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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Holography principle and Moishezon twistor spaces }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf 
Conference on The Geometry, Topology and Physics \\
of Moduli Spaces of Higgs Bundles\\
National University of Singapore, 4-8 Aug 2014
}
\end{center}


\newpage

{\bf \blue Plan}

1. Hyperk\"ahler manifolds. Twistor spaces.

2. Holography principle

3. Moishezon twistor spaces. Applications of
the holography principle to the local structure of twistor spaces.

4. Proof of holography principle.

* 5. Hyperk\"ahler reduction.

\newpage

{\bf \blue Questions}


\definition
A {\bf \blue hyperk\"ahler structure} on a manifold $M$
is a Riemannian structure $g$ and a triple of complex
structures $I,J,K$, satisfying quaternionic relations
$I\circ J = - J \circ I =K$, such that $g$ is K\"ahler
for $I,J,K$.

\remark 
Any compact, K\"ahler, holomorphically symplectic manifold,
such as a K3 surface, admits a hyperk\"ahler metric, which is
unique in is K\"ahler class.


\question Let $(M,I,J,K)$ be a hyperk\"ahler manifold.
{\bf \purple Define when \\ $(M,I,J,K)$ is ``algebraic''.}

\remark This question is motivated by [HKLR] definition
of hyperk\"ahler manifolds in terms of twistor spaces,
which makes it possible to define a general ``hyperk\"ahler space''
with singularities and nilpotents.

\question
{\bf \purple Can an open ball in a K3 surface (or some other compact
hyperk\"ahler manifold)
with its hyperk\"ahler structure be equivalent to an open ball
in a quiver space or a Hitchin moduli space with its hyperk\"ahler structure?}



\newpage

{\bf \blue Twistor spaces}

\definition 
{\bf \color{blue} Induced complex structures} 
on a hyperk\"ahler manifold are 
complex structures of form $L:= aI + bJ +c K$. {\bf \purple It is a sphere:
$S^2 \cong \{ L:= aI + bJ +c K, \ \ \ a^2+b^2+c^2=1.\}$}

\definition
A {\bf\blue twistor space} $\Tw(M)$ of a hyperk\"ahler manifold
is {\bf \green a complex manifold obtained by gluing these complex structures into
a holomorphic family over $\C P^1$.} More formally:

 Let $\Tw(M) := M \times S^2$. Consider the complex structure $I_m:T_mM \to T_mM$ 
on $M$ induced by $J \in S^2 \subset {\Bbb H}$. Let $I_J$
denote the complex structure on $S^2 = \C P^1$.

The operator $I_{\Tw} = I_m \oplus I_J:T_x\Tw(M) \to T_x\Tw(M)$ 
satisfies $I_{\Tw} ^2 = -\Id$. {\bf \purple It defines 
an almost complex structure on $\Tw(M)$.} This almost 
complex structure is known to be integrable (Obata)


\newpage

{\bf \blue Geometry of twistor spaces}

\example If $M={\Bbb H^n}$, $\Tw(M)= \Tot (\calo(1)^{\oplus n})
\cong \C P^{2n+1} \backslash \C P^{2n-1}$
(total space of a vector bundle $(\calo(1)^{\oplus n}$).

\remark 
(Deligne, Simpson)\\
The quaternionic structure on a hyperk\"ahler manifold
{\bf \purple can be reconstructed
from the geometry of rational lines on its twistor space}.

\remark Twistor spaces can be defined for any
conformally semiflat 4-dimensional Riemannian manifold $M$.
These twistor space are often Moishezon (birational
to projective).  Results
of today's talk (``holography principle'') are true
for such twistor spaces. 

\theorem {\bf \blue (Hitchin)}
For compact 4-dimensional $M$, its twistor space is
K\"ahler only if $M=S^4$ or $M=\C P^2$.

\newpage

{\bf \blue Twistor spaces are non-K\"ahler}


\claim Suppose that $(M,I,J,K)$ is a hyperk\"ahler manifold 
such that $(M,I)$ contains a compact, odd-dimensional complex
subvariety $Z$. {\bf \red Then $\Tw(M)$ is non-K\"ahler.}

{\bf \green Proof:} Consider the fundamental class $[Z_I]\in H_*(M)$ of
$Z$ in $(M,I)\subset \Tw(M)$.
Then $[Z_I]+ [Z_{-I}]=0$, giving $\int_{Z_I\cup
  Z_{-I}}\alpha=0$
for each closed form $\alpha\in H^2(M)$. This is
impossible if $\alpha$ is a K\"ahler form, because
this integral is Riemannian volume of $Z_I\cup Z_{-I}$. \endproof


\claim {\bf 
\red When $M$ is compact and hyperk\"ahler, 
 $\Tw(M)$ never admits a K\"ahler structure.}

{\bf\blue Proof:}
Let $\omega$ be the standard Hermitian form of $\Tw(M)$.
Then $dd^c\omega$ is a positive (2,2)-form (a calculation due
to Kaledin-V.) For any K\"ahler form
$\omega_0$, {\bf \purple this would imply 
\[ 
 \int_{\Tw(M)}d\left(\omega_0^{\dim_\C M-1} \wedge d^c\omega\right)=
 \int_{\Tw(M)}\omega_0^{\dim_\C M-1} \wedge dd^c\omega>0,
\]
which is impossible by Stokes' theorem}. \endproof


\newpage 

{\bf \blue Rational curves on $\Tw(M)$.}

\definition
{\bf\blue An ample rational curve} on a complex manifold $M$ is
a smooth curve $S \cong \C P^1\subset M$ such that 
$NS=\bigoplus_{k=1}^{n-1}\calo(i_k)$, with $i_k >0$.
It is called {\bf \blue a quasiline} if all
$i_k=1$.

\claim
Let $M$ be a compact complex manifold containing a 
an ample rational line. {\bf \purple Then any $N$ points $z_1, ..., z_N$ can
be connected by an ample rational curve.}

\claim
Let $M$ be a hyperk\"ahler
manifold, $\Tw(M)\stackrel \sigma\arrow M$ its twistor
space, $m\in M$ a point, and $S_m=\C P^1 \times \{m\}$
the corresponding rational curve in $\Tw(M)$. {\bf \red Then $S_m$
  is a quasiline.}

{\bf \blue Proof:} Since the claim is essentially
infinitesimal, it suffices to check it when $M$ is flat.
{\bf \purple Then $\Tw(M)= \Tot (\calo(1)^{\oplus 2p})
\cong \C P^{2p+1} \backslash \C P^{2p-1}$, and
$S_m$ is a section of $\calo(1)^{\oplus 2p}$.}
\endproof


\newpage 

{\bf \blue Holography principle}

{\bf \green Theorem 1:} {\blue \bf (Holography principle for line bundles)}\\
Let $S\subset M$ be an ample curve in a simply connected, 
connected complex manifold, which is covered by deformations of $S$.
Consider a tubular neighbourhood $U\supset S$. 
Then, {\bf \red for any holomorphic
line bundle $L$ on $M$, the space
$H^0(U,L)$ is independent from the
choice of $U$ and $S$ in its deformation class.}



\remark In these assumptions, {\bf \purple $H^0(U,L)$
is always finite-dimensional} (Hartshorne). 


\theorem
{\blue \bf (Holography principle for meromorphic functions)}\\
In assumptions of Theorem 1, {\bf \red the space of meromorphic functions on
$U$ is equal to the space of meromorphic functions on $M$.}

\definition
Given a complex manifold $Z$, denote by $\Mer(Z)$
the field of global meromorphic functions on $Z$,
and let {\bf\blue the algebraic dimension} $a(Z)$
be the transcendence degree of $\Mer(Z)$.

\remark 
{\bf \purple It could be infinite!}

\corollary
{\bf \purple $a(\Tw(M))=a(U)$
for any connected neighbourhood $U$ of a quasiline}
in a connected, simply connected $M$.

\newpage 

{\bf \blue Moishezon twistor spaces}


\definition
A compact complex variety $Z$ is called {\bf \blue Moishezon}
if the ring of meromorphic functions on $Z$ has
algebraic dimension $\dim Z$, or, equivalently, if
$Z$ is bimeromorphic to a projective manifold.


\claim
Let $M$ be a simply connected hyperka\"ahler manifold,
and $\Tw(M)$ its twistor space. {\bf \purple Then 
$a(\Tw(M))\leq \dim_\C \Tw(M)$.}

\definition
A twistor space satisfying $a(\Tw(M))=\dim_\C \Tw(M)$
is called {\bf \blue Moishezon}.

\claim {\bf \purple All Moishezon twistor spaces 
are bimeromorphic to open subsets of 
projective manifolds.}


\theorem
Let $V$ be a quaterionic Hermitian vector space, and
$G\subset \Sp(V)$ a compact Lie group acting on $V$ by quaternionic
isometries. Denote by $M$ the hyperk\"ahler reduction of $V$.
{\bf \red Then $\Tw(M)$ is Moishezon.} 


\newpage 

{\bf \blue Local structure of hyperk\"ahler manifolds}

\theorem
(Fujiki, 1987) Let $M$ be a compact hyperk\"ahler manifold,
and $L\in S^2 \subset {\Bbb H}$ a generic induced complex structure.
{\bf \red Then $M$ contains no divisors.}

\corollary
Let $M$ be a compact hyperk\"ahler manifold,
and $D\subset \Tw(M)$ a divisor. {\bf \purple Then $D$ is a union
of several fibers of the projection $\pi:\; \Tw(M) \arrow \C P^1$.}

{\bf \green Proof:} Suppose that $D$ contains a component
which intersects generic fiber of $\pi$. By transversality,
this intersection is a divisor, contradicting Fujiki's theorem.
\endproof


\corollary 
Let $M$ be a compact, simply connected hyperk\"ahler manifold,
$M'$ a hyperk\"ahler manifold obtained by hyperk\"ahler
reduction (such as Nakajima quiver variety), and $U\subset M$,
$U'\subset M'$ open subsets. {\bf \red Then $U$ is never equivalent to
$U'$ as a hyperk\"ahler manifold.}

{\bf \green Proof:} $a(\Tw(U))=a(\Tw(M))=1$, and\\ 
$a(\Tw(U'))=a(\Tw(M'))=\dim(\Tw(M'))$. \endproof


\newpage 

{\bf \blue Complex manifolds and quasilines}

\remark
Let $S\subset M$ be a quasiline. Then, for an appropriate
tubular neighbourhood $U\subset M$ of $S$, {\bf
  \red ``for every two points
$x, y\in U$ close to $S$ and far from each other,  there is a 
 unique deformation of $S$ containing $X$ and $Y$.''}

More precisely:

\claim
Let $S\subset M$ be a quasi-line. Then, for any sufficiently small
tubular neighbourhood $U\subset M$ of $S$, there exists
a smaller tubular neighbourhood $W\subset U$, satisfying the
following condition. Let $\Delta_S$ be
the image of the diagonal embedding $\Delta_S:\; S \arrow W \times W$.
Then there exists an open neighbourhood $V$ of
$\Delta_S$, properly contained in $W\times W$, such that {\bf \purple 
for any pair
 $(x,y) \in W\times W \backslash V$, there exists a unique deformation
$S'\subset U$ of $S$ containing $x$ and $y$.}

\corollary For any quasiline in $M$, its deformation space
is $2(\dim M-1)$-dimensional.



\newpage 

{\bf \blue Proof of holography principle}


\theorem 
{\bf \blue (Holography Principle)}\\
Let $S\subset M$ be a quasiline in a simply connected 
complex manifold, which is covered by deformations of $S$.
Assume that $M$ is equipped with a projection 
$\pi:\; M \arrow S$ inducing identity on $S$.
Consider a tubular neighbourhood $U\supset S$,
and assume that $\pi:\; U \arrow S$ has 
connected fibers. Then, {\bf \purple for any holomorphic
line bundle $L$ on $M$, the space
$H^0(U,L)$ is independent from the
choice of $U$ and $S$ in its deformation class.}

{\bf \green A (slighly) weaker statement.}

{\bf \green THEOREM 1:}
Let $S\subset M$ be a quasiline, and $L$ a holomorphic
bundle on $M$. Assume that $M$ is equipped with a projection 
$\pi:\; M \arrow S$ inducing identity on $S$.
Consider a sufficiently small tubular neighbourhood $U\supset S$,
and a smaller tubular neighbourhood
$V\subset U$. {\bf \red Then the restriction map
$H^0(U,L)\arrow H^0(V,L)$ is an isomorphism.}

{\bf \blue We deduce the ``Holography principle''
from Theorem 1.}


\newpage 

{\bf \blue Deducing the holography principle from Theorem 1}


{\bf \green Step 1:} Choose a 
continuous, connected family $S_b$ of quasilines parametrized
by $B$ such that $\bigcup_{b\in B} S_b = M$.
Find a tubular neighbourhood $U_b$ for each $S_b$
in such a way that an intersection $U_b\cap U_{b'}$
for sufficiently close $b,b'$ always contains $S_b$ and
$S_{b'}$. {\bf \purple
By Theorem 1, $H^0(U_b\cap U_{b'},L)=H^0(U_b,L)=H^0( U_{b'},L)$.}

{\bf \green Step 2:} Since $B$ is connected, {\bf \purple all 
the spaces $H^0(U_b,L)$ are isomorphic,} and these isomorphisms
are compatible with the restrictions to the intersections 
$U_b\cap U_{b'}$. 

{\bf \green Step 3:} Let now $f\in H^0(U_b,L)$, and let
$\tilde M_f$ be the {\bf \blue domain of holomorphy} for $f$,
that is, a maximal domain (non-ramified over $M$) such that
$f$ admits a holomorphic extension to $\tilde M_f$. 
Since $\cup U_b=M$, and $f$ can be holomorphically
extended to any $U_b$, the domain $\tilde M_f$
is a covering of $M$. {\bf \purple Now, ``holography principle''
follows, because $M$ is simply connected.}
\endproof


\newpage 

{\bf \blue Proof of Theorem 1}


{\bf \green THEOREM 1:}
Let $S\subset M$ be a quasiline, and $L$ a holomorphic
bundle on $M$. Assume that $M$ is equipped with a projection 
$\pi:\; M \arrow S$ inducing identity on $S$.
Consider a sufficiently small tubular neighbourhood $U\supset S$,
and a smaller tubular neighbourhood
$V\subset U$. {\bf \red Then the restriction map
$H^0(U,L)\arrow H^0(V,L)$ is an isomorphism.}

{\bf \green Proof. Step 1:}
Fix a point $x_0=\infty$ in $\C P^1$. A section of
$L\restrict{\C P^1}=\calo(d)$ is the same as meromorphic function having a pole
of degree $\leq d$ at $\pi^{-1}(\infty)$.

{\bf \green  Step 2:} For each section
$S_1:\; \C P^1 \arrow M$ of $\pi$, a degree $d$ meromorphic
function on $S_1$  is uniquely determined
by its values at any $d+1$ points of $S_1$.

{\bf \green  Step 3:} Given a meromorphic
function $f\in H^0(V,L)$
and a quasiline $S_1$ intersecting $V$ in an open set,
we can extend $f$ to a meromorphic function $\tilde f$ on
$S_1$ by computing its values at $d+1$ distinct points
$z_1, ..., z_{d+1}$ of $S_1\cap V$. Whenever $S_1$ is in $V$, this 
procedure gives $f\restrict {S_1}$. 
{\bf \purple By analytic continuation,
the values of $\tilde f(z)$ at any $z\in S_1$ are independent
from the choice of $z_i$ and $S_1$.}

{\bf \green  Step 4:} Therefore, {\bf \purple
$\tilde f$ is a well-defined
meromorphic function on the union $V_1$ of all quasilines intersecting
$V$. } For $U$ sufficiently small, $V_1$ contains $U$.
Therefore, any $f \in H^0(V,L)$ can be extended to $U$.
\endproof


\newpage 

{\bf \blue Hamiltonians}

{\bf \green Let's define the hyperk\"ahler reduction.}

We denote the Lie derivative along a vector field
as $\Lie_x:\; \Lambda^i M \arrow \Lambda^i M$,
and contraction with a vector field by 
$i_x:\; \Lambda^i M \arrow \Lambda^{i-1} M$.

{\bf \blue Cartan's formula:} $d\circ i_x + i_x \circ d =\Lie_x$.

\remark
Let $(M,\omega)$ be a symplectic manifold, $G$ a 
Lie group acting on $M$ by symplectomorphisms, and $\goth g$
its Lie algebra. For any $g\in {\goth g}$,
denote by $\rho_g$ the corresponding vector field.
Then $\Lie_{\rho_g}\omega=0$, giving 
$d(i_{\rho_g}(\omega))=0$. {\bf \purple We obtain that 
$i_{\rho_g}(\omega)$ is closed, for any $g\in {\goth g}$.}

\definition
{\bf\blue A Hamiltonian} of $g\in {\goth g}$ is a function
$h$ on $M$ such that $dh=i_{\rho_g}(\omega)$.


\newpage

{\bf \blue Moment maps}

\definition
$(M,\omega)$ be a symplectic manifold, $G$ a 
Lie group acting on $M$ by symplectomorphisms.
{\bf \blue A moment map} $\mu$ of this action is a linear map
${\goth g}\arrow C^\infty M$
associating to each $g\in G$ its Hamiltonian.

\remark
It is more convenient to consider $\mu$
as an element of ${\goth g}^* \otimes_\R C^\infty M$,
or (and this is most standard) {\bf \red as a function
with values in ${\goth g}^*$}. 

\remark
Moment map {\bf \purple always exists} if $M$ 
is simply connected.

\definition 
A moment map $M \arrow  {\goth g}^*$
is called {\bf \blue equivariant}
if it is equivariant with respect to the 
coadjoint action of $G$ on ${\goth g}^*$.


\remark
$M\stackrel\mu \arrow  {\goth g}^*$ is a moment map
iff for all $g\in {\goth g}$, 
$\langle d\mu,g\rangle= i_{\rho_g}(\omega)$.
Therefore, {\bf \purple a moment map is defined up to 
a constant ${\goth g}^*$-valued function.}
An equivariant moment map is is defined up to 
{\bf \purple a constant ${\goth g}^*$-valued function
which is $G$-invariant}. 

\definition A $G$-invariant $c\in \goth g^*$ is called
{\bf\blue central}.

\claim
{\bf \red
An equivariant moment map exists whenever $H^1(G, {\goth g}^*)=0$.}
In particular, when $G$ is reductive and $M$ is simply connected,
an equivariant moment map exists.

\newpage

\newcommand{\3}{{/\!\!/\!\!/}} 
\newcommand{\2}{{/\!\!/}}

\newpage

{\bf\blue Hyperk\"ahler reduction}

\definition
Let $G$ be a compact Lie group, $\rho$ its action 
on a hyperk\"ahler manifold $M$ by hyperk\"ahler isometries, and
$\g^*$ a dual space to its Lie algebra. {\bf \blue A 
hyperk\"ahler moment map} is a $G$-equivariant 
smooth map $\mu: M\to\g^*\otimes\R^3$ such that
 $\langle \mu_i(v),g \rangle = \omega_i(v,d\rho(g))$, 
for every $v\in TM$, $g\in\g$ and $i=1,2,3$,
where $\omega_i$ is one three K\"ahler forms associated 
with the hyperk\"ahler structure.

\definition
Let $\xi_1,\xi_2,\xi_3$ be three $G$-invariant vectors
in $\g^*$.
The quotient manifold $M\3 G := \mu^{-1}(\xi_1,\xi_2,\xi_3)/G$ 
is called {\bf\blue  the hyperk\"ahler quotient} of $M$.

\theorem (Hitchin, Karlhede, 
 Lindstr\"om, Ro\v cek)\\ {\bf \red  The quotient $M\3 G$
is hyperkaehler}.

\newpage


{\bf \blue Holomorphic moment map}

Let $\Omega:=\omega_J+ \1\omega_K$. This is a holomorphic
symplectic (2,0)-form on $(M,I)$.

{\bf \green The proof of HKLR theorem. Step 1:}
Let $\mu_J, \mu_K$ be the moment map associated with 
$\omega_J, \omega_K$, and $\mu_\C:=  \mu_J+ \1\mu_K$.
Then $\langle d\mu_\C,g\rangle= i_{\rho_g}(\Omega)$
Therefore, $d\mu_\C\in \Lambda^{1,0}(M,I)\otimes {\goth g}^*$.

{\bf \green Step 2:} This implies that the map $\mu_\C$ is
holomorphic. It is called {\bf \blue the holomorphic moment map.}

{\bf \green Step 3:} 
By definition, $M\3 G=\mu_{\C}^{-1}(c)\2 G$, 
where $c\in {\goth g}^*\otimes_\R \C$
is a central element.
{\bf \red This is a K\"ahler manifold,} because it is a K\"ahler
quotient of a K\"ahler manifold.

{\bf \green Step 4:} We obtain 3 complex structures
$I,J,K$ on the hyperk\"ahler quotient $M\3 G$.
{\bf \purple They are compatible in the usual way} 
(an easy exercise).
\endproof

\newpage

{\bf \blue Twistor spaces and hyperk\"ahler reduction}

\theorem
Let $V$ be a quaterionic Hermitian vector space, and
$G\subset \Sp(V)$ a compact Lie group acting on $V$ by quaternionic
isometries. Denote by $M$ the hyperk\"ahler reduction of $V$.
{\bf \red Then $\Tw(M)$ is Moishezon.} 

{\bf \green Proof:} $\Tw(M)$ is obtained
as the space of stable $G_\C$-orbits in
$\mu_\C^{-1}(0)\subset \Tw(V)$. The space 
$\Tw(V)=\C P^{2n+1}\backslash \C P^{2n-1}$
is algebraic. Averaging over $G$, we obtain 
that the field $G_\C$-invariant rational
functions on $\Tw(M)$ has dimension $\dim \Tw(M)$,
and $\Tw(M)$ is Moishezon. \endproof

\corollary
Let $U$ be an open subset of a compact, simply
connected hyperk\"ahler manifold, and $U'$
an open subset of a hyperk\"ahler manifold obtained
as $V\3 G$, where $V$ is flat and $G$ reductive.
{\bf \red Then $U$ is not isomorphic to $U'$ as
hyperk\"ahler manifold}.

{\bf \green Proof:} $\Tw(U')$ has many meromorphic
functions, and $a(\Tw(U))=1$.
\endproof


%\newpage 
%
%{\bf \blue Infinite-dimensional vector bundles}
%
%\question
%Is there a ``normal form'' for a 
%formal neighbourhood of a quasiline?
%
%{\bf \purple Apparently, no.}
%
%\definition
%{\bf \blue An infinite-dimensional vector bundle} on $S$
%is a $\calo_S$-sheaf $A$ equipped with a filtration
%$F^iA\subset F^{i-1}A \subset ... \subset F^0 A=A$ 
%with all associated graded components $F^iA/F^{i+1}A$
%vector bundles.
%
%\definition
%{\bf \blue A morphism of infinite-dimensional vector bundles}
%is a map $A\arrow B$ of the corresponding  $\calo_S$-sheaves,
%mapping $F^iA$ to $F^{i+j}B$.
%
%\example
%Let $S\subset M$ be a quasiline, 
%$U$ its tubular neighbourhood, and
%$\pi:\; U \arrow S$ a projection with connected Stein fibers.
%Then $\pi_*\calo_U$ is a ring object in the category of
%infinite-dimensional vector bundles.
%
%Indeed: let $I_S$ be the ideal sheaf of $S\subset U$.
%Then $I^k_S/I^{k+1}_S$ is $\Sym^k (NS)$, hence
%the quotients $\pi_*\calo_U/\pi_* I^{k+1}_S$ are vector
%bundles, and we can take the filtration
%$F^k\pi_*\calo_U=\pi_*I_s^k$.
%
%\remark We are interested in the local structure
%of a twistor space around a quasiline. It turns
%out that the infinite-dimensional vector bundle
%$\pi_*\calo_U$ with its ring structure is very 
%rich in invariants, and uniquely determines the 
%twistor space if it is Moishezon.
%
%
%\newpage 
%
%{\bf \blue Formal hyperk\"ahler manifold}
%
%
%
%\definition
%Let $A$ be an infinite-dimensional vector bundle over $A$.
%The inverse limit $\lim\limits_{\leftarrow} (A/F^iA)$
%is called {\bf\blue the formal completion of $A$}.
%



\end{document}

