数理解析研究所講究録1163 巻 2000 年 117-125

117

On the Rumin complex
Akahori, Takao (赤堀隆夫)

The moduli of complex structures seems to be related to the hyperbolicity.
Here, we treat the moduli of isolated singularities. Let (V, ) be a normal. Consider the link $M$ ,isolated singularity in a complex euclidean space, cendefined by the intersection of (V, ) and the real hyperspheretered at the origin with the radius of . Over this link $M$ , a CR structure
is naturally induced from . Concerning the moduli problem of isolated
singularities, Kuranishi initiated the deformation theory of CR structures.
Of course, the problem of constructing a versal family of deformations of
exisolated singularities is done by several authors, almost 30 years ago(),
, in algebraic
more direct
ample, Grauert,Douady,geometry). However, our approach( CR geometry) has one geometric aspect. In order to introduce ”several new methods in topology (Symplectic
structure, Seiberg-Witten invariants,etc..)” to the moduli of isolated singularities, our method is definitely more accessible( are directly treating
links, contact structures and CR structures).
The purpose of this survey is to introduce the brief sketch of the Rumethod in the deformation theory of CR structures and establish a
precise proof will be publishedversal family in the 5-dimensional case(elsewhere). Obviously, the moduli of CR structures should be related to”Seiberg-Witten equation in CR strucures” In the future, I wolud like to-Witten equatin in CR strucdiscuss the relation of our settingtures” But, at the present time, our project in this direction is in process.
, briefly. Anyway,
Here, we only sketch our joint work
we start with the deformation theory of CR structures.
$\mathit{0}$$C^{N}$$S_{\epsilon}^{2N-1}(\mathit{0})$$\mathit{0}$$\mathit{0}$$\epsilon$$V$$\mathrm{f}_{\mathrm{o}\mathrm{r}}$$\mathrm{e}\mathrm{t}\mathrm{c}..$$\mathrm{W}\mathrm{i}\mathrm{t}\mathrm{h}$$\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{h}_{0}\mathrm{d}(\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}$$\mathrm{i}\mathrm{n}$$\mathrm{w}\mathrm{e}$$\min^{)}\mathrm{s}$$\mathrm{t}\mathrm{h}\mathrm{e}$)’$\mathrm{a}\mathrm{n}\mathrm{d}$$\mathrm{S}\mathrm{e}\mathrm{i}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{g}$$([\mathrm{A}-\mathrm{G}- \mathrm{L}[1]],[\mathrm{A}_{-\mathrm{G}\mathrm{L}}-[2]])$

1Standard Deformation Complex

differenLet $(M, 0T”)$ be a CR structure. This means that $M$ is atiable manifold with real odd dimension and $0T’$ is a sub-vector bundle ofthe complexfied tangent bundle $C\otimes TM$ satisfying:$C^{\infty}$

118
(1)

$0_{T’’}\cap 0_{T’}=0,$ $dimC \frac{C\otimes TM}{0T’+^{0}T’},=1$$[\Gamma(M, 0_{T}//), \mathrm{r}(M, 0\tau’’)]\subset\Gamma(M, 0Tl/)$(2),

where $0T’=\overline{0T’’}$ .
Now we assume that there is a global real vector fieldevery point of $M$ ,$\xi$satisfying; for$p$$\xi_{p}\not\in 0\tau//+0T\prime pp$(3).

We set$T’:=^{0}\tau’+C\otimes\xi$(4),means the line bundle generated by . For brevity, we use
wherevectorthe notation $F$ for this line bundle. By using this $T’$ , we set abundle decomposition$\xi$$C\otimes\xi$$C^{\infty}$(5)$C\otimes TM=\tau/+T0//$ .

from
Following [A1], we introduce a first order differential operator
means$\Gamma(M, T’)$ to), by: for $u\in\Gamma(M, T’)$ ,sections of (resp.
the space of$\overline{\partial}_{T’}$$\Gamma(M, T^{l}\otimes(^{0}T’’)*)(\mathrm{r}(M, T^{;})(\mathrm{r}\mathrm{e}\mathrm{S}\mathrm{p}.

$\tau’$$C^{\infty}$$X\in 0T_{\mathrm{P}’}’,$$p$\Gamma(M, T’\otimes(^{0}T^{\prime l})*))$$T’\otimes(^{0}T^{\prime/})*$

is a point of $M$ ,$\overline{\partial}_{T};u(X)=[X, u]_{T^{l}}$,(6)

to $T’$ according to (5), andwhere $[X, u]_{T’}$ means the projection of
doesn’t depend on the extenextension of $X$ tomeans asion). Like the case for scalar valued differential forms, we can introduce, operators, $p=1,2,$(see [A1]) and we have a differential complex$[\tilde{X}, u]$$M(\overline{\partial}_{\tau^{lu}}(X)$$C^{\infty}$$\overline{\partial}_{T}^{(p)}$$\tilde{X}$$\cdots$$0arrow\Gamma(M, \tau’)^{\overline{\partial}\prime}arrow^{T}\Gamma(M, \tau’\otimes(^{0}\tau\prime\prime)^{*},)^{\overline{\partial}_{\tau’}^{(1}}\overline{\partial}_{T}^{(}p)arrow)\Gamma(M, T/\otimes\wedge(20\tau/l)*)arrow$$arrow\Gamma(M, T’\otimes\wedge p(_{\backslash }^{0\prime}T’)^{*})arrow\Gamma(M, T/\otimes\wedge p+1(0\tau\prime\prime)*)arrow\cdots$

(7)

.
with
This complex is called the standard deformation complex. We, briefly recall
the deformation theory of CR structures. Let $(M, 0T”)$ be a CR structure.
$\overline{\partial}_{T}(p,+1)(p\overline{\partial}_{\tau},)=0$

119Definition 1 Let $E$ be a complex subbundle of the complexfied tangent bundle $C\otimes TM$ satisfying:$E\cap\overline{E}=0$.

We call this pair $(M, E)$ an almost $CR$ structure.

Almost CR structures satisfying a certain condition can be parametrized
by elements of $\Gamma(M, T’\otimes(^{0}T’’)*)$ as follows.
Definition 2 An almost $CR$ structure $(M, E)$ is at finite distance from
if and only if the composition map of the inclusion map of $E$ into
$C\otimes TM$ , and the projection map
of $C\otimes TM$ to $0T”$ according to (5),isomorphic.
$(M, 0T^{l\prime})$$is$Then we have
Theorem 1 (Proposition 1.1 in $[A]$). The almost $CR$ structure, which is at
finite distance from $(M, 0T”)$ , corresponds to an element in $\Gamma(M, Hom(0T’l, T’))$$=\Gamma(M, T^{l}\otimes(^{0}T’’)*)$ , bijectively.
The correspondence is that; forin,$\phi$$\phi$$\Gamma(M, T^{l_{\otimes}}(0\tau\prime\prime)^{*})$$\emptyset\tau’’=\{X’ : X’=X+\phi(X), x\in^{0_{T’}\prime} \}$.

And the following theorem explains when this almost CR structure
is really a CR structure.

$(M, \emptyset T’’)$

Theorem 2 (Theorem 2.1 in [A]). An almost $CR$ structure, which$(M,
0\tau/;)$
is at finite distance fromis a $CRstruCture(this$ means that oursatisfies the integrability condition), if and only if our is a solution of the non linear partial differential equations$(M, \emptyset T^{ll})$$(M, \emptyset T’’)$$\phi$$P(\emptyset)$

$=$

$=$

$\overline{\partial}_{\tau}^{(1)},\emptyset+R2(\phi)+R_{3}(\emptyset)$$0$.

Namely the linear term of $P(\emptyset)=0$ is. And the real difficult
problem, in soloving this non-linear partial differential equation, is that:the non-linear term
includes the first order derivatives of , and our
cpmplex is subelliptic(the Neumann operator gains only 1). So, in CR
case, the mehod in complex manifolds is not available. In order to overcome
this difficulty, we introduce
structure in [A1]. Henceforth, we assume
strongly convexity.
$\overline{\partial}_{T}^{(1)},\phi$$R_{2}(\phi)$$\phi$$\overline{\partial}_{T’}$$E_{j}$

1202$E_{j}$

structure

be a strongly
We recall several results obtained in [A1]. Let
is an abstract of my talk, so we omit the
speudo convex CR structure(
notion of strongly pseudo convex CR structures). We set$(M, 0T^{\prime/})$$\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{S}$$\Gamma_{j}=\{u:u\in \mathrm{r}(M, 0\tau’\otimes\wedge j(0_{T’’)^{*})}, (\overline{\partial}^{(}\tau j,)_{u)_{F\otimes\wedge}j+1(^{0}T^{lJ})*}=0\}$Hereaccording to the$(\overline{\partial}_{T}^{(j)},u)F\otimes\wedge^{j+}1(0\tau l’)^{*}$$C^{\infty}$means the projection of$\overline{\partial}_{T}^{(j)},u$

to

(8)

$F\otimes\wedge^{j}(^{0}\tau’’)^{*}$

vector bundle decomoposition$T’\otimes\wedge j+1(^{0}T’’)^{*}=T’0\otimes\wedge j+1(0_{T}/l)^{*}+F\otimes\wedge^{j+}(^{0}1\tau’’)^{*}$

induced by (4) and (5). In the definition of , the first order derivative ofappears. But, actually as takes its value in $0T’,$is an algebraic condition. We see this more precisely. We consider a bundle$u$$\Gamma_{j}$$(\overline{\partial}_{T}^{(j)},u)F\otimes\wedge^{j}+1(0\tau\prime\prime)^{*}=0$

$u$

map:
$0_{T’\wedge(^{0}T^{l})}\otimes j’*arrow F\otimes\wedge^{j+1}(0T’’)^{*}$

defined by; for

$u$

in

$0T’\otimes\wedge^{j}(^{0}\tau’’)^{*}$,$(\overline{\partial}_{T}^{(j)},u)_{F}\otimes\wedge^{j}+1(^{0}\tau\prime\prime)*$.

Then, we have the following theorems.

). There is a sub-vector bundleof$\Gamma(M, E_{j})=\Gamma_{j}$ . And especially, our $CR$ structure

Theorem 3 (Proposition 2.1 in, satisfying
is strongly pseudo convex, $E_{0}=0$ .

$E_{j}$$[\mathrm{A}l]$$0T’\otimes\wedge^{j}(^{0*}\tau^{l/})$Theorem 4 (Theorem 2.2 in).

$[\mathrm{A}l]$$\overline{\partial}_{T}^{(j)},\Gamma_{j}\subset\Gamma_{j+1}$

that is to say,$(\Gamma_{j}, \overline{\partial}_{j})$,, where$\overline{\partial}_{j}=\overline{\partial}_{T}^{(j)},|_{\Gamma_{j}}$is a $sub-\acute{d}ifferential$ complex of the standard deformation complex of $(\Gamma(M,$$\wedge^{j}(0\tau’’)*),$$\overline{\partial}_{T}^{(},)j)$$T’\otimes$.

So we have a sub-differential complex of the standard deformation complex;$0arrow 0arrow\Gamma(M, E_{1})^{\underline{\overline{\partial}}}\mathrm{f}\Gamma(M, E_{2})\underline{\overline{\partial}?},$

And for this complex, we have$\mathrm{r}(M, E_{3})$

.

(9)

121
Theorem 5 (Theorem 2.4 in $[Al]$). This subcomplex
recovers the cohomology group of the standard deformation complex. Moreprecisely, the inclusion map induces the isomorphim map$(\Gamma(M, E_{j}),\overline{\partial}j)$$\frac{Ker\overline{\partial}_{j}}{Im\overline{\partial}_{j-1}}\simeq\frac{Ker\overline{\partial}_{T}\prime(j)}{Im\overline{\partial}_{T}^{(j1)}-}$

,

if

,

$j\geq 2$

and if $j=1$ , the inclusion map induces the surjecive map.

$Ker \overline{\partial}_{1}arrow\frac{Ker\overline{\partial}_{T}^{(1)}\prime}{Im\overline{\partial}_{T}},arrow 0$

Furthermore for this complex,

Theorem 6 (Theorem 4.1 in $[Al]$) If $dim_{R}M=2n-1\geq 7$ , then at,
there is a sub- elliptic estimate. So this estimate insures the Kodaira-Hodge
decomposition theorem over
. That is to say, we put thenormon
and complete these spaces. We denote
for these
completed hilbert spaces. Let.
Then, there is a Neumann type operator $N$ fromto itself and theharmonic operator $H$ fromtosatisfying;$E_{2}$$\Gamma(M, E_{2})$$L^{2}$$\Gamma(M, E_{i})$$\Gamma_{2}(M, E_{i})$$\mathrm{H}=\{u;u\in\Gamma(M, E_{2}),\overline{\partial}_{1}^{*}u=0, \overline{\partial}_{2}u=0\}$$\Gamma_{2}(M, E_{2})$$\Gamma_{2}(M, E_{2})$(1)$NH=HN=\mathit{0}$

(2)

for

$\mathrm{H}$,.

$u\in\Gamma_{2}(M, E_{2}),$(3) $||Nu||\leq c||u||’’$ ,$u=Hu+\overline{\partial}_{2}^{*}\overline{\partial}_{2}Nu+\overline{\partial}_{1}\overline{\partial}_{1}^{*}Nu$for$u\in\Gamma_{2}(M, E_{2})f$

where

$c$

is a positive constant,independent of .
$u$

For the norm$||||’’$, see [A1].

The new complex inspired by Rumin

3
Let$H_{0}=\{v:v\in\Gamma(M, T’), (\overline{\partial}\tau\prime v)F\otimes(^{\mathit{0}}\tau\prime l)^{*}=0\}$

(10)

.

Instead of (7), we introduce the following differential complex.
$0arrow H_{0}-\overline{\partial}_{T’}\Gamma(M, E_{1})-^{\overline{\partial}_{1}}\Gamma(M, E_{2})$

.

(11)

(because
is in
By the definition of)is obvious. So, (9) makes sense as aterm vanishes). And
like. First, we notedifferential complex. We have to explain what
sections of a vector bundle
does not come from the space ofthatis a graph of the following first order differential operator. For
over M.
$H_{0},$$\overline{\partial}_{T}\prime u$$\Gamma(M, 0T’\otimes(^{0}\tau’’)^{\star})$$\overline{\partial}_{\tau’ T’}^{(1)}\overline{\partial}u=0$$H_{0}\mathrm{i}\mathrm{s}$$C^{\infty}$$H_{0}$$H_{0}$$F\otimes(^{0}T’’)^{*}$

122
in $\Gamma(M, F)$ , where$\Gamma(M, 0T’)$ by:$u\cdot\xi$

$u$

is a

$C^{\infty}$function on $M$ , we set an element$(\overline{\partial}_{\tau\prime}(\psi u+u\cdot\xi))_{F\otimes}(^{0}T’’)*=0$

of$\psi_{u}$

(12)

,

that is to say,$[X, \psi_{u}]F+(Xu)\xi+u[X, \xi]_{F}=0$ ,(13)

for $X\in\Gamma(M, 0T’’)$ .

isThis is possible. Since our CR structure is strongly pseudo convex,
uniquely determined. Hence we have a first order differetial operator from, and so we
in $\Gamma(M, F),$ $\psi u+u\cdot\xi$ in$\Gamma(M, F)$ to
defined by; forhave a differeital complex.
$\psi_{u}$$H_{0}$$u\cdot\xi$$H_{0}$$\Gamma(M, F)\rho\downarrow$$0arrow$$arrow\overline{\partial}_{T’}\Gamma(M, E_{1})\underline{\overline{\partial}_{1\backslash }},$

$H_{0}$$\Gamma(M, E_{2})$.

includes a first derivative of . We see this more explicitly.
We note that, satisfying:of
By using a moving frame$u$$\psi_{u}$$\{e_{1}^{\lambda}, e_{2}^{\lambda}\}$$0T”|_{U_{\lambda}}$$-\sqrt{-1}[e_{i}^{\lambda},\overline{e}^{\lambda}]_{F}j=\delta_{ij}\xi$

where
down

$[e_{i}^{\lambda\lambda}, \overline{e}_{j}]_{F}$$\psi_{u}$means the$F$part of$[e_{i}^{\lambda},\overline{e}_{j}^{\lambda}]$

(14)

,

according to (2.4), we write. Set$\psi_{u}=\psi_{1}^{\lambda}\overline{e}1+\psi_{22}\lambda\lambda\overline{e}^{\lambda}$

(15)

.

Then,

$[e_{i’\psi_{1}\overline{e}}^{\lambda\lambda\lambda}1+\psi^{\lambda}2\overline{e}2]\lambda+F(eiu\lambda)\xi+u[e_{i}^{\lambda}, \xi]_{F}=0$

,

(16)

that is to say,$\sqrt{-1}\psi_{i}^{\lambda}\xi+(e_{i}^{\lambda})\xi+u[e_{i}, \xi\lambda]F=0,$$i=1,2$ .

(17)

So

$\psi_{i}^{\lambda}$$=\sqrt{-1}e_{i}^{\lambda}u+0$-th order term of .
$u$

(18)

123
We set a second order linear differential operator, by: for, where is afunction,$\Gamma(M, E_{1})$$u\cdot\xi$$D$from $\Gamma(M, F)$ to$C^{\infty}$$u$$D(u\cdot\xi)=\overline{\partial}_{T’}(\rho(u\cdot\xi))$

(19)

.

We would like to explain about this second order differential operator
and Rumin’s one. For this, we set avector bundle decomposition overcomplex valued differential
we write this space by$TM)^{*})$ as follows.
$C^{\infty}$$\Lambda^{k}(C\otimes$$k\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{s}(\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$$\Lambda^{k}(c\otimes^{\tau M})*$$=$$r+S=k-1,rs \geq\sum_{0 ,)}\theta\wedge\wedge r(^{0}\tau/)*\wedge\wedge s(^{0/\prime}T)^{*}$(20)$+ \sum_{rr+s=k,,S\geq 0}\theta\wedge\wedge^{r}(0T’)*\wedge\wedge^{s}(^{0}T^{l\prime})^{*}$

Here$\theta$is defined by:$\theta(\xi)=1$$\{$,$\theta(X)=0$,$X\in^{0}T/$,$d\theta(\xi, X)=0X\in^{0}T’$.

(if necessary, we change ). By using this decomposition, we introduce; for$i\geq 1$ ,$\xi$$F^{n-2,i}=\{u:u\in\Gamma(M, \theta \wedge\wedge^{n-2}(^{0}\tau’)^{*}\wedge\wedge^{i}(^{0}T’’)^{*})$,$(du)_{\wedge^{n}(}-10T’)^{*}\wedge\wedge i+1(\mathrm{o}T’’)*)=0\}$means theHereof $du$ according to the above decomposition. For $i=0$ , we set$(du)_{\wedge^{n}(}-10T’)*\wedge\wedge i+1(^{\mathit{0}}T’’)*)$$\Gamma(M, \wedge^{n-}1(\tau’)^{*})$

.

(21)

$\wedge^{n-1}(^{0}\tau’)^{*}\wedge\wedge^{i+1}(^{0}\tau^{l}’)*)$

part

.

(22)

Then we have a complex version of the Rumin complex$\Gamma(M, \wedge^{n-1}(\tau’)^{*})arrow\Gamma(M, F^{n-2}’ 1)Darrow\Gamma(Md’’, F^{n-}2,2)$.

We note that: if the canonical line bundle is trivial, this is nothing but our
new complex.
Our main result is as follows.
Theorem 7 For this complex, a subelliptic estimate on $\Gamma(M, F^{n-2},1)$ holds.
(see$[\mathrm{A}- G-Ll^{\mathit{2}}\mathit{1}]j$

And so, this assures that:

124
Theorem 8 The following family is parametrized by a finite dimensional
analytic space.
$P(\emptyset)=0$,$\{$$\overline{\partial}_{\tau’\emptyset=}^{*}0$, for$\phi\in \mathrm{r}(M, E_{1})$.

means the adjoint operator in the our setting ($not$ the standard$one)(see[A- G-Ll\mathit{2}]\mathit{1})$ . And this family is parametrized by a complex analytic
space locally at , complex analytically and versal in the sence of Kuranishi.

Here$\overline{\partial}_{T}^{*}$,$\mathit{0}$We note that the linear term of these non-linear equations is$\overline{\partial}_{1}\phi=0$,$\{$$\overline{\partial}_{T}^{*},\phi=0$, for $\phi\in\Gamma(M, E_{1})$ .

This is the dual of the Dirac operator in our setting.

$*,\vee^{\prime\doteqdot}\mathrm{X}$

[A]

ffl

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Kyoto Univ.14. No.3, pp615-641 (1978).

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$E_{j}$[A2]

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[A-M2] T.Akahori,K.Miyajima, A remark on a Tian-Todorov type theorem
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12 (1994).
[

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$\mathrm{A}- \mathrm{G}_{-}\mathrm{L}[1]$125[

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