\documentclass{slides} \usepackage{amssymb, amsmath, amscd, color, epsfig} %\usepackage[matrix,arrow]{xy} \newcommand{\green}{\color[rgb]{0,0.4,0}} \newcommand{\purple}{\color[rgb]{0.4,0,0.4}} \newcommand{\red}{\color[rgb]{0.7,0,0}} \newcommand{\blue}{\color{blue}} \def\eqref#1{(\ref{#1})} \newcommand{\goth}{\mathfrak} \newcommand{\g}{{\frak g}} \newcommand{\arrow}{{\:\longrightarrow\:}} \newcommand{\Z}{{\Bbb Z}} \newcommand{\C}{{\Bbb C}} \newcommand{\R}{{\Bbb R}} \newcommand{\Q}{{\Bbb Q}} \renewcommand{\H}{{\Bbb H}} \newcommand{\6}{\partial} \def\1{\sqrt{-1}\:} \newcommand{\restrict}[1]{{\left|_{{#1}}\right.}} \newcommand{\cntrct} % contraction with a vector field {\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}} \def\Bbb#1{\mathbb #1} \newcommand{\calo}{{\cal O}} \newcommand{\cac}{{\cal C}} % Correcting TeX... %\let\oldtilde=\tilde %\renewcommand{\tilde}{\widetilde} \renewcommand{\bar}{\overline} \renewcommand{\phi}{\varphi} \renewcommand{\epsilon}{\varepsilon} \renewcommand{\geq}{\geqslant} \renewcommand{\leq}{\leqslant} % Operatornames \newcommand{\even}{{\rm even}} \newcommand{\odd}{{\rm odd}} \newcommand{\const}{{\it const}} \newcommand{\fl}{{\rm fl}} \newcommand{\im}{\operatorname{im}} \newcommand{\End}{\operatorname{End}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\Hol}{\operatorname{{\cal H}ol}} \newcommand{\Tot}{\operatorname{Tot}} \newcommand{\Id}{\operatorname{Id}} \newcommand{\id}{\operatorname{\text{\sf id}}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\Mat}{\operatorname{Mat}} \newcommand{\Alt}{\operatorname{Alt}} \newcommand{\Iso}{\operatorname{Iso}} \newcommand{\Sec}{\operatorname{Sec}} \newcommand{\Can}{\operatorname{Can}} \newcommand{\Sing}{\operatorname{Sing}} \newcommand{\Spin}{\operatorname{Spin}} \newcommand{\codim}{\operatorname{codim}} \newcommand{\coim}{\operatorname{coim}} \newcommand{\coker}{\operatorname{coker}} \newcommand{\slope}{\operatorname{slope}} \newcommand{\rk}{\operatorname{rk}} \newcommand{\Def}{\operatorname{Def}} \newcommand{\Lie}{\operatorname{Lie}} \newcommand{\Tw}{\operatorname{Tw}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\Diff}{\operatorname{Diff}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\inbfpare}[1]{{% \mbox{\tt (}\hspace{-5pt}\mbox{\tt (} #1 % \mbox{\tt )}\hspace{-5pt}\mbox{\tt )}% }} \newcommand{\comment}[1]{{}} \def\blacksquare{\hbox{\vrule width 10pt height 10pt depth 0pt}} \def\endproof{\blacksquare} \def\shortdash{\mbox{\vrule width 4.5pt height 0.55ex depth -0.5ex}} \makeatletter %\@ifundefined{Bbb} % {\newcommand{\Bbb}[1]{{\mathbb #1}}}% %{}% {\edef\Bbb#1{{\Bbb #1}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Pagestyle % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\ps@verbit}{% \renewcommand{\@oddhead}{% \scriptsize {\it \small Trisymplectic manifolds \hfil \tiny M. Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \setlength\paperheight {9.5in}% \setlength\paperwidth {13.0in} \setlength{\textwidth}{0.8\paperwidth} \setlength{\textheight}{0.8\paperheight} \setlength{\pdfpageheight}{\paperheight} \setlength{\pdfpagewidth}{\paperwidth} \addtolength{\topmargin}{-20mm} \addtolength{\leftmargin}{-25mm} \addtolength{\rightmargin}{-25mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Lemma, sublemma, corollary, proposition, theorem, % % definition,example defined there: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcounter{section} \newcounter{Mycounter}[section] \newcounter{lemma}[section] \setcounter{lemma}{0} \renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}} \newcommand{\lemma}{% \setcounter{lemma}{\value{Mycounter}} \refstepcounter{lemma} \stepcounter{Mycounter} {\bf \green LEMMA:\ }} \newcounter{claim}[section] \setcounter{claim}{0} \renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}} \newcommand{\claim}{% \setcounter{claim}{\value{Mycounter}} \refstepcounter{claim} \stepcounter{Mycounter} {\bf \green CLAIM:\ }} \newcounter{corollary}[section] \setcounter{corollary}{0} \renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}} \newcommand{\corollary}{% \setcounter{corollary}{\value{Mycounter}} \refstepcounter{corollary} \stepcounter{Mycounter} {\bf \green COROLLARY:\ }} \newcounter{theorem}[section] \setcounter{theorem}{0} \renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}} \newcommand{\theorem}{% \setcounter{theorem}{\value{Mycounter}} \refstepcounter{theorem} \stepcounter{Mycounter} {\bf \green THEOREM:\ }} \newcounter{conjecture}[section] \setcounter{conjecture}{0} \renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}} \newcommand{\conjecture}{% \setcounter{conjecture}{\value{Mycounter}} \refstepcounter{conjecture} \stepcounter{Mycounter} {\bf \green CONJECTURE:\ }} \newcounter{proposition}[section] \setcounter{proposition}{0} \renewcommand{\theproposition} {\noindent{Proposition \thesection.\arabic{proposition}}} \newcommand{\proposition}{% \setcounter{proposition}{\value{Mycounter}} \refstepcounter{proposition} \stepcounter{Mycounter} {\bf \green PROPOSITION:\ }} \newcounter{definition}[section] \setcounter{definition}{0} \renewcommand{\thedefinition} {\noindent{Definition~\thesection.\arabic{definition}}} \newcommand{\definition}{% \setcounter{definition}{\value{Mycounter}} \refstepcounter{definition} \stepcounter{Mycounter} {\bf \green DEFINITION:\ }} \newcounter{example}[section] \setcounter{example}{0} \renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}} \newcommand{\example}{% \setcounter{example}{\value{Mycounter}} \refstepcounter{example} \stepcounter{Mycounter} {\bf \green EXAMPLE:\ }} \newcounter{remark}[section] \setcounter{remark}{0} \renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}} \newcommand{\remark}{% \setcounter{remark}{\value{Mycounter}} \refstepcounter{remark} \stepcounter{Mycounter} {\bf \green REMARK:\ }} \newcounter{observation}[section] \setcounter{observation}{0} \renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcounter{question}[section] \setcounter{question}{0} \renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}} \newcommand{\question}{% \setcounter{question}{\value{Mycounter}} \refstepcounter{question} \stepcounter{Mycounter} {\bf \green QUESTION:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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}