is a
${\mathbb{C}}^{*}$-equivariant bundle. Since
$\iota\circ\rho(z)=\rho(\overline{z})\circ\iota$, the identification
$\operatorname{Lin}(M)=(M,I)\times(M,I)$ is compatible with
${\mathbb{C}}^{*}$-action. Let ${\mathbf{r}}=\frac{d}{dr}$ be the holomorphic
vector field on $(M,I)$ corresponding to the ${\mathbb{C}}^{*}$-action. To
prove 8.2, we have to show that the operator

$[\nabla_{\mathbf{r}},\nabla_{\overline{\mathbf{r}}}]:\;F_{I}{\>\longrightarrow\>}F_{I}\otimes\Lambda^{1,1}(M,I)$

vanishes.

Consider the equivariant structure operator

$\rho(z)^{F}:\rho_{I}(z)^{*}F_{I}{\>\longrightarrow\>}F_{I}.$

Let $U$ be a ${\mathbb{C}}^{*}$-invariant Stein subset of $(M,I)$. Consider
$\rho(z)^{F}$ an an endomorphism of the space of global holomorphic sections
$\Gamma_{U}(F_{I})$. Let

$D_{r}(f):=\lim\limits_{\varepsilon\rightarrow
0}\frac{\rho_{I}(1+\varepsilon)}{\varepsilon},$

for $f\in\Gamma_{U}(F_{I})$. Clearly, $D_{r}$ is a well defined sheaf
endomorphism of $F_{I}$, satisfying

$D_{r}(\alpha\cdot f)=\frac{d}{dr}\alpha\cdot f+\alpha\cdot D_{r}(f),$

for all $\alpha\in{\cal O}_{(M,I)}$. We say that a holomorphic section $f$ of
$F_{I}$ is ${\mathbb{C}}^{*}$-invariant if $D_{r}(f)=0$. Clearly, the ${\cal
O}_{(M,I)}$-sheaf $F_{I}$ is generated by ${\mathbb{C}}^{*}$-invariant
sections. Therefore, it suffices to check the equality

$[\nabla_{\mathbf{r}},\nabla_{\overline{\mathbf{r}}}](f)=0$

for holomorphic ${\mathbb{C}}^{*}$-invariant $f\in F_{I}$.

Since $f$ is holomorphic, we have $\nabla_{\overline{\mathbf{r}}}f=0$. Thus,

$[\nabla_{\mathbf{r}},\nabla_{\overline{\mathbf{r}}}](f)=\nabla_{\overline{\mathbf{r}}}\nabla_{\mathbf{r}}(f).$

We obtain that 8.2 is implied by the following lemma.

Lemma 8.9:  In the above assumptions, let $f$ be a
${\mathbb{C}}^{*}$-invariant section of $F_{I}$. Then
$\nabla_{\mathbf{r}}(f)=0$.

Proof: Return to the notation we used in the beginning of the proof of 8.2.
Then, $\nabla(f)=\overline{\partial}(f)+\partial^{\Psi}(f)$. Since $f$ is
holomorphic, $\overline{\partial}(f)=0$, so we need to show that
$\partial^{\Psi}(f)({\mathbf{r}})=0$. By definition of $\partial^{\Psi}$, this
is equivalent to proving that

$\partial\Psi_{1,-1}(f)({\mathbf{r}})=0.$

Consider the ${\mathbb{C}}^{*}$-action on $\operatorname{Lin}(M)$ which is
induced by the ${\mathbb{C}}^{*}$-action on $\operatorname{Tw}(M)$. Since the
maps $p_{I}$, $p_{-I}$ are compatible with the ${\mathbb{C}}^{*}$-action, the
sheaves $F_{1}$, $F_{-1}$ are ${\mathbb{C}}^{*}$-equivariant. We can repeat
the construction of the operator $D_{r}$ for the sheaf $F_{-I}$. This allows
one to speak of holomorphic ${\mathbb{C}}^{*}$-invariant sections of $F_{-I}$.
Pick a ${\mathbb{C}}^{*}$-invariant Stein subset $U\subset(M,-I)$. Since the
statement of 8.2 is local, we may assume that $M=U$. Let $g_{1},...,g_{n}$ be
a set of ${\mathbb{C}}^{*}$-invariant sections of $F_{I}$ which generated
$F_{I}$. Then, the sections $p_{-I}^{*}(g_{1}),...,p_{-I}^{*}(g_{n})$ generate
$F_{-1}$. Consider the section $\Psi_{1,-1}(f)$ of $F_{-1}$. Clearly,
$\Psi_{1,-1}$ commutes with the natural ${\mathbb{C}}^{*}$-action. Therefore,
the section $\Psi_{1,-1}(f)$ is ${\mathbb{C}}^{*}$-invariant, and can be
written as

$\Psi_{1,-1}(f)=\sum\alpha_{i}p_{-I}^{*}(g_{i}),$

where the functions $\alpha_{i}$ are ${\mathbb{C}}^{*}$-invariant. By
definition of $\partial$ we have

$\partial\left(\sum\alpha_{i}p_{-I}^{*}(g_{i})\right)=\sum\partial(\alpha_{i}p_{-I}^{*}(g_{i}))+\sum\alpha_{i}\partial(p_{-I}^{*}(g_{i})).$

On the other hand, $g_{i}$ is a holomorphic section of $F_{-I}$, so $\partial
p_{-I}^{*}(g_{i})=0$. We obtain

$\partial\left(\sum\alpha_{i}p_{-I}^{*}\cdot(g_{i})\right)=\sum\partial\alpha_{i}p_{-I}^{*}(g_{i}).$

Thus,

$\partial\Psi_{1,-1}(f)({\mathbf{r}})=\sum\frac{\partial\alpha_{i}}{\partial
r}p_{-I}^{*}(g_{i}),$

but since the functions $\alpha_{i}$ are ${\mathbb{C}}^{*}$-invariant, their
derivatives along ${\mathbf{r}}$ vanish. We obtain
$\partial\Psi_{1,-1}(f)({\mathbf{r}})=0$. This proves 8.2. 8.2 is proven.

###  8.3 Twistor transform and the ${\mathbb{H}}^{*}$-action

For the duration of this Subsection, we fix a hyperkähler manifold $M$,
equipped with a special ${\mathbb{H}}^{*}$-action $\rho$, and the
corresponding quaternionic-Kähler manifold $Q=M/{\mathbb{H}}^{*}$. Denote the
natural quotient map by $\varphi:\;M{\>\longrightarrow\>}Q$. Clearly, 8.1 is
an immediate consequence of the following theorem.

Theorem 8.10:  Let ${\cal F}$ be a ${\mathbb{C}}^{*}$-equivariant holomorphic
bundle over $\operatorname{Tw}(M)$, which is compatible with the twistor
transform. Consider the natural connection $\nabla_{{\cal F}}$ on ${\cal F}$.
Then $\nabla_{{\cal F}}$ is flat along the leaves of
${\mathbb{H}}^{*}$-action.

Proof: The leaves of ${\mathbb{H}}^{*}$-action are parametrized by the points
of $q\in Q$. Consider such a leaf $M_{q}:=\varphi^{-1}(q)\subset M$. Clearly,
$M_{q}$ is a hyperkähler submanifold in $M$, equipped with a special action of
${\mathbb{H}}^{*}$. Moreover, the restriction of ${\cal F}$ to
$\operatorname{Tw}(M_{q})\subset\operatorname{Tw}(M)$ satisfies assumptions of
8.3. To prove that $\nabla_{{\cal F}}$ is flat along the leaves of
${\mathbb{H}}^{*}$-action, we have to show that ${{\cal
F}}{\left|{}_{{\phantom{|}\\!\\!}_{\operatorname{Tw}(M_{q})}}\right.}$ is
flat, for all $q$. Therefore, it suffices to prove 8.3 for
$\dim_{\mathbb{H}}M=1$.

Lemma 8.11:  We work in notation and assumptions of 8.3. Assume that
$\dim_{\mathbb{H}}M=1$. Then the connection $\nabla_{{\cal F}}$ is flat.

Proof: Let $I$ be an induced complex structure, and
$F_{I}:=F{\left|{}_{{\phantom{|}\\!\\!}_{(M,I)}}\right.}$ the corresponding
holomorphic bundle on $(M,I)$. Denote by $z_{I}$ the vector field
corresponding to the ${\mathbb{C}}^{*}$-action $\rho_{I}$ on $(M,I)$. By
definition, the connection
$\nabla{\left|{}_{{\phantom{|}\\!\\!}_{F_{I}}}\right.}$ has $SU(2)$-invariant
curvature $\Theta_{I}$. On the other hand,
$\Theta_{I}(z_{I},\overline{z}_{I})=0$ by 8.2. Since $\nabla_{{\cal
F}}=\sigma^{*}\nabla$ is a pullback of an autodual connection $\nabla$ on $M$,
its curvature is a pullback of $\Theta_{I}$. In particular,
$\Theta=\Theta_{I}$ is independent from the choice of induced complex
structure $I$. We obtain that $\Theta(z_{I},\overline{z}_{I})=0$ for all
induced complex structures $I$ on $M$.

Now 8.3 is implied by the following linear-algebraic claim.

Claim 8.12:  Let $M$ be a hyperkähler manifold equipped with a special
${\mathbb{H}}^{*}$-action, $\dim_{\mathbb{H}}M=1$. Consider the vectors
$z_{I}$, $\overline{z}_{I}$ defined above. Let $\Theta$ be a smooth
$SU(2)$-invariant 2-form, such that for all induced complex structures, $I$,
we have $\Theta(z_{I},\overline{z}_{I})=0$. Then $\Theta=0$.

Proof: The proof of 8.3 is an elementary calculation. Fix a point $m_{0}\in
M$. Consider the flat hyperkähler manifold ${\mathbb{H}}\backslash 0$,
equipped with a natural special action of ${\mathbb{H}}^{*}$. From the
definition of a special action, it is clear that the map $\rho$ defines a
covering ${\mathbb{H}}\backslash 0{\>\longrightarrow\>}M$,
$h{\>\longrightarrow\>}\rho(h)m_{0}$ of hyperkähler manifolds, and this
covering is compatible with the special action. Therefore, the hyperkähler
manifold $M$ is flat, and the ${\mathbb{H}}^{*}$-action is linear in the flat
coordinates.

Let

$\Lambda^{2}(M)=\Lambda^{+}(M)\oplus\Lambda^{-}(M)$

be the standard decomposition of $\Lambda^{2}(M)$ according to the eigenvalues
of the Hodge $*$ operator. Consider the natural Hermitian metric on
$\Lambda^{2}(M)$. Then $\Lambda^{-}(M)$ is the bundle of $SU(2)$-invariant
2-forms (see, e. g., [V1]), and $\Lambda^{+}(M)$ is its orthogonal complement.
Consider the corresponding orthogonal projection
$\Pi:\;\Lambda^{2}(M){\>\longrightarrow\>}\Lambda^{-}(M)$. Denote by
$dz_{I}\wedge d\overline{z}_{I}$ the differential form which is dual to the
bivector $z_{I}\wedge\overline{z}_{I}$. Let $R\subset\Lambda^{-}(M)$ be the
$C^{\infty}(M)$-subsheaf of $\Lambda^{-}(M)$ generated by $\Pi(dz_{I}\wedge
d\overline{z}_{I})$, for all induced complex structures $I$ on $M$. Clearly,
$\Theta\in\Lambda^{-}(M)$ and $\Theta$ is orthogonal to
$R\subset\Lambda^{-}(M)$. Therefore, to prove that $\Theta=0$ it suffices to
show that $R=\Lambda^{-}(M)$. Since $M$ is covered by ${\mathbb{H}}\backslash
0$, we may prove $R=\Lambda^{-}(M)$ in assumption $M={\mathbb{H}}\backslash
0$.

Let $\gamma$ be the real vector field corresponding to dilatations of
$M={\mathbb{H}}\backslash 0$, and $d\gamma$ the dual 1-form. Clearly,

$dz_{I}\wedge d\overline{z}_{I}=2\sqrt{-1}\>d\gamma\wedge I(d\gamma).$

Averaging $d\gamma\wedge I(d\gamma)$ by $SU(2)$, we obtain

$\Pi(dz_{I}\wedge d\overline{z}_{I})=\sqrt{-1}\>\bigg{(}d\gamma\wedge
I(d\gamma)-J(d\gamma)\wedge K(d\gamma)\bigg{)}$

where $I$, $J$, $K$ is the standard triple of generators for quaternion
algebra. Similarly,

$\Pi(dz_{J}\wedge d\overline{z}_{J})=\sqrt{-1}\>\bigg{(}d\gamma\wedge
J(d\gamma)+K(d\gamma)\wedge I(d\gamma)\bigg{)}$

and

$\Pi(dz_{K}\wedge d\overline{z}_{K})=\sqrt{-1}\>\bigg{(}d\gamma\wedge
K(d\gamma)+I(d\gamma)\wedge J(d\gamma)\bigg{)}$

Thus, $\Pi(R)$ is a 3-dimensional sub-bundle of $\Lambda^{-}(M)$. Since
$\dim\Lambda^{-}(M)=3$, we have $\Pi(R)=\Lambda^{-}(M)$. This proves 8.3. 8.3
and 8.3 is proven.

###  8.4 Hyperholomorphic sheaves and ${\mathbb{C}}^{*}$-equivariant bundles
over $M_{\rm fl}$

Let $M$ be a hyperkähler manifold, $I$ an induced complex structure and $F$ a
reflexive sheaf over $(M,I)$, equipped with a hyperholomorphic connection.
Assume that $F$ has an isolated singularity in $x\in M$. Consider the sheaf
${\cal F}$ on $\operatorname{Tw}(M)$ corresponding to ${\cal F}$ as in the
proof of 3.3. Let $s_{x}\subset\operatorname{Tw}(M)$ be the horizontal twistor
line corresponding to $x$, and $\mathfrak{m}$ its ideal. Consider the
associated graded sheaf of $\mathfrak{m}$. Denote by $\operatorname{Tw}^{gr}$
the spectre of this associated graded sheaf. Clearly, $\operatorname{Tw}^{gr}$
is naturally isomorphic to $\operatorname{Tw}(T_{x}M)$, where $T_{x}M$ is the
flat hyperkähler manifold corresponding to the space $T_{x}M$ with induced
quaternion action. Consider the natural ${\mathbb{H}}^{*}$-action on $T_{x}M$.
This provides the hyperkähler manifold $T_{x}M\backslash 0$ with a special
${\mathbb{H}}^{*}$-action.

Let $s_{0}\subset\operatorname{Tw}^{gr}$ be the horisontal twistor line
corresponding to $s_{x}$. The space $\operatorname{Tw}^{gr}\backslash s_{0}$
is equipped with a holomorphic ${\mathbb{C}}^{*}$-action (8.1). Denote by
${\cal F}^{gr}$ the sheaf on $\operatorname{Tw}^{gr}$ associated with ${\cal
F}$. Clearly, ${\cal F}^{gr}$ is ${\mathbb{C}}^{*}$-equivariant. In order to
be able to apply 8.1 and 8.3 to ${\cal
F}^{gr}{\left|{}_{{\phantom{|}\\!\\!}_{\operatorname{Tw}^{gr}\backslash
s_{0}}}\right.}$, we need only to show that ${\cal F}^{gr}$ is compatible with
twistor transform.

Proposition 8.13:  Let $M$ be a hyperkähler manifold, $I$ an induced complex
structure and $F$ a reflexive sheaf over $(M,I)$, equipped with a
hyperholomorphic connection. Assume that ${\cal F}$ has an isolated
singularity in $x\in M$. Let ${\cal F}^{gr}$ be the
${\mathbb{C}}^{*}$-equivariant bundle on $\operatorname{Tw}^{gr}\backslash
s_{0}$ constructed above. Then

(i)

the bundle ${\cal F}^{gr}$ is compatible with twistor transform.

(ii)

Moreover, the natural connection $\nabla_{{\cal F}^{gr}}$ (7.1) is Hermitian.

Proof: The argument is clear, but cumbersome, and essentially hinges on taking
associate graded quotients everywhere and checking that all equations remain
true. We give a simplified version of the proof, which omits some details and
notation.

Consider the bundle ${\cal F}{\left|{}_{{\phantom{|}\\!\\!}_{M\backslash
s_{x}}}\right.}$. This bundle is compatible with twistor transform, and
therefore, is equipped with a natural connection $\nabla_{{\cal F}}$. This
connection is constructed using the isomorphism
$\Psi_{1,-1}:\;F_{1}{\>\longrightarrow\>}F_{-1}$ (see the proof of 8.2). We
apply the same consideration to ${\cal
F}^{gr}{\left|{}_{{\phantom{|}\\!\\!}_{(T_{x}M,I)}}\right.}$, and show that
the resulting connection $\nabla_{{\cal F}^{gr}}$ is hyperholomorphic. This
implies that ${\cal F}^{gr}$ admits a $(1,1)$-connection which is a pullback
of some connection on ${\cal
F}^{gr}{\left|{}_{{\phantom{|}\\!\\!}_{(T_{x}M,I)}}\right.}$. This argument is
used to prove that ${\cal F}^{gr}$ is compatible with the twistor transform.

We use the notation introduced in the proof of 8.2. Let
$\operatorname{Lin}^{gr}$ be the space of twistor maps in
$\operatorname{Tw}^{gr}$. Consider the maps $p_{\pm
I}^{gr}:\;\operatorname{Lin}^{gr}{\>\longrightarrow\>}(T_{x}M,\pm I)$ and the
sheaves $F^{gr}_{\pm 1}:=(p_{\pm I}^{gr})^{*}{\cal F}^{gr}_{\pm I}$ obtained
in the same way as the maps $p_{\pm I}$ and the sheaves $F_{\pm 1}$ from the
corresponding associated graded objects. Taking the associated graded of
$\Psi_{1,-1}$ gives an isomorphism
$\Psi^{gr}_{1,-1}:\;F^{gr}_{1}{\>\longrightarrow\>}F^{gr}_{-1}$. Using the
same construction as in the proof of 8.2, we obtain a connection operator

$\overline{\partial}^{gr}+\partial^{\Psi^{gr}}=\nabla^{gr}_{I}:\;F_{1}^{gr}{\>\longrightarrow\>}F_{1}^{gr}\otimes\bigg{(}(p^{gr}_{-I})^{*}\Omega^{1}(T_{x}M,I)\oplus(p^{gr}_{I})^{*}\Omega^{1}(T_{x}M,-I)\bigg{)}.$

Since $(\overline{\partial}^{gr})^{2}=(\partial^{\Psi^{gr}})^{2}=0$, the
curvature of $\nabla^{gr}_{I}$ has Hodge type $(1,1)$ with respect to $I$. To
prove that $\nabla^{gr}_{I}$ is hyperholomorphic, we need to show that the
curvature of $\nabla^{gr}_{I}$ has type $(1,1)$ with respect to every induced
complex structure. Starting from another induced complex structure $J$, we
obtain a connection $\nabla^{gr}_{J}$, with the curvature of type $(1,1)$ with
respect to $J$. To prove that $\nabla^{gr}_{J}$ is hyperholomorphic it remains
to show that $\nabla^{gr}_{J}=\nabla^{gr}_{I}$.

Let $\nabla_{I}$, $\nabla_{J}$ be the corresponding operators on $F_{1}$. From
the construction, it is clear that $\nabla^{gr}_{I}$, $\nabla^{gr}_{J}$ are
obtained from $\nabla_{I}$, $\nabla_{J}$ by taking the associated graded
quotients. On the other hand, $\nabla_{I}=\nabla_{J}$. Therefore, the
connections $\nabla^{gr}_{I}$ and $\nabla^{gr}_{J}$ are equal. We proved that
the bundle ${\cal
F}^{gr}{\left|{}_{{\phantom{|}\\!\\!}_{\operatorname{Tw}^{gr}\backslash
s_{0}}}\right.}$ is compatible with the twistor structure. To prove 8.4, it
remains to show that the natural connection on ${\cal F}^{gr}$ is Hermitian.

The bundle ${{\cal
F}}{\left|{}_{{\phantom{|}\\!\\!}_{\operatorname{Tw}(M\backslash{x_{0}})}}\right.}$
is by definition Hermitian. Consider the corresponding isomorphism ${\cal
F}{\>\longrightarrow\>}(\iota^{*}\overline{{\cal F}})^{*}$ (7.2). Taking an
associate graded map, we obtain an isomorphism

${\cal F}^{gr}\tilde{\rightarrow}(\iota^{*}\overline{{\cal F}}^{gr})^{*}.$

This gives a non-degenerate semilinear form $h^{gr}$ on ${\cal F}^{gr}$. It
remains only to show that $h^{gr}$ is pseudo-Hermitian (i. e. satisfies
$h(x,y)=\overline{h(y,x)}$) and positive definite.

Let $M^{gr}_{\mathbb{C}}$ be a complexification of $M^{gr}=T_{x}M$,
$M^{gr}_{\mathbb{C}}=\operatorname{Lin}(M^{gr})$. Consider the corresponding
complex vector bundle ${\cal F}^{gr}_{\mathbb{C}}$ over $M^{gr}_{\mathbb{C}}$
underlying ${\cal F}^{gr}$. The metric $h^{gr}$ can be considered as a
semilinear form ${\cal F}^{gr}_{\mathbb{C}}\times{\cal
F}^{gr}_{\mathbb{C}}{\>\longrightarrow\>}{\cal O}_{M^{gr}_{\mathbb{C}}}$. This
semilinear form is obtained from the corresponding form $h$ on ${\cal F}$ by
taking the associate graded quotients. Since $h$ is Hermitian, the form
$h^{gr}$ is pseudo-Hermitian. To prove that $h^{gr}$ is positive semidefinite,
we need to show that for all $f\in{\cal F}^{gr}_{\mathbb{C}}$, the function
$h^{gr}(f,\overline{f})$ belongs to ${\cal
O}_{M^{gr}_{\mathbb{C}}}\cdot\overline{\cal O}_{M^{gr}_{\mathbb{C}}}$, where
${\cal O}_{M^{gr}_{\mathbb{C}}}\cdot\overline{\cal O}_{M^{gr}_{\mathbb{C}}}$
denotes the ${\mathbb{R}}^{>0}$-semigroup of ${\cal O}_{M^{gr}_{\mathbb{C}}}$
generated by $x\cdot\overline{x}$, for all $x\in{\cal
O}_{M^{gr}_{\mathbb{C}}}$. A similar property for $h$ holds, because $h$ is
positive definite. Clearly, taking associated graded quotient of the semigroup
${\cal O}_{M_{\mathbb{C}}}\cdot\overline{\cal O}_{M_{\mathbb{C}}}$, we obtain
${\cal O}_{M^{gr}_{\mathbb{C}}}\cdot\overline{\cal O}_{M^{gr}_{\mathbb{C}}}$.
Thus,

$h^{gr}(f,\overline{f})\in\left({\cal O}_{M_{\mathbb{C}}}\cdot\overline{\cal
O}_{M_{\mathbb{C}}}\right)^{gr}={\cal
O}_{M^{gr}_{\mathbb{C}}}\cdot\overline{\cal O}_{M^{gr}_{\mathbb{C}}}$

This proves that $h^{gr}$ is positive semidefinite. Since $h^{gr}$ is non-
degenerate, this form in positive definite. 8.4 is proven.

Remark 8.14:  Return to the notations of 6. Consider the bundle
$\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}$, where
$C={\mathbb{P}}T_{x}M$ is the blow-up divisor. Clearly, this bundle
corresponds to the graded sheaf $F_{I}^{gr}={\cal
F}^{gr}{\left|{}_{{\phantom{|}\\!\\!}_{(M,I)}}\right.}$ on $(T_{x}M,I)$. By
8.4 (see also 8.3), the bundle
$\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}$ is equipped with a
natural ${\mathbb{H}}^{*}$-invariant connection and Hermitian structure.222As
usually, coherent sheaves over projective variety $X$ correspond to finitely
generated graded modules over the graded ring $\oplus\Gamma({\cal O}_{X}(i))$.
The sheaf $\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{\widetilde{M}\backslash
C}}\right.}$ is a hyperholomorphic bundle over $\widetilde{M}\backslash C\cong
M\backslash x_{0}$. Therefore,
$\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{\widetilde{M}\backslash C}}\right.}$
is equipped with a natural metric and a hyperholomorphic connection. It is
expected that the natural connection and metric on
$\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{\widetilde{M}\backslash C}}\right.}$
can be extended to $\pi^{*}F$, and the rectriction of the resulting connection
and metric to $\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}$ coincides
with that given by 8.4 and 8.3. This will give an alternative proof of 8.4
(ii), because a continuous extension of a positive definite Hermitian metric
is a positive semidefinite Hermitian metric.

###  8.5 Hyperholomorphic sheaves and stable bundles on ${\mathbb{C}}P^{2n+1}$

The purpose of the current Section was to prove the following result, which is
a consequence of 8.4 and 8.3.

Theorem 8.15:  Let $M$ be a hyperkähler manifold, $I$ an induced complex
structure and $F$ a reflexive sheaf on $(M,I)$ admitting a hyperholomorphic
connection. Assume that $F$ has an isolated singularity in $x\in M$, and is
locally trivial outside of $x$. Let
$\pi:\;\widetilde{M}{\>\longrightarrow\>}(M,I)$ be the blow-up of $(M,I)$ in
$x$. Consider the holomorphic vector bundle $\pi^{*}F$ on $\widetilde{M}$ (6).
Let $C\subset(M,I)$ be the blow-up divisor, $C={\mathbb{P}}T_{x}M$. Then the
holomorphic bundle $\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}$
admits a natural Hermitian connection $\nabla$ which is flat along the leaves
of the natural ${\mathbb{H}}^{*}$-action on ${\mathbb{P}}T_{x}M$. Moreover,
the connection $\nabla$ is Yang-Mills, with respect to the Fubini-Study metric
on $C={\mathbb{P}}T_{x}M$, the degree $\deg
c_{1}\left(\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}\right)$
vanishes, and the holomorphic vector bundle
$\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}$ is polystable.

Proof: By definition, coherent sheaves on $C={\mathbb{P}}T_{x}M$ correspond
bijectively to ${\mathbb{C}}^{*}$-equivariant sheaves on $T_{x}M\backslash 0$.
Let $F^{gr}$ be the associated graded sheaf of $F$ (Subsection 8.4). Consider
$F^{gr}$ as a bundle on $T_{x}M\backslash 0$. In the notation of 8.4,
$F^{gr}={\cal F}^{gr}{\left|{}_{{\phantom{|}\\!\\!}_{(M,I)}}\right.}$. By 8.4,
the sheaf ${\cal F}^{gr}$ is ${\mathbb{C}}^{*}$-equivariant and compatible
with the twistor structure. According to the Swann’s formalism for bundles
(8.3), the bundle $F^{gr}{\left|{}_{{\phantom{|}\\!\\!}_{T_{x}M\backslash
0}}\right.}$ is equipped with a natural Hermitian connection $\nabla_{F^{gr}}$
which is flat along the leaves of ${\mathbb{H}}^{*}$-action. Let
$(B,\nabla_{\mathbb{H}})$ be the corresponding $B_{2}$-bundle on

${\mathbb{P}}_{\mathbb{H}}T_{x}M:=\bigg{(}T_{x}M\backslash
0\bigg{)}/{\mathbb{H}}^{*}\cong{\mathbb{H}}P^{n}.$

Then $\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}$ is a holomorphic
bundle over the corresponding twistor space
$C=\operatorname{Tw}({\mathbb{P}}_{\mathbb{H}}T_{x}M)$, obtained as a pullback
of $(B,\nabla_{\mathbb{H}})$ as in 7.3. The natural Kähler metric on the
twistor space $C=\operatorname{Tw}({\mathbb{P}}_{\mathbb{H}}T_{x}M)$ is the
Fubini-Study metric (7.3). By 7.3, the bundle
$\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}$ is Yang-Mills and has
$\deg
c_{1}\left(\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}\right)=0$.
Finally, by Uhlenbeck-Yau theorem (2.5), the bundle
$\pi^{*}F{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}$ is polystable.

##  9 Moduli spaces of hyperholomorphic sheaves and bundles

###  9.1 Deformation of hyperholomorphic sheaves with isolated singularities

The following theorem is an elementary consequence of 8.5. The proof uses well
known results on stability and reflexization (see, for instance, [OSS]). The
main idea of the proof is the following. Given a family of hyperholomorphic
sheaves with an isolated singularity, we blow-up this singularity and restrict
the obtained family to a blow-up divisor. We obtain a family of coherent
sheaves $\mathfrak{V}_{s}$, $s\in S$ over ${\mathbb{C}}P^{2n+1}$, with fibers
semistable of slope zero. Assume that for all $s\in S$, $s\neq s_{0}$, the
sheaf $\mathfrak{V}_{s}$ is trivial. Then the family $\mathfrak{V}$ is also
trivial, up to a reflexization.

We use the following property of reflexive sheaves.

Definition 9.1: Let $X$ be a complex manifold, and $F$ a torsion-free coherent
sheaf. We say that $F$ is normal if for all open subvarieties $U\subset X$,
and all closed subvarieties $Y\subset U$ of codimension 2, the restriction

$\Gamma_{U}(F){\>\longrightarrow\>}\Gamma_{U\backslash Y}(F)$

is an isomorphism.

Lemma 9.2:  Let $X$ be a complex manifold, and $F$ a torsion-free coherent
sheaf. Then $F$ is reflexive if and only if $F$ is normal.

Proof: [OSS], Lemma 1.1.12.

Theorem 9.3:  Let $M$ be a hyperkähler manifol, $I$ an induced complex
structure, $S$ a complex variety and $\mathfrak{F}$ a family of coherent
sheaves over $(M,I)\times S$. Consider the sheaf
$F_{s_{0}}:={\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{(M,I)\times\\{s_{0}\\}}}\right.}$.
Assume that the sheaf $F_{s_{0}}$ is equipped with a filtration $\xi$. Let
$F_{i}$, $i=1,...,m$ denote the associated graded components of $\xi$, and
$F_{i}^{**}$ denote their reflexizations. Assume that $\mathfrak{F}$ is
locally trivial outside of $(x_{0},s_{0})\in(M,I)\times S$. Assume, moreover,
that all sheaves $F_{i}^{**}$, $i=1,...,m$ admit a hyperholomorphic
connection. Then the reflexization ${\mathfrak{F}}^{**}$ is locally trivial.

Proof: Clearly, it suffices to prove 9.1 for ${\mathfrak{F}}$ reflexive. Let
$\widetilde{X}$ be the blow-up of $(M,I)\times S$ in $\\{x_{0}\\}\times S$,
and $\widetilde{\mathfrak{F}}$ the pullback of $\mathfrak{F}$ to
$\widetilde{X}$. Clearly, $\widetilde{X}=\widetilde{M}\times S$, where
$\widetilde{M}$ is a blow-up of $(M,I)$ in $x_{0}$. Denote by
$C\subset\widetilde{M}$ the blow-up divisor of $\widetilde{M}$. Taking $S$
sufficiently small, we may assume that the bundle
$\mathfrak{F}{\left|{}_{{\phantom{|}\\!\\!}_{\\{x_{0}\\}\times(S\backslash\\{s_{0}\\})}}\right.}$
is trivial. Thus, the bundle
$\widetilde{\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{(C\times
S)\backslash(C\times\\{s_{0}\\})}}\right.}$, which is a pullback of
$\mathfrak{F}{\left|{}_{{\phantom{|}\\!\\!}_{\\{x_{0}\\}\times(S\backslash\\{s_{0}\\})}}\right.}$
under the natural projection $(C\times
S)\backslash(C\times\\{s_{0}\\}){\>\longrightarrow\>}\\{x_{0}\\}\times(S\backslash\\{s_{0}\\}$
is trivial. To prove that $\mathfrak{F}$ is locally trivial, we have to show
that $\widetilde{\mathfrak{F}}$ is locally trivial, and that the restriction
of $\widetilde{\mathfrak{F}}$ to $C\times S$ is trivial along the fibers of
the natural projection $C\times S{\>\longrightarrow\>}S$. Clearly, to show
that $\widetilde{\mathfrak{F}}$ is locally trivial we need only to prove that
the fiber
$\widetilde{\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{z}}\right.}$ has
constant dimension for all $z\in C\times S$. Thus, $\widetilde{\mathfrak{F}}$
is locally trivial if and only if
$\widetilde{\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{C\times S}}\right.}$
is locally trivial. This sheaf is reflexive, since it corresponds to an
associate graded sheaf of a reflexive sheaf, in the sense of Footnote to 8.4.
It is non-singular in codimension 2, because all reflexive sheaves are non-
singular in codimension 2 ([OSS], Ch. II, Lemma 1.1.10). By 8.5, the sheaf
$\widetilde{\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s\\}}}\right.}$
is semistable of slope zero. 9.1 is implied by the following lemma, applied to
the sheaf $\widetilde{\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{C\times
S}}\right.}$.

Lemma 9.4:  Let $C$ be a complex projective space, $S$ a complex variety and
$\mathfrak{F}$ a torsion-free sheaf over $C\times S$. Consider an open set
$U\stackrel{{\scriptstyle j}}{{\hookrightarrow}}C\times S$, which is a
complement of $C\times\\{s_{0}\\}\subset C\times S$. Assume that the sheaf
${\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{U}}\right.}$ is trivial:
${\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{U}}\right.}\cong{\cal
O}_{U}^{n}$. Assume, moreover, that $\mathfrak{F}$ is non-singular in
codimension 2, the sheaf
$\left({\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}\right)^{**}$
is semistable of slope zero and

$\operatorname{rk}{\mathfrak{F}}=\operatorname{rk}{\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}.$

Then the reflexization ${\mathfrak{F}}^{**}$ of $\mathfrak{F}$ is a trivial
bundle.

Proof: Using induction, it suffices to prove 9.1 assuming that it is proven
for all $\mathfrak{F}^{\prime}$ with
$\operatorname{rk}{\mathfrak{F}}^{\prime}<\operatorname{rk}{\mathfrak{F}}$. We
may also assume that $S$ is Stein, smooth and 1-dimensional.

Step 1: We construct an exact sequence

$0{\>\longrightarrow\>}\mathfrak{F}_{2}{\>\longrightarrow\>}\mathfrak{F}{\>\longrightarrow\>}\operatorname{im}p_{O_{1}}{\>\longrightarrow\>}0$

of sheaves of positive rank, which, as we prove in Step 3, satisfy assumptions
of 9.1.

Consider the pushforward sheaf $j_{*}{\cal O}_{U}^{n}$. From the definition of
$j_{*}$, we obtain a canonical map

${\mathfrak{F}}{\>\longrightarrow\>}j_{*}{\cal O}_{U}^{n},$ (9.1)

and the kernel of this map is a torsion subsheaf in ${\mathfrak{F}}$.

Let $f$ be a coordinate function on $S$, which vanishes in $s_{0}\in S$.
Clearly,

$j_{*}{\cal O}_{U}^{n}\cong{\cal O}_{C\times S}^{n}\left[\frac{1}{f}\right].$

On the other hand, the sheaf ${\cal O}_{C\times S}\left[\frac{1}{f}\right]$ is
a direct limit of the following diagram:

${\cal O}_{C\times S}^{n}\stackrel{{\scriptstyle\cdot
f}}{{{\>\longrightarrow\>}}}{\cal O}_{C\times
S}^{n}\stackrel{{\scriptstyle\cdot f}}{{{\>\longrightarrow\>}}}{\cal
O}_{C\times S}^{n}\stackrel{{\scriptstyle\cdot
f}}{{{\>\longrightarrow\>}}}...,$

where $\cdot f$ is the injection given by the multiplication by $f$. Thus, the
map (9.1) gives an embedding

${\mathfrak{F}}\stackrel{{\scriptstyle p}}{{\hookrightarrow}}{\cal O}_{C\times
S}^{n},$

which is idenity outside of $(x_{0},s_{0})$. Multiplying $p$ by $\frac{1}{f}$
if necessary, we may assume that the restriction
$p{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}$ is non-
trivial. Thus, $p$ gives a map

${\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}{\>\longrightarrow\>}{\cal
O}^{n}_{C\times\\{s_{0}\\}}.$ (9.2)

with image of positive rank. Since both sides of (9.2) are semistable of slope
zero, and ${\cal O}^{n}_{C\times\\{s_{0}\\}}$ is polystable, the map (9.2)
satisfies the following conditions. (see [OSS], Ch. II, Lemma 1.2.8 for
details).

Let
$F_{1}:=\operatorname{im}p{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}$,
and $F_{2}:=\ker
p{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}$. Then the
reflexization of $F_{1}$ is a trivial bundle ${\cal
O}^{k}_{C\times\\{s_{0}\\}}$, and $p$ maps $F_{1}$ to the direct summand
$O_{1}^{\prime}={\cal O}^{k}_{C\times\\{s_{0}\\}}\subset{\cal
O}^{n}_{C\times\\{s_{0}\\}}$.

Let $O_{1}={\cal O}^{k}_{C\times S}\subset{\cal O}^{n}_{C\times S}$ be the
corresponding free subsheaf of ${\cal O}^{n}_{C\times S}$. Consider the
natural projection $\pi_{O_{1}}$ of ${\cal O}^{n}_{C\times S}$ to $O_{1}$. Let
$p_{O_{1}}$ be the composition of $p$ and $\pi_{O_{1}}$, $\mathfrak{F}_{1}$
the image of $p_{O_{1}}$, and $\mathfrak{F}_{2}$ the kernel of $p_{O_{1}}$.

Step 2: We show that the sheaves $\mathfrak{F}_{2}$ and $\mathfrak{F}_{1}$ and
non-singular in codimension 2.

Consider the exact sequence

$Tor^{1}({\cal
O}_{C\times\\{s_{0}\\}},{\mathfrak{F}}_{1}){\>\longrightarrow\>}\mathfrak{F}_{2}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}{\>\longrightarrow\>}\mathfrak{F}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}{\>\longrightarrow\>}{\mathfrak{F}}_{1}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}{\>\longrightarrow\>}0$

obtained by tensoring the sequence

$0{\>\longrightarrow\>}\mathfrak{F}_{2}{\>\longrightarrow\>}\mathfrak{F}{\>\longrightarrow\>}{\mathfrak{F}}_{1}{\>\longrightarrow\>}0$

with ${\cal O}_{C\times\\{s_{0}\\}}$. From this sequence, we obtain an
isomorphism
${\mathfrak{F}}_{1}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}\cong
F_{1}$.

A torsion-free coherent sheaf over a smooth manifold is non-singular in
codimension 1 ([OSS], Ch. II, Corollary 1.1.8).

Since $\mathfrak{F}$ is non-singular in codimension 2, the restriction
$\mathfrak{F}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}$ is
non-singular in codimension 1. Therefore, the torsion of
$\mathfrak{F}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}$ has
support of codimension at least 2 in $C\times\\{s_{0}\\}$. Since the sheaf
$F_{2}$ is a subsheaf of
$\mathfrak{F}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}$,
its torsion has support of codimension at least 2. Therefore, the singular set
of $F_{2}$ has codimension at least 2 in $C\times\\{s_{0}\\}$. The rank of
$F_{2}$ is by definition equal to $n-k$.

Since $F_{1}$ has rank $k$, the singular set of $\mathfrak{F}_{1}$ coincides
with the singular set of $F_{1}$. Since the restriction
${\mathfrak{F}}_{1}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}=F_{1}$,
is a subsheaf of a trivial bundle of dimension $k$ on $C\times\\{s_{0}\\}$, it
is torsion-free. Therefore, the singularities of ${\mathfrak{F}}_{1}$ have
codimension at least 2 in $C\times\\{s_{0}\\}$.

We obtain that the support of $Tor^{1}({\cal
O}_{C\times\\{s_{0}\\}},{\mathfrak{F}}_{1})$ has codimension at least 2 in
$C\times\\{s_{0}\\}$. Since the quotient sheaf

$\mathfrak{F}_{2}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}\bigg{/}Tor^{1}({\cal
O}_{C\times\\{s_{0}\\}},{\mathfrak{F}}_{1})\cong F_{2}$ (9.3)

is isomorphic to the sheaf $F_{2}$, this quotient is non-singular in
codimension 1. Since we proved that $F_{2}$ is non-singular in codimension 1,
the sheaf
$\mathfrak{F}_{2}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}$
is also non-singular in codimension 1, and its rank is equal to the rank of
$F_{2}$.

Let $R$ be the union of singular sets of the sheaves $\mathfrak{F}_{2}$,
$\mathfrak{F}$, ${\mathfrak{F}}_{1}$. Clearly, $R$ is contained in
$C\times\\{s_{0}\\}$, and $R$ coincides with the set of all $x\in
C\times\\{s_{0}\\}$ where the dimension of the fiber of the sheaves
$\mathfrak{F}_{2}$, $\mathfrak{F}$, ${\mathfrak{F}}_{1}$ is not equal to
$n-k$, $n$, $k$. We have seen that the restrictions of $\mathfrak{F}_{2}$,
${\mathfrak{F}}_{1}$ to ${C\times\\{s_{0}\\}}$ have ranks $n-k$, $k$.
Therefore, the singular sets of $\mathfrak{F}_{2}$, ${\mathfrak{F}}_{1}$
coincide with the singular sets of
$\mathfrak{F}_{2}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}$,
${\mathfrak{F}}_{1}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}$.
We have shown that these singular sets have codimension at least 2 in
$C\times\\{s_{0}\\}$. On the other hand, $\mathfrak{F}$ is non-singular in
codimension 2, by the conditions of 9.1. Therefore, $R$ has codimension at
least 3 in $C\times S$.

Step 3: We check the assumptions of 9.1 applied to the sheaves
$\mathfrak{F}_{2}$, ${\mathfrak{F}}_{1}$.

Since the singular set of ${\mathfrak{F}}_{1}$ has codimension 2 in
$C\times\\{s_{0}\\}$, the ${\cal O}_{C\times\\{s_{0}\\}}$-sheaf $Tor^{1}({\cal
O}_{C\times\\{s_{0}\\}},{\mathfrak{F}}_{1})$ is a torsion sheaf with support
of codimension 2 in $C\times\\{s_{0}\\}$. By (9.3), the reflexization of
$\mathfrak{F}_{2}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}$
coincides with the reflexization of $F_{2}$. Thus, the sheaf
$\left(\mathfrak{F}_{2}{\left|{}_{{\phantom{|}\\!\\!}_{C\times\\{s_{0}\\}}}\right.}\right)^{**}$
is semistable. On the other hand, outside of ${C\times\\{s_{0}\\}}$, the sheaf
$\mathfrak{F}_{2}$ is a trivial bundle. Thus, $\mathfrak{F}_{2}$ satisfies
assumptions of 9.1. Similarly, the sheaf $\mathfrak{F}_{1}$ is non-singular in
codimension 2, its restriction to $C\times\\{s_{0}\\}$ has trivial
reflexization, and it is free outside of $C\times\\{s_{0}\\}$.

Step 4: We apply induction and prove 9.1.

By induction assumption, the reflexization of $\mathfrak{F}_{2}$ is isomorphic
to a trivial bundle ${\cal O}^{n-k}_{C\times S}$. and reflexization of
$\mathfrak{F}_{1}$ is ${\cal O}^{k}_{C\times S}$. We obtain an exact sequence

$0{\>\longrightarrow\>}\mathfrak{F}_{2}{\>\longrightarrow\>}\mathfrak{F}{\>\longrightarrow\>}{\mathfrak{F}}_{1}{\>\longrightarrow\>}0,$
(9.4)

where the sheaves $\mathfrak{F}_{2}$ and ${\mathfrak{F}}_{1}$ have trivial
reflexizations.

Let $V:=C\times S\backslash R$. Restricting the exact sequence (9.4) to $V$,
we obtain an exact sequence

$0{\>\longrightarrow\>}{\cal O}^{n-k}_{V}\stackrel{{\scriptstyle
a}}{{{\>\longrightarrow\>}}}{\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{V}}\right.}\stackrel{{\scriptstyle
b}}{{{\>\longrightarrow\>}}}{\cal O}^{k}_{V}{\>\longrightarrow\>}0.$ (9.5)

Since $V$ is a complement of a codimension-3 complex subvariety in a smooth
Stein domain, the first cohomology of a trivial sheaf on $V$ vanish.
Therefore, the sequence (9.5) splits, and the sheaf
${\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{V}}\right.}$ is a trivial
bundle. Consider the pushforward
$\zeta_{*}{\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{V}}\right.}$, where
$\zeta:\;V{\>\longrightarrow\>}C\times S$ is the standard map. Then
$\zeta_{*}{\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{V}}\right.}$ is a
reflexization of ${\mathfrak{F}}$ (a pushforward of a reflexive sheaf over a
subvariety of codimension 2 or more is reflexive – see 9.1). On the other
hand, since the sheaf
${\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{V}}\right.}$ is a trivial
bundle, its push-forward over a subvariety of codimension at least 2 is also a
trivial bundle over $C\times S$. We proved that the sheaf
${\mathfrak{F}}^{**}=\zeta_{*}{\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{V}}\right.}$
is a trivial bundle over $C\times S$. The push-forward
$\zeta_{*}{\mathfrak{F}}{\left|{}_{{\phantom{|}\\!\\!}_{V}}\right.}$ coincides
with reflexization of $\mathfrak{F}$, by 9.1. This proves 9.1 and 9.1.

###  9.2 The Maruyama moduli space of coherent sheaves

This Subsection is a compilation of results of Gieseker and Maruyama on the
moduli of coherent sheaves over projective manifolds. We follow [OSS], [Ma2].

To study the moduli spaces of holomorphic bundles and coherent sheaves, we
consider the following definition of stability.

Definition 9.5: (Gieseker–Maruyama stability) ([Gi], [OSS]) Let $X$ be a
projective variety, ${\cal O}(1)$ the standard line bundle and $F$ a torsion-
free coherent sheaf. The sheaf $F$ is called Gieseker–Maruyama stable (resp.
Gieseker–Maruyama semistable) if for all coherent subsheaves $E\subset F$ with
$0<\operatorname{rk}E<\operatorname{rk}F$, we have

$p_{F}(k)<p_{E}(k)\ \ (\text{resp.},\ \ p_{F}(k)\leqslant p_{E}(k))$

for all sufficiently large numbers $k\in{\mathbb{Z}}$. Here

$p_{F}(k)=\frac{\dim\Gamma_{X}(F\otimes{\cal O}(k))}{\operatorname{rk}F}.$

Clearly, Gieseker–Maruyama stability is weaker than the Mumford-Takemoto
stability. Every Gieseker–Maruyama semistable sheaf $F$ has a so-called
Jordan-Hölder filtration $F_{0}\subset F_{1}\subset...\subset F$ with
Gieseker–Maruyama stable successive quotients $F_{i}/F_{i-1}$. The
corresponding associated graded sheaf

$\oplus F_{i}/F_{i-1}$

is independent from a choice of a filtration. It is called the associate
graded quotient of the Jordan-Hölder filtration on $F$.

Definition 9.6: Let $F$, $G$ be Gieseker–Maruyama semistable sheaves on $X$.
Then $F$, $G$ are called $S$-equivalent if the corresponding associate graded
quotients $\oplus F_{i}/F_{i-1}$, $\oplus G_{i}/G_{i-1}$ are isomorphic.

Definition 9.7: Let $X$ be a complex manifold, $F$ a torsion-free sheaf on
$X$, and $Y$ a complex variety. Consider a sheaf ${\cal F}$ on $X\times Y$
which is flat over $Y$. Assume that for some point $s_{0}\in Y$, the sheaf
${\cal F}{\left|{}_{{\phantom{|}\\!\\!}_{X\times\\{s_{0}\\}}}\right.}$ is
isomorphic to $F$. Then ${\cal F}$ is called a deformation of $F$ parametrized
by $Y$. We say that a sheaf $F^{\prime}$ on $X$ is deformationally equivalent
to $F$ if for some $s\in Y$, the restriction ${\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{X\times\\{s\\}}}\right.}$ is isomorphic to
$F^{\prime}$. Slightly less formally, such sheaves are called deformations of
$F$. If $F^{\prime}$ is a (semi-)stable bundle, it is called a (semi-)stable
bundle deformation of $F$.

Remark 9.8:  Clearly, the Chern classes of deformationally equivalent sheaves
are equal.

Definition 9.9:  Let $X$ be a complex manifold, and $F$ a torsion-free sheaf
on $X$, and ${\cal M}_{mar}$ a complex variety. We say that ${\cal M}_{mar}$
is a coarse moduli space of deformations of $F$ if the following conditions
hold.

(i)

The points of $s\in{\cal M}_{mar}$ are in bijective correspondence with
$S$-equivalence classes of coherent sheaves $F_{s}$ which are deformationally
equivalent to $F$.

(ii)

For any flat deformation ${\cal F}$ of $F$ parametrized by $Y$, there exists a
unique morphism $\varphi:\;Y{\>\longrightarrow\>}{\cal M}_{mar}$ such that for
all $s\in Y$, the restriction ${\cal
F}{\left|{}_{{\phantom{|}\\!\\!}_{X\times\\{s\\}}}\right.}$ is $S$-equivalent
to the sheaf $F_{\varphi(s)}$ corresponding to $\varphi(s)\in{\cal M}_{mar}$.

Clearly, the coarse moduli space is unique. By 9.2, the Chern classes of
$F_{s}$ are equal for all $s\in{\cal M}_{mar}$.

It is clear how to define other kinds of moduli spaces. For instance,
replacing the word sheaf by the word bundle throughout 9.2, we obtain a
definition of the coarse moduli space of semistable bundle deformations of
$F$. Further on, we shall usually omit the word “coarse” and say “moduli
space” instead.

Theorem 9.10:  (Maruyama) Let $X$ be a projective manifold and $F$ a coherent
sheaf over $X$. Then the Maruyama moduli space ${\cal M}_{mar}$ of
deformations of $F$ exists and is compact.

Proof: See, e. g., [Ma2].

###  9.3 Moduli of hyperholomorphic sheaves and $C$-restricted comples
structures

Usually, the moduli space of semistable bundle deformations of a bundle $F$ is
not compact. To compactify this moduli space, Maruyama adds points
corresponding to the deformations of $F$ which are singular (these
deformations can be non-reflexive and can have singular reflexizations). Using
the desingularization theorems for hyperholomorphic sheaves, we were able to
obtain 9.1, which states (roughly speaking) that a deformation of a semistable
hyperholomorphic bundle is again a semistable bunlde, assuming that all its
singularities are isolated. In Section 5, we showed that under certain
conditions, a deformation of a hyperholomorphic sheaf is again
hyperholomorphic (5.3). This makes it possible to prove that a deformation of
a semistable hyperholomorphic bundle is locally trivial.

In [V5], we have shown that a Hilbert scheme of a K3 surface has no non-
trivial trianalytic subvarieties, for a general hyperkähler structure.

Theorem 9.11:  Let $M$ be a compact hyperkähler manifold without non-trivial
trianalytic subvarieties, $\dim_{\mathbb{H}}\geqslant 2$, and $I$ an induced
complex structure. Consider a hyperholomorphic bundle $F$ on $(M,I)$ (3.2).
Assume that $I$ is a $C$-restricted complex structure, $C=\deg_{I}c_{2}(F)$.
Let ${\cal M}$ be the moduli space of semistable bundle deformations of $F$
over $(M,I)$. Then ${\cal M}$ is compact.

Proof: The complex structure $I$ is by definition algebraic, with unique
polarization. This makes it possible to speak of Gieseker–Maruyama stability
on $(M,I)$. Denote by ${\cal M}_{mar}$ the Maruyama moduli of deformations of
$F$. Then ${\cal M}$ is naturally an open subset of ${\cal M}_{mar}$. Let
$s\in{\cal M}_{mar}$ be an arbitrary point and $F_{s}$ the corresponding
coherent sheaf on $(M,I)$, defined up to $S$-equivalence. According to 9.2,
the Chern classes of $F$ and $F_{s}$ are equal. Thus, by 5.3, the sheaf
$F_{s}$ is hyperholomorphic. Therefore, $F_{s}$ admits a filtration with
hyperholomorphic stable quotient sheaves $F_{i}$, $i=1,...,m$. By 3.3, the
singular set $S$ of $F_{s}$ is trianalytic. Since $M$ has no proper
trianalytic subvarieties, $S$ is a collection of points. We obtain that
$F_{s}$ has isolated singularities. Let $\mathfrak{F}$ be a family of
deformations of $F$, parametrized by $Y$. The points $y\in Y$ correspond to
deformations $F_{y}$ of $F_{s}$. Assume that for all $y\in Y$, $y\neq s$, the
sheaf $F_{y}$ is a bundle. Since ${\cal M}$ is open in ${\cal M}_{mar}$, such
a deformation always exists.

The sheaf $F_{s}$ has isolated singularities and admits a filtration with
hyperholomorphic stable quotient sheaves. This implies that the family
$\mathfrak{F}$ satisfies the conditions of 9.1. By 9.1, the reflexization
$\mathfrak{F}^{**}$ is locally trivial. To prove that ${\cal M}={\cal
M}_{mar}$, we have to show that for all $s\in{\cal M}_{mar}$, the
corresponding coherent sheaf $F_{s}$ is locally trivial. Therefore, to finish
the proof of 9.3, it remains to prove the following algebro-geometric claim.

Claim 9.12:  Let $X$ be a compact complex manifold, $\dim_{\mathbb{C}}X>2$,
and $\mathfrak{F}$ a torsion-free coherent sheaf over $X\times Y$ which is
flat over $Y$. Assume that the reflexization of $\mathfrak{F}$ is locally
trivial, $\mathfrak{F}$ has isolated singularities, and for some point $s\in
Y$, the restriction of $\mathfrak{F}$ to the complement $(X\times
Y)\backslash(X\times\\{s\\})$ is locally trivial. Then the reflexization
$\left(\mathfrak{F}{\left|{}_{{\phantom{|}\\!\\!}_{X}}\right.}\times\\{s\\}\right)^{**}$
is locally trivial.

Remark 9.13: We say that a kernel of a map from a bundle to an Artinian sheaf
is a bundle with holes. In slightly more intuitive terms, 9.3 states that a
flat deformation of a bundle with holes is again a bundle with holes, and
cannot be smooth, assuming that $\dim_{\mathbb{C}}X>2$.

Proof of 9.3: 9.3 is well known. Here we give a sketch of a proof. Consider a
coherent sheaf
$F_{s}=\mathfrak{F}{\left|{}_{{\phantom{|}\\!\\!}_{X\times\\{s\\}}}\right.}$,
and an exact sequence

$0{\>\longrightarrow\>}F_{s}{\>\longrightarrow\>}F_{s}^{**}{\>\longrightarrow\>}k{\>\longrightarrow\>}0,$

where $k$ is an Artinian sheaf. By definition, the sheaf $F_{s}^{**}$ is
locally trivial. The flat deformations of $F_{s}$ are infinitesimally
classified by $Ext^{1}(F_{s},F_{s})$. Replacing $F_{s}$ by a quasi-isomorphic
complex of sheaves $F_{s}^{**}{\>\longrightarrow\>}k$, we obtain a spectral
sequence converging to $Ext^{\bullet}(F_{s},F_{s})$. In the $E_{2}$-term of
this sequence, we observe the group

$Ext^{1}(F_{s}^{**},F_{s}^{**})\oplus Ext^{1}(k,k)\oplus
Ext^{2}(k,F_{s}^{**})\oplus Ext^{0}(F_{s}^{**},k).$

which is responsible for $Ext^{1}(F_{s},F_{s})$. The term
$Ext^{1}(F_{s}^{**},F_{s}^{**})$ is responsible for deformations of the bundle
$F_{s}^{**}$, the term $Ext^{0}(F_{s}^{**},k)$ for the deformations of the map
$F_{s}^{**}{\>\longrightarrow\>}k$, and the term $Ext^{1}(k,k)$ for the
deformations of the Artinian sheaf $k$. Thus, the term $Ext^{2}(k,F_{s}^{**})$
is responsible for the deformations of $F_{s}$ which change the dimension of
the cokernel of the embedding $F_{s}{\>\longrightarrow\>}F_{s}^{**}$. We
obtain that whenever $Ext^{2}(k,F_{s}^{**})=0$, all deformations of $F_{s}$
are singular. On the other hand, $Ext^{2}(k,F_{s}^{**})=0$, because the $i$-th
$Ext$ from the skyscraper to a free sheaf on a manifold of dimension more than
$i$ vanishes (this is a basic result of Grothendieck’s duality, [H-Gro]).

##  10 New examples of hyperkähler manifolds

###  10.1 Twistor paths

This Subsection contains an exposition and further elaboration of the results
of [V3-bis] concerning the twistor curves in the moduli space of complex
structures on a complex manifold of hyperkähler type.

Let $M$ be a compact manifold admitting a hyperkähler structure. In 5.2, we
defined the coarse, marked moduli space of complex structures on $M$, denoted
by $Comp$. For the duration of this section, we fix a compact simple
hyperkähler manifold $M$, and its moduli $Comp$.

Further on, we shall need the following fact.

Claim 10.1:  Let $M$ be a hyperkähler manifold, $I$ an induced complex
structure of general type, and $B$ a holomorphic vector bundle over $(M,I)$.
Then $B$ is stable if an only if $B$ is simple.111Simple sheaves are coherent
sheaves which have no proper subsheaves

Proof: By 2.5, for all $\omega\in Pic(M,I)$, we have $\deg_{I}(\omega)=0$.
Therefore, every subsheaf of $B$ is destabilising.

Remark 10.2: In assumptions of 10.1, all stable bundles are hyperholomorphic
(2.5). Therefore, 10.1 implies that $B$ is hyperholomorphic if it is simple.

In Subsection 5.2, we have shown that every hyperkähler structure ${\cal H}$
corresponds to a holomorphic embedding

$\kappa({\cal H}):\;{\mathbb{C}}P^{1}{\>\longrightarrow\>}Comp,\ \
L{\>\longrightarrow\>}(M,L).$

Definition 10.3: A projective line $C\subset Comp$ is called a twistor curve
if $C=\kappa({\cal H})$ for some hyperkähler structure ${\cal H}$ on $M$.

The following theorem was proven in [V3-bis].

Theorem 10.4:  ([V3-bis], Theorem 3.1) Let $I_{1},I_{2}\in Comp$. Then there
exist a sequence of intersecting twistor curves which connect $I_{1}$ with
$I_{2}$.

Definition 10.5: Let $P_{0}$, …, $P_{n}\subset Comp$ be a sequence of twistor
curves, supplied with an intersection point $x_{i+1}\in P_{i}\cap P_{i+1}$ for
each $i$. We say that $\gamma=P_{0},...,P_{n},x_{1},...,x_{n}$ is a twistor
path. Let $I$, $I^{\prime}\in Comp$. We say that $\gamma$ is a twistor path
connecting $I$ to $I^{\prime}$ if $I\in P_{0}$ and $I^{\prime}\in P_{n}$. The
lines $P_{i}$ are called the edges, and the points $x_{i}$ the vertices of a
twistor path.

Recall that in 2.3, we defined induced complex structures which are generic
with respect to a hyperkähler structure.

Given a twistor curve $P$, the corresponding hyperkähler structure ${\cal H}$
is unique (5.2). We say that a point $x\in P$ is of general type, or generic
with respect to $P$ if the corresponding complex structure is generic with
respect to ${\cal H}$.

Definition 10.6:  Let $I$, $J\in Comp$ and $\gamma=P_{0},...,P_{n}$ be a
twistor path from $I$ to $J$, which corresponds to the hyperkähler structures
${\cal H}_{0}$, …, ${\cal H}_{n}$. We say that $\gamma$ is admissible if all
vertices of $\gamma$ are of general type with respect to the corresponding
edges.

Remark 10.7: In [V3-bis], admissible twistor paths were defined slightly
differently. In addition to the conditions above, we required that $I$, $J$
are of general type with respect to ${\cal H}_{0}$, ${\cal H}_{n}$.

10.1 proves that every two points $I$, $I^{\prime}$ in $Comp$ are connected
with a twistor path. Clearly, each twistor path induces a diffeomorphism
$\mu_{\gamma}:\;(M,I){\>\longrightarrow\>}(M,I^{\prime})$. In [V3-bis],
Subsection 5.2, we studied algebro-geometrical properties of this
diffeomorphism.

Theorem 10.8:  Let $I$, $J\in Comp$, and $\gamma$ be an admissible twistor
path from $I$ to $J$. Then

(i)

There exists a natural isomorphism of tensor cetegories

$\Phi_{\gamma}:\;Bun_{I}({\cal H}_{0}){\>\longrightarrow\>}Bun_{J}({\cal
H}_{n}),$

where $Bun_{I}({\cal H}_{0})$, $Bun_{J}({\cal H}_{n})$ are the categories of
polystable hyperholomorphic vector bundles on $(M,I)$, $(M,J)$, taken with
respect to ${\cal H}_{0}$, ${\cal H}_{n}$ respectively.

(ii)

Let $B\in Bun_{I}({\cal H}_{0})$ be a stable hyperholomorphic bundle, and

${\cal M}_{I,{\cal H}_{0}}(B)$

the moduli of stable deformations of $B$, where stability is taken with
respect to the Kähler metric induced by ${\cal H}_{0}$. Then $\Phi_{\gamma}$
maps stable bundles which are deformationally equivalent to $B$ to the stable
bundles which are deformationally equivalent to $\Phi_{\gamma}(B)$. Moreover,
obtained this way bijection

$\Phi_{\gamma}:\;{\cal M}_{I,{\cal H}_{0}}(B){\>\longrightarrow\>}{\cal
M}_{J,{\cal H}_{n}}(\Phi_{\gamma}(B))$

induces a real analytic isomorphism of deformation spaces.

Proof: 10.1 (i) is a consequence of [V3-bis], Corollary 5.1. Here we give a
sketch of its proof.

Let $I$ be an induced complex structure of general type. By 10.1, a bundle $B$
over $(M,I)$ is stable if and only if it is simple. Thus, the category
$Bun_{I}({\cal H})$ is independent from the choice of ${\cal H}$ (10.1).

In 3.5, we constructed the equivalence of categories $\Phi_{I,J}$, which gives
the functor $\Phi_{\gamma}$ for twistor path which consists of a single
twistor curve. This proves 10.1 (i) for $n=1$. A composition of isomorphisms
$\Phi_{I,J}\circ\Phi_{J,J^{\prime}}$ is well defined, because the category
$Bun_{I}({\cal H})$ is independent from the choice of ${\cal H}$. Taking
successive compositions of the maps $\Phi_{I,J}$, we obtain an isomorphism
$\Phi_{\gamma}$. This proves 10.1 (i).

The variety ${\cal M}_{I,{\cal H}}(B)$ is singular hyperkähler ([V1]), and the
variety ${\cal M}_{J,{\cal H}}(B)$ is the same singular hyperkähler variety,
taken with another induced complex structure. By definition of singular
hyperkähler varieties, this implies that ${\cal M}_{I,{\cal H}}(B)$, ${\cal
M}_{J,{\cal H}}(B)$ are real analytic equivalent, with equivalence provided by
$\Phi_{I,J}$. This proves 10.1 (ii).

For $I\in Comp$, denote by $Pic(M,I)$ the group $H^{1,1}(M,I)\cap
H^{2}(M,{\mathbb{Z}})$, and by $Pic(I,{\mathbb{Q}})$ the space
$H^{1,1}(M,I)\cap H^{2}(M,{\mathbb{Q}})\subset H^{2}(M)$. Let $Q\subset
H^{2}(M,{\mathbb{Q}})$ be a subspace of $H^{2}(M,{\mathbb{Q}})$, and

$Comp_{Q}:=\\{I\in Comp\;\;|\;\;Pic(I,{\mathbb{Q}})=Q\\}.$

Theorem 10.9:  Let ${\cal H}$, ${\cal H}^{\prime}$ be hyperkähler structures,
and $I$, $I^{\prime}$ be complex structures of general type to and induced by
${\cal H}$, ${\cal H}^{\prime}$. Assume that
$Pic(I,{\mathbb{Q}})=Pic(I^{\prime},{\mathbb{Q}})=Q$, and $I$, $I^{\prime}$
lie in the same connected component of $Comp_{Q}$. Then $I$, $I^{\prime}$ can
be connected by an admissible path.

Proof: This is [V3-bis], Theorem 5.2.

For a general $Q$, we have no control over the number of connected components
of $Comp_{Q}$ (unless global Torelli theorem is proven), and therefore we
cannot directly apply 10.1 to obtain results from algebraic
geometry.222Exception is a K3 surface, where Torelli holds. For K3, $Comp_{Q}$
is connected for all $Q\subset H^{2}(M,{\mathbb{Q}})$. However, when $Q=0$,
$Comp_{Q}$ is clearly connected and dense in $Comp$. This is used to prove the
following corollary.

Corollary 10.10:  Let $I$, $I^{\prime}\in Comp_{0}$. Then $I$ can be connected
to $I^{\prime}$ by an admissible twistor path.

Proof This is [V3-bis], Corollary 5.2.

Definition 10.11:  Let $I\in Comp$ be a complex structure, $\omega$ be a
Kähler form on $(M,I)$, and ${\cal H}$ the corresponding hyperkähler metric,
which exists by Calabi-Yau theorem. Then $\omega$ is called a generic
polarization if any of the following conditions hold

(i)

For all $a\in Pic(M,I)$, the degree $\deg_{\omega}(a)\neq 0$, unless $a=0$.

(ii)

For all $SU(2)$-invariant integer classes $a\in H^{2}(M,{\mathbb{Z}})$, we
have $a=0$.

The conditions (i) and (ii) are equivalent by 2.5.

Claim 10.12:  Let $I\in Comp$ be a complex structure, $\omega$ be a Kähler
form on $(M,I)$, and ${\cal H}$ the corresponding hyperkähler structure, which
exists by Calabi-Yau theorem. Then $\omega$ is generic if and only if for all
integer classes $a\in H^{1,1}(M,I)$, the class $a$ is not orthogonal to
$\omega$ with respect to the Bogomolov-Beauville pairing.

Proof: Clearly, the map
$\deg_{\omega}:\;H^{2}(M){\>\longrightarrow\>}{\mathbb{R}}$ is equal (up to a
scalar multiplier) to the orthogonal projection onto the line
${\mathbb{R}}\cdot\omega$. Then, 10.1 is equivalent to 10.1, (i).

From 10.1 it is clear that the set of generic polarizations is a complement to
a countable union of hyperplanes. Thus, generic polarizations are dense in the
Kähler cone of $(M,I)$, for all $I$.

Claim 10.13:  Let $I,J\in Comp$, and $a$, $b$ be generic polarizations on
$(M,I)$. Consider the corresponding hyperkähler structures ${\cal H}_{0}$ and
${\cal H}_{n}$ inducing $I$ and $J$. Then there exists an admissible twistor
path starting from $I,{\cal H}_{0}$ and ending with ${\cal H}_{n},J$.

Proof: Consider the twistor curves $P_{0}$, $P_{n}$ corresponding to ${\cal
H}_{0}$, ${\cal H}_{n}$. Since $a$, $b$ are generic, the curves $P_{0}$,
$P_{n}$ intersect with $Comp_{0}$. Applying 10.1, we connect the curves
$P_{0}$ and $P_{n}$ by an admissible path.

Putting together 10.1 and 10.1, we obtain the following result.

Theorem 10.14:  Let $I$, $J\in Comp$ be complex structures, and $a,b$ be
generic polarizations on $(M,I)$, $(M,J)$. Then

(i)

There exist an isomorphism of tensor cetegories

$\Phi_{\gamma}:\;Bun_{I}(a){\>\longrightarrow\>}Bun_{J}(a),$

where $Bun_{I}(a)$, $Bun_{J}(b)$ are the categories of polystable
hyperholomorphic vector bundles on $(M,I)$, $(M,J)$, taken with respect to the
hyperkähler structures defined by the Kähler classes $a$, $b$ as in 2.1.

(ii)

Let $B\in Bun_{I}(a)$ be a stable hyperholomorphic bundle, and

${\cal M}_{I,a}(B)$

the moduli of stable deformations of $B$, where stability is taken with
respect to the polarization $a$. Then $\Phi_{\gamma}$ maps stable bundles
which are deformationally equivalent to $B$ to the stable bundles which are
deformationally equivalent to $\Phi_{\gamma}(B)$. Moreover, obtained this way
bijection

$\Phi_{\gamma}:\;{\cal M}_{I,a}(B){\>\longrightarrow\>}{\cal
M}_{J,b}(\Phi_{\gamma}(B))$

induces a real analytic isomorphism of deformation spaces.

Lemma 10.15:  In assumptions of 10.1, let $B$ be a holomorphic tangent bundle
of $(M,I)$. Then $\Phi_{\gamma}(B)$ is a holomorphic tangent bundle of
$(M,J)$.

Proof: Clear.

Corollary 10.16:  Let $I,J\in Comp$ be complex structures, and $a$, $b$
generic polarizations on $(M,I)$, $(M,J)$. Assume that the moduli of stable
deformations ${\cal M}_{I,a}(T(M,I))$ of the holomorphic tangent bundle
$T^{1,0}(M,I)$ is compact. Then the space ${\cal M}_{J,b}(T(M,J))$ is also
compact.

Proof: Let $\gamma$ be the twistor path of 10.1. By 10.1,
$\Phi_{\gamma}(T(M,I))=T(M,J)$. Applying 10.1, we obtain a real analytic
equivalence from ${\cal M}_{I,a}(T(M,I))$ to ${\cal M}_{J,b}(T(M,J))$.

###  10.2 New examples of hyperkähler manifolds

Theorem 10.17:  Let $M$ be a compact hyperkähler manifold without non-trivial
trianalytic subvarieties, $\dim_{\mathbb{H}}M\geqslant 2$, and $I$ an induced
complex structure. Consider a hyperholomorphic bundle $F$ on $M$ (3.5). Let
$F_{I}$ be the corresponding holomorphic bundle over $(M,I)$. Assume that $I$
is a $C$-restricted complex structure, $C=\deg_{I}c_{2}(F)$. Assume, moreover,
that all semistable bundle deformations of $F_{I}$ are stable.333This may
happen, for instance, when $\operatorname{rk}F=\dim_{\mathbb{C}}M=n$, and the
number $c_{n}(F)$ is prime. Denote by ${\cal M}_{F}^{I}$ the moduli of stable
bundle deformations of $F_{I}$ over $(M,I)$. Then

(i)

the normalization $\widetilde{{\cal M}}_{F}^{I}$ is a compact and smooth
complex manifold equipped with a natural hyperkähler structure.

(ii)

Moreover, for all induced complex structures $J$ on $M$, the the variety
${\cal M}_{F}^{J}$ is compact, and has a smooth normalization
$\widetilde{{\cal M}}_{F}^{J}$, which is also equipped with a natural
hyperkähler structure.

(iii)

Finally, the hyperkähler manifolds $\widetilde{{\cal M}}_{F}^{J}$,
$\widetilde{{\cal M}}_{F}^{I}$ are naturally isomorphic.

Proof: The variety ${\cal M}_{F}^{I}$ is compact by 9.3. In [V1], it was
proven that the space ${\cal M}_{F}^{I}$ of stable deformations of $F$ is a
singular hyperkähler variety (see also [KV] for an explicit construction of
the twistor space of ${\cal M}_{F}^{I}$). Then 10.2 is a consequence of the
Desingularization Theorem for singular hyperkähler varietiess (2.3).

The assumptions of 10.2 are quite restrictive. Using the technique of twistor
paths, developed in Subsection 10.1, it is possible to prove a more accessible
form of 10.2.

Let $M$ be a hyperkähler manifold, and $I$, $J$ induced complex structures.
Given an admissible twistor path from $I$ to $J$, we obtain an equivalence
$\Phi_{\gamma}$ between the category of hyperholomorphic bundles on $(M,I)$
and $(M,J)$.

Theorem 10.18:  Let $M$ be a compact simple hyperkähler manifold,
$\dim_{\mathbb{H}}M>1$, and $I$ a complex structure on $M$. Consider a generic
polarization $a$ on $(M,I)$. Let ${\cal H}$ be the corresponding hyperkähler
structure, and $F$ a hyperholomorphic bundle on $(M,I)$. Fix a hyperkähler
structure ${\cal H}^{\prime}$ on $M$ admitting $C$-restricted complex
structures, such that $M$ has no trianalytic subvarieties with respect to
${\cal H}^{\prime}$. Assume that for some $C$-restricted complex structure $J$
induced by ${\cal H}^{\prime}$, $C=\deg_{I}c_{2}(F)$, all admissible twistor
paths $\gamma$ from $I$ to $J$, and all semistable bundles $F^{\prime}$ which
are deformationally equivalent to $\Phi_{\gamma}(F)$, the bundle $F^{\prime}$
is stable. Then the space of stable deformations of $F$ is compact.

Remark 10.19: The space of stable deformations of $F$ is singular hyperkähler
([V1]) and its normalization is smooth and hyperkähler (2.3).

Proof of 10.2: Clearly, $F^{\prime}$ satisfies assumptions of 10.2, and the
moduli space of its stable deformations is compact. Since $\Phi_{\gamma}$
induces a homeomorphism of moduli spaces (10.1), the space of stable
deformations of $F$ is also compact.

Applying 10.2 to the holomorphic tangent bundle $T(M,I)$, we obtain the
following corollary.

Theorem 10.20:  Let $M$ be a compact simple hyperkähler manifold,
$\dim_{\mathbb{H}}(M)>1$. Assume that for a generic hyperkähler structure
${\cal H}$ on $M$, this manifold admits no trianalytic subvarieties.444This
assumption holds for a Hilbert scheme of points on a K3 surface. Assume,
moreover, that for some $C$-restricted induced complex structure $I$, all
semistable bundle deformations of $T(M,I)$ are stable, for
$C>\deg_{I}c_{2}(M)$. Then, for all complex structures $J$ on $M$ and all
generic polarizations $\omega$ on $(M,J)$, the deformation space ${\cal
M}_{J,\omega}(T(M,J))$ is compact.

Proof: Follows from 10.2 and 10.1.

###  10.3 How to check that we obtained new examples of hyperkähler manifolds?

A. Beauville [Bea] described two families of compact hyperkähler manifolds,
one obtained as the Hilbert scheme of points on a K3-surface, another obtained
as the Hilbert scheme of a 2-dimensional torus factorized by the free torus
action.

Conjecture 10.21:  There exist compact simple hyperkähler manifolds which are
not isomorphic to deformations of these two fundamental examples.

Here we explain our strategy of a proof of 10.3 using results on compactness
of the moduli space of hyperholomorphic bundles.

The results of this subsection are still in writing, so all statements below
this line should be considered as conjectures. We give an idea of a proof for
each result and label it as “proof”, but these “proofs” are merely sketches.

First of all, it is possible to prove the following theorem.

Theorem 10.22:  Let $M$ be a complex K3 surface without automorphisms. Assume
that $M$ is Mumford-Tate generic with respect to some hyperkähler structure.
Consider the Hilbert scheme $M^{[n]}$ of points on $M$, $n>1$. Pick a
hyperkähler structure ${\cal H}$ on $M^{[n]}$ which is compatible with the
complex structure. Let $B$ be a hyperholomorphic bundle on $(M^{[n]},{\cal
H})$, $\operatorname{rk}B=2$. Then $B$ is a trivial bundle.

Proof: The proof of 10.3 is based on the same ideas as the proof of 2.3.

For a compact complex manifold $X$ of hyperkähler type, denote its coarse,
marked moduli space (5.2) by $Comp(X)$.

Corollary 10.23:  Let $M$ be a K3 surface, $I\in Comp(X)$ an arbitrary complex
structure on $X=M^{[n]}$, $n>1$, and $a$ a generic polarization on $(X,J)$.
Consider the hyperkähler structure ${\cal H}$ which corresponds to ($I$, $a$)
as in 2.1. Let $B$, $\operatorname{rk}B=2$ be a hyperholomorphic bundle over
$(X,{\cal H})$. Then $B$ is trivial.

Proof: Follows from 10.3 and 10.1.

Corollary 10.24:  Let $M$ be a K3 surface, $I\in Comp(X)$ an arbitrary complex
structure on $X=M^{[n]}$, $n>1$, and $a$ a generic polarization on $(X,I)$.
Consider the hyperkähler structure ${\cal H}$ which corresponds to ($J$, $a$)
(2.1). Let $B$, $\operatorname{rk}B\leqslant 6$ be a stable hyperholomorphic
bundle on $(X,{\cal H})$. Assume that the Chern class
$c_{\operatorname{rk}B}(B)$ is non-zero. Assume, moreover, that $I$ is
$C$-restricted, $C=\deg_{I}(c_{2}(B))$. Let $B^{\prime}$ be a semistable
deformation of $B$ over $(X,I)$. Then $B^{\prime}$ is stable.

Proof: Consider the Jordan–Hölder serie for $B^{\prime}$. Let $Q_{1}\oplus
Q_{2}\oplus...$ be the associated graded sheaf. By 5.3, the stable bundles