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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Hodge Theory  On Nearly \\[4mm] K\"ahler Manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\tiny March 17, 2006, K\"uhlungsborn}
\\[14mm]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\color{blue}Nearly  K\"ahler Manifolds: an introduction}
\label{_Nije_non-dege_Subsection_}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{center}
\newpage


{\color{red}
Classification of holonomies.}

{\small
\begin{tabular}{|l|l|}
\hline
\multicolumn{2}{|c|}{\bf \color[rgb]{0,0,0.6}Berger's list}\\[1mm]
\hline
\it Holonomy  & \it Geometry\\[1mm]
\hline
$SO(n)$ acting on $\R^n$ & Riemannian manifolds\\[1mm]
\hline
$U(n)$ acting on $\R^{2n}$ & K\"ahler manifolds\\[1mm]
\hline
$SU(n)$ acting on $\R^{2n}$, $n>2$ & Calabi-Yau manifolds\\[1mm]
\hline
$Sp(n)$ acting on $\R^{4n}$ & hyperk\"ahler manifolds\\[1mm]
\hline
$Sp(n)\times Sp(1)/\{\pm 1\}$ & 
quaternionic-K\"ahler\\[1mm] acting on $\R^{4n}$, $n>1$ &  manifolds\\[1mm]
\hline
$G_2$ acting on $\R^7$ & $G_2$-manifolds \\[1mm]
\hline
$Spin(7)$ acting on $\R^8$ & $Spin(7)$-manifolds\\[1mm]
\hline
\end{tabular}
}

{\color{blue} \it ``Hodge theory on manifolds with special
  holonomy''}

{
1. $\Lambda^*(M)= \oplus \Lambda^*_\epsilon(M)$
($\epsilon$ - weights of representations of holonomy).
Then $\cal H^*(M)= \oplus \cal H^*_\epsilon(M)$
{\color[rgb]{.4,0,0}(Chern)}.

2. $dd^c$-lemma, implying restrictions on topology
{\color[rgb]{.4,0,0}(``formality'': Deligne-Griffiths-Morgan-Sullivan)}.

3. Solutions of Maurer-Cartan equation $\bar\6\gamma =
   \frac1 2 \{\gamma,\gamma\}$ 
{\color[rgb]{.4,0,0}(Kodaira-Spencer, Bogomolov-Tian-Todorov)}.
}

\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\Large Holonomy of Riemannian cones}
\label{_Nije_non-dege_Subsection_}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\definition 
Let $(M, g)$ be a Riemannian manifold. The {\bf Riemannian cone}
of $M$ is \[C(M):=(M\times \R^{>0}, t^2g+ dt^2),\] where $t$ denotes
the coordinate on the half-line $\R^{>0}$.


{\color{blue}
Conical singularities of manifolds with special holonomy
give special holonomy on Riemannian cones.}

Suppose $C(M)$ has special holonomy. What can we say about
geometry of $M$?

{\small
\begin{tabular}{|l|l|l|}
\hline
\multicolumn{3}{|c|}{\bf \color{red} Riemannian cones with
special holonomy}\\[1mm]
\hline
\it \color{blue} Holonomy of $C(M)$  &\it\color{blue} Geometry of $C(M)$  & \it\color{blue} Geometry of $M$\\[1mm]
\hline
$SO(n)$ &  Riemannian & --- \\[1mm]
\hline
$U(n)$ &  K\"ahler & Sasakian\\[1mm]
\hline
$SU(n)$  & Calabi-Yau &
Sasaki-Einstein \\[1mm]
\hline
$Sp(n)$ &  hyperk\"ahler & 3-Sasakian\\[1mm]
\hline
$Sp(n) Sp(1)$ & 
quaternionic-K\"ahler & --- \\[1mm]
\hline
$G_2$  & $G_2$-manifolds & nearly K\"ahler \\[1mm]
\hline
$Spin(7)$  & $Spin(7)$-manifolds & nearly $G_2$-manifolds \\[1mm]
\hline
\end{tabular}
}\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\Large \it Killing spinors on Riemannian cones}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\it \color{blue}\small Not essential for understanding of today's talk,
because the spinor interpretation will not be used}


Recall that we have a ``Clifford multiplication map''
$TM\otimes \goth S \arrow \goth S$, where $TM$ is a bundle of
tangent vectors on a manifold $M$, and $\goth S$ the bundle
of spinors. 

``Killing spinor'' on $M$ is $\Psi\in \goth S$ which
satisfies \[ \nabla_X(\Psi)= \lambda X\Psi\] for all
tangent fields $X\in TM$.


{\bf \color{blue} Fact 1:} Killing spinors on $M$ 
correspong uniquely to parallel spinors on $C(M)$.

{\bf \color{blue} Fact 2:} Killing spinors on $M$ exist
only if $M$ is an Einstein manifold, with 
Einstein constant $|\lambda|^2\geq 0$.

{\bf \hspace{0pt}\color{blue}Remark:} Similarly, if $M$ admits a {\it parallel}
spinor, $M$ is Ricci-Flat  (follows from Weitzenb\"ock formula).

{\bf \color{blue}Remark:} In Berger's list, the following
holonomies correspond to Ricci-flat manifolds:
$SU(n)$, $Sp(n)$, $G_2$, $\Spin(7)$.

{\bf \color{blue} Fact 3:} $SU(n)$, $Sp(n)$, $G_2$, $\Spin(7)$
admit parallel spinors.

{\bf \color{red} COROLLARY:} Sasaki-Einstein,
3-Sasakian, nearly K\"ahler and nearly $G_2$-manifolds
admit Killing spinors; hence Einstein.

{\bf Proof:} Their cones admit a parallel spinor.
\newpage

{\bf \Large Nearly K\"ahler manifolds}\\

{\it\color{red}
The name is confusing.
}

{\bf \color{blue} The original definition:}
(Alfred Gray). Let $(M, g, I)$ be a Hermitian almost
complex manifold, $\omega\in \Lambda^{1,1}(M)$ its Hermitian form,
$\nabla$ the Levi-Civita connection. Then $\nabla\omega$
lies in $\Lambda^1(M)\otimes \Lambda^2(M)$.
Gray defined {\color{blue}\it ``nearly K\"ahler manifolds''} as those
that satisfy
\[
\nabla\omega \in \Lambda^3(M)\subset \Lambda^1(M)\otimes \Lambda^2(M)
\]
{\color[rgb]{0.6,0,0}($\nabla\omega$ is skew-symmetric).}

{\bf \color{blue}Trivial remark:}
In this case $d\omega= \nabla\omega$, because $\nabla$ is
torsion-free.

\newpage

{\color{blue}\it ``Strictly nearly K\"ahler''} means that
the 3-form $\rho=d\omega$ is non-degenerate, that is, the map
\[ TM\stackrel\rho\arrow \Lambda^2M, 
\] defined as $X\arrow \rho(X, \cdot, \cdot)$, is 
injective. It is {\color{red} \it much 
more restrictive condition} than the K\"ahler condition
$d\omega=0$. There is a {\it ``splitting theorem''}
in nearly K\"ahler geometry (due to P.-A. Nagy).
It follows that strictly nearly K\"ahler manifolds
are products of Einstein ones, hence they are 
real analytic and have finite-dimensional moduli,
if compact. 


{\large \color[rgb]{0,0,0.5} 
\bf Some examples}

1. 6-manifolds with parallel $G_2$ cones.

2. Twistor spaces of positive quaternionic-K\"ahler manifolds 
with non-standard complex structure due to Eels and Salamon.


\newpage

{\large \color[rgb]{0,0,0.5} \bf Connections with totally antisymmetric torsion}

Let $\nabla_0:\;\Lambda^1(M) \arrow \Lambda^1(M)\otimes \Lambda^1(M)$ 
be an orthogonal connection on a Riemannian manifold. Its  torsion $T$
lies in
\[
T\in \Lambda^1(M)^* \otimes \goth{so}(M)=\Lambda^1(M)^* \otimes  \Lambda^2(M).
\]
Using the metric, we identify $\Lambda^1(M)^*$ with
$\Lambda^1(M)$ and consider $T$ as a 3-form. 

{\bf Definition:} $\nabla_0$ is a 
{\it \color{blue} connection with totally anti-symmetric torsion}
if $T$ is totaly anti-symmetric.

\newpage

{\large \color[rgb]{0,0,0.5} 
\bf Connection with totally antisymmetric torsion
on a nearly K\"ahler manifold}


Let $M$ be a nearly K\"ahler manifold (in the sense of Gray),
$\rho=d\omega=\nabla\omega$ the corresponding
3-form. Consider the operator 
\[
\theta:\; \Lambda^1(M) \arrow \Lambda^2(M)\subset 
 \Lambda^1(M)\otimes \Lambda^1(M).
\]
mapping $\xi$ to $\rho(\xi^\sharp, \cdot, \cdot)$, where
$\xi^\sharp$ is the dual vector field. Let $\nabla_T$ be a 
new connection.
\[
  \nabla_T:= \nabla+ \frac12\theta:\; 
  \Lambda^1(M) \arrow \Lambda^1(M)\otimes \Lambda^1(M).
\]
{\bf  \color{blue} A trivial  observation.}\\
$\nabla_T$ preserves the Hermitian structure. 

{\bf \color{blue} Another trivial observation.}\\
The torsion of $\nabla_T$ is $\rho$.\\
\\
{\it \color{red} This gives...}

{\bf \  \!\!\! \!\! \color{blue} A simple theorem:}
Let $M$ be a nearly K\"ahler manifold (in the sense of Gray).
Then $\nabla_T$ is a Hermitian connection with totally 
antisymmetric torsion.

{\bf\color{blue} A difficult theorem}
(V. Kirichenko). \\
On nearly K\"ahler manifolds,
\[
\nabla_T(T)=0.
\]
{\it \small\color{red} The torsion is parallel.}


This is used to obtain a splitting theorem.

\newpage


{\Large \color[rgb]{0,0,0.5} 
\bf Manifolds with parallel \\[4mm]antisymmetric
torsion}.

Let $M$ be a Riemannian manifold, $\nabla_T$ a connection
with totally antisymmetric parallel torsion $T$. Assume its
local holonomy is irreducible.


{\bf \color{blue}  THEOREM:}
(R. Cleyton and A. Swann, 2002) \\ Any such 
manifold is either locally homogeneous, has vanishing torsion,
is nearly $G_2$ (in dimension 7) or nearly K\"ahler 
(in dimension 6).


{\bf \color{blue} Remark:} The last two cases are ones we have
seen in the classification of cones with special holonomy.

{\color{red}\it \small
This is an antisymmetric torsion analogue of Berger's theorem
on irreducible holonomies.}

\newpage

This is used to obtain splitting for nearly K\"ahler manifolds.

{\bf \color{blue}COROLLARY:} (P.-A. Nagy, 2002)\\
Let $M$ be a nearly K\"ahler manifold, in the sense of Gray.
Then $M$ is locally a product of the following nearly
K\"ahler types.\\
{\it \small \color[rgb]{0,.5,0}
1.  Homogeneous (classified by J.B. Butruille in 2004)\\
2. Twistor spaces of positive quaternionic-K\"ahler manifolds\\
3. 6-dimensional nearly K\"ahler
}

{\color{blue} Remark:} The positive quaternionic-K\"ahler manifolds
and their twistors are (conjecturally) symmetric. Hence the
only interesting example of ``nearly K\"ahler'' is 6-dimensional
nearly K\"ahler manifolds.

{\color[rgb]{.7,0,0} \it \small In modern literature,
``nearly K\"ahler'' usually denotes a 6-dimensional Hermitian manifold
with $\nabla\omega$ antisymmetric. We shall always assume ``6-dimensional''.}


{\bf \Large \hspace{1pt}\color[rgb]{0,0,0.7}Nearly K\"ahler manifolds}\\
{\it\small\color[rgb]{0,0.4,0} The many definitions of
  NK-manifolds}\\[2mm]
{\bf \color[rgb]{0,0,0.7} ``A well-known theorem:''} \\
Let $(M, I, \omega)$ be a Hermitian almost complex 6-manifold.
Then the following conditions are equivalent.

{\it \small \color[rgb]{0,.3,0}
{\bf 1.}  
The form $\nabla\omega\in \Lambda^1(M)\otimes \Lambda^2(M)$
is non-zero and totally skew-symmetric (that is, $\nabla\omega$ is a 
3-form).

{\bf 2.}  The structure group of 
$M$ admits a reduction to $SU(3)$, that is,
there is $(3,0)$-form $\Omega$ with $|\Omega|=1$, and 
\[ 
d\omega =  3\lambda \Re\Omega,\ \ \ 
d \Im \Omega =   -2 \lambda\omega^2
\]
where $\lambda$ is a non-zero real constant.
}\\[2mm]
{\bf\hspace{0pt}\color[rgb]{0,0,0.7}Another well-known theorem:} \\
Let $M$ be a Riemannian 6-manifold. Then the following 
conditions are equivalent.

{\it \small \color[rgb]{0,.3,0}
{\bf 1.}  
$M$ admits a nearly K\"ahler Hermitian structure.

{\bf 2.}   $M$ admits a  Killing spinor.

{\bf 3.} The Riemannian cone $C(M)$ has holonomy
  $G_2$.}



{\bf \hspace{0pt}\color[rgb]{0,0,0.7} Remark:} Let $M$ be nearly K\"ahler.
Unless $C(M)$ is flat, and $M$ is $S^6$,
the almost complex structure is uniquely determined 
by the metric (Friedrich). Conversely, the metric is 
uniquely determined by the almost complex structure.



{\hspace{1pt}\large\color[rgb]{0,0,0.7} 
\bf Examples of nearly K\"ahler manifolds \\[1mm](all four of them)}

{\bf \it 1.} The sphere $S^6$. Its cone is $\R^7$.

{\bf \it 2 and 3.} \\$\C P^3$ and the flag variety $F(2,1)$.
These are twistor spaces for self-dual Einstein
manifolds $S^4$ and $\C P^2$,
we take the Eels-Salamon almost complex structure.

{\bf \it 4.} $S^3\times S^3$.

{\color{red}\it
No non-homogeneous compact examples (so far).}

\newpage 

{\bf \hspace{0pt}\color{blue} \Large Geometry of NK-manifolds}

{\bf \color{blue} A trivial remark:}
An NK-manifold is never integrable. 
Indeed, $d\omega^{1,1}=3\lambda \Re\Omega^{3,0}$.
In fact, the Nijenhuis tensor
\[ 
N:\; \Lambda^{0,1}(M)\arrow \Lambda^{2,0}(M)
\]
is {\it \color{red} invertible} (unless $\lambda=0$).


{\bf \color{blue} Another trivial remark:}
If $\lambda=0$, the NK-equations {\it \color{red}  degenerate to
equations defining Calabi-Yau.}

Let 
\[
d = d^{2,-1}+ d^{1,0} + d^{0,1}+ d^{-1,2},
\]
be the Hodge decomposition of de Rham differential.
Clearly, {\it \color{red} $d^{2,0}$ is the Nijenhuis tensor.}

{\color{blue} Notation:} We use
{\color[rgb]{0,0.4,0} \[ d^{2,-1}=: N, \ \ d^{-1, 2}=: \bar N,
   \ \ d^{1,0}=: \6, \ \ d^{0,1}=: \bar \6.
\]}


\newpage 

{\bf \hspace{0pt}\color{blue} \large K\"ahler identities on NK-manifolds}

The usual K\"ahler identities take a form
{\it \color[rgb]{0.7,0,0}``a commutator of some Hodge component of de Rham
differential with the Hodge operator $\Lambda$
is proportional to a Hermitian adjoint of some
other Hodge component of de Rham
differential''.}

On nearly K\"ahler, we have the same set of identities,
{\color{blue} \[
[\Lambda_\omega, \6] = \1 \bar\6^*, \ \  
[\Lambda_\omega, \bar\6] = -\1\6^*
\]}
plus
{\color{blue}\[
[\Lambda_\omega, N] = \1 2 \bar N^*, \ \  
[\Lambda_\omega, \bar N] = -\1 2N^*
\]}
In addition, we have
{\color{blue}\[ [L_\Omega, \Lambda_\omega]=\lambda N.
\]}
{\color[rgb]{0,.3,0} \it \small The proof is similar to the one used
  to obtain the usual K\"ahler identities.}

\newpage


{\bf \hspace{0pt}\color{blue} \large Laplacian operators on NK-manifolds}

{\color{blue} Notation:}
For an operator $v$ on a Hermitian space, let
$\Delta_v$ denote $v v^* + v^* v$. 


{\bf \color{blue}\small THEOREM:} \\
On a nearly K\"ahler manifold, we have
{\color[rgb]{0,0,0.6} 
\[ 
   \Delta_\6-\Delta_{\bar\6}= R
\]}
where $R$ is a scalar operator, acting on $(p,q)$-forms
as a multiplication with $\lambda^2 (p-q) (3-p-q)$.


{\bf \color{blue}\small THEOREM:} \\ 
On a nearly K\"ahler manifold, we have
{\color[rgb]{0,0,0.6}
\[  \Delta_d-\Delta_{\6-\bar\6}=  \Delta_N + \Delta_{\bar
   N}=\Delta_{N+\bar N}
\] 
}

{\color{blue} \it Compare:} On a K\"ahler manifold
{\color[rgb]{0,0,0.6}\[
2\Delta_\6=2\Delta_{\bar\6}=\Delta_d=\Delta_{\6-\bar\6}
\]}
{\color[rgb]{0,.3,0} \it \small The relations between
Laplacians follow from the K\"ahler identities in a standard way.}


\newpage

{\bf \hspace{0pt}\color{blue} \large Hodge theory on NK-manifolds}

{\it \small \color{red} Use the comparison formulas for Laplacians to obtain
a Hodge decomposition theorem.}


{\bf \color{blue}THEOREM:} 
Let $M$ be a compact nearly K\"ahler manifold,
and $\eta=\oplus \eta^{p,q}$ a differential form. 
Then $\eta$ is harmonic if and only if all its Hodge
components $\eta^{p,q}$ are harmonic and primitive.

{\bf \color{blue} Proof:}
Let $\eta\in \ker \Delta_v$ (for some $v$).
Then
\[ 
   0=(\Delta_v\eta, \eta) =(v v^* \eta + v^* v\eta, \eta) =
   (v\eta, v\eta) + (v^*\eta, v^*\eta).
\]
In other words, {\color[rgb]{0.6,0,0} \it $\eta\in \ker \Delta_v$
if and only if $\eta \in \ker v \cap \ker v^*,$
and $(\Delta_v\eta, \eta)>0$ otherwise.}

From the second comparison formula, the same 
positivity argument gives
\[ 0= (\Delta_d\eta,\eta) = (\Delta_{\6-\bar\6}\eta,\eta)+
(\Delta_{N}\eta,\eta)+(\Delta_{\bar N}\eta,\eta)\geq 0
\]
hence
{\color{blue}\[ \eta \in \ker N, \ \ \eta \in \ker \bar N, \ \ \eta \in
   \ker(\6-\bar \6).
\]}
Then 
{\color{blue}\[
0 = d\eta= (N + \6 + \bar\6+\bar N)\eta= (\6 + \bar\6)\eta=0
\]}
Adding $(\6-\bar\6)\eta=0$, obtain $\6\eta=0$.
Then all Hodge components of $d$ vanish on $\eta$:
{\color{red}
\[
d^{2,-1}\eta= d^{1,0}\eta=d^{0,1}\eta = d^{-1,2}\eta=0.
\]
} 
\!However, the $(p+i, q+j)$-th component of $d^{i,j}\eta$
is $d^{i,j}\eta^{p,q}$, hence \[ d^{i,j}\eta^{p,q}=0\]
for all $i,j,p,q$. Then $d\eta^{p,q}=0$. The same argument is
used to show $d^*\eta^{p,q}=0$.

\newpage


{\bf \hspace{0pt}\color{blue} \large Possible applications}

{\it \color[rgb]{0,0.5,0} ``Algebraic geometry of nearly
  K\"ahler manifolds.''}

0. Hitchin's functional on the moduli of 
almost complex manifolds.

1. Yang-Mills bundles (Hermitian bundles with curvature 
two-form which satisfies $\Lambda\Theta=0$). 

2. The space of pseudoholomorphic curves in NK-manifold.
Its connected components are {\it
  \it \color{red} 
compact}.


3. Finding solutions of 
Maurer-Cartan equation with applications to the 
moduli of nearly K\"ahler manifolds.

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