# Monge-Ampère equation, hyperkähler structure and adapted complex structure

Su-Jen Kan 2020 Mathematics Subject Classification. 32C09, 32Q15, 53C26

( April. 26, 2024. )

###### Abstract

In the tangent bundle of $(M,g)$, it is well-known that the Monge-Ampère
equation $({\partial}\bar{\partial}\sqrt{\rho})^{n}=0$ has the asymptotic
expansion $\rho(x+iy)=\sum_{ij}g_{ij}(x)y_{i}y_{j}+O(y^{4})$ near $M$. Those
4th order terms are made explicit in this article:

$\rho(x+iy)=\sum_{i}y_{i}^{2}-\frac{1}{3}\sum_{pqij}R_{ipjq}(0)x_{p}x_{q}y_{i}y_{j}+O(5).$

At $M$, sectional curvatures of the Kähler metric
$2i{\partial}\bar{\partial}\rho$ can be computed. This has enabled us to find
a family of Kähler manifolds whose tangent bundles have admitted complete
hyperkähler structures whereas the adapted complex structure can only be
partially defined on the tangent bundles.

In these cases, the study of the adapted complex structure is equivalent to
the study of some gauge transformations on the baby Nahm’s equation
$\dot{T}_{1}+[T_{0},T_{1}]=0.$

##  1 Introduction

For an open $\mathfrak{D}_{M}\subset TM$ containing $M$, the adapted complex
structure is the unique complex structure in $\mathfrak{D}_{M}$ turning each
leaf $\gamma^{\mathbb{C}}\cap\mathfrak{D}_{M}$ into a holomorphic curve. It is
equivalent to the solving of the Monge-Ampère equation
$({\partial}\bar{\partial}\sqrt{\rho})^{n}=0$ on $\mathfrak{D}_{M}-M$
subjected to the initial condition that the restriction of the Kähler metric
$2i{\partial}\bar{\partial}\rho$ to $M$ has coincided with the original
Riemannian metric $g$ on $M$. Near $M$, the asymptotic expansion
$\rho(x+iy)=\sum_{ij}g_{ij}(x)y_{i}y_{j}+O(y^{4})$ is well-known for quite a
long while. Any attempt to reach the 4th order terms will encounter tedious
computations on complicated curvature tensors. As far as we know, there is no
further development on this yet. In Theorem 3.5, we complete the expansion up
to 4th order terms

$\rho(x+iy)=\sum_{i}y_{i}^{2}-\frac{1}{3}\sum_{pqij}R_{ipjq}(0)x_{p}x_{q}y_{i}y_{j}+O(5).$

Despite the vanishing of the holomorphic sectional curvature along any Riemann
foliation $\gamma^{\mathbb{C}},$ no other sectional curvature is known.

With these 4th order terms at hand, we are able to compute any sectional
curvature of the Kähler metric $2i{\partial}\bar{\partial}\rho$ at $M$. It
turns out that if $(M,g)$ is positively curved some sectional curvature of
$2i{\partial}\bar{\partial}\rho$ at $M$ must be negative.

If a non-flat $(M,g)$ has non-negative sectional curvature, an immediate
application of the above result is that the adapted complex structure cannot
be globally defined on the whole tangent bundle of the Kähler manifold
$(\mathfrak{D}_{M},2i{\partial}\bar{\partial}\rho).$

A hyperkähler structure in a differentiable manifold $M$ consists of two anti-
commuting complex structures $I,J$ and a Riemannian metric $\kappa$, which is
Kählerian with respect to all $I,J$ and $IJ$.

It is natural to expect the tangent bundle of a Kähler manifold might be able
to produce an extra complex structure anti-commuting with the original one to
produce such a hyperkähler structure. A local existence was proved in [Fe] and
in [Kal]. In other words, near the zero section of the tangent bundle, the
adapted complex structure and hyperkähler structures have existed
simultaneously provided the underlying Kähler manifold is either compact or
homogeneous. For a compact Hermitian symmetric $M$, both the hyperkähler and
the adapted complex structures have existed on the whole $TM$. It is thus
interesting to see examples where one of the structure has existed but not the
other one.

We show that the hyperkähler structure on $TG^{\mathbb{C}}/H^{\mathbb{C}}$
constructed in [Da-Sw] can be viewed as a hyperkähler structure on the tangent
bundle of $(T(G/H,g_{n}),2i{\partial}\bar{\partial}\rho),$ the Kähler manifold
obtained from doing the adapted complexification on the tangent bundle of the
Riemannian manifold $(G/H,g_{n})$. However, the adapted complex structure
cannot exist on the whole tangent bundle of
$(T(G/H,g_{n}),2i{\partial}\bar{\partial}\rho)$.

We briefly review the adapted complexification and prove the existence of it
on $T(G/H,g_{n})$ in the coming section. In §3, the 4th order terms of the
Monge-Ampère equation $({\partial}{\bar{\partial}}\sqrt{\rho})^{n}=0$ are
computed and are used to decide the sign of sectional curvatures at the zero
section in §4. The last section is devoted to those hyperkähler metrics on
$TG^{\mathbb{C}}/H^{\mathbb{C}}$ obtained from solving Nahm’s equations. It
turns out the adapted complexification on $T(G/H,g_{n})$ is reduced to the
solving of the baby Nahm’s equation $\dot{T}_{1}+[T_{0},T_{1}]=0.$

##  2 Adapted complex structure $TG/H$

Let $\gamma:\mathbb{R}\to M$ be a geodesic in a complete real-analytic
Riemannian manifold $(M,g)$ parametrized by the arclength. We denote its
complexification as the immersion

$\gamma^{\mathbb{C}}:\mathbb{C}\to TM;\ \
\gamma^{\mathbb{C}}(t+is)=(\gamma(t),s\dot{\gamma}(t))\in T_{\gamma(t)}M.$
(2.1)

The adapted complex structure, first established in [Gu-Sten] and in [Le-Sz],
is the unique complex structure $J$ in a domain $\mathfrak{D}_{M}\subset TM$
containing $M$ such that, for any geodesic $\gamma$, the immersion
$\gamma^{\mathbb{C}}:\
\mathbb{C}\cap{\gamma^{\mathbb{C}}}^{-1}(\mathfrak{D}_{M})\to(\mathfrak{D}_{M},J)$
is holomorphic. In other words, it is the unique complex structure in
$\mathfrak{D}_{M}$ turning each leaf of the foliation
$\cup_{\gamma}(\gamma^{\mathbb{C}}\cap\mathfrak{D}_{M})$ into a holomorphic
curve.

For generic $(M,g)$, the domain $\mathfrak{D}_{M}$ is properly contained in
$TM$. One necessary condition for the adapted complex structure to exist on
the whole tangent bundle is that $(M,g)$ is non-negatively curved. Since any
bi-invariant metric of a compact Lie group $G$ has positive sectional
curvature, it is natural to invest the definition on the tangent bundle of a
compact Lie group. The following construction has borrowed ideas from [Sz1,2].

The tangent bundle $TG$ of a Lie group is identified with
$G\times\mathfrak{g}$ through the mapping
$(a,v)\overset{\phi}{\longrightarrow}(a,(L_{a^{-1}})_{*}v)$ where $L_{b}$ is
the left translation by $b$. For compact $G$, any bi-invariant metric
$\langle,\rangle$ can be picked up to make the following $\varphi$

$\varphi:G\times\mathfrak{g}\longrightarrow G^{\mathbb{C}};\ (g,\eta)\to g\exp
i\eta$ (2.2)

a diffeomorphism. That is, the tangent bundle $TG$ can be identified with
$G^{\mathbb{C}}$ through $\varphi\cdot\phi$.

A geodesic in $(G,\langle,\rangle)$ is of the form $\gamma(t)=a\exp tX$ for
some $a\in G,X\in\mathfrak{g}$ and
$\phi(\gamma(t),s\cdot\gamma(t))=(\gamma(t),sX)\in G\times\mathfrak{g}$.

The mapping
$\mathbb{C}\overset{\eta}{\longrightarrow}TG\overset{\phi}{\longrightarrow}G\times\mathfrak{g}\overset{\varphi}{\longrightarrow}G^{\mathbb{C}}$
defined by

$\displaystyle\varphi\cdot\phi\cdot\eta(t+is)$
$\displaystyle=\varphi\cdot\phi(\gamma(t),s\dot{\gamma}(t))$ (2.3)
$\displaystyle=\varphi(\gamma(t),sX)$ $\displaystyle=a\exp tX\exp isX$
$\displaystyle=a\exp(t+is)X$

is holomorphic. Thus, $G^{\mathbb{C}}$ is the adapted complexification on the
tangent bundle of $(G,\langle,\rangle).$ The adapted complex structure is
defined on the whole $T(G,g)$ for any bi-invariant metric $g$ in a compact Lie
group $G$ and the resulting complex manifold is $G^{\mathbb{C}}$.

The next step is to extend the above from compact Lie groups to compact
homogeneous spaces. Let $H$ be a closed subgroup of a compact Lie group $G$, a
bi-invariant metric $g$ will decompose
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$ where $\mathfrak{h}$ is the Lie
algebra of $H$ and $\mathfrak{m}=T_{[e]}G/H$. The homogeneous space $G/H$ has
inherited the induced metric $g_{n}$, a normal metric, and $TG/H$ is
diffeomorphic to $G\times_{H}\mathfrak{m}$ through the mapping
$(a,v)\overset{\phi}{\longrightarrow}(a,(L_{a^{-1}})_{*}v)$ for any $a\in G/H$
and $v\in T_{[a]}G/H$ where the left translation $L_{a}:G/H\to G/H;xH\to axH$
and ${L_{a}}_{*}:T_{[x]}G/H\to T_{[ax]}G/H.$

Any geodesic of $(G/H,g_{n})$ is of the form $\gamma(t)=a\exp tY,a\in
G,Y\in\mathfrak{m}$, p.192 Cor. 2.5 [Ko-No]. $(\gamma(t),s\dot{\gamma}(t))\in
T_{\gamma(t)}G/H$. We compute

$\displaystyle(L_{{\gamma(t)}^{-1}})_{*}(s\dot{\gamma}(t))$
$\displaystyle=s(L_{{\gamma(t)}^{-1}})_{*}\frac{d}{d\sigma}|_{\sigma=t}\gamma(\sigma)$
(2.4)
$\displaystyle=s\frac{d}{d\sigma}|_{\sigma=t}{\gamma(t)}^{-1}\gamma(\sigma)$
$\displaystyle=s\frac{d}{d\sigma}|_{\sigma=t}\exp(\sigma-t)Y$
$\displaystyle=sY.$

As in the $G^{\mathbb{C}}$ case, to see the adapted complex structure on
$T(G/H,g_{n})$, a crucial step is to set a diffeomorphism between
$G\times_{H}\mathfrak{m}$ and $G^{\mathbb{C}}/H^{\mathbb{C}}$.

For this to work, we need a decomposition theorem for $G^{\mathbb{C}}$. For a
compact semi-simple $G$ and a closed subgroup $H<G$, the Lie algebra
$\mathfrak{g}$ can be decomposed as $\mathfrak{g}=\mathfrak{h}\oplus m$ with
respect to the Killing metric. Mostow’s decomposition theorem, [Mo], has
implied $G^{\mathbb{C}}=G\cdot\exp{im}\cdot H^{\mathbb{C}}.$ One crucial step
in the proof is that the non-degenerate Killing form $B$ of
$\mathfrak{g}^{\mathbb{C}}$ is $Aut(\mathfrak{g}^{\mathbb{C}})-$invariant
which has implied $B(\mathfrak{g},i\mathfrak{g})=0$ to arrange a good basis to
achieve the decomposition.

If $G$ is compact but not semi-simple, $H<G$ is a closed subgroup, a bi-
invariant metric on $G$ can decompose $\mathfrak{g}=\mathfrak{h}\oplus m$ as
well. However, it is not clear how to build up a non-degenerate scalar product
$\langle\ ,\ \rangle$ in $\mathfrak{g}^{\mathbb{C}}$ to play the role of the
above Killing form $B$ so that $\langle\mathfrak{g},i\mathfrak{g}\rangle=0$ to
extend the decomposition to a generic compact Lie group $G$.

Nevertheless, as a byproduct of their work about Lie group actions on complex
spaces, Heinzner-Schwarz, p.212 Remark 9.4 [He-Sch], have confirmed the
decomposition

$G^{\mathbb{C}}=G\cdot\exp{im}\cdot H^{\mathbb{C}}$ (2.5)

has also worked for generic compact Lie group $G$. It follows, the mapping

$\psi:G\times_{H}\mathfrak{m}\to G^{\mathbb{C}}/H^{\mathbb{C}};\ \ (g,v)\to
g\exp(iv)H^{\mathbb{C}},$ (2.6)

is a $G$-equivariant diffeomorphism. The mapping
$\mathbb{C}\overset{\eta}{\longrightarrow}TG/H\overset{\phi}{\longrightarrow}G\times_{H}\mathfrak{m}\overset{\psi}{\longrightarrow}G^{\mathbb{C}}/H^{\mathbb{C}}$
defined by

$\displaystyle\psi\cdot\phi\cdot\eta(t+is)$
$\displaystyle=\psi\cdot\phi(\gamma(t),s\dot{\gamma}(t))$ (2.7)
$\displaystyle=\psi(\gamma(t),sY)$ $\displaystyle=a\exp sY\exp itY$
$\displaystyle=a\exp(s+it)Y$

is holomorphic. Thus, $G^{\mathbb{C}}/H^{\mathbb{C}}$ is the adapted
complexification on the tangent bundle of $(G/H,g_{n}).$

We summarize the above as a proposition.

######  Proposition 2.1.

Let $G$ be a compact Lie group, $H<G$ be a closed subgroup and $g_{n}$ be a
normal metric on $G/H$. The diffeomorphism (2.6) has pulled back the complex
structure of $G^{\mathbb{C}}/H^{\mathbb{C}}$ to the adapted complex structure
on the whole $T(G/H,g_{n})$.

##  3 4-th order terms in $\rho$

The adapted complexification $(\mathfrak{D}_{M},J)$ is a Bruhat-Whitney type
of complexification of $M$ in the sense that $M$ can be embedded as a totally
real submanifold of $(\mathfrak{D}_{M},J)$. Such kind of complexifications is
unique up to biholomorphisms near $M$. That is, for two complexifications
$\Omega_{j}$ of $M$ and embeddings $i_{j}:M\to\Omega_{j},j=1,2$, there exist
neighborhoods $\Theta_{j}$ of $M$ in $\Omega_{j}$ and a biholomorphism
$\phi:\Theta_{1}\to\Theta_{2}$ such that $i_{2}=\phi\cdot i_{1}$.

In [Gu-St], the authors have treated this problem by solving
$({\partial}\bar{\partial}\sqrt{\rho})^{n}=0$. In $\mathfrak{D}_{M}$, the
Monge-Ampère equation $({\partial}\bar{\partial}\sqrt{\rho})^{n}=0$ has a
solution unique up to a scalar. The preferred initial condition is that the
restriction of the Kähler metric $h$, induced from the Kähler form
$2\sqrt{-1}{\partial}\bar{\partial}\rho$, to $M$ has coincided with the
original Riemannian metric $g$ of $M$. In other words, for any vector fields
$X$ and $Y$ in $M$,

$h(X,Y)=2\sqrt{-1}{\partial}\bar{\partial}\rho(X,JY)=g(X,Y).$ (3.1)

This metric is called the canonical Kähler metric of the adapted
complexification over $(M,g).$ The solution $\rho$ is strictly
plurisubharmonic and

$\displaystyle\rho(p,v)=\frac{|v|^{2}}{2}$ (3.2)

for any tangent vector $(p,v)\in\mathfrak{D}_{M},v\in T_{p}M$, cf. p.693 [Le-
Sz].

Near the zero section $M$, the solution is rather concrete in terms of local
coordinates. Let $p\in M$ and let $\\{x_{1},\cdots,x_{n}\\}$ be a geodesic
normal coordinate for a neighborhood $U$ of $p$ in $M$ such that $TU\simeq
U\times\mathbb{R}^{n}$ is trivial. Taking $\epsilon>0$ small enough so that
$T^{\epsilon}U\simeq U\times\mathbb{R}_{\epsilon}^{n}$ for an open subset
$\mathbb{R}_{\epsilon}^{n}$ of $\mathbb{R}^{n}$ is a Bruhat-Whitney
complexification which is biholomorphic to the adapted complexification
$(T^{\epsilon}U,J)$.

A holomorphic coordinates $\\{z_{j}=x_{j}+\sqrt{-1}y_{j}:j=1,\cdots,n\\}$ can
thus be formed for the complex manifold $(T^{\epsilon}U,J)$ where
$\\{x_{1},\cdots,x_{n}\\}$ is a geodesic normal coordinates for $U$ and
$\\{y_{1},\cdots,y_{n}\\}$ is a coordinate system of
$\mathbb{R}_{\epsilon}^{n}$ with
$J(\frac{{\partial}}{{\partial}x_{j}})=\frac{{\partial}}{{\partial}y_{j}}$.

Solving the defining Monge-Ampère equation in $T^{\epsilon}U$ subjected to the
initial condition (3.1), the solution $\rho$ has the asymptotic expansion near
$M$, Prop.3.1 in [Kan],

$\rho(x+\sqrt{-1}y)=\sum_{ij}g_{ij}(x)y_{i}y_{j}+O(y^{4})$ (3.3)

where the higher order terms have depended upon the curvature behavior of the
metric $g$. Locally, the Kähler potential $\rho$ has agreed with the Euclidean
potential up to third order terms. Thus, $(M,g)$ is a complete totally
geodesic submanifold of $(\mathfrak{D}_{M},h)$. Throughout this article, the
notation $O(y^{4})$ in (3.3) means

$\lim_{|y|\to
0}\frac{\rho(x+\sqrt{-1}y)-\sum_{ij}g_{ij}(x)y_{i}y_{j}}{|y|^{4}}<\infty.$

In a Riemannian manifold $(M,g)$ with the Levi-Civita connection
$\triangledown$, the curvature tensor $R$ is defined, for any vector fields
$X,Y,Z,W$, to be

$\displaystyle R(X,Y,Z,W)$
$\displaystyle=g(\triangledown_{X}\triangledown_{Y}W-\triangledown_{Y}\triangledown_{X}W-\triangledown_{[X,Y]}W,Z)$
(3.4) $\displaystyle:$ $\displaystyle=g(R(X,Y)W,Z).$

The sectional curvature of the plan $\Pi$ spanned by the vectors $X$ and $Y$
is defined as

$\displaystyle S(\Pi)=\frac{R(X,Y,X,Y)}{\|X\wedge Y\|^{2}}$ (3.5)

which is independent of any choice of two independent vectors in the plane
$\Pi$.

In a local coordinate chart $\\{x_{j}\\}_{j=1}^{n}$, the following notation is
commonly adapted:

$\displaystyle
R_{ijkl}(p):=R(\frac{{\partial}}{{\partial}x_{i}},\frac{{\partial}}{{\partial}x_{j}},\frac{{\partial}}{{\partial}x_{k}},\frac{{\partial}}{{\partial}x_{l}})(p).$
(3.6)

Furthermore, if $\\{x_{j}\\}_{j=1}^{n}$ is a geodesic normal coordinate
centered at 0, the following asymptotic expansion works for small $x$,

$g_{ij}(x)=\delta_{ij}-\frac{1}{3}\sum_{p,q}R_{ipjq}(0)x_{p}x_{q}+O(x^{3}).$
(3.7)

Substituting this into (3.3), $\rho$ has the following asymptotic expansion:

$\displaystyle\rho(x+\sqrt{-1}y)$
$\displaystyle=\sum_{ij}g_{ij}(x)y_{i}y_{j}+O(y^{4})$ (3.8)
$\displaystyle=\sum_{i}y_{i}^{2}-\frac{1}{3}\sum_{ijpq}R_{ipjq}(0)x_{p}x_{q}y_{i}y_{j}+\sum_{ij}O(x^{3})y_{i}y_{j}+O(y^{4})$
$\displaystyle=\sum_{i}y_{i}^{2}-\frac{1}{3}\sum_{ijpq}R_{ipjq}(0)x_{p}x_{q}y_{i}y_{j}+\sum_{i\leq
j\leq\gamma\leq\delta}\mathcal{A}_{ij\gamma\delta}y_{i}y_{j}y_{\gamma}y_{\delta}+O(5),$

where $O(5)$ is the sum of polynomials of orders, in $x$ and $y$, greater than
or equal to 5. The aim of this section is to show those constant
$\mathcal{A}_{ij\gamma\delta}$ have vanished. We’ll need a general lemma on
the inverse matrix to start with.

######  Lemma 3.1.

Let $A=\left[\delta_{ij}+a_{ij}(y)\right]$ be a $n\times n$ matrix where
$a_{ij}(y)$ is a homogeneous polynomial of degree 2 in
$y=(y_{1},\cdots,y_{n}).$ Then $A^{-1}=\left[b^{ij}(y)\right]$ with

$b^{ij}(y)=\left\\{\begin{array}[]{ll}-a_{ij}(y)+O(y^{4})&\mbox{if $i\neq
j$};\\\ 1-a_{jj}(y)+O(y^{4})&\mbox{if $i=j$}.\end{array}\right.$

###### Proof.

The determinant

$\displaystyle\triangle=\det A=1+\sum_{j=1}^{n}a_{jj}(y)+O(y^{4}),$ (3.9)

then

$\displaystyle\triangle^{-1}=1-\sum_{j=1}^{n}a_{jj}(y)+O(y^{4}).$ (3.10)

Let $A^{ij}$ denote the $(n-1)\times(n-1)$ matrix obtaining from deleting the
$i$-th column and the $j$-th row from $A$. Denoting the inverse matrix
$A^{-1}=\left[b^{ij}(y)\right]$, then

$\displaystyle b^{ij}(y)=\frac{1}{\triangle}(-1)^{i+j}\det A^{ij}.$ (3.11)

$A^{jj}$ is an $(n-1)\times(n-1)$ with diagonals
$a_{11}(y),..,a_{j-1,j-1}(y),a_{j+1,j+1}(y),\cdots,a_{nn}(y)$ then

$\displaystyle\det A^{jj}$ $\displaystyle=1+\sum_{k\neq j}a_{kk}(y)+O(y^{4});$
(3.12) $\displaystyle b^{jj}(y)$
$\displaystyle=\frac{1}{\triangle}(-1)^{j+j}\det A^{jj}=1-a_{jj}(y)+O(y^{4}).$

For $i<j$, the components $a_{ii}(y)$ and $a_{jj}(y)$ were missing in the
$(n-1)\times(n-1)$ matrix $A^{ij}$. In the computation of $\det A^{ij}$, the
only possibility with order $\leq 2$ comes from the term
$(-1)^{j-i+1}a_{ij}(y)\Pi_{k\neq i,j}a_{kk}(y)$. Thus

$\displaystyle\det A^{ij}$ $\displaystyle=(-1)^{j-i+1}a_{ij}(y)+O(y^{4});$
(3.13) $\displaystyle b^{ij}(y)$
$\displaystyle=\frac{1}{\triangle}(-1)^{i+j}\det A^{ij}$
$\displaystyle=-(1-\sum_{j=1}^{n}a_{jj}(y))a_{ij}(y)+O(y^{4})$
$\displaystyle=-a_{ij}(y)+O(y^{4}).$

The situation $i>j$ goes exactly the same line. ∎

In (3.8), we have asked the permutation of the indexes in $\mathcal{A}$ is in
a non-decreasing order. We will use the same notation for a general
permutation, that is, $\mathcal{A}_{ij\gamma\delta}\neq 0$ only if $i\leq
j\leq\gamma\leq\delta$. Let’s denote the following constants $B_{\alpha ijk}$
and $C_{\alpha\beta kl}$ as:

$\displaystyle B_{\alpha ijk}$ $\displaystyle:=\mathcal{A}_{\alpha
ijk}+\mathcal{A}_{i\alpha jk}+\mathcal{A}_{ij\alpha
k}+\mathcal{A}_{ijk\alpha};$ (3.14) $\displaystyle C_{\alpha\beta kl}$
$\displaystyle:=B_{\beta\alpha kl}+B_{\beta k\alpha l}+B_{\beta kl\alpha}.$

######  Lemma 3.2.

Let $i\leq j\leq k\leq l,\\{\alpha,\beta,\gamma,\delta\\}$ be a permutation of
$\\{i,j,k,l\\}.$ Then

$\sum_{\alpha\beta\gamma\delta\in\\{i,j,k,l\\}}(B_{\alpha\beta\gamma\delta}-\frac{1}{2}C_{\delta\gamma\alpha\beta})=-2\mathcal{A}_{ijkl}.$

###### Proof.

As $\alpha\beta\gamma\delta$ run through $i,j,k,l$,
$\sum_{\delta\gamma\alpha\beta}C_{\delta\gamma\alpha\beta}$ will run through
all $B_{\alpha\beta\gamma\delta}$ three times. Similarly,
$\sum_{\delta\gamma\alpha\beta}B_{\alpha\beta\gamma\delta}$ will run through
all $\mathcal{A}_{\alpha\beta\gamma\delta}$ four times. However,
$\mathcal{A}_{\alpha\beta\gamma\delta}\neq 0$ only if
$\alpha\leq\beta\leq\gamma\leq\delta.$

$\displaystyle-\frac{1}{2}\sum_{\alpha\beta\gamma\delta}C_{\delta\gamma\alpha\beta}+\sum_{\alpha\beta\gamma\delta}B_{\alpha\beta\gamma\delta}$
$\displaystyle=-\frac{3}{2}\sum_{\alpha\beta\gamma\delta}B_{\alpha\beta\gamma\delta}+\sum_{\alpha\beta\gamma\delta}B_{\alpha\beta\gamma\delta}$
(3.15)
$\displaystyle=\frac{-1}{2}\sum_{\alpha\beta\gamma\delta}B_{\alpha\beta\gamma\delta}=-2\mathcal{A}_{ijkl}.$

∎

The Monge-Ampère equation $({\partial}\bar{\partial}\sqrt{\rho})^{n}=0$ is
equivalent to

$\displaystyle\sum_{\alpha}\rho^{\alpha}\rho_{\alpha}=2\rho$ (3.16)

where

$\displaystyle\rho^{\alpha}=\sum_{\beta}\rho^{\alpha\bar{\beta}}\rho_{\bar{\beta}}$
(3.17)

and $(\rho^{\alpha\bar{\beta}})$ is the inverse matrix to
$(\rho_{\alpha\bar{\beta}})$ so that
$\sum_{\beta}\rho^{\alpha\bar{\beta}}\rho_{\gamma\bar{\beta}}=\delta_{\alpha\gamma}.$

We make a computation on
$\rho_{\alpha}=\frac{1}{2}(\frac{{\partial}\rho}{{\partial}x_{\alpha}}-\sqrt{-1}\frac{{\partial}\rho}{{\partial}y_{\alpha}})$.
The expression of $\rho$ in (3.8) has ensured that
$\frac{{\partial}\rho}{{\partial}x_{\alpha}}(\sqrt{-1}y)$ lies in $O(y^{4})$
terms. Then

$\displaystyle\rho_{\alpha}(\sqrt{-1}y)=\frac{{\partial}\rho}{{\partial}z_{\alpha}}(\sqrt{-1}y)$
$\displaystyle=-\sqrt{-1}y_{\alpha}-\frac{\sqrt{-1}}{2}\sum_{jkl}B_{\alpha
jkl}y_{j}y_{k}y_{l}+O(y^{4});$ (3.18)
$\displaystyle\rho_{\bar{\alpha}}(\sqrt{-1}y)=\frac{{\partial}\rho}{{\partial}\bar{z}_{\alpha}}(\sqrt{-1}y)$
$\displaystyle=\sqrt{-1}y_{\alpha}+\frac{\sqrt{-1}}{2}\sum_{jkl}B_{\alpha
jkl}y_{j}y_{k}y_{l}+O(y^{4}).$

To find the inverse matrix $(\rho^{\alpha\bar{\beta}})$, we take derivatives
to $\rho$ and then take the value at the point $\sqrt{-1}y$,

$\displaystyle\frac{{\partial}^{2}\rho(\sqrt{-1}y)}{{\partial}z_{\alpha}{\partial}z_{\bar{\beta}}}$
$\displaystyle=\frac{1}{4}\left((\frac{{\partial}^{2}\rho}{{\partial}x_{\alpha}{\partial}x_{\beta}}+\frac{{\partial}^{2}\rho}{{\partial}y_{\alpha}{\partial}y_{\beta}})-\sqrt{-1}(\frac{{\partial}^{2}\rho}{{\partial}x_{\alpha}{\partial}y_{\beta}}-\frac{{\partial}^{2}\rho}{{\partial}x_{\beta}{\partial}y_{\alpha}})\right)(\sqrt{-1}y).$
(3.19)

The third and the fourth terms are in $O(y^{3})$, so it left to check the
first and the second terms.

$\displaystyle\frac{{\partial}^{2}\rho}{{\partial}x_{\alpha}{\partial}x_{\beta}}(\sqrt{-1}y)$
$\displaystyle=\frac{-1}{3}\sum_{ij}(R_{i\beta j\alpha}(0)+R_{i\alpha
j\beta}(0))y_{i}y_{j};$ (3.20)
$\displaystyle\frac{{\partial}^{2}\rho}{{\partial}y_{\alpha}{\partial}y_{\beta}}(\sqrt{-1}y)$
$\displaystyle=2\delta_{\alpha\beta}+\frac{{\partial}}{{\partial}y_{\alpha}}\sum_{l\gamma\delta}B_{\beta
l\gamma\delta}y_{l}y_{\gamma}y_{\delta}$
$\displaystyle=2\delta_{\alpha\beta}+\sum_{k,l}C_{\alpha\beta kl}y_{k}y_{l}.$

Plugging (3.20) to (3.19), we have

$\displaystyle
2\rho_{\alpha\bar{\beta}}(\sqrt{-1}y)=\delta_{\alpha\beta}+D_{\alpha\beta}(y)+O(y^{3})$
(3.21)

where

$\displaystyle D_{\alpha\beta}(y):=\sum_{i,j}\left(\frac{1}{2}C_{\alpha\beta
ij}-\frac{1}{6}(R_{i\beta j\alpha}(0)+R_{i\alpha j\beta}(0))\right)y_{i}y_{j}$
(3.22)

is a homogeneous polynomial of degree 2 in $y$ and the $O(y^{3})$ term is the
sum of polynomials in $y$ of degree $\geq 3$. To find the inverse matrix up to
order 2, we may throw away those $O(y)$ terms away. So, the matrix
$\left[2\rho_{\alpha\bar{\beta}}\right]$ has fit the form in Lemma 3.1.
Denoting the inverse matrix as
$\left[2\rho_{\alpha\bar{\beta}}(\sqrt{-1}y)\right]^{-1}:=\left[G^{\alpha\bar{\beta}}(y)\right].$
By Lemma 3.1,

$G^{\alpha\bar{\beta}}(y)=\left\\{\begin{array}[]{ll}-D_{\alpha\beta}(y)+O(y^{4})&\mbox{if
$\alpha\neq\beta$};\\\ 1-D_{\alpha\beta}(y)+O(y^{4})&\mbox{if
$\alpha=\beta$}.\end{array}\right.$

Then

$\displaystyle\rho^{\alpha\bar{\beta}}(\sqrt{-1}y)=2\delta_{\alpha\beta}-2D_{\alpha\beta}(y)+O(y^{4}).$
(3.23)

######  Lemma 3.3.

$\mathcal{A}_{ijkl}=\frac{1}{18}\sum_{\alpha\beta\gamma\delta\in\\{i,j,k,l\\}}(R_{\alpha\gamma\beta\delta}(0)+R_{\alpha\delta\beta\gamma}(0)),\
i\leq\j\leq k\leq l.$

###### Proof.

The idea is to equating those 4th order terms in both sides of

$\displaystyle\sum_{\alpha\beta}\rho^{\alpha\bar{\beta}}(\sqrt{-1}y)\rho_{\bar{\beta}}(\sqrt{-1}y)\rho_{\alpha}(\sqrt{-1}y)=2\rho(\sqrt{-1}y).$
(3.24)

By (3.23) and (3.18), 4th order terms of the left hand side come from

$\displaystyle\sum_{\alpha\beta}(2\delta_{\alpha\beta}-2D_{\alpha\beta}(y))(\sqrt{-1}y_{\beta}+\frac{\sqrt{-1}}{2}\sum_{jkl}B_{\beta
jkl}y_{j}y_{k}y_{l})(-\sqrt{-1}y_{\alpha}-\frac{\sqrt{-1}}{2}\sum_{jkl}B_{\alpha
jkl}y_{j}y_{k}y_{l}).$ (3.25)

They are, by (3.22),

$\displaystyle 2\sum_{\alpha jkl}B_{\alpha
jkl}y_{\alpha}y_{j}y_{k}y_{l}-2\sum_{\alpha\beta}D_{\alpha\beta}(y)y_{\alpha}y_{\beta}$
(3.26) $\displaystyle=2\sum_{\alpha jkl}B_{\alpha
jkl}y_{\alpha}y_{j}y_{k}y_{l}-2\sum_{\alpha\beta
ij}\left(\frac{1}{2}C_{\alpha\beta ij}-\frac{1}{6}(R_{i\beta
j\alpha}(0)+R_{i\alpha j\beta}(0))\right)y_{i}y_{j}y_{\alpha}y_{\beta}$
$\displaystyle=2\sum_{\alpha\beta\gamma\delta}\left(B_{\alpha\beta\gamma\delta}-\frac{1}{2}C_{\delta\gamma\alpha\beta}+\frac{1}{6}(R_{\alpha\delta\beta\gamma}(0)+R_{\alpha\gamma\beta\delta}(0))\right)y_{\alpha}y_{\beta}y_{\gamma}y_{\delta}.$

Equating these 4th order terms with those 4th order terms from the right hand
side, we get: for $i\leq\j\leq k\leq l$, by Lemma 3.2

$\displaystyle\mathcal{A}_{ijkl}$
$\displaystyle=\sum_{\alpha\beta\gamma\delta\in\\{i,j,k,l\\}}\left(B_{\alpha\beta\gamma\delta}-\frac{1}{2}C_{\delta\gamma\alpha\beta}+\frac{1}{6}(R_{\alpha\delta\beta\gamma}(0)+R_{\alpha\gamma\beta\delta}(0))\right)$
(3.27)
$\displaystyle=-2\mathcal{A}_{ijkl}+\frac{1}{6}\sum_{\alpha\beta\gamma\delta}(R_{\alpha\delta\beta\gamma}(0)+R_{\alpha\gamma\beta\delta}(0)).$

Then

$\mathcal{A}_{ijkl}=\frac{1}{18}\sum_{\alpha\beta\gamma\delta\in\\{i,j,k,l\\}}(R_{\alpha\gamma\beta\delta}(0)+R_{\alpha\delta\beta\gamma}(0)),\
i\leq j\leq k\leq l.$

∎

######  Lemma 3.4.

$\mathcal{A}_{ijkl}=0,\forall i\leq j\leq k\leq l.$

###### Proof.

When we sum up all those $R_{\alpha\gamma\beta\delta}(0)$, for example, for a
fixed pair of $\alpha=i,\gamma=j$, there must contain both the pair
$\beta=k,\delta=l$ and the pair $\beta=l,\delta=k$. So, for any fixed $\alpha$
and $\gamma$, $\sum_{\beta\delta}R_{\alpha\gamma\beta\delta}(0)=0.$

$\displaystyle\sum_{\alpha\beta\gamma\delta\in\\{i,j,k,l\\}}(R_{\alpha\gamma\beta\delta}(0)+R_{\alpha\delta\beta\gamma}(0))$
$\displaystyle=\sum_{\alpha\gamma}\sum_{\beta\delta}R_{\alpha\gamma\beta\delta}(0)+\sum_{\alpha\delta}\sum_{\beta\gamma}R_{\alpha\delta\beta\gamma}(0)=0.$
(3.28)

∎

The following Theorem has been proved.

######  Theorem 3.5.

For $x$ small in a normal coordinate in $M$ centered at 0 and $y$ the
corresponding small imaginary part, $\rho$ has the following asymptotic
expansion:

$\rho(x+\sqrt{-1}y)=\sum_{i}y_{i}^{2}-\frac{1}{3}\sum_{pqij}R_{ipjq}(0)x_{p}x_{q}y_{i}y_{j}+O(5).$

##  4 Estimate on some sectional curvatures

The manifold $\mathfrak{D}_{M}$ is equipped with the Kähler metric $h$
obtained from the Kähler form $\omega=-2i{\partial}\bar{\partial}\rho$ and the
adapted complex structure $J$. We use $K$ to denote the curvature tensor of
this Kähler metric, i.e.,

$\displaystyle
K(X,Y,Z,W)=h(\hat{\triangledown}_{X}\hat{\triangledown}_{Y}W-\hat{\triangledown}_{Y}\hat{\triangledown}_{X}W-\hat{\triangledown}_{[X,Y]}W,Z)=h(K(X,Y)W,Z)$
(4.1)

where $\hat{\triangledown}$ is the corresponding Levi-Civita connection of the
Riemannian manifold $(\mathfrak{D}_{M},h).$ The holomorphic sectional
curvature of the $J$-invariant plane $\Pi$ spanned by the vectors $X$ and $JX$
is given as

$S(\Pi)=\frac{K(X,JX,X,JX)}{\|X\wedge JY\|^{2}}.$ (4.2)

From the fact that the connection is almost complex
$K(JX,JY)=K(X,Y);K(X,Y)\cdot J=J\cdot K(X,Y)$. The sectional curvature of the
plane spanned by $JX$ and $JY$ is the same as the sectional curvature of the
plane spanned by $X$ and $Y$:

$\displaystyle K(JX,JY,JX,JY)$ $\displaystyle=\langle K(JX,JY)JY,JX\rangle$
(4.3) $\displaystyle=\langle K(X,Y)JY,JX\rangle$ $\displaystyle=\langle
JK(X,Y)Y,JX\rangle$ $\displaystyle=\langle K(X,Y)Y,X\rangle$
$\displaystyle=K(X,Y,X,Y).$

In terms of local coordinate $z_{j}=x_{j}+\sqrt{-1}y_{j}$, the curvature
tensors on vectors
$\\{\frac{{\partial}}{{\partial}z_{j}},\frac{{\partial}}{{\partial}\bar{z}_{k}}:j,k=1,\cdots,n\\}$
can be computed with $K_{ij\bar{k}l}$ defined in a natural way. For instance,

$\displaystyle K_{i\bar{j}k\bar{l}}$
$\displaystyle:=K(\frac{{\partial}}{{\partial}z_{i}},\frac{{\partial}}{{\partial}\bar{z}_{j}},\frac{{\partial}}{{\partial}z_{k}},\frac{{\partial}}{{\partial}\bar{z}_{l}});$
(4.4) $\displaystyle K_{ij\bar{k}l}$
$\displaystyle:=K(\frac{{\partial}}{{\partial}z_{i}},\frac{{\partial}}{{\partial}z_{j}},\frac{{\partial}}{{\partial}\bar{z}_{k}},\frac{{\partial}}{{\partial}z_{l}}).$

It is well-known, [Ko-No] IX.5, that only the following types can be different
from 0:

$K_{i\bar{j}k\bar{l}},\ K_{i\bar{j}\bar{k}l},\ K_{\bar{i}jk\bar{l}},\
K_{\bar{i}j\bar{k}l},$ (4.5)

which can be interpreted by derivatives of the potential function of the
Kähler metric. Since the Kähler form
$\omega=2\sqrt{-1}{\partial}\bar{\partial}\rho,$

$\displaystyle K_{i\bar{j}k\bar{l}}$
$\displaystyle=\frac{{\partial}^{2}\rho_{k\bar{l}}}{{\partial}z_{i}{\partial}z_{\bar{j}}}-\sum_{\nu\mu}\rho^{\nu\bar{\mu}}\frac{{\partial}\rho_{k\bar{\mu}}}{{\partial}z_{i}}\frac{{\partial}\rho_{\nu\bar{l}}}{{\partial}\bar{z}_{j}}$
(4.6)
$\displaystyle=\frac{{\partial}^{4}\rho}{{\partial}z_{i}{\partial}z_{\bar{j}}{\partial}z_{k}{\partial}z_{\bar{l}}}-\sum_{\nu\mu}\rho^{\nu\bar{\mu}}\frac{{\partial}^{3}\rho}{{\partial}z_{i}{\partial}z_{k}{\partial}\bar{z}_{\mu}}\frac{{\partial}^{3}\rho}{{\partial}\bar{z}_{j}{\partial}\bar{z}_{l}{\partial}z_{\nu}}.$

As

$\frac{{\partial}}{{\partial}x_{j}}=\frac{{\partial}}{{\partial}z_{j}}+\frac{{\partial}}{{\partial}\bar{z}_{j}};\
\
\frac{{\partial}}{{\partial}y_{j}}=\sqrt{-1}(\frac{{\partial}}{{\partial}z_{j}}-\frac{{\partial}}{{\partial}\bar{z}_{j}}),$
(4.7)

the sectional curvature of the plane spanned by
$\frac{{\partial}}{{\partial}x_{i}}$ and $\frac{{\partial}}{{\partial}y_{j}}$
can be computed as following:

$\displaystyle
K(\frac{{\partial}}{{\partial}x_{i}},\frac{{\partial}}{{\partial}y_{j}},\frac{{\partial}}{{\partial}x_{i}},\frac{{\partial}}{{\partial}y_{j}})$
$\displaystyle=-K(\frac{{\partial}}{{\partial}z_{i}}+\frac{{\partial}}{{\partial}\bar{z}_{i}},\frac{{\partial}}{{\partial}z_{j}}-\frac{{\partial}}{{\partial}\bar{z}_{j}},\frac{{\partial}}{{\partial}z_{i}}+\frac{{\partial}}{{\partial}\bar{z}_{i}},\frac{{\partial}}{{\partial}z_{j}}-\frac{{\partial}}{{\partial}\bar{z}_{j}})$
(4.8)
$\displaystyle=-(K_{i\bar{j}i\bar{j}}+2K_{i\bar{j}j\bar{i}}+\overline{K_{i\bar{j}i\bar{j}}}).$

Plugging the expression of $\rho$ in Theorem 3.5 it is clear to see, when
evaluated at the point $0$, the second part of (4.6) has vanished since there
is no third order terms in $\rho$. It is sufficient to take care of the first
part.

$\displaystyle\frac{{\partial}^{4}\rho}{{\partial}z_{i}{\partial}z_{\bar{j}}{\partial}z_{k}{\partial}z_{\bar{l}}}=\frac{1}{16}$
$\displaystyle\left((\frac{{\partial}^{2}}{{\partial}x_{k}{\partial}x_{l}}+\frac{{\partial}^{2}}{{\partial}y_{k}{\partial}y_{l}})+\sqrt{-1}(\frac{{\partial}^{2}}{{\partial}x_{k}{\partial}y_{l}}-\frac{{\partial}^{2}}{{\partial}x_{l}{\partial}y_{k}})\right)$
(4.9)
$\displaystyle\left((\frac{{\partial}^{2}}{{\partial}x_{i}{\partial}x_{j}}+\frac{{\partial}^{2}}{{\partial}y_{i}{\partial}y_{j}})+\sqrt{-1}(\frac{{\partial}^{2}}{{\partial}x_{i}{\partial}y_{j}}-\frac{{\partial}^{2}}{{\partial}x_{j}{\partial}y_{i}})\right)\rho$
$\displaystyle=\frac{1}{16}$
$\displaystyle\left(\frac{{\partial}^{4}\rho}{{\partial}x_{i}{\partial}x_{j}{\partial}x_{k}{\partial}x_{l}}+\frac{{\partial}^{4}\rho}{{\partial}x_{i}{\partial}x_{j}{\partial}y_{k}{\partial}y_{l}}+\frac{{\partial}^{4}\rho}{{\partial}x_{k}{\partial}x_{l}{\partial}y_{i}{\partial}y_{j}}+\frac{{\partial}^{4}\rho}{{\partial}y_{i}{\partial}y_{j}{\partial}y_{k}{\partial}y_{l}}\right)$
$\displaystyle-\frac{1}{16}$
$\displaystyle\left(\frac{{\partial}^{4}\rho}{{\partial}x_{i}{\partial}x_{k}{\partial}y_{j}{\partial}y_{l}}+\frac{{\partial}^{4}\rho}{{\partial}x_{j}{\partial}x_{l}{\partial}y_{i}{\partial}y_{k}}-\frac{{\partial}^{4}\rho}{{\partial}x_{i}{\partial}x_{l}{\partial}y_{j}{\partial}y_{k}}-\frac{{\partial}^{4}\rho}{{\partial}x_{j}{\partial}x_{k}{\partial}y_{i}{\partial}y_{l}}\right)$
$\displaystyle+\frac{i}{16}$
$\displaystyle\left(\frac{{\partial}^{4}\rho}{{\partial}x_{i}{\partial}x_{k}{\partial}x_{l}{\partial}y_{j}}+\frac{{\partial}^{4}\rho}{{\partial}x_{i}{\partial}y_{j}{\partial}y_{k}{\partial}y_{l}}-\frac{{\partial}^{4}\rho}{{\partial}x_{j}{\partial}x_{k}{\partial}x_{l}{\partial}y_{i}}-\frac{{\partial}^{4}\rho}{{\partial}x_{j}{\partial}y_{i}{\partial}y_{k}{\partial}y_{l}}\right)$
$\displaystyle+\frac{i}{16}$
$\displaystyle\left(\frac{{\partial}^{4}\rho}{{\partial}x_{i}{\partial}x_{j}{\partial}x_{k}{\partial}y_{l}}+\frac{{\partial}^{4}\rho}{{\partial}x_{k}{\partial}y_{i}{\partial}y_{j}{\partial}y_{l}}-\frac{{\partial}^{4}\rho}{{\partial}x_{i}{\partial}x_{j}{\partial}x_{l}{\partial}y_{k}}-\frac{{\partial}^{4}\rho}{{\partial}x_{l}{\partial}y_{i}{\partial}y_{j}{\partial}y_{k}}\right).$

Evaluated at the point $0$, only terms of the type
$\frac{{\partial}^{4}\rho}{{\partial}x_{i}{\partial}x_{j}{\partial}y_{k}{\partial}y_{l}}$
could be non-vanished.

$\displaystyle\frac{{\partial}^{4}\rho(0)}{{\partial}x_{i}{\partial}x_{j}{\partial}y_{k}{\partial}y_{l}}$
$\displaystyle=\frac{-1}{3}(R_{kilj}+R_{kjli}+R_{likj}+R_{ljki})(0)$ (4.10)
$\displaystyle=\frac{-2}{3}(R_{kilj}+R_{likj})(0).$

$\displaystyle\frac{{\partial}^{4}\rho(0)}{{\partial}z_{i}{\partial}z_{\bar{j}}{\partial}z_{k}{\partial}z_{\bar{l}}}$
$\displaystyle=\frac{1}{16}\left(\frac{{\partial}^{4}\rho(0)}{{\partial}x_{i}{\partial}x_{j}{\partial}y_{k}{\partial}y_{l}}+\frac{{\partial}^{4}\rho(0)}{{\partial}x_{k}{\partial}x_{l}{\partial}y_{i}{\partial}y_{j}}\right)$
(4.11) $\displaystyle\ \
-\frac{1}{16}\left(\frac{{\partial}^{4}\rho(0)}{{\partial}x_{i}{\partial}x_{k}{\partial}y_{j}{\partial}y_{l}}+\frac{{\partial}^{4}\rho(0)}{{\partial}x_{j}{\partial}x_{l}{\partial}y_{i}{\partial}y_{k}}-\frac{{\partial}^{4}\rho(0)}{{\partial}x_{i}{\partial}x_{l}{\partial}y_{j}{\partial}y_{k}}-\frac{{\partial}^{4}\rho(0)}{{\partial}x_{j}{\partial}x_{k}{\partial}y_{i}{\partial}y_{l}}\right)$
$\displaystyle=\frac{-1}{24}(R_{kilj}+R_{likj}+R_{ikjl}+R_{jkil})(0)$
$\displaystyle\ \
-\frac{1}{24}(-R_{jilk}-R_{lijk}-R_{ijkl}-R_{kjil}+R_{jikl}+R_{kijl}+R_{ijlk}+R_{ljik})(0)$
$\displaystyle=\frac{1}{6}(R_{ijkl}+R_{ilkj})(0).$

Taking $k=i,l=j$ and then taking $k=j,l=i$

$\displaystyle K_{i\bar{j}i\bar{j}}(0)$
$\displaystyle=\frac{{\partial}^{4}\rho(0)}{{\partial}z_{i}{\partial}z_{\bar{j}}{\partial}z_{i}{\partial}z_{\bar{j}}}$
(4.12) $\displaystyle=\frac{1}{6}\left(R_{ijij}+R_{ijij}\right)(0)$
$\displaystyle=\frac{1}{3}R_{ijij}(0);$

$\displaystyle K_{i\bar{j}j\bar{i}}(0)$
$\displaystyle=\frac{{\partial}^{4}\rho(0)}{{\partial}z_{i}{\partial}z_{\bar{j}}{\partial}z_{j}{\partial}z_{\bar{i}}}$
(4.13) $\displaystyle=\frac{1}{6}\left(R_{ijji}+R_{iijj}\right)(0)$
$\displaystyle=\frac{-1}{6}R_{ijij}(0)$

######  Proposition 4.1.

At $p\in M$, the sectional curvature of the metric $h$ on the plane spanned by
$\frac{{\partial}}{{\partial}x_{i}}$ and $\frac{{\partial}}{{\partial}y_{j}}$
is $\frac{-1}{3}R_{ijij}(p)$. All holomorphic sectional curvatures of the
metric $h$ have vanished at $p$.

###### Proof.

The sectional curvature for the plane spanned by
$\frac{{\partial}}{{\partial}x_{i}}$ and $\frac{{\partial}}{{\partial}y_{j}}$
is
$K(\frac{{\partial}}{{\partial}x_{i}},\frac{{\partial}}{{\partial}y_{j}},\frac{{\partial}}{{\partial}x_{i}},\frac{{\partial}}{{\partial}y_{j}}).$
Take a normal coordinate centered at $p$ as discussed in the previous section.
At the point 0, applying (4.8)–(4.13),

$\displaystyle
K(\frac{{\partial}}{{\partial}x_{i}},\frac{{\partial}}{{\partial}y_{j}},\frac{{\partial}}{{\partial}x_{i}},\frac{{\partial}}{{\partial}y_{j}})(0)$
$\displaystyle=-(K_{i\bar{j}i\bar{j}}(0)+2K_{i\bar{j}j\bar{i}}(0)+\overline{K_{i\bar{j}i\bar{j}}}(0))$
(4.14) $\displaystyle=-2(\frac{1}{3}-\frac{1}{6})R_{ijij}(0)$
$\displaystyle=\frac{-1}{3}R_{ijij}(0).$

Holomorphic sectional curvatures are sectional curvatures of planes spanned by
$\frac{{\partial}}{{\partial}x_{j}}$ and
$J\frac{{\partial}}{{\partial}x_{j}}=\frac{{\partial}}{{\partial}y_{j}}$.
Taking $i=j$,

$\displaystyle
K(\frac{{\partial}}{{\partial}x_{i}},\frac{{\partial}}{{\partial}y_{i}},\frac{{\partial}}{{\partial}x_{i}},\frac{{\partial}}{{\partial}y_{i}})=\frac{-1}{3}R_{iiii}=0.$
(4.15)

∎

As $(M,g)$ is a totally geodesic submanifold of $(\mathfrak{D}_{M},h)$, the
two corresponding Levi-Civita connections $\triangledown$ and
$\hat{\triangledown}$ have coincided when applying to vector fields of $M$.
Thus for vector fields in $M$, the two curvature tensors computed from either
connection will be the same. This can be easily checked following lines of the
above computations.

$\displaystyle
K(\frac{{\partial}}{{\partial}x_{i}},\frac{{\partial}}{{\partial}x_{j}},\frac{{\partial}}{{\partial}x_{i}},\frac{{\partial}}{{\partial}x_{j}})$
$\displaystyle=K(\frac{{\partial}}{{\partial}z_{i}}+\frac{{\partial}}{{\partial}\bar{z}_{i}},\frac{{\partial}}{{\partial}z_{j}}+\frac{{\partial}}{{\partial}\bar{z}_{j}},\frac{{\partial}}{{\partial}z_{i}}+\frac{{\partial}}{{\partial}\bar{z}_{i}},\frac{{\partial}}{{\partial}z_{j}}+\frac{{\partial}}{{\partial}\bar{z}_{j}})$
(4.16)
$\displaystyle=K_{i\bar{j}i\bar{j}}-2K_{i\bar{j}j\bar{i}}+\overline{K_{i\bar{j}i\bar{j}}}.$

Thus

$\displaystyle
K(\frac{{\partial}}{{\partial}x_{i}},\frac{{\partial}}{{\partial}x_{j}},\frac{{\partial}}{{\partial}x_{i}},\frac{{\partial}}{{\partial}x_{j}})(0)$
$\displaystyle=2K_{i\bar{j}i\bar{j}}(0)-2K_{i\bar{j}j\bar{i}}(0)$ (4.17)
$\displaystyle=2(\frac{1}{3}+\frac{1}{6})R_{ijij}(0)=R_{ijij}(0).$

The fact that
$J\frac{{\partial}}{{\partial}x_{j}}=\frac{{\partial}}{{\partial}y_{j}}$
together with the $J$-invariance of $K$ has implied that

$\displaystyle
K(\frac{{\partial}}{{\partial}y_{i}},\frac{{\partial}}{{\partial}y_{j}},\frac{{\partial}}{{\partial}y_{k}},\frac{{\partial}}{{\partial}y_{l}})(0)$
$\displaystyle=K(J\frac{{\partial}}{{\partial}x_{i}},J\frac{{\partial}}{{\partial}x_{j}},J\frac{{\partial}}{{\partial}x_{k}},J\frac{{\partial}}{{\partial}x_{l}})(0)$
(4.18)
$\displaystyle=K(\frac{{\partial}}{{\partial}x_{i}},\frac{{\partial}}{{\partial}x_{j}},\frac{{\partial}}{{\partial}x_{k}},\frac{{\partial}}{{\partial}x_{l}})(0)$
$\displaystyle=R_{ijij}(0).$

If the adapted complex structure is defined on the whole tangent bundle of
$(M,g)$, the metric $g$ must be non-negatively curved, cf. [Theorem 4.3, Le-
Sz]. Although their theorem was stated for compact $M$ only, the arguments
used in the proof have solely relied on local curvature properties which are
irrelevant to the compactness of the manifold. We denote the resulting complex
manifold by $T(M,g)$; the canonical Kähler metric by $h$. The following
proposition is immediate.

######  Proposition 4.2.

Suppose the adapted complex structure is defined on the whole tangent bundle
of a non-flat $(M,g)$ with the canonical Kähler metric $h$. Then the adapted
complex structure can at most be partially defined on the tangent bundle of
the Riemannian manifold $(T(M,g),h)$.

###### Proof.

Since the adapted complexification is globally defined and $g$ is not flat,
some sectional curvature $R_{ijij}$ must be positive. By Proposition 4.1, the
metric $h$ must be negatively curved at the corresponding plane. Thus the
adapted complex structure can not be defined on the whole tangent bundle of
$(T(M,g),h)$.

∎

##  5  Hyperkähler structure on tangent bundles

A hyperkähler manifold, a manifold modeled on the quaternions, is a
$4n$-dimensional Riemannian manifold $(M,\kappa)$ equipped with anti-
commutative complex structures $I$ and $J$ so that the metric $\kappa$ is tri-
Kählerian with respect to $I,J$ and $IJ$. A hyperkähler metric is naturally
Ricci-flat and real-analytic.

Through the adapted complexification the tangent bundle of a Riemannian
manifold can be partially complexified to a Kähler manifold, it is expected
that the tangent(cotangent) bundle of a Kähler manifold may be quaternionised
to a hyperkähler manifold, at least near the zero section. A local existence
theorem has been proved in [Fe] and in [Kal]: there exists a hyperkähler
structure in a neighborhood of the zero section in the cotangent(tangent)
bundle of a real-analytic Kähler manifold.

For a compact Hermitian symmetric space $M$, both the hyperkähler and the
adapted complex structures have existed in the whole $TM$, cf. [Da-Sz].

For a generalized flag manifold $G/K$, i.e., $G$ is compact semi-simple, $K<G$
is the centralizer of some $\eta\in\mathfrak{g}$, a family of complete
hyperkähler structures on the coadjoint orbit
$G^{\mathbb{C}}/K^{\mathbb{C}}=T(G/K,g_{n})$ have been successfully
constructed in [Kr2] [Kov] and [Biq]. That is to say, one of the complex
structure comes from the adapted complexification on the tangent bundle of
$(G/K,g_{n})$. The hyperkähler metric $\kappa$ is $G$-invariant and
$\kappa|_{G/K}$ is $G$-invariant in the homogeneous space $G/K$. If $G/K$ is
isotropy irreducible, $\kappa|_{G/K}$ is a normal metric and both the
hyperkähler and adapted complex structures have existed on the total space of
the tangent bundle of the Riemannian manifold $(G/K,g_{n})$.

A natural question arisen is: will the two structures exist simultaneously? It
is attempting to find examples where one of the structure has existed but not
the other one. We’ll work on some global hyperkähler structures on tangent
bundles, obtained from a hyperkähler quotient from studying Nahm’s equations.
We show at Theorem 5.5 that none of those tangent bundles can possess a
globally defined adapted complex structure.

###  5.1  Hyperkähler structure in $TG^{\mathbb{C}}$.

The coming two sections have basically followed from [Kr1] and [Da-Sw] with
some mirror changes in the notations. Given a compact Lie group $G$ with Lie
algebra $\mathfrak{g}$, let’s denote the associate infinite-dimensional
quaternion $\mathcal{A}$ as

$\mathcal{A}=\\{T_{0}+iT_{1}+jT_{2}+kT_{3};\
T_{l}:[0,1]\overset{C^{\infty}}{\longrightarrow}\mathfrak{g},\forall l\\}.$
(5.1)

$\mathcal{A}$ is an infinite-dimensional vector space endowed with three anti-
commuting complex structures $I,J,K$ given by left multiplications by $i,j,k$
respectively. An $Ad_{G}$-invariant inner product $\langle\ ,\ \rangle$ in
$\mathfrak{g}$ is specified to produce a natural $L^{2}$-metric in $\cal A$
defined as, for tangent vectors $(X_{0},X_{1},X_{2},X_{3})$ and
$(Y_{0},Y_{1},Y_{2},Y_{3})$ of $\cal A$

$\langle(X_{0},X_{1},X_{2},X_{3}),(Y_{0},Y_{1},Y_{2},Y_{3})\rangle:=\int_{0}^{1}\sum_{j=0}^{3}\langle
X_{j}(t),Y_{j}(t)\rangle dt.$ (5.2)

This metric together with the complex structures $I,J,K$ has endowed $\cal A$
with a hyperkähler structure. The three relative Kähler forms are

$\displaystyle\omega_{I}=dT_{0}dT_{1}+dT_{2}dT_{3};\text{ and cyclically.}$
(5.3)

The function

$\displaystyle
f(T):=\frac{1}{2}T_{1}^{2}+\frac{1}{4}T_{2}^{2}+\frac{1}{4}T_{3}^{2}$ (5.4)

is a Kähler potential for $\omega_{I}$ since

$\displaystyle 2i{\partial}{\bar{\partial}}f=dIdf$
$\displaystyle=dI(T_{1}dT_{1}+\frac{1}{2}T_{2}dT_{2}+\frac{1}{2}T_{3}dT_{3})$
(5.5)
$\displaystyle=d(-T_{1}dT_{0}+\frac{1}{2}T_{2}dT_{3}-\frac{1}{2}T_{3}dT_{2})$
$\displaystyle=dT_{0}\wedge dT_{1}+dT_{2}\wedge dT_{3}=\omega_{I}.$

Because $T_{0}+iT_{1}+jT_{2}+kT_{3}=T_{0}+iT_{1}+(T_{2}+iT_{3})j,$ the vector
space $(\mathcal{A},I)$ can be interpreted by
$\alpha=T_{0}+iT_{1},\beta=T_{2}+iT_{3}.$

The gauge group $\mathcal{G}$ consists of smooth maps $g:[0,1]\to G$ has acted
on $\mathcal{A}$ by

$\displaystyle g\cdot T_{0}=gT_{0}g^{-1}-\frac{dg}{dt}g^{-1};\;\ g\cdot
T_{j}=gT_{j}g^{-1},\;j=1,2,3.$ (5.6)

The $\mathcal{G}$-action has kept all $I,J,K$ and the relative Kähler forms.
Let $\mathcal{G}_{0}:=\\{g\in\mathcal{G}:g(0)=g(1)=1\\}$, the corresponding
moment map of this $\mathcal{G}_{0}$-action is denoted by
$\mu=(\mu_{I},\mu_{J},\mu_{K})$. The zero locus $\mu^{-1}(0)$ is exactly the
solution space of the Nahm’s equations

$\displaystyle\frac{dT_{1}}{dt}+[T_{0},T_{1}]+[T_{2},T_{3}]=0\;\;\text{ and
cyclically.}$ (5.7)

The moduli space $\mu^{-1}(0)/\mathcal{G}_{0}$ is indeed a hyperkähler
manifold. Thus, given a compact Lie group $(G,\langle\ ,\ \rangle)$, a
hyperkähler manifold $\mu^{-1}(0)/\mathcal{G}_{0}$ is associated to it through
the above construction.

Fixing one complex structure $I$ and set
$\alpha=T_{0}+iT_{1},\beta=T_{2}+iT_{3}$, the $\mathcal{G}_{0}$-action on
$\mu^{-1}(0)$ can be extended to a holomorphic $\mathcal{G}_{0}^{c}$-action on
$\mu_{c}^{-1}(0):=(\mu_{J}+i\mu_{K})^{-1}(0)$ to make
$\mu_{c}^{-1}(0)/\mathcal{G}_{0}^{c}$ a Kähler manifold biholomorphic to
$({\mu}^{-1}(0)/\mathcal{G}_{0},I)$.

One advantage of taking the complex manifold
$\mu_{c}^{-1}(0)/\mathcal{G}_{0}^{c}$ is to develop a concrete biholomorphism
with $TG^{\mathbb{C}}$.

$\displaystyle({\mu}^{-1}(0)/\mathcal{G}_{0},I)\overset{\eta}{\longrightarrow}{\mu_{c}}^{-1}(0)/\mathcal{G}_{0}^{c}\overset{\phi}{\longrightarrow}TG^{\mathbb{C}}=G^{\mathbb{C}}\times\mathfrak{g}^{c}$
(5.8)
$\displaystyle(T_{0},T_{1},T_{2},T_{3})\longrightarrow(\alpha=T_{0}+iT_{1},\beta=T_{2}+iT_{3})\longrightarrow(g(0)^{-1},\beta(1))$

where $g\in\mathcal{G}^{c}$ such that $g\cdot\alpha=0$ and $g(1)=id.$ The
requirement $g\cdot\alpha=0$ is equivalent to solving the linear ordinary
differential equation $\frac{dg}{dt}=g\alpha$ and the initial condition
$g(1)=id$ will determine such a $g$ uniquely.

Restricted to the subspace $\\{T_{2}=T_{3}=0\\}$, we define

$\displaystyle\varphi:{\mu}^{-1}(0)/\mathcal{G}_{0}|_{T_{2}=T_{3}=0}$
$\displaystyle\to G\times\mathfrak{g}$ (5.9) $\displaystyle(T_{0},T_{1})$
$\displaystyle\to(\xi(0)^{-1},T_{1}(1))$

where $\xi$ is the unique element in $\mathcal{G}$ gauging $T_{0}$ to 0 with
the initial condition $\xi(1)=id.$

######  Lemma 5.1.

$\varphi$ in (5.9) is a diffeomorphism whose inverse can be explicitly written
down as

$\displaystyle\varphi^{-1}:G\times\mathfrak{g}$
$\displaystyle\to{\mu}^{-1}(0)/\mathcal{G}_{0}|_{T_{2}=T_{3}=0}$ (5.10)
$\displaystyle(a,v)$ $\displaystyle\to(-\frac{dh}{dt}h^{-1},hvh^{-1})$

for some $h\in\mathcal{G}$ with the boundary condition $h(0)=a,h(1)=id.$

###### Proof.

Given $a\in G$, it is always possible to find an $h\in\mathcal{G}$ with
boundary conditions $h(0)=a,h(1)=id.$ A direct computation verifies both
$-\frac{dh}{dt}h^{-1}$ and $hvh^{-1}$ are smooth mappings from $[0,1]$ to
$\mathfrak{g}$ satisfying

$\displaystyle\frac{d}{dt}(hvh^{-1})+[-\frac{dh}{dt}h^{-1},hvh^{-1}]=0$ (5.11)

which is the equation $\dot{T}_{1}+[T_{0},T_{1}]=0$. So,
$(-\frac{dh}{dt}h^{-1},hvh^{-1})\in{\mu}^{-1}(0)/\mathcal{G}_{0}|_{T_{2}=T_{3}=0}.$

It remains to show the mapping is well-defined. Suppose there are two
$h,l\in\mathcal{G}$ sharing the same boundary conditions. Then
$lh^{-1}\in\mathcal{G}$ as well which has gauged the point
$(-\frac{dh}{dt}h^{-1},hvh^{-1})$ to the point
$(-\frac{dl}{dt}l^{-1},lvl^{-1})$. In other words, they have represented the
same point in the moduli space; the mapping $\varphi^{-1}$ is well-defined. ∎

Restricted to the subspace $T_{2}=T_{3}=0$ in (4.10) along with the
surjectivity proved in Lemma 5.1, the following diffeomorphisms are well-
defined.

$\displaystyle G\times\mathfrak{g}\overset{\varphi^{-1}}{\longrightarrow}$
$\displaystyle({\mu}^{-1}(0)/\mathcal{G}_{0}|_{T_{2}=T_{3}=0},I)$
$\displaystyle\overset{\eta}{\longrightarrow}{\mu_{c}}^{-1}(0)/\mathcal{G}^{c}|_{\beta=0}\overset{\phi}{\longrightarrow}G^{\mathbb{C}}$
(5.12) $\displaystyle(\xi(0)^{-1},T_{1}(1))\longrightarrow$
$\displaystyle\;\;\;\;\;\;\;\;\;(T_{0},T_{1})$
$\displaystyle\longrightarrow\alpha=T_{0}+iT_{1}\longrightarrow g(0)^{-1}$

where $\xi\in\mathcal{G}$ such that $\xi\cdot T_{0}=0$ and
$\xi(1)=id;g\in\mathcal{G}^{c}$ such that $g\cdot\alpha=0$ and $g(1)=id$. When
$T_{2}\equiv T_{3}\equiv 0$, the criteria for $(T_{0},T_{1})$ is simply the
baby Nahm’s equation $\frac{dT_{1}}{dt}+[T_{0},T_{1}]=0.$ The
$\mathcal{G}$-invariance of the equation together with the fact $\xi\cdot
T_{0}=0$ has implied $\frac{d(\xi\cdot T_{1})}{dt}=0$. That is, $\xi\cdot
T_{1}$ is a constant which can be taken as

$\displaystyle(\xi\cdot T_{1})(t)=\xi(1)T_{1}(1)\xi(1)^{-1}=T_{1}(1),\ \forall
t\in[0,1].$ (5.13)

######  Lemma 5.2.

$\displaystyle\phi\cdot\eta\cdot\varphi^{-1}:$ $\displaystyle
G\times\mathfrak{g}\to G^{\mathbb{C}}$ (5.14) $\displaystyle(a,v)\to a\exp
iv.$

This diffeomorphism has pulled back the complex structure of $G^{\mathbb{C}}$
to the adapted complex structure on $T(G,\langle\ ,\ \rangle)$.

###### Proof.

By Lemma 5.1, every point $(a,v)\in G\times\mathfrak{g}$ is of the form
$(\xi(0)^{-1},T_{1}(1))$ as in described in (5.9). Take

$\displaystyle\hat{g}(t)=\exp{(i(t-1)T_{1}(1))}\xi(t):[0,1]\overset{C^{\infty}}{\longrightarrow}G^{c}.$
(5.15)

Then $\hat{g}(1)=\xi(1)=id,\hat{g}\in\mathcal{G}^{c}$ and

$\displaystyle\hat{g}\cdot\alpha$
$\displaystyle=\exp{(i(t-1)T_{1}(1))}\xi(t)\cdot(T_{0}+iT_{1})$ (5.16)
$\displaystyle=\exp{(i(t-1)T_{1}(1))}\cdot(\xi\cdot T_{0}+\xi\cdot iT_{1})$
$\displaystyle=\exp{(i(t-1)T_{1}(1))}\cdot(iT_{1}(1))$
$\displaystyle=e^{(i(t-1)T_{1}(1))}iT_{1}(1)e^{(-i(t-1)T_{1}(1))}-\frac{d}{dt}(e^{(i(t-1)T_{1}(1))})e^{(-i(t-1)T_{1}(1))}$
$\displaystyle=iT_{1}(1)-iT_{1}(1)=0.$

Both $g$ and $\hat{g}$ have gauged $\alpha$ to 0 with the same initial
condition at $t=1$, by the uniqueness of ODE, $\hat{g}\equiv g$.

$\displaystyle g^{-1}(0)={\hat{g}}^{-1}(0)=\xi(0)^{-1}\exp(iT_{1}(1)).$ (5.17)

Identifying $(\xi(0)^{-1},T_{1}(1))$ with $(a,v)$, the diffeomorphism is
proved. The last statement is a consequence of Proposition 2.1 by taking
$H=id$. ∎

In (4.10), the diffeomorphism $\phi\cdot\eta$ has pushed forward the Kähler
form $\omega_{I}$ to a Kähler form $(\phi\cdot\eta)_{*}\omega_{I}$ in
$G^{\mathbb{C}}\times\mathfrak{g}^{c}$ determining the hyperkähler metric
$\kappa$ in $G^{\mathbb{C}}\times\mathfrak{g}^{c}$. Restricted to the subspace
$\\{T_{2}=T_{3}=0\\}$, $\phi\cdot\eta$ at $\eqref{e17}$ is still a
biholomorphism from $({\mu}^{-1}(0)/\mathcal{G}_{0}|_{T_{2}=T_{3}=0},I)$ to
$G^{\mathbb{C}}$ sending the Kähler form $\hat{\omega}_{I}=dT_{0}dT_{1}$ to
the Kähler form $(\phi\cdot\eta)_{*}{\hat{\omega}_{I}}$ in $G^{\mathbb{C}}$
corresponding to the Kähler metric $\kappa|_{G^{c}}$.

The diffeomorphisms in (5.29) has pulled back the Kähler form
$(\phi\cdot\eta)_{*}{\hat{\omega}_{I}}$ to the Kähler form
$\varphi^{-1*}\hat{\omega}_{I}$ on $G\times\mathfrak{g}$. By (5.4), the Kähler
potential of $\hat{\omega}_{I}$ is
$\hat{f}(T_{0},T_{1})=\frac{1}{2}\int_{0}^{1}\langle T_{1}(t),T_{1}(t)\rangle
dt$. Since $\langle\ ,\ \rangle$ is $Ad_{G}$-invariant, $\langle
Ad_{g}v,Ad_{g}v\rangle=\langle v,v\rangle$ for any $g\in G$. That is, for
$h\in\mathcal{G}$

$\displaystyle\hat{f}(-\frac{dh}{dt}h^{-1},hvh^{-1})=\frac{1}{2}\int_{0}^{1}\langle
h(t)vh(t)^{-1},h(t)vh(t)^{-1}\rangle dt=\frac{|v|^{2}}{2}.$ (5.18)

The identification at (5.10) shows that the corresponding Kähler potential
$\tilde{f}$ in the $G\times\mathfrak{g}$ model is, for $(a,v)\in
G\times\mathfrak{g}$, $\tilde{f}(a,v)=\frac{|v|^{2}}{2}$, which is the
standard Kähler potential at (3.2).

Thus, the corresponding Kähler metric is the canonical Kähler metric of the
adapted complexification on $T(G,\langle\ ,\ \rangle).$ We conclude the above
discussions as the following.

######  Proposition 5.3.

Let $g$ be a bi-invariant metric in the compact Lie group $G$, and let $h$
denote the canonical Kähler metric of the adapted complexification on
$T(G,g)$. A complete hyperkähler metric $\kappa$ has existed on the tangent
bundle of the Kähler manifold $(TG,h)$ whose restriction to $TG$ is $h$.

###  5.2  Hyperkähler structure in $TG^{\mathbb{C}}/H^{\mathbb{C}}$.

Throughout this section, $H<G$ is a closed subgroup of the compact Lie group
$G$ and the notation $\mathcal{M}$ is used to denote the hyperkähler manifold
$\mu^{-1}(0)/\mathcal{G}_{0}$ developed in §5.1. The $H$-action on
$\mathcal{M}$ is given by the gauge group $\mathcal{H}=\\{g:[0,1]\to
G;g(0)=id,g(1)\in H\\}\subset\mathcal{G}$ via

$\displaystyle g\cdot T_{0}=gT_{0}g^{-1}-\frac{dg}{dt}g^{-1};\;\ g\cdot
T_{j}=gT_{j}g^{-1},\;j=1,2,3.$ (5.19)

It is clear this $H$-action has preserved the hyperkähler structure in
$\mathcal{M}$. The notation $\pi_{\mathfrak{h}}$ is used to denote the
projection of $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$ to the
$\mathfrak{h}$ component. The hyperkähler moment map $\Phi$ for the $H$-action
on $\mathcal{M}$ is, cf. Lemma 2 in [Da-Sw],

$\Phi(T)=(\Phi_{I}(T),\Phi_{J}(T),\Phi_{K}(T))=(\pi_{\mathfrak{h}}T_{1}(1),\pi_{\mathfrak{h}}T_{2}(1),\pi_{\mathfrak{h}}T_{3}(1)).$

Thus,
$\Phi^{-1}(0)=\\{(T_{0},T_{1},T_{2},T_{3})\in\mathcal{M};T_{j}(1)\in\mathfrak{m},j=1,2,3\\}$
and
$\Phi_{c}^{-1}(0)=\\{(\alpha,\beta)\in\mathcal{M};\beta(1)\in\mathfrak{m}^{c}\\}$.
The hyperkähler quotient $\Phi^{-1}(0)/\mathcal{H}$ has created a new
hyperkähler manifold; Kähler forms and Kähler potentials are descending from
those of $\mathcal{M}$. The $H$-action can be extended to a holomorphic
$H^{c}$-action via the gauge group $\mathcal{H}^{c}=\\{g:[0,1]\to
G^{\mathbb{C}};g(0)=id,g(1)\in H^{\mathbb{C}}\\}$.

A necessary condition to identify $(\Phi^{-1}(0)/\mathcal{H},I)$ with
$\Phi_{c}^{-1}(0)/\mathcal{H}^{c}$ is the stable points $\mathcal{M}^{s}$ of
the $H^{c}$-action must be $\Phi_{c}^{-1}(0)$ where

$\displaystyle\mathcal{M}^{s}:=\\{T\in\mathcal{M}:\mathcal{H}^{c}\cdot
T\cap\Phi^{-1}(0)\neq\emptyset\\}.$ (5.20)

Since $\mathcal{M}$ is of finite dimensional, the $H$-action is free and the
potential function $f$ at (5.4) is bounded above, the properness of $f$ is the
crucial ingredient remained to assure the identification of
$(\Phi^{-1}(0)/\mathcal{H},I)$ with $\Phi_{c}^{-1}(0)/\mathcal{H}^{c}$, cf.
p.199 [May1] and Theorem 3.6 [May3]. The function $F(k,X)$ listed in [May2] is
the Kähler potential of $\omega_{I}$ computed in the model $\mathcal{W}\subset
G\times{\mathfrak{g}}^{3}$, which is diffeomorphic to $\mathcal{M}$ by Theorem
3 of [Da-Sw]. The corresponding potential in $\mathcal{M}$ is $f$ whose
properness has inherited from that of $F$ and the following biholomorphism is
established:

$\displaystyle(\Phi^{-1}(0)/\mathcal{H},I)\to\Phi_{c}^{-1}(0)/\mathcal{H}^{c};\;\;(T_{0},T_{1},T_{2},T_{3})\to(\alpha=T_{0}+iT_{1},\beta=T_{2}+iT_{3}).$
(5.21)

The next attempt is to interpret $\Phi_{c}^{-1}(0)/\mathcal{H}^{c}$ in terms
of $G$ and $H$. The homogeneous space $G^{\mathbb{C}}/H^{\mathbb{C}}$ is
reductive and
$TG^{\mathbb{C}}/H^{\mathbb{C}}=G^{\mathbb{C}}\times_{H^{\mathbb{C}}}\mathfrak{m}^{c}$.
Defining

$\displaystyle\phi:{\Phi_{c}}^{-1}(0)/\mathcal{H}^{c}\longrightarrow$
$\displaystyle
TG^{\mathbb{C}}/H^{\mathbb{C}}=G^{\mathbb{C}}\times_{H^{\mathbb{C}}}\mathfrak{m}^{c}$
(5.22) $\displaystyle(\alpha,\beta)\longrightarrow$
$\displaystyle(g(0)^{-1},\beta(1))$

where $g$ is the unique element in $\mathcal{G}$ with $g(1)=id$ and
$g\cdot\alpha=0$. For any $\zeta\in\mathcal{H}^{c}$,
$(\zeta\cdot\beta)(1)=\zeta(1)\beta(1)\zeta(1)^{-1}=Ad_{\zeta(1)}\beta(1).$ On
the other hand, $\zeta(1)g\zeta^{-1}$ is the identity at $t=1$ and has gauged
$\zeta\cdot\alpha$ to zero since

$\zeta(1)g\zeta^{-1}\cdot(\zeta\cdot\alpha)=\zeta(1)g\cdot\alpha=\zeta(1)\cdot
0=0.$

Thus,
$\phi(\zeta\cdot(\alpha,\beta))=(g(0)^{-1}\zeta(1)^{-1},Ad_{\zeta(1)}\beta(1))$
which is equivalent to $(g(0)^{-1},\beta(1))$ in the space
$G^{\mathbb{C}}\times_{\mathfrak{h}^{c}}\mathfrak{m}^{c}$. The mapping $\phi$
is well-defined and settles the diffeomorphism. Thus,
$TG^{\mathbb{C}}/H^{\mathbb{C}}$ has a complete hyperkähler structure coming
from that of $\Phi^{-1}(0)/\mathcal{H}$ and the restriction of the hyperkähler
metric $\kappa$ to $G/H$ is the original $Ad_{G}$-invariant inner product
$\langle\ ,\ \rangle$ in $\mathfrak{g}$, cf. Proposition 6 in [Da-Sw].

The hyperkähler manifold $(TG^{\mathbb{C}}/H^{\mathbb{C}},\kappa)$ may be
viewed as a hyperkähler structure on the tangent bundle of the Kähler manifold
$(G^{\mathbb{C}}/H^{\mathbb{C}},\kappa|_{G^{\mathbb{C}}/H^{\mathbb{C}}}).$

We consider the following $S^{1}$-action on $\mathcal{A}$ at (5.1), for
$\theta\in[0,2\pi)$

$\displaystyle
e^{i\theta}\cdot(T_{0},T_{1},T_{2},T_{3}):=(T_{0},T_{1},\cos\theta\
T_{2}-\sin\theta\ T_{3},\sin\theta\ T_{2}+\cos\theta\ T_{3}).$ (5.23)

It is direct to see this $S^{1}$-action has preserved both $\mu^{-1}(0)$ and
$\Phi^{-1}(0)$; it also interchanges with any $\mathcal{G}$-action. Briefly,
we’ll show this action has preserved the complex structure $I$ and the Kähler
form $\omega_{I}$ and hence has preserved the hyperkähler metric $\kappa$ of
$\Phi^{-1}(0)/\mathcal{H}.$ Let $X$ denote the vector field generated by this
$S^{1}$-action.

$\displaystyle X(T)=\frac{d}{d\theta}|_{\theta=0}e^{i\theta}\cdot
T=(0,0,-T_{3},T_{2})=-T_{3}\frac{{\partial}}{{\partial}T_{2}}+T_{2}\frac{{\partial}}{{\partial}T_{3}}.$
(5.24)

######  Lemma 5.4.

The $S^{1}$-action has preserved the hyperkähler metric $\kappa$; both
$L_{X}\omega_{I}$ and $L_{X}I$ have vanished.

###### Proof.

$\displaystyle
L_{X}\omega_{I}=L_{X}(dT_{2}dT_{3})=d(i_{X}(dT_{2}dT_{3}))=-d(T_{2}dT_{2}+T_{3}dT_{3})=0.$
(5.25)

Applying the formula

$\displaystyle(L_{X}I)(Y)=L_{X}(IY)-I(L_{X}Y)=[X,IY]-I[X,Y]$ (5.26)

to see $(L_{X}I)(\frac{{\partial}}{{\partial}T_{j}})=0,\forall j.$ Since both
$I$ and $\omega_{I}$ are preserved by the $S^{1}$-action, the hyperkähler
metric is an isometry with respect to this action. ∎

The $S^{1}$-action by scalar multiplication on the fibers is an isometry, by
the uniqueness of Theorem A in [Fe], we conclude that near the zero section of
$TG^{\mathbb{C}}/H^{\mathbb{C}}$, the hyperkähler metric $\kappa$ has
coincided with the one constructed over the tangent bundle of the Kähler
manifold
$(G^{\mathbb{C}}/H^{\mathbb{C}},\kappa|_{G^{\mathbb{C}}/H^{\mathbb{C}}})$ by
Feix using the twistor method.

The next step is to have a close look at the Kähler manifold
$(G^{\mathbb{C}}/H^{\mathbb{C}},\kappa|_{G^{\mathbb{C}}/H^{\mathbb{C}}}).$
Analogous to the $TG^{\mathbb{C}}$ case, we work in the subspace
$\\{T_{2}=T_{3}=0\\}$ to set a diffeomorphism

$\displaystyle\varphi:{\Phi}^{-1}(0)/\mathcal{H}|_{T_{2}=T_{3}=0}$
$\displaystyle\to G\times_{H}\mathfrak{m}$ (5.27) $\displaystyle(T_{0},T_{1})$
$\displaystyle\to(\xi(0)^{-1},T_{1}(1))$

where $\xi\in\mathcal{G},\xi(1)=id,\ \xi\cdot T_{0}=0.$ This $\varphi$ is
well-defined as the mapping in (5.22).

$\displaystyle\varphi^{-1}:G\times_{H}\mathfrak{m}$
$\displaystyle\to{\Phi}^{-1}(0)/\mathcal{H}|_{T_{2}=T_{3}=0};$ (5.28)
$\displaystyle(a,v)$ $\displaystyle\to(-\frac{dh}{dt}h^{-1},hvh^{-1})$

for some $h\in\mathcal{G}$ with the boundary condition $h(0)=a,h(1)\in H.$

The following diffeomorphisms are well-defined.

$\displaystyle G\times_{H}\mathfrak{m}\overset{\varphi^{-1}}{\longrightarrow}$
$\displaystyle({\Phi}^{-1}(0)/\mathcal{H}|_{T_{2}=T_{3}=0},I)$
$\displaystyle\overset{\eta}{\longrightarrow}{\Phi}_{c}^{-1}(0)/\mathcal{H}^{c}|_{\beta=0}\overset{\phi}{\longrightarrow}G^{\mathbb{C}}\times_{H^{\mathbb{C}}}0$
(5.29) $\displaystyle(\xi(0)^{-1},T_{1}(1))\longrightarrow$
$\displaystyle\;\;\;\;\;\;\;\;\;(T_{0},T_{1})$
$\displaystyle\longrightarrow\alpha=T_{0}+iT_{1}\longrightarrow g(0)^{-1}$

where $\xi\in\mathcal{G}$ such that $\xi\cdot T_{0}=0$ and
$\xi(1)=id;g\in\mathcal{G}^{c}$ such that $g\cdot\alpha=0$ and $g(1)=id$. For
any $\zeta\in\mathcal{H}^{c}$, $\zeta(1)g\zeta^{-1}$ is the identity at $t=1$
and has gauged $\zeta\cdot\alpha$ to zero,
$\phi(\zeta\cdot\alpha)=\zeta(0)g^{-1}(0)\zeta^{-1}(1)=g^{-1}(0)\zeta^{-1}(1)$.
The mapping $\phi$ is well-defined since $\zeta(1)\in H^{\mathbb{C}}$.

Following the same lines in the $TG^{\mathbb{C}}$ case in Lemma 5.2, we’ll
obtain a mapping similar to (5.14)

$\displaystyle\phi\cdot\eta\cdot\varphi^{-1}:G\times_{H}\mathfrak{m}\to
G^{\mathbb{C}}/H^{\mathbb{C}};\ \ \ (a,v)\to a\exp ivH^{\mathbb{C}}.$ (5.30)

By Proposition 2.1, $G^{\mathbb{C}}/H^{\mathbb{C}}$ is the adapted
complexification on the tangent bundle of the Riemannian manifold
$(G/H,\mathfrak{b})$ where $\mathfrak{b}$ is the normal metric in $G/H$
induced from $\langle\ ,\ \rangle$.

As computed at (5.18), the Kähler potential of the Kähler metric
$(\phi\eta\varphi^{-1})^{*}\kappa|_{G^{\mathbb{C}}/H^{\mathbb{C}}}$ at the
point $(a,v)\in G\times_{H}\mathfrak{m}$ is $\frac{|v|^{2}}{2}$. Thus,
$\kappa|_{G^{\mathbb{C}}/H^{\mathbb{C}}}$ is the canonical Kähler metric of
the adapted complexification on $T(G/H,\mathfrak{b})$.

Let $G$ be a compact Lie group; $H<G$ be a closed subgroup; $g_{n}$ be the
normal metric on $G/H$ induced from a bi-invariant metric $g$ in $G$. Let
$\kappa$ be the hyperkähler metric on $TG^{\mathbb{C}}/H^{\mathbb{C}}$
associated to $g$ obtained in this section. We have the following theorem.

######  Theorem 5.5.

(I). The adapted complex structure has existed on the whole tangent bundle of
$(G/H,g_{n})$ with the canonical Kähler metric $h$. The resulting Kähler
manifold is denoted as $(T(G/H,g_{n}),h)$ which is biholomorphically isometric
to the Kähler manifold
$(G^{\mathbb{C}}/H^{\mathbb{C}},\kappa|_{G^{c}/H^{c}})$.

(II). Taking the tangent bundle to the Riemannian manifold $(T(G/H,g_{n}),h)$,
a hyperkähler metric $\kappa$ is obtained in
$T(T(G/H,g_{n}),h)=T(G^{\mathbb{C}}/H^{\mathbb{C}},\kappa|_{G^{\mathbb{C}}/H^{\mathbb{C}}})$
with $\kappa|_{G^{\mathbb{C}}/H^{\mathbb{C}}}=h;h_{G/H}=g_{n}$. All the three
metrics $g_{n},h$ and $\kappa$ are complete.

(III). Though the hyperkähler structure is defined on the whole
$T(G^{\mathbb{C}}/H^{\mathbb{C}},\kappa|_{G^{\mathbb{C}}/H^{\mathbb{C}}})$,
the adapted complex structure can at most be partially defined on it.

###### Proof.

Since
$(G^{\mathbb{C}}/H^{\mathbb{C}},\kappa|_{G^{\mathbb{C}}/H^{\mathbb{C}}})=(T(G/H,g_{n}),h)$
is the adapted complexification over the Riemannian manifold $(G/H,g_{n})$,
the statement (III) has followed from Proposition 4.2. ∎

Remark. Given a generalized flag manifold $G^{\mathbb{C}}/P$, using Nahm’s
equations with fast decay at the infinity, Biquard [Biq] has constructed a
complete hyperkähler metric $\kappa$ on $T^{*}G^{\mathbb{C}}/P$. After this
work is completely done, we realized that $\kappa|_{G^{\mathbb{C}}/P}$ is the
given Kähler metric in viewing $G^{\mathbb{C}}/P$ as the orbit $Ad_{G}(\tau)$
with the Kirillov-Kostant symplectic form. Though this fact was not mentioned
in [Biq], it turns out to be true, cf. p.7 in [Bie]. Thus, for those non-
positively curved $G^{\mathbb{C}}/P$, there are plenty of them, the adapted
complex structure is not defined on the whole tangent bundle whereas a
hyperkähler structure does exist in $T^{*}G^{\mathbb{C}}/P$.

## References

  * [Bie] R. Bielawski Lie groups, Nahm’s equations and hyperkähler manifolds.  Algebraic groups. Universitätsverlag Göttingen, Göttingen (2005) 1-17.
  * [Biq] O. Biquard Sur les équations de Nahm et la structure de Poisson des algèbres de Lie semi-simples complexes. Math. Ann. 304 (1996) 253-276.
  * [Da-Sw] A. Dancer $\&$ A. Swann Hyperkähler metrics associated to compact Lie groups.  Math. Proc. Cambridge Philos. Soc. 120, no.1 (1996) 61-69.
  * [Da-Sz] A. Dancer $\&$ R. Szőke Symmetric spaces, adapted complex structures and hyperkähler structures. Quart. J. Math. Oxford 48, no.2 (1997) 27-38.
  * [Fe] B. Feix Hyperkähler metrics on cotangent bundles. J. reine angew. Math. 532 (2001) 33-46.
  * [Gu-St] V. Guillemin & M. Stenzel Grauert tubes and the homogeneous Monge-Ampère equation. J.D.G. 34 (1991) 561-570.
  * [He-Sch] P. Heinzner & G. Schwarz Cartan decomposition of the moment map. Math. Ann. 337 (2007), no.1, 197-232.
  * [Kal] D. Kaledin A canonical hyperkähler metric on the total space of a cotangent bundle. Univ. Studi Roma “La Sapienza”, Roma (1999) 195-230.
  * [Kan] S.-J. Kan The asymptotic expansion of a CR invariant and Grauert tubes. Math. Ann. 304 (1996) 63-92.
  * [Ko-No] S. Kobayashi & K. Nomizu Foundations of differential geometry Volume II (1996).
  * [Kov] A. Kovalev Nahm’s equations and complex adjoint orbits. Quart. J. Math. 47 (1996) 41-58.
  * [Kr1] P. Kronheimer A hyperkähler structure on the cotangent bundle of a complex Lie group. 1988 preprint; arXiv:math 0409253(DG).
  * [Kr2] P. Kronheimer A hyperkähler structure on coadjoint orbits of a semisimple complex group. J. Lond. Math. Soc. 42 (1990) 193-208.
  * [Le-Sz] L. Lempert & R. Szőke Global solutions of the homogeneous complex Monge-Ampére equation and complex structures on the tangent bundle of Riemannian manifolds. Math. Ann. 290 (1991) 689-712.
  * [May1] M. Mayrand Stratified hyperkähler spaces from semisimple Lie algebras. Transform. Groups 25 (2020) no.1, 191-215.
  * [May2] M. Mayrand Erratum to “Stratified hyperkähler spaces from semisimple Lie algebras”. Transform. Groups 25 (2020) no.2, 645-646.
  * [May3] M. Mayrand Nahm’s equations in hyperkähler geometry. Lecture notes, The Fields institute, Canada (2020).
  * [Mo] G.D. Mostow Some new decomposition theorems for semisimple groups. Memoirs of the AMS 14 (1955) 31-54.
  * [Sz1] R. Szőke Automorphism groups of certain Stein manifolds. Math. Z. 219 (1995) 357-385.
  * [Sz2] R. Szőke Adapted complex structures and Riemannian homogeneous spaces. Ann. Polon. Math. Z. 70 (1998) 215-220.

Institute of Mathematics, Academia Sinica

6F, Astronomy-Mathematics Building

No. 1, Sec. 4, Roosevelt Road

Taipei 10617, TAIWAN
E-mail address<EMAIL_ADDRESS>