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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Hypercomplex manifolds with holonomy $SL(n, {\Bbb H})$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf
Workshop
"Special Geometries in Mathematical Physics", \\
April 3, 2008,
K\"uhlungsborn}
\end{center}
\newpage


\newpage

{\bf \blue Hypercomplex manifolds}

{\bf\green DEFINITION:} Let $M$ be a smooth
manifold equipped with endomorphisms
$I, J, K:\; TM\arrow TM$, satisfying the quaternionic relation
$I^2=J^2=K^2=IJK=-\Id.$  Suppose that $I$, $J$, $K$ are
integrable almost complex structures. Then $(M, I, J, K)$ 
is called {\bf \blue a  hypercomplex manifold}.

{\bf \purple
Compact hypercomplex manifolds (examples)}

0. {\red\bf Hyperk\"ahler manifolds.}

1.  {\red\bf In dimension 1 (real dimension 4), 
we have a complete classification,} due to C. P. Boyer (1988)

2. {\red\bf Many homogeneous examples}, due to D. Joyce and physicists 
Ph. Spindel, A. Sevrin, W. Troost, A. Van Proeyen (1980-ies, 
early 1990-ies).

3. {\red\bf Some nilmanifolds} admit homogeneous hypercomplex structure
(M. L. Barberis, I. Dotti, A. Fino) (1990-ies).

4. {\red\bf Some inhomogeneous examples} are constructed by deformation
(G. Grantcharov, H. Pedersen, Y.-S. Poon) or as fiber bundles (V.).


\newpage

    {\bf {}\color{blue}  OBATA CONNECTION}


{\small\bf \red Hypercomplex manifolds can 
be characterized in terms of holonomy}

{\bf \green Theorem:} (M. Obata, 1955) Let $(M,I,J,K)$ be 
a hypercomplex manifold. Then $M$ admits a unique torsion-free
affine connection preserving $I, J, K$.

{\bf \color{blue} Converse is also true}. {
Suppose that $I,J,K$ are operators defining quaternionic structure
on $TM$, and $\nabla$ a torsion-free, affine connection 
preserving $I$, $J$, $K$. Then $I$, $J$, $K$ are integrable
almost complex structures, and $(M,I,J,K)$ is hypercomplex.}

{\bf \purple Holonomy of Obata connection lies in $GL(n, {\Bbb H})$.}
{\em\color[rgb]{0,0.4,0} A manifold equipped with an affine,
torsion-free  connection with holonomy in $GL(n, {\Bbb H})$ 
is hypercomplex}.

{\bf \color{blue} This can be used as a definition of a hypercomplex 
structure}.

\newpage

{\bf \blue Geometry of hypercomplex manifolds}

{\bf \color{red} QUESTIONS}

1. Given a complex manifold $M$, {\bf \purple when $M$ admits a
   hypercomplex structure?} How many?

2. {\bf \purple What are the possible holonomies} of Obata connection,
for a compact hypercomplex manifold? What are the special properties
of reducible holonomies in this case? Irreducicble holonomies
are $GL(n, {\Bbb H})$, $SL(n, {\Bbb H})$ and $Sp(n)$ 
only (Merkulov, Schwachh\"ofer).

3. {\bf \purple Describe the structure of automorphism group} 
of a hypercomplex manifold.

Partial answer to Question 1 is known.

{\bf \color[rgb]{0,0.4,0} THEOREM:} Let $(M,I,J,K)$ be a compact hypercomplex
manifold. Assume that the complex manifold $(M,I)$ admits a 
K\"ahler structure. Then $(M,I)$ is hyperk\"ahler (V., 2004).


\newpage 

{\bf \color{blue} Quaternionic Hermitian structures}

{\bf \green DEFINITION:}
Let $(M,I,J,K)$ be a hypercomplex manifold, and $g$ a Riemannian
metric. We say that $g$ is {\bf \blue quaternionic Hermitian} if $I,J,K$ are
orthogonal with respect to $g$.


Given a quaternionic Hermitian metric $g$ on $(M,I,J,K)$,
consider its Hermitian forms \[ \omega_I(\cdot,\cdot)=g(\cdot,I\cdot),\ \ 
\omega_J=g(\cdot,J\cdot),\ \  \omega_K=g(\cdot,K\cdot)\] 
(real, but {\em \red not closed}).
Then $\Omega=\omega_J+\1\omega_K$ is of Hodge 
type (2,0) with respect to $I$. 

{\red \bf If $d\Omega=0,$
$(M,I,J,K,g)$ is hyperk\"ahler} (this is one of the definitions).

{\bf \color[rgb]{0,0.4,0} 
Consider a weaker condition:}
\[ \6\Omega=0,\  \ \ \6:\; \Lambda^{2,0}(M,I) \arrow \Lambda^{3,0}(M,I)\]

\newpage


{\bf\color{blue}  DEFINITION:} (Howe, Papadopoulos, 1998)\\
Let $(M,I,J,K)$ be a hypercomplex manifold, 
$g$ a quaternionic Hermitian metric, and $\Omega=\omega_J+\1\omega_K$ 
the corresponding $(2,0)$-form. We say that $g$ is {\bf \color{blue} HKT (``weakly hyperk\"ahler
with torsion'')} if $\6\Omega=0.$

{\bf \red
HKT-metrics play in hypercomplex geometry the same role
as K\"ahler metrics play in complex geometry.}

1. {\color[rgb]{0,0.4,0}\bf They admit a smooth potential (locally). }
There is a notion of an ``HKT-class'' (similar to K\"ahler
class) in a certain finite-dimensional 
coholology group. Two metrics in the same
HKT-class differ by a potential, which is a function.

2. When $(M,I)$ has trivial canonical bundle,
{\color[rgb]{0,0.4,0} \bf a version of Hodge theory is established,
giving an ${\goth {sl}}(2)$-action on cohomology.}


\newpage

{\bf \color{blue} 
Canonical bundle of a hypercomplex manifold.}

0. Quaternionic Hermitian structure always exists.

1. {\color[rgb]{0,0.4,0} \bf Complex dimension is even.}

2. {\color[rgb]{0,0.4,0}\bf The canonical line bundle $\Lambda^{n,0}(M,I)$
of $(M,I)$ is always
trivial topologically.} Indeed, a non-degenerate section 
of canonical line bundle is provided
by top power of a form $\Omega$ associated with some
quaternionic Hermitian strucure. In particular, $c_1(M,I)=0.$

3. Canonical bundle {\bf \red is non-trivial holomorphically} in many cases.
However, $\Lambda^{n,0}(M,I)$ is trivial and holonomy of
Obata connection lies in $SL(n,{\Bbb H})$ {\bf \blue when 
$M$ is a nilmanifold } (Barberis-Dotti-V., 2007)

4. If $\Hol(M)$ lies in $SL(n,{\Bbb H})$, canonical bundle
is trivial. The converse is true when $M$ is compact and HKT
(V., 2004): {\bf \red an HKT manifold with holomorphically trivial
canonical bunlde has $\Hol(M)\subset SL(n,{\Bbb H})$}.

\newpage

{\bf\blue $SU(2)$-action on $\Lambda^*(M)$}

The group $SU(2)$ of unitary quaternions acts on
$TM$, because quaternion algebra acts. By multilinearity,
this action is extended to $\Lambda^*(M)$.

{\bf \green 1. The Hodge decomposition 
$\Lambda^*(M)=\bigoplus\Lambda^{p,q}(M)$} is recovered from this
$SU(2)$-action. {\bf \purple 
``Hypercomplex analogue of the Hodge decomposition''.}

{\bf \green 2. $\langle \omega_I, \omega_J, \omega_K\rangle$}
is an irreducible 3-dimensional representation of $SU(2)$,
for any quaternionic Hermitian structure (``representation
of weight 2'').


{\bf \red WEIGHT of a representation.}\\
We say that an irreducible $SU(2)$-representation
$W$ {\bf \green has weight $i$} if $\dim W=i+1$.
A representation is said to be {\bf \green pure of weight $i$}
if all its irreducible components have weight $i$.
If all irreducible components of a representation $W_1$
have weight $\leq i$, we say that $W_1$ {\bf\green is a
  representation of weight $\leq i$}.
In a similar fashion one defines representations
of weight $\geq i$.

\newpage

{\bf \blue Quaternionic Dolbeault algebra}

{\bf \green The weight is multiplicative,} in the
following sense: a tensor product of representations of
weights $\leq i$ and $\leq j$ has weight
$\leq i+j$. 

Clearly, $\Lambda^1(M)$ has weight 1. Therefore, 
{\bf\green  $\Lambda^i(M)$ has weight $\leq i$.}

Let $V^i\subset \Lambda^i(M)$ be the maximal
$SU(2)$-invariant subspace of weight $<i$.

By multiplicativity, {\bf \red $V^*= \bigoplus_i V^i$
is an ideal in $\Lambda^*(M)$}. We also have
$V^i = \Lambda^i(M)$ for $i> 2n$. Also, 
$dV^i\subset V^{i+1}$, hence $V^*\subset \Lambda^*(M)$
is a differential ideal in $(\Lambda^i(M), d)$.

{\bf \blue Denote by $(\Lambda^*_+(M), d_+)$ the quotient algebra
$\Lambda^*(M)/V^*$.} \\We call it 
{\bf\green the quaternionic Dolbeault algebra (qD-algebra) } of $M$.

A similar construction was given by Salamon in a more general situation.

\newpage

{\blue \bf The Hodge decomposition of quaternionic Dolbeault algebra.}

The Hodge decomposition is induced from the $SU(2)$-action,
hence it is \\{\bf \red compatible with weights}: 
$\Lambda^i_+(M)=\bigoplus_{p+q=i}\Lambda^{p,q}_{+,I}(M)$.

Let $\1{\cal  I}$ be an element of the Lie algebra ${\goth {su}}(2)\otimes \C$
acting as $\1(p-q)$ on $\Lambda^{p,q}(M)$. This vector generates
the Cartan algebra of ${\goth {su}}(2)$. The ${\goth {su}}(2)$-
action induces an isomorphism of $\Lambda^{p,q}_{+,I}(M)$
for all $\{ p, q \ | \ p+q =k, \ p, q \geq 0\}$. 
This gives

{\bf \green Theorem:} \ \ \ $\Lambda^{p,q}_{+,I}(M)\cong \Lambda^{0,p+q}(M,I).$

This isomorphism is provided by the 
${\goth {su}}(2)\otimes \C$-action.

\newpage

{\bf \blue Differentials in the qD-complex}


 We extend $J:\; \Lambda^1(M) \arrow \Lambda^1(M)$
to $\Lambda^*(M)$ by multiplicativity. Since
$I$ and $J$ anticommute on $\Lambda^1(M)$, {\bf \red we have
$J(\Lambda^{p,q}(M,I))=\Lambda^{q,p}(M,I).$} 

{\bf\green Denote by 
$\6_J:\;  \Lambda^{p,0}(M,I)\arrow \Lambda^{p,0}(M,I)$
the operator $J\circ \bar \6 \circ J$,} where
$\bar\6:\;  \Lambda^{0,p}(M,I)\arrow \Lambda^{0,p}(M,I)$
is the standard Dolbeault differential. Then $\6$, $\6_J$
anticommute. Moreover, 
{\bf \red there exists a multiplicative isomorphism
of bicomplexes}.


\begin{equation*}\label{_bicomple_XY_isomo_Equation}
\begin{minipage}[m]{0.85\linewidth}
{ \small $
\xymatrix @C+10mm @R+25mm@!0  { 
%
  && \Lambda^0_+(M) \ar[dl]^{d^{1,0}_+} \ar[dr]^{d^{0,1}_+} 
   && && && \Lambda^{0,0}_I(M) \ar[dl]^{\6_J} \ar[dr]^{ \6}
   &&  \\
%
 & \Lambda^{1,0}_+(M) \ar[dl]^{d^{1,0}_+} \ar[dr]^{d^{0,1}_+} &
 & \Lambda^{0,1}_+(M) \ar[dl]^{d^{1,0}_+} \ar[dr]^{d^{0,1}_+}&& 
\text{\large \ \ \ \ $\cong$\ \ \ \ } &
 &\Lambda^{1,0}_I(M)\ar[dl]^{ \6_J} \ar[dr]^{ \6}&  &
 \Lambda^{1,0}_I(M)\ar[dl]^{ \6_J} \ar[dr]^{ \6}&\\
%
 \Lambda^{2,0}_+(M) && \Lambda^{1,1}_+(M) 
   && \Lambda^{0,2}_+(M)& \ \ \ \ \ \ & \Lambda^{2,0}_I(M)& & 
\Lambda^{2,0}_I(M) & & \Lambda^{2,0}_I(M) \\
%
}
$
}
\end{minipage}
\end{equation*}
\centerline {...  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ...\ \ \ \ \ \ \ \ \ \ \ \ \ \ \  }

\newpage

{\bf\blue Potentials for HKT-metrics}

A quaternionic Hermitian metric {\green \bf can be recovered from
the corresponding $(2,0)$-form:} $\omega_I(x,\bar y) = 
\frac{1}{2}\Omega(x, J(\bar y))$, where $x, y \in T^{1,0}(M)$.
The HKT-structures uniquely correspond to $(2,0)$-forms which are

{\bf \green 1. Real:} $J(\Omega)=\bar\Omega$

{\bf \green 2. Closed:} $\6\Omega=0$.

{\bf \green 2. Positive:} $\Omega(x, J(\bar x))>0$,
for any non-zero $x\in T^{1,0}(M)$

{\bf \blue Locally, any HKT-metric is given by a potential:}
$\Omega = \6\6_J \phi$ where $\phi$ is a smooth function.

{\bf \purple Any convex, and any 
strictly plurisubharmonic function
is a potential of some HKT-structure}. Therefore,
HKT-structures locally always  exist (Grantcharov, Poon).

\newpage

{\bf\blue HKT-manifolds with holonomy in $SL(n,{\Bbb H})$}

Let $M$ be a compact 
HKT-manifold with holonomy in $SL(n,{\Bbb H})$,
and $\Delta_{\bar\6}:= \bar\6\bar\6^* + \bar\6^*\bar\6$ be the 
{\bf \blue antiholomorphic Laplacian} with 
\[ \ker \Delta_{\bar\6}\restrict{\Lambda^{0,*}(M)}=H^*(M, \calo_{(M,I)}).
\]

{\bf \green Theorem:}  $\Delta_{\bar\6}$ commutes with the
multiplication by the HKT-form $\bar\Omega$, and with the operator
$\eta \arrow J(\bar\eta)$. In particular, {\bf \red there is a Lefschetz-like
${\goth sl}(2)$-action on $H^*(M, \calo_{(M,I)})$.}

{\bf \green Theorem (``$\6\6_J$-lemma'')} Let $\Omega, \Omega'$ be HKT-forms on a
compact  HKT-manifold with holonomy in $SL(n,{\Bbb H})$.
Assume that the cohomology classes of $\bar\Omega, \bar\Omega'$ in 
$H^*(M, \calo_{(M,I)})$ are equal. Then $\Omega-\Omega'=\6\6_J\phi$
for some smooth function $\phi$ on $M$.

{\bf \green Example:} Any hypercomplex nilmanifold has holonomy
in $SL(n,{\Bbb H})$ (Barberis, Dotti, V.).

\newpage 

{\bf\blue Quaternionic Monge-Ampere equation}

Let $M$ be an HKT-manifold with holonomy in $SL(n, {\Bbb H})$.
{\green (this is equivalent to having trivial canonical bundle).}
Then the canonical bundle is trivialized by a 
form $\Phi_I\in \Lambda^{2n,0}$, non-degenerate, closed and satisfying
$J(\Phi_I)=\bar \Phi_I$. 

{\bf \red Quaternionic Monge-Ampere equation:
\[
(\Omega+\6\6_J\phi)^n=A_f e^f \Phi_I   \ \ \ (*)
\]
where $\Omega+\6\6_J\phi$ is an HKT-form.} Here
$\phi$ is unknown, and $A_f$ is a number determined from 
\[
\int_M \Omega^n\wedge \bar \Phi_I = A_f \int_M e^f \Phi_I\wedge \bar \Phi_I
\]

{\bf \green Theorem: (Alesker, V., 2008)} {\bf \purple The solution $\phi$ of 
(*) is unique, if exists.} Moreover, any solution of (*) admits 
a $C^0$-estimation in terms of $f, \Phi_I, \Omega$.

{\bf \green Conjecture:} (``hypercomplex Calabi-Yau'')\\
{\bf \red The equation (*) has a solution for all $f, \Phi_I, \Omega$.}


\newpage

{\bf \blue Uniqueness of solutions of Monge-Ampere equations}

Suppose $\Omega_1, \Omega_2$ are HKT-forms which are solutions of M-A,
$\Omega_1-\Omega_2= \6\6_J\phi$.
Then $\Omega_1^n -\Omega_2^n=0$. This gives
\[
0=\Omega_1^n -\Omega_2^n= \6\6_J\phi\wedge \sum_{i=0}^{n-1}\Omega_1^i\wedge\Omega_2^{n-1-i}.
\]
Denote by $P$ the form $\sum_{i=0}^{n-1}\Omega_1^i\wedge\Omega_2^{n-1-i}$
and consider the differential operator $D:\; C^\infty(M)\arrow C^\infty(M)$
\[ 
  \phi \arrow \frac{\6\6_J\phi\wedge P}{\Omega^n}.
\]
{\bf \red Then $D$ is a second order operator 
with positive symbol.} 

{\bf\purple Solutions of
$D(f)=0$  cannot have local maxima} (``generalized maximum principle'').
Since $M$ is compact, {\bf \green all solutions of $D(f)=0$ are constant.}

\newpage

{\bf \blue Calabi-Yau HKT manifolds}

{\bf \green Definition:} Let $(M,I,J,K, \Omega)$ be an HKT-manifold with
holonomy in $SL(n, {\Bbb H})$, and $\Phi_I\in \Lambda^{n,0}_I(M)$
the parallel section of the canonical class. We say that
$M$ is {\bf\blue  a Calabi-Yau HKT manifold} if $\Omega^n=\Phi_I$.

{\bf \green Example:} Let $G$ be a nilpotent Lie group
with a left-invariant hypercomplex HKT-structure.
Then $\Hol(G)\subset SL(n, {\Bbb H})$ (Barberis, Dotti, V.).
Since the forms $\Phi_I$ and $\Omega$ are $G$-invariant,
the quotient $\frac{\Omega^n}{\Phi_I}$ is constant.
Rescaling, we obtain that {\bf\blue all HKT-nilmanifolds are
Calabi-Yau HKT }.

{\bf\green Claim:} Let $(M,I,J,K, g)$ be a Calabi-Yau HKT-manifold.
{\bf \red Then $(M,J,g)$ is balanced}, that is, $d(\omega^{2n-1}_J)=0$.

% Taken from extarrows.sty by Chim Cut Canh
\makeatletter
\def\x@arrow{\DOTSB\Relbar}
\def\xlongequalsignfill@{\arrowfill@\x@arrow\Relbar\x@arrow}
\newcommand{\xlongequal}[2][]{%
\ext@arrow 0099\xlongequalsignfill@{#1}{#2}} \makeatother

{\bf \green Proof:} $\omega^{2n-1}_J= \Re(\Omega^{n-1}\wedge \Omega^n)$.
However,
\begin{align*}
d(\Omega^{n-1}\wedge \bar\Omega^n) &
\xlongequal{\text{Hodge decomposition}} \6(\Omega^{n-1}\wedge \bar\Omega^n)\\=&
(n-1)\6\Omega\wedge \Omega^{n-2}\wedge \bar\Omega^n +\Omega^{n-1}\wedge \6(\bar\Omega^n)
\end{align*}
The first term vanishes because $\Omega$ is HKT, the second because
$\bar\Omega^n$ is closed. \endproof


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