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\begin{document}
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\begin{center}
{\Large\bf Trisymplectic manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\small\bf Advances in hyperk\"ahler and holomorphic symplectic geometry }\\[10mm]

{\tiny\bf BIRS, Canada, March  16, 2012\\[10mm]
}

\end{center}



\newcommand{\bOmega}{{\boldsymbol{\boldsymbol{\Omega}}}}
\newcommand{\bMu}{{\boldsymbol{\boldsymbol \mu}}}
\newcommand{\4}{{/\!\!/\!\!/\!\!/}} 
\newcommand{\3}{{/\!\!/\!\!/}} 
\newcommand{\2}{{/\!\!/}}



\newpage



{\bf \blue Complexification of a manifold}

\definition Let $M$ be a complex manifold, equipped with
an anticomplex involution $\iota$. The fixed point set $M_\R$
of $\iota$ is called {\bf \blue a real analytic manifold},
and a germ of $M$ in $M_\R$ is called
{\bf \blue a complexification} of $M_\R$.

\question {\bf \purple 
What is a complexification of a K\"ahler manifold}
(considered as real analytic variety)?

\theorem (D. Kaledin, B. Feix) 
Let $M$ be a real analytic
K\"ahler manifold, and $M_\C$ its complexification.
{\bf \red Then $M_\C$ admits a hyperk\"ahler structure,} determined
uniquely and functorially by the K\"ahler structure on $M$.
\begin{center}
\question {\bf \red What is a complexification of a 
hyperk\"ahler manifold?}

THIS IS THE MAIN SUBJECT OF TODAY'S TALK.

{\bf \blue (A joint work with Marcos Jardim).}
\end{center}

\newpage 

{\bf \blue Plan of the talk:}

1. Trisymplectic structures on a vector space (linear algebra).

2. Trisymplectic structures on a manifold (differential geometry).

3. Trisymplectic structure on the space 
of rational lines in the twistor space (hyperk\"ahler geometry).

4. Applications to the instanton spaces.


\newpage

{\bf \blue Trisymplectic structure on a vector space}

\definition
A {\bf \blue trisymplectic structure} on a complex 
vector space of dimension $2n$ is a 3-dimensional
space $\bOmega\subset \Lambda^2V$ of complex linear 2-forms,
such that any $\eta\in \bOmega$ has rank 
$2n$, $n$ or 0.

\remark It is easy to see that
{\bf \purple $\bOmega$ contains a symplectic form.}

\proposition
Given two symplectic forms $\omega_1, \omega_2\in \bOmega$,
consider the map $\phi_{\Omega_1, \Omega_2}:=
\omega_1\circ\omega_2^{-1}\in \End(V)$. Then
$\phi_{\Omega_1, \Omega_2}$ 
can be expressed in an appropriate basis by the matrix
{\scriptsize\[
\phi_{\omega_1, \omega_2}=\begin{pmatrix}
\lambda &0&0 &\hdotsfor{1} &0&0&0\\
0&\lambda &0 &\hdotsfor{1} &0&0&0\\
0&0&\lambda &\hdotsfor{1} &0&0&0\\
\vdots&\vdots&\vdots&
\ddots
&\vdots&\vdots&\vdots\\
0&0&0 &\hdotsfor{1} &\lambda'&0&0\\
0&0&0 &\hdotsfor{1} &0&\lambda'&0\\
0&0&0 &\hdotsfor{1} &0&0&\lambda'
\end{pmatrix},
\]}
with the eigenspaces of equal dimension.

\theorem
Let $(V, \bOmega)$ be a be a trisymplectic 
vector space, and $H\subset \End(V)$ the algebra
generated by $\phi_{\Omega_1, \Omega_2}$,
for all $\omega_1, \omega_2\in \bOmega$.
{\bf \red Then $H$ is isomorphic to the matrix algebra $\Mat(2)$,}
acting on $V$ in a standard way.

\newpage

{\bf \blue Trisymplectic structures as $\Mat(2)$-representations}


\definition
Let $V$ be a complex vector space with the 
{\bf \blue standard action} of the matrix algebra $\Mat(2)$, 
i.e. $V \cong V_0\otimes \C^2$ and $\Mat(2)$ acts 
only through the second factor.

\claim
Consider the natural $SL(2)$-action on $V$ induced by $\Mat(2)$,
and extend it multiplicatively to all tensor powers of $V$.
Let $g\in \Sym^2_\C(V)$ be an $SL(2)$-invariant, non-degenerate
2-form on $V$, and $\{I,J,K\}$ a quaternionic basis in
$\Mat(2)$ Then 
\[
g(x, Iy) = g(Ix, I^2y) = - g(Ix, y)
\]
hence the form $\Omega_I(\cdot, \cdot):= g(\cdot, I\cdot)$
is a symplectic form, obviously non-degenerate; the forms
$\Omega_J$, $\Omega_K$
have the same properties. Let $\bOmega:=\langle\Omega_I,
\Omega_J, \Omega_K\rangle$.
 It turns out that 
{\bf \red this construction gives a trisymplectic
structure, and all trisymplectic structures can be obtained in this way.}


\newpage

{\bf \blue Trisymplectic structures as $\Mat(2)$-representations II}


\theorem
Let $V$ be a vector space equipped with a standard action of
the matrix algebra $H\cong \Mat(2)$, and
$\{I,J,K\}$ a quaternionic basis in
$\Mat(2)$. Consider the corresponding action of $SL(2)$
on the tensor powers of $V$. Then, for any 
$SL(2)$-invariant symmetric form $g$, denote by $\bOmega$ the
space generated by 
$\Omega_I:= g(\cdot, I\cdot)$, $\Omega_J$, $\Omega_K$
Then {\bf \purple 
$\bOmega$ is a trisymplectic structure on $V$, with the operators 
$\Omega_K^{-1}\circ \Omega_J$, $\Omega_K^{-1} \circ \Omega_I$  generating 
$H$.} Moreover, {\bf \purple for each trisymplectic structure
$\bOmega$  on $V$, there exists a unique (up to a constant) 
$SL(2)$-invariant non-degenerate quadratic form 
$g$ inducing $\bOmega$ as above.}


\newpage

{\bf \blue Trisymplectic manifold}

\definition
A {\bf \blue trisymplectic structure} on a complex $2n$-manifold 
$M$ is a  triple of holomorphic symplectic forms $\Omega_1$, $\Omega_2$, 
$\Omega_3$, such that any
linear combination of these forms has rank $2n$, $n$ or 0.
We denote by $\bOmega$ the
3-dimensional space generated by $\Omega_i$.
Obviously, $\bOmega$ defines a trisymplectic
structure at each point of $M$.

\remark
Let $\Omega_1, \Omega_2\in \bOmega$. Consider
$P(t):=\det(\Omega_1 + t\Omega_2)$ as a polynomial of 
$t$. Since the eigenvalues of $\Omega_1 + t\Omega_2$
occur in $n$-tuples, {\bf \red $P(t)=Q(t)^{n/2}$,
where $Q$ is a quadratic polynomial.}

\claim
There exists a non-degenerate
quadratic form $Q$ on $\bOmega$, unique up to a 
constant, such that {\bf \red $\Omega\in \bOmega$ is
 degenerate if and only if $Q(\Omega,\Omega)=0$.}

\corollary 
For each degenerate $\Omega\in \bOmega$,
its radical $\ker \Omega$ is a sub-bundle
of codimension $n$ in $TM$. Moreover,
{\bf \purple for all non-proportional degenerate $\Omega, \Omega'\in \bOmega$,
one has $TM=\ker \Omega\oplus \ker \Omega'$.}

\remark Since $\Omega$ is closed,
$\ker \Omega$ is {\bf \blue involutive:}
$[\ker \Omega, \ker \Omega]\subset \ker \Omega$.

\remark 
Similar to web geometry!



\newpage 

{\bf \blue Holomorphic 3-webs.}

\definition 
Let $M$ be a complex manifold, and
$S_1$, $S_2$, $S_3$ integrable, pairwise 
transversal holomorphic  sub-bundles in $TM$, 
of dimension $\frac 1 2 \dim M$. Then $(S_1, S_2, S_3)$
is called {\bf \blue a holomorphic 3-web} on $M$.

\remark On smooth manifolds, the theory of 3-webs 
is due to Chern and Blaschke (1930-ies).

\theorem (Ph. D. thesis of Chern, 1936)
Let $S_1, S_2, S_3$ be a holomorphic 3-web on a
complex manifold $M$. {\bf \blue Then there exists a unique
holomorphic connection $\nabla$ on $M$ which 
preserves the sub-bundles $S_i$,} and such that its
torsion $T$ satisfies $T(S_1, S_2)=0$.

\newpage 

{\bf \blue Holomorphic $SL(2)$-webs.}

\definition
A holomorphic 3-web $S_1$, $S_2$, $S_3$ on a
complex manifold $M$ is 
called {\bf\blue an $SL(2)$-web}
if \\
\hphantom{.} $\ \bullet$ the projection operators $P_{i,j}$ of $TM$ to $S_i$ along
$S_j$ {\bf \red generate the standard action of $\Mat(2)$} on $\C^2 \otimes \C^n$,
\\
\hphantom{.} $\ \ \ \bullet$ for any nilpotent $v\in \Mat(2)$, 
{\bf \red the bundle $v(TM)\subset TM$
is involutive.}

\remark The set of $v\in \Mat(2)$ with $\rk v=1$
satisfies ${\Bbb P} V= \C P^1$, hence the
sub-bundles $v(TM)\subset TM$ are parametrized by $\C P^1$.
{\bf \purple An $SL(2)$-web is determined by a set of sub-bundles 
$S_t\subset TM$,  $t\in \C P^1$, which are
pairwise transversal and involutive.}

\theorem (Jardim--V.) 
Let $S_t\subset TM$,  $t\in \C P^1$ be an $SL(2)$-web
on $M$, and $t_1, t_2, t_3 \in \C P^1$ distinct points. 
Then the Chern connection of a 3-web $S_{t_1}$,
$S_{t_2}$, $S_{t_3}$ {\bf \red is a torsion-free
affine holomorphic connection with holonomy in
$GL(n, \C)$} acting on $\C^{2n}= \C^n \otimes \C^2$,
and {\bf \red independent from the choice of $t_i$.}

\hfill


{\bf \blue Trisymplectic manifolds}

\theorem (Jardim--V.) 
For any trisymplectic  structure on $M$,
{\bf \red the bundles $\ker \Omega\subset TM$ 
define an $SL(2)$-web.} Moreover, the Chern
connection of this $SL(2)$-web {\bf \red 
preserves all forms in $\bOmega$}.

\remark In this case, {\bf \purple the Chern connection
has holonomy in $Sp(n,\C)$ acting 
on $\C^{2n}\otimes \C^2$.}

\remark For a trisymplectic structure $\bOmega$,
it is just the {\bf \purple Levi-Civita connection
of the holomorphic Riemannian form associated with $\bOmega$.}


{\bf \green THE REST OF TODAY'S TALK IS EXAMPLES AND
APPLICATIONS OF TRISYMPLECTIC GEOMETRY}

\newpage

{\bf \blue Hyperk\"ahler manifolds}

\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 A hyperk\"ahler manifold {\bf \purple has three symplectic forms\\
$\omega_I:=  g(I\cdot, \cdot)$, $\omega_J:=  g(J\cdot, \cdot)$,
$\omega_K:=  g(K\cdot, \cdot)$.}


{\bf\green REMARK:} This is equivalent to $\nabla I=\nabla J = \nabla K=0$:
the parallel translation along the connection preserves $I, J,K$.

{\bf\green DEFINITION:} 
Let $M$ be a Riemannian manifold, $x\in M$ a point.
The subgroup of $GL(T_xM)$ generated by parallel 
translations (along all paths) is called {\bf \blue 
the holonomy group} of $M$.

{\bf\green REMARK:}
{\bf \purple A hyperk\"ahler manifold can be defined as a manifold which has
holonomy in $Sp(n)$ } (the group of all endomorphisms preserving
$I,J,K$).

\newpage 

{\bf \blue Twistor space}


\definition 
{\bf \color{blue} Induced complex structures} 
on a hyperk\"ahler manifold are 
complex structures of form 
$S^2 \cong \{ L:= aI + bJ +c K, \ \ \ a^2+b^2+c^2=1.\}$
{\bf \red They are usually non-algebraic}. Indeed,
if $M$ is compact, for generic $a, b, c$, $(M,L)$ has no divisors (Fujiki).

\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 $I_{\Tw} ^ = -\Id$. {\bf \purple It defines 
an almost complex structure on $\Tw(M)$.} This almost 
complex structure is known to be integrable (Obata, Salamon)

\example If $M={\Bbb H^n}$, $\Tw(M)= \Tot (\calo(1)^{\oplus n})
\cong \C P^{2n+1} \backslash \C P^{2n-1}$

\remark {\bf 
\red For $M$ compact,  $\Tw(M)$ never admits a K\"ahler structure.}

\newpage 

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

\remark The twistor space {\bf \red has many rational curves}.
In fact, it is {\bf \red rationally connected} (Campana).

\definition
Denote by $\Sec(M)$ {\bf \blue the
space of holomorphic sections} of the
twistor fibration ${\rm Tw}(M)\stackrel\pi\arrow\C P^1$.


\definition
For each point $m \in M$, one has {\bf \blue a horizontal section}
$C_m:=\{m\} \times \C P^1$ of $\pi$. The space of horizontal
sections is denoted $\Sec_{hor}(M)\subset 
\Sec(M)$

\remark The space of
horizontal sections of $\pi$ is identified with $M$.
The normal bundle  $N C_m= {\cal O}(1)^{\dim M}$.
Therefore, {\bf \purple some neighbourhood of $\Sec_{hor}(M)\subset 
\Sec(M)$ is a smooth manifold of
dimension $2\dim M$.}

\definition A twistor section $C\subset \Tw(M)$
is called {\bf \blue regular}, \\ if $N C= {\cal O}(1)^{\dim M}$.

\claim For any $I\neq J\in \C P^n$, consider the
evaluation map $\Sec(M) \stackrel {E_{I,J}}
\arrow (M,I)\times (M,J)$, $s\arrow s(I)\times s(J)$.
Then {\bf \red $E_{I,J}$ is an isomorphism around the set 
$\Sec_0(M)$ of regular
sections.} 


\newpage 

{\bf \blue Complexification of a hyperk\"ahler manifold.}

\remark Consider an anticomplex 
involution $\Tw(M) \stackrel \iota \arrow \Tw(M)$
mapping $(m,t)$ to $(m, i(t))$, where $i:\; \C P^1\arrow \C P^1$
is a central symmetry. Then 
{\bf \purple $\Sec_{hor}(M)=M$ is a component of the
fixed set of $\iota$.}

\corollary {\bf \red $\Sec(M)$ is a complexification of $M$.}

\question What are geometric structures on $\Sec(M)$?

{\bf \green Answer 1:} For compact $M$,
$\Sec(M)$ {\bf \purple is holomorphically convex} (Stein
if $\dim M=2$). 

{\bf \green Answer 2:} . Let $I\in \C P^1$, and 
$ev_I:\; \Sec_0(M)\arrow (M,I)$ be an evaluation map
putting $S\in \Sec_0(M)$ to $S(I)$. Then {\bf \purple the 2-forms
 $ev_I^*\Omega_I$, $I\in \C P^1$ generate a trisymplectic
structure on $\Sec_0(M)$.}


{\bf \green Answer 3:} The space $\Sec_0(M)$
{\bf \purple admits a holomorphic, torsion-free connection} with holonomy
$Sp(n,\C)$ acting on $\C^{2n}\otimes \C^2$.

\newpage

{\bf\blue Holomorphic bundles on $\C P^3$ and twistor sections}


\definition
{\bf \blue An instanton} on $\C P^2$
is a stable bundle $B$ with $c_1(B)=0$.
{\bf\blue A framed
instanton} is an instanton equipped with a trivialization
$B\restrict C$ for a line $C \subset \C P^2$.


\theorem
(Nahm, Atiyah, Hitchin) 
The space ${\cal M}_{r,c}$ of framed instantons on $\C P^2$
is {\bf \red smooth, connected, hyperk\"ahler.}


{\bf \green THEOREM:}
{\bf \purple There is a correspondence}
between the holomorphic bundles on $\Tw({\Bbb H})=\C
P^3\backslash \C P^1$, with appropriate stability and
framing conditions,  
and twistor sections in $\Sec({\cal M}_{r,c})$.


It is used to prove the following longstanding conjecture.


\theorem
(Jardim--V.) 
{\bf \red The space ${\Bbb M}_{r,c}$ of framed mathematical
instantons on $\C P^3$ is smooth.}

\remark
To prove that ${\cal M}_{r,c}$ is smooth, one could
use hyperk\"ahler reduction. To prove that ${\Bbb M}_{r,c}$
is smooth,
we develop {\bf \blue trihyperk\"ahler reduction},
which is {\bf \purple a reduction defined on trisymplectic
manifolds.}


\newpage

{\bf\blue Mathematical instantons}


\definition A
{\bf\blue mathematical
instanton bundle} on $\C P^n$ is a locally free coherent sheaf $E$ on 
$\C P^n$ with $c_1(E)=0$ satisfying the following cohomological conditions:\\
1.  for $n\geq2$, $H^0(E(-1))=H^n(E(-n))=0$;\\
2. for $n\geq3$, $H^1(E(-2))=H^{n-1}(E(1-n))=0$;\\
3. for $n\geq4$, $H^p(E(k))=0$, $2\leq p\leq n-2$ and $\forall k$;\\
The integer $c=-\chi(E(-1))=h^1(E(-1))=c_2(E)$ is called
 {\bf\blue the
 charge} of $E$.
{\bf\blue A framed instanton} is a
mathematical instanton equipped with a trivialization
of $B\restrict \ell$ for some fixed line $\ell =\C P^1 \subset \C P^n$.

\remark Mathematical instantons of rank 2 {\bf \red are always
  stable} (follows from the monad description below).

\remark The space ${\Bbb M}_{r,c}$
of framed instantons with charge $c$ and rank $r$  
{\bf \purple is a principal $SL(2)$-bundle } over the space
of all mathematical instantons trivial on $\ell$.


\theorem 
(Jardim--V.) 
The space ${\Bbb M}_{c}$ of framed rank $r$ mathematical
instantons on $\C P^3$ {\bf \red is naturally
identified with the space of twistor sections
$\Sec({\cal M}_{r,c})$.}


\newpage

{\bf\blue Monads and mathematical instantons}

\definition
{\bf \blue A monad} is a sequence of vector bundles
$0 \arrow A \stackrel{i}\arrow B \stackrel j 
\arrow C\arrow 0$ which is exact in the
first and the last term. {\bf \blue The cohomology} of a monad
is $\ker j/\im i$.

\theorem 
Let $B$ be a holomorphic bundle of rank 2 on $\C P^n$,
$c_1(B)=0$, $c_2(B)=c$.
{\bf \red Then the following conditions are equivalent.}

(i) $B$ is a mathematical instanton.

(ii) $B$ is a cohomology of a monad
\[
0 \arrow V \otimes_\C \calo_{\C P^k}(-1) 
\arrow W\otimes_\C \calo_{\C P^k} \arrow U \otimes_\C \calo_{\C P^k}(1) 
\arrow 0
\]
with $\dim V =\dim U =c$ and $\dim W=2c+2$.



\newpage

{\bf\blue ADHM construction}

\definition
Let $V$ and $W$ be complex vector spaces, with dimensions
$c$ and $r$, respectively. The {\bf \blue ADHM data}
is maps
\[ A,B \in {\rm End}(V),
I \in {\rm Hom}(W,V), J \in {\rm Hom}(V,W). 
\]
We say that ADHM data is \\
 \phantom{MMM}  {\bf\blue stable}, \\ if there is no subspace $S\subsetneq V$ such that
$A(S),B(S)\subset S$ and $I(W)\subset S$;\\
 \phantom{MMM} {\bf\blue costable},\\ 
 if there is no nontrivial subspace $S\subset V$ such that
$A(S),B(S)\subset S$ and $S\subset \ker J$;\\
\phantom{MMM}  {\bf\blue regular}, \\  if it is both stable and costable.

{\bf\blue The  ADHM equation} is $[A,B] + IJ =0$.

\theorem (Atiyah, Drinfeld, Hitchin, Manin)
Framed rank $r$, charge $c$
instantons on $\C P^2$ are in bijective correspondence
with the set of equivalence classes of regular ADHM solutions.
In other words, {\bf \red the moduli of instantons on $\C P^2$
is identified with moduli of the corresponding 
quiver representation.}


\newpage

{\bf\blue The multi-dimensional ADHM construction}

\definition
Let $V$ and $W$ be complex vector spaces, with dimensions
$c$ and $r$, respectively. The {\bf \blue $d$-dimensional ADHM data}
is   maps 
\[ A_k,B_k \in {\rm End}(V),
I_k \in {\rm Hom}(W,V), J_k \in {\rm Hom}(V,W), (k=0,\dots,d)
\]
Choose homogeneous coordinates $[z_0:\dots:z_d]$ on $\C P^d$ and define
\[ \tilde{A} := A_{0}\otimes z_0 + \cdots + A_{d}\otimes
z_d  \ \ \ {\rm and} \ \ \
\tilde{B} := B_{0}\otimes z_0 + \cdots + B_{d}\otimes z_d.
\]
We say that $d$-dimensional ADHM data is \\
{\bf\blue globally regular}, if  
$(\tilde{A}_p, \tilde B_p, \tilde I_p, \tilde J_p)$ 
is regular for every $p\in\C P^d$.
The {\bf\blue $d$-dimensional ADHM equation} is 
$[\tilde A_p,\tilde B_p] + \tilde I_p\tilde J_p =0$,
for all $p\in \C P^d$ 

\theorem
(Marcos Jardim, Igor Frenkel)
Let $C_d(r,c)$ denote the set of globally 
regular solutions of the $d$-dimensional ADHM equation.
Then {\bf \red there exists a 1-1 correspondence between equivalence
classes of globally regular solutions of the
$d$-dimensional  ADHM equations and isomorphism 
classes of rank $r$ instanton bundles} on $\C\mathbb{P}^{d+2}$ 
framed at a fixed line $\ell$, where $\dim W={\rm rk}(E)$
and $\dim V=c_2(E)$.

\newpage

{\bf\blue The multi-dimensional ADHM construction for $d=1$}

For $d=1$, we obtain that the $d$-dimensional ADHM
solutions are families of solutions of ADHM parametrized
by $\C P^3$. Also, the space of 1-dimensional ADHM data is the space
of sections of 
\[ \calo(1)\otimes_\C \bigg[{\rm Hom}(W,V)\oplus{\rm
    Hom}(V,W)
   \oplus {\rm
    End}(V)\oplus {\rm
    End}(V)\bigg]
\]
over $\C P^1$, that is, the twistor space of a 
qquaternionic vector space
$U={\rm Hom}(W,V)\oplus{\rm
    Hom}(V,W)
   \oplus {\rm
    End}(V)\oplus {\rm
    End}(V)$.
Now, the hyperk\"ahler structure on 0-dimensional 
ADHM solutions for each $p\in \C P^1$ 
is compatible with the hyperk\"ahler structure on $U$,
because the space of 0-dimensional 
ADHM solutions is obtained from $U$ by hyperk\"ahler
reduction. {\bf \purple This is used to prove the theorem
about instantons on $\C P^3$ and twistor sections.}



\end{document}

