数理解析研究所講究録
1502 巻 2006 年 160-172

160

Symbolic calculus of pseudo-differential operatorsand curvature of manifolds

兵庫県立大学大学院物質理学研究科
岩崎千里 (Chisato Iwasaki)
Depart. of Math. University of Hyogo
Abstract

The method of construction of the fundamental solution for a heat equations as pseudodifferential operators with parameter time variable is discussed, which is applicable to calculate
traces of oPerators. This gives extensions of a local version of both Gaus\S Bonnet-Chem
Theorem and Riemann-Rpch Theorem. Moreover a characterization of complex manifolds
which hold a local version of Riemann-Roch Theorem is obtained.
$\cdot$Introduction

1

In this paper we give, by means of symbolic calculus of pseudo-differential operators, both anextension theorem of a local version of Gauss-Bonnet-Chem theorem given in C.Iwasaki[10] andthat of a local version of Riemann-Roch theorem given in C.Iwasaki[ll]. We give also a characterization of complex manifolds where a local version of Riemann-Roch theorem holds. For more
precise discussion see C.Iwasaki[12] and C.Iwasaki[13].

Let $M$ be a Riemannian manifold of dimension
theorem is stated as follows:$n$without boundary. The Gauss-Bonnet-Chea$\sum_{p=0}^{n}(-1)^{p}\dim H_{\mathrm{p}}(M)=\int_{M}C_{n}(x, M)dv$

where
$0$

if

$n$

,

is the set of harmonic p–forms, $C_{n}(x, M)dv$ is the Euler form if is even and $C_{n}(x, M)dv=$
is odd. Its analytical proof based on the following formula$H_{p}$$n$,$\sum_{\mathrm{p}=0}^{n}(-1)^{\mathrm{p}}\dim H_{\mathrm{p}}(M)=\int_{M}\sum_{\mathrm{p}=0}^{n}(-1)^{\mathrm{p}}\mathrm{t}\mathrm{r}e_{p}(t,x,x)dv$

where $e_{p}(t, x, y)$ denotes the kernel of the fundamental solution
heat equation of
on differental p–forms;$\Delta_{\mathrm{p}}$$E_{p}(t)$of Cauchy problem for the$\Gamma(\wedge^{\mathrm{p}}T^{*}(M))$$E_{p}(t) \varphi(x)=\int_{M}e_{\mathrm{p}}(t,x,y)\varphi(y)dv_{y},$$\varphi\in\Gamma(\wedge^{p}T^{*}(M))$

satisfies$( \frac{d}{dt}+\Delta_{\mathrm{p}})E_{\mathrm{p}}(t)$

$=$

$0$

in $(0,T)\mathrm{x}M$ ,$E_{p}(0)$

$=$

$I$

in $M$ .

161

So, we may call a local version of Gauss-Bonnet-Chern theorem holds, if we have
(1.1)$\sum_{\mathrm{p}=0}^{n}(-1)^{p}\mathrm{t}\mathrm{r}e_{p}(t, x, x)=C_{n}(x, M)+\mathrm{O}(\sqrt{t})$as tends to .
$0$

$t$

The author has proov\’e (1.1) in [10], using both algebraic theorem on linear spaces stated in
H.l.Cycon,R.G.Froese,W.Kirsch and B. Simon[3] and the method of construction of the fundamental
solution by technique of pseudodifferential operators of new weights on symbols. In this paper, agenaralization of a local version of Gauss-Bonnet-Chem theorem is obtained. Before stating ourtheorems, we introduce notations.
We denote$\mathcal{I}$the set of index$\mathcal{I}=\{I=(i_{1},i_{2}, \cdots, i_{r}) :0\leq r\leq n, 1\leq i_{1}<\cdots<i_{\ell}\leq n\}$,and$=0\mathrm{i}\mathrm{f}a<b$$=1$ .

, or $b<0$ ,Fix an integer such that $0\leq l\leq n$ in the rest of this paper.
$\ell$

Set the following constants(1.2)$\{f_{p}\}_{\mathrm{p}=0,1,\cdots,n}$of the form with arbitrary constants$f_{\mathrm{p}}=(_{n}^{n}=_{\ell}^{p})+ \sum_{j=\max\{\mathrm{p}\ell+1\}}^{n}.k_{j}(_{n}^{n}=_{j}^{p})(0\leq p\leq n)$$\{k_{j}\}_{j=\ell+1,\cdots,n}$.

Theorem 1.1 (Main Theorem I) Let $M$ be a Riemannian manifold rvithout boundary of ;. Suppose are of the form
menssion and let
be the fundamental solution on(1.2). Then we have
$d$

$n$

$\Gamma(\wedge^{p}T"(M))$$E_{p}(t)$$\sum_{p=0}^{n}(-1)^{\mathrm{p}}f_{\mathrm{p}}\mathrm{t}\mathrm{r}e_{p}(t, x, x)=C_{\ell}(x)t^{-\#+\neq}+0(t^{-}\#+\mathrm{z}+\tau)\ell 1$$C\ell(x)$ and is given as follow;(1) If is odd, $C_{\ell}(x)=0$(2)$Ifl$ is even $(l=2m),$$f_{\mathrm{p}}$as$tarrow \mathrm{O}$

,

where

$\ell$$C_{\ell}(x)= \sum_{I\in \mathcal{I},\#(I)=}{}_{p}C_{I}(x)$, for$I=(i_{1},i_{2}, \cdots,i\ell)\in \mathcal{I}$$C_{I}(x)=( \frac{1}{2\sqrt{\pi}})^{n}\frac{1}{m!}(\frac{1}{2})^{m}\sum_{rr.\sigma\in S_{\ell}}sign(\pi)sign(\sigma)$$\mathrm{x}R;\pi(1\rangle \mathfrak{i}.(2\rangle:\sigma(1):\sigma(2)‘.$

Remark 1.2 Assume $\ell=n$ . Then
Gauss-Bonnet-Chefn theorem.

$f_{\mathrm{p}}=1$

.

. ..

$R\iota*(\ell-1)i.(\ell)::\sigma(\ell-1)\sigma(\ell)$of (1.2) for $dlp$ . Theorem 1.1$\dot{u}$.

a local version of$f_{p}=0(\ell+1\leq p\leq n)$ . So
Remark 1.3 Assume $k_{j}=0$ for all . Then
Theorem 1.1 coinside with the result in P. Gunther and $R.Schimming[\mathit{4}J$.
$j$$f_{p}=(^{n}n=^{\mathrm{p}}\ell)(0\leq p\leq\ell),$162

Now consider the similar problem for Dolbeault complex on a Kaehler manifold $M$ , that is, adenotes the kernel of$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}1\mathrm{s}\mathrm{o}1\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}E_{\mathrm{p}}(t)\mathrm{o}\mathrm{f}\mathrm{C}\mathrm{a}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}1\mathrm{e}\mathrm{m}\Gamma(\wedge^{p}T^{\mathrm{r}(0,1)}(M)))\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}1\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{f}\mathrm{a}1\mathrm{o}\mathrm{c}\mathrm{a}1\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{f}\mathrm{R}\mathrm{e}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{n}- \mathrm{R}\mathrm{o}\mathrm{c}\mathrm{h}\mathrm{t}\mathrm{h}\mathrm{m}\mathrm{r}\mathrm{e}\mathrm{m}.\mathrm{L}\mathrm{e}\mathrm{t}e_{p}(.t, x, y)$$E_{p}(t) \varphi(x)=\int_{M}e_{\mathrm{p}}(t, x,y)\varphi(y)dv_{y}$,$\varphi\in\Gamma(\wedge^{p}T^{(0,1\rangle}"(M))$

satisfleswhere$( \frac{d}{dt}+\mathrm{L}_{\mathrm{p}})E_{p}(t)$

$=$

$0$

$E_{\mathrm{p}}(0)$

$=$

$I$

$L_{p}=\overline{\partial}_{p}^{*}\overline{\partial}_{p}+\overline{\partial}_{\mathrm{p}-1}\overline{\partial}_{p-1}^{l}$in $(0,T)\mathrm{x}M$,
in $M$ ,.

The author in [11] have given a proof of a local version of Riemann-Roch theorem, constructing
the fundamental solution according to the method of symbolic calculus for a degenerate parabolic
operator in C.Iwasaki and N.Iwasaki[9]. There are several papers about a local version of RiemannRoch theorem. T.Kotake[15] proved this formula for manifolds of dimension 1. V.K.Patodi[17] hasproved for Kaehler manifolds of any dimension. P.B.Gilkey[8] also has shown, using invariant
theory. E.Getzler[6] treated this problem by different approach. We obtain an extension of thisproblem as follows:
Theorem 1.4 (Main Theorem II) Let $M$ be a compact Kaefder manifold whose complex dibe the fundamental
mension is , and let
on $A^{0,p}(M)=\Gamma(\wedge^{p}T^{*(0.1)}(M))$ . Suppose
are of the form
. Then we have$n$$sol\mathrm{u}t:on$$E_{\mathrm{p}}(t)$$(\mathit{1}.l)$$f_{p}$$\sum_{p=0}^{n}(-1)^{p}f_{p}\mathrm{t}\mathrm{r}e_{p}(t,x,x)dv=(\frac{1}{2\pi i})^{n}C_{\ell}^{D}(x)t^{-n+\ell}+0(t^{-n+\ell+1})$

where$D_{\ell}^{D}(x)$are defined as follows:, where for$C_{p}^{D}(x)= \sum_{I\in \mathcal{I},\#(I)=\ell}C_{I}^{D}(x)$$I=(i_{1},i_{2}, \cdots,i_{\ell})$El$C_{I}^{D}(x)=[ \det(\frac{\Omega}{e^{\Omega}-Id})]_{2\ell}$

Here$\Omega$as $tarrow 0$ ,A$dv^{I}$is a matrix whose $(j, k)$ element is 2-form defined as$( \Omega)_{jk}=\sum_{a,b=1}^{n}R_{\mathrm{k}jab}\omega^{a}\wedge\varpi^{b}$

and$dv^{I^{\iota}}=\overline{\omega}^{j_{1}}$where$I^{\epsilon}=(j_{1},j_{2}, \cdots,j_{n-}\ell)\in$A$\omega^{j_{1}}$A

I such that$j-2\wedge\omega^{j_{2}}\cdots$

A$\dot{d}^{-n-\ell}$$I\cup I^{\mathrm{c}}=\{1,2, \cdots, n\}$Rmark 1.5 Assume $\ell=n$ . Then $f_{p}=1$ of (1.2) for all
version of Riemann-Roch theorern.

$\mathrm{p}$A$\dot{d}^{\hslash-\ell}$,.

. In this case Theorem 1.4 is a local

It is known that a local version of Riemann-Roch theorem does not hold on complex manifolds
by P.B.Gilkey[7]. A characterization of complex manifolds where a local version of Reimann-Rochtheorem holds is given by the Kaelher form of complex manifolds as follows.
$\Phi$

163

Theorem 1.6 (Main Theorem III) (1) If$n$is even and$\partial\overline{\partial}\Phi\neq 0$, then we have$\sum_{p=0}^{n}(-1)^{p}\mathrm{t}\mathrm{r}e_{p}(t, x, x)dv_{x}=(2\pi)^{-n}(-1)^{\mathrm{v}}\frac{(i\partial\overline{\partial}\Phi)\mathrm{v}n}{(\frac{n}{2})!}nt^{-\tau}\mathfrak{n}+O(t^{-\mathrm{g}+8_{)}}$(2) If$\partial\overline{\partial}\Phi=0$.

, then we have$\sum_{p=0}^{n}(-1)^{\mathrm{p}}\mathrm{t}\mathrm{r}e_{p}(t,x, x)dv_{\Phi}=(\frac{1}{2\pi i})^{n}[\sqrt{\det(\frac{7\Lambda}{\sinh(\frac{\Lambda}{2})})}e^{-\}tr\Omega^{S}}]_{2n}+0(t)$where A
precise$\dot{u}$a$2n\mathrm{x}2n’ \mathrm{t}d$).

,anti-symmetric matrix whose $(p,q)$ element is 2-form(See (5.2) for the$defini\hslash on$

Our point is that one can prove the above theorems by only calculating the main term of thesymbol of the fundamental solution, introducing a new weight of symbols of pseudodifferential
operators.
The plan of this papaer is following. In section 2 an algebraic theorem, which is the key of theproof, is stated. The sketch of proof is given in section 3, section 4 and section 5.

Algebraic properties for the calculation of the trace2

be its anti-symmetricbe a vector space of dimension with an inner product and let
be an orthonormal basis for . Let “ be. Lettensors. Set
defined by $a.”.v=v_{i}$ A and let : be the adjoint operator of “
a linear transformation on

Let

$V$

$\wedge^{p}(V)$$n$$\wedge^{l}(V)=\sum_{p=0}^{n}\wedge^{\mathrm{p}}(V\rangle$

$p$

$V$

$\{v_{1}, \cdots, v_{n}\}$$\wedge"(V)$

$a$

$v$

$a_{i}$$a_{i}$

on A“ (V).

Deflnition 2.1 Set $A=\{(\mu_{1}, \cdots,\mu_{k}) : 1 \leq k\leq 2n, 1\leq\mu_{1}<\cdots<\mu_{k}\leq 2n\},$ $\gamma 2k-1=$$a_{k}+a_{k}",$and$\gamma_{2k}=i^{-1}(a_{k}-a_{k}^{*})$$\gamma_{\phi}=1$.

for$k\in\{1,2, \cdots , n\},$ $\gamma_{A}=i^{\frac{k(k-1\rangle}{2}}\gamma_{\mu_{1}}\cdots\gamma_{\mu_{k}}$We have$\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=2\delta_{\mu\nu}$,for$A=(\mu_{1}, \cdots, \mu_{k})\in A$$1\leq\mu,$ $\nu\leq 2n$

and

for any $A\in A$ .
The following propositions are shown in [3] under the above assumptions.
$\gamma_{A}^{2}=1$

Proposition 2.2 We have the follo wing equality for transfomation$0$,$\mathrm{t}\mathrm{r}(\gamma_{A})=\{$$2^{n}$

,

$0$

,

if
if

$A\neq\phi$;$A=\emptyset$.

Corollary 2.3 For any $A,$ $B\in A$$\mathrm{t}\mathrm{r}(\gamma_{A}\gamma_{B})=\{$$2^{n}$,

if $A\neq B$ ;if $A=B$ .

$on\wedge^{\mathrm{r}}(V)$

.

164

Definition 2.4 Set
I.
$\Gamma_{0}=1,$$\beta_{\phi}=1,\beta_{j}=i\gamma_{2j-1}\gamma_{2j}$$\Gamma_{k}=\sum_{I\in \mathcal{I},\#(I)=k}\beta_{I}$It holds that for

.

for

and

$1\leq j\leq n$$\beta_{I}=\beta_{i_{1}}\cdots\beta_{i_{k}}$for$I=(i_{1}, \cdots, i_{k})\in$$I=(i_{1}, i_{2}, \cdots,i_{k})\in \mathcal{I}$$\beta_{I}=\gamma_{\overline{I}}$,where $\tilde{I}=(2i_{1}-1,2i_{1},2i_{2}-1,2i_{2}, \cdots,2i_{k}-1,2i_{k})\in A$ . It is clear that$a_{k}" a_{k}= \frac{1}{2}(1+\beta_{k})$by the ProPertiu of$\gamma_{j}$,,$\beta_{j}\beta_{k}=\beta_{k}\beta_{j}$$\beta_{j}^{2}=1$.

ProPosition 2.5 We have for any$I=(i_{1}, \cdots, i_{k})\in \mathcal{I}$(1)$Ifp<k$the follouring assertionsj$\mathrm{t}\mathrm{r}[\beta \mathrm{r}a_{j_{1}}a_{j_{2}}\cdots a_{j_{\mathrm{p}}}a‘ 1a" 2\ldots a_{h_{p}}^{l}]=0$(2) $Sup\mathrm{p}osep=k$ andwe have$\mathrm{b}1,j_{2},$$\cdots,j_{k}$}$\neq\{i_{1}, i_{2}, \cdots, i_{k}\}$or.

$\{h_{1}, h_{2}, \cdots, h_{k}\}\neq\{i_{1},i_{2}, \cdots,\dot{i}k\}$. Then.

$\mathrm{t}\mathrm{r}[\beta_{I}a_{j},a_{j_{2}}\cdots a_{j_{p}}a_{h_{1}}^{*}a_{h_{2}}"\cdots a_{h,}"]=0$(3) Let , a be dements of the permutation group of degree
$\pi$

$k$

.

we have$?7\iota en$$\mathrm{t}\mathrm{r}[\beta Ia_{\dot{2}_{\pi(1)}}^{*}ai_{\sigma(1)i_{\pi(2\rangle}\dot{\cdot}.:_{\sigma(k)}}a^{*}ai_{\sigma(2)(k)}\ldots a^{*}a]=2^{n-k}sign(\pi)sign(\sigma)$Letbe the projection ofonargument of the proof of this section.
$\Psi_{p}$$\wedge"(V)$$\wedge^{\mathrm{p}}(V)$. The folowing Proposition is the key algebraic

Proposition 2.6 For any $p(0\leq p\leq n)$ we have the folloutng equation$\Psi_{p}=\sum_{q=0}^{n}\mathcal{M}_{p\mathrm{q}}\Gamma_{q}$

,

where
$\mathcal{M}_{pq}=\sum_{\mathrm{p},q\leq j\leq n}(-1)^{p+j}2^{-j}(_{n}^{n}=_{j}^{q})$Note that a $(n+1)\mathrm{x}(n+1)$ matrix.

$\mathcal{M}=(\mathcal{M}_{pq})_{0\leq p,q\leq n}$.

is regular because$( \mathcal{M}^{-1})_{pq}=\sum_{0\leq j\leq p,q}(-1)^{p+j}2^{j}(_{n}^{n}=_{p}^{j})$Then we have

Theorem 2.7 Let $\alpha_{p}(\ell+1\leq p\leq n)$ be constants. The equation$\sum_{q=0}^{n}f_{\mathfrak{g}}\Psi_{q}=(-1)^{\ell}\not\simeq^{-n}\Gamma_{\ell}+\sum_{\mathrm{p}=\ell+1}^{n}\alpha_{\mathrm{p}}\Gamma_{p}$

.

165

has solution as follows:$\sum n$$f_{p}=(-1)^{p}\{(_{n}^{n}=_{l}^{\mathrm{P}})+$$k_{j}(_{n}^{n}=_{j}^{p})\}$

for any$\mathrm{p}$$j=\mathrm{m}\mathrm{B}(\ell+1,p)$with constants $k_{j}(\ell+1\leq j\leq n)$ defined by$k_{j}=(-1)^{j}2^{n-j} \{2^{\ell-n}(-1)^{\ell}+\sum_{p=\ell+1}^{j}\alpha_{p}\}$.

Especidly(1) If $\ell=n$ , then$\sum_{q=0}^{n}f_{q}\Psi_{q}=(-1)^{n}\Gamma_{n}$holds if and only if

for any .

$f_{p}=(-1)^{p}$$p$(2) If $\alpha_{p}=(-1)^{p}2^{\ell-n}(_{\ell}^{p})(\ell+1\leq p\leq n)$ , we have$(-1)^{p}(^{n}n=^{\mathrm{p}}\ell)$$f_{\mathrm{p}}=\{$

$0$

,

$0$

3

of the following form$(0\leq p\leq\ell)$,;$(l+1\leq p\leq n)$(3) If $\alpha_{p}=(-1)^{\ell}2^{p-n}(\ell+1\leq p\leq n)$ , we have$f_{\mathrm{p}}=\{$$f_{p}$,$f_{p}$.

of the following $fom$$(0\leq p\leq n-\ell-1)$$(-1)^{n-\ell+p}$ ;$(n-\ell\leq p\leq n)$;.

The proof of Main Theorem I (Riemannian manifolds)

Let $M$ be a smooth Riemannian manifold of dimension with a Riemannian metric . Let
be a local orthonormal frame of $T(M)$ in a lokal path . And let
are written as follows,using
be its dual. The differential and its dual acting on
the Levi-Civita connection V (See Appendix A of S.Murakami[16]):
$n$

$g$

$U$

$X_{1},X_{2},$ $\cdots,X_{n}$$d$$\Gamma(\wedge^{p}T^{*}(M))$$\theta$$d= \sum_{j=1}^{n}e(\dot{d})\nabla_{X_{f}}$,$\{v^{1},\omega^{2},$ $\cdots,\omega^{n}$$\theta=-\sum_{j=1}^{n}\iota(X_{j})\nabla_{X_{\mathrm{j}}}$,where we use the following notations.
Notations.
$e(j)\omega=\dot{d}\wedge\omega,$Let$R(X, \mathrm{Y})$$\iota(X_{j})\omega(Y_{1}, \cdots, \mathrm{Y}_{p-1})=\omega(X_{j}, Y_{1}, \cdots,\mathrm{Y}_{p-1})$

be the curvature transformation, that is$R(X,\mathrm{Y})=[\nabla_{X}, \nabla_{Y}]-\nabla_{[X.Y]}$.

Set$R(X_{\backslash }, X_{j})X_{k}= \sum_{\ell=1}^{n}R_{\ell kij}X\ell 1\leq i,j,$ $k,$ $\leq n$

.

.

166

The Laplacian(3.1)$\Delta=d\theta+\theta d$on$\sum_{p=0}^{n}\Gamma(\wedge^{p}T"(M))$

has the following Weitzenb\"ock’s formula:$\Delta=-\{\sum_{j=1}^{n}\nabla_{X_{j}}\nabla_{X_{f}} -- \sum_{j=1}^{n}\nabla_{(\nabla_{X_{j}}X,)}+\sum_{i,j=1}^{n}e(\omega))_{t}(X_{j})R(X_{l}, X_{j})\}$.

We use the following notations in the rest of this section.
$a_{j}"=e(\omega*)$,.

$a_{k}=\iota(X_{k})$The fundamental solution $E(t)$ has a expansion, due to [10].
$E(t) \sim\sum_{j=0}u_{j}(t,x,D)$,

where $u_{j}(t,x,D)$ are pseudodifferential operators with parameter .
$t$

The following statement is obtained in p.255 of [10]. The kemel of pseudo-differential operator
is obtained aswith symbol$\mathrm{u}_{0}(t,x,\xi)$$\tilde{u}_{0}(t, x,x)=(2\pi)^{-n}\int_{\mathrm{R}^{n}}u_{0}(t,x,\xi)d\xi$$=( \frac{1}{2\sqrt{\pi}t})^{n}\sqrt{\det g}e^{-\ell R}(1+0(\sqrt{t}))$

,

where
$R= \sum_{:,j,k,q=1}^{n}R_{qkij}a^{l}.\cdot a_{j}a_{k}^{*}a_{q}$.

We shall calculate
tr

(3.2)

for

$I\in \mathcal{I},$$( \beta_{I}\tilde{\mathrm{u}}_{0}(t,x, x))dx=(\frac{1}{2\sqrt{\pi}t})^{n}\mathrm{t}\mathrm{r}(\beta_{I}e^{-tR})dv(1+0(\sqrt{t}))$,$\#(I)=r$ .

Using$e^{-tR}= \sum_{k=0}^{\infty}\{\frac{(-1)^{k}}{k!}R^{k}t^{k}\}$,and by Proposition 2.5 we have

(3.3)

tr

$(\beta \mathrm{r}e^{-tR})=\{$ $0(t^{r} \mathrm{t}\mathrm{r}(\beta_{I,+}\frac{(-1)^{m}}{1),m!}R^{m})t^{m}+0(t^{m+1})$

, if
if

;
is odd.

$\mathrm{r}=2m$$r$

We have the following proposition.

Proposition 3.1 For(3.4)$I=(i_{1}, i_{2}, \cdots,\mathrm{i}_{f})\in \mathcal{I}(r=2m)$$\mathrm{t}\mathrm{r}(\beta_{I}(-1)^{[] n}R^{m})=2^{n-\tau-n}’\sum_{n,\sigma\in S_{f}}sign(\pi)sign(\sigma)$$\mathrm{x}R_{i::,:}\cdots R\pi(1\rangle n(2\rangle(1\rangle\sigma\langle 2)\pi(r-1):\pi(r):\sigma\{r-1\rangle:\sigma(r)$

.

167

By (3.2),(3.3) and Proposition 3.1 we have
(3.5)

with

tr
$C_{I}(x)$$2^{n-\tau}t^{-\eta+\epsilon_{C_{I}(x)dv}}n+0(t^{-_{\mathrm{F}}^{n}+\xi+1})$$(\beta I\tilde{u}\mathrm{o}(t, x,x))dx=\{$$0(t^{-T^{+}}n.+:)$

if $r=2m$ ;,,if is odd$r$defined in Defiffiition 1.1. Similarly we have

(3.6)

tr

.

$(\beta_{I}\tilde{u}_{j}(t,x,x))dx=0(t^{-\mathrm{g}+\frac{r}{2}+\mathrm{i}})$

By Theorem 2.7 we obtaintr$(\beta\tau e(t,x, x))$$=$$( \sum_{p=0}^{n}f_{p}e_{\mathrm{p}}(t,x,x))$$(-1)^{p}2^{\ell-n} \sum_{I\in \mathcal{I},\#(I)=\ell}\mathrm{t}\mathrm{r}$$+ \sum_{p=^{p+1}}^{n}\alpha_{p}\mathrm{t}\mathrm{r}(\Gamma_{p}e(t,x,x))$.

By (3.5) and (3.6) we havetr$(\Gamma_{p}e(t, x,x))=0(t^{-\S+_{\mathrm{z}^{+}\tau)}}\ell 1$.

Applying (3.5) (3.6), we have
$C_{I}(x)t^{-\mathrm{I}^{+}}n\ell,+0(t^{-l^{+_{\mathrm{I}}^{p}+4_{)}}}n$tr

4$( \sum_{\mathrm{p}=0}^{n}f_{p}e_{\mathrm{p}}(t,x,x))=\{$$0(t^{-}\tau+\mathrm{n}\mathrm{f}+:)$,, if is even ;
ifodd.
$\ell$$l\mathrm{i}\mathrm{a}$The proof of Main Theorem II (Kaehler manifolds)

Let $M$ be a compact Kaehler manifold whose complex dimension is with a hermitian metricbe a local orthonormal frame of $T^{1,0}(M)$ in a local patch of chart . And
9. Setletbe its dual. The differential and its dual “ acting on $A^{0,p}(M)$ are given as
follows, using the Levi-Civita connection V:$n$$Z_{1},Z_{2},$ $\cdots,$$\iota v^{1},\omega^{2},$$U$$Z_{\mathrm{n}}$$\overline{\partial}$$\overline{\partial}$$\cdots,\omega^{n}$$\overline{\partial}=\sum_{i=1}^{n}e(-\dot{d})\nabla_{Z_{f}},\overline{\partial}\cdot=-\sum_{j=1}^{n}\iota(Z_{j})\nabla_{Z_{J’}}$

where we use the following notations.

Notations.
$Z_{\overline{j}}=\overline{Z}_{j}$,$e(\omega^{\alpha})\omega=\omega^{\alpha}\wedge\omega$,$0=1,$ $\cdots,n))$$\omega\overline{j}=\overline{\omega}^{j}$$t(Z_{\alpha})\omega(\mathrm{Y}_{1}, \cdots, Y_{p-1})=\omega(Z_{\alpha}, Y_{1}, \cdots, \mathrm{Y}_{p-1})$$(\alpha\in\Lambda=\{1, \cdots, n, \overline{1}, \cdots,\overline{n}\})$

Let$R(Z_{\alpha},\overline{Z}_{\beta})$.

be the curvature transformation;$R(Z_{\alpha}, Z_{\beta})=[\nabla_{Z_{a}}, \nabla_{Z\rho}]-\nabla_{[Z_{\alpha},Z\rho]}$$(\alpha, \beta\in\Lambda)$

The curvarute transformations satisfy$R(\mathrm{z}_{:}, Z_{j})=0$,$R(Z_{l},Z_{j})=0$

,

.

,

168

because of $M$ is a Kaehler manifold. Set$(\beta\in\Lambda)$$R(Z_{1}, \overline{Z}_{j})Z_{\beta}=\sum_{\gamma\in\Lambda}R_{\gamma\beta_{\dot{\mathrm{t}}J}^{\neg}}Z_{\gamma}$

The Laplacian
formula:$L=\overline{\partial}"\overline{\partial}+\overline{\partial}\overline{\partial}^{*}$

on$A^{0}$,“$(M)= \sum_{p=0}^{n}A^{0,p}(M)$.

has the following Bochner-Kodaira.

$L=- \frac{1}{2}\{\sum_{j=1}^{n}(\nabla_{Z_{f}}\nabla_{Z_{f}}+\nabla_{Z_{f}}\nabla_{Z_{f}})-\sum_{j=1}^{n}\nabla_{(\nabla z_{f}Z_{f}+\nabla_{B_{j}}Z_{f})}-\sum_{j=1}^{n}R(Z_{j}, Z_{j})\}$We use the following notations in the rest of this section.
$e(-\dot{d})=a_{j}",$ $\iota(\overline{Z}_{k})=a_{k}$.

The fundamental solution has a expansion, due to [11]$E(t) \sim\sum_{j=0}u_{j}(t,x,D)$,

where $u_{j}(t, x, D)$ are pseudodifferential operators with parameter
symbols $u_{0}(t,x, \xi)$ is represented of the pricise form. The kernel
operator with symbol $v_{0}(t,x,\xi)$ is obtain\’e as in p.90 [11]$t$

and the main part of theirof pseudodifferential$\tilde{u}_{0}(t,x,x)$.

$\tilde{u}_{0}(t,x,x)=(2\pi t)^{-n}\det(\frac{t\mathcal{M}_{0}}{\exp(t\mathcal{M}0)-Id})\sqrt{\det g}$

Note that(4.1)$e(t,x,x)dv=\tilde{u}_{0}(t, x,x)dx(1+0(t))$

We shall calculate
(4.2)

for

tr $(\beta_{I}\tilde{u}_{0}(t,x, x))dx=(2\pi t)^{-n}$ tr$I\in \mathcal{I},$$\#(I)=r$, where$= \sum_{p,q=1}^{n}R_{jk\overline{\mathrm{q}}p^{a_{q}^{*}a_{p}}}$,$( \beta_{I}\det(\frac{t\mathcal{M}_{0}}{\exp(t\mathcal{M}_{0})-Id}))dv$$( \mathcal{M}_{0})_{jk}=R(Z_{j}, Z_{k})=-\sum_{p,q=1}^{n}R_{pqj\overline{k}}a_{q}^{*}a_{p}$

.

Set
.

$\det(\frac{t\mathcal{M}_{0}}{\exp(t\mathcal{M}_{0})-Id})=\sum_{j=0}^{n}A_{j}t^{j}$

Then we have by proposition 2.5
(4.3)

Set

$\Omega$tr$( \beta_{I}\det(\frac{t\mathcal{M}_{0}}{\exp(t\mathcal{M}_{0})-Id}))=$

tr $(\beta_{I}A_{r})t^{f}+0(t^{r+1})$ .

a matrix whose $(j, k)$ element is 2-forn defined by$( \Omega)_{jk}=-\sum^{n}R_{j\overline{k}q_{\mathrm{P}}}\varpi^{q}$$p,q=1$

Then we haveA$\omega^{p}=\sum \mathrm{n}R_{\overline{k}jp\mu}$$p,q=1$A$\overline{\omega}^{q}$

.

169

Proposition 4.1
(4.4)

tr

$( \beta_{I}A_{r})dv=(-1)^{r}(\frac{1}{i})^{n}2^{n-r}[\det(\frac{\Omega}{\exp\Omega-Id})]_{2r}\wedge dv^{I^{\mathrm{c}}}$From (4.2),(4.3) and Proposition 4.1 the following equation holds.
tr$(\beta_{I}\overline{u}_{0}(t,x, x))dx$$=(-1)’( \frac{1}{2\pi i})^{n}2^{n-f}t^{-n+\prime}$$[ \det(\frac{\Omega}{\exp\Omega-Id})]_{2r}\wedge dv^{P}+0(t^{-n+\tau+1})$.

Now by Theorem 2.7 we obtaintr$( \sum_{p=0}^{n}f_{p}e_{p}(t,x,x))$$=$$(-1)^{\ell}2^{\ell-n} \sum_{I\in \mathcal{T},\#(I)=\ell}\mathrm{t}\mathrm{r}(\beta_{I}e(t,x,x))$$+ \sum_{p=\ell+1}^{n}\alpha_{p}\mathrm{t}\mathrm{r}(\Gamma_{p}e(t,x,x))$with some constants $a_{j}(\ell+1\leq j\leq n)$ . Applying (4.5) we have$\sum_{p=\ell+1}^{n}\alpha_{p}$tr $(\Gamma_{p}e(t,x,x))dv=0(t^{-n+\ell+1})$ .

By (4.1) and (4.5) we obtaintr$=$$( \sum_{\mathrm{r}=0}^{n}f_{p}e_{p}(t,x,x))dv$$( \frac{1}{2\pi i})^{n}t^{-n+\ell}\sum_{I\in \mathcal{I},\#(I\rangle=^{p}}[\det(\frac{\Omega}{\mathrm{a}\mathrm{p}\Omega-Id})]_{2\ell}\wedge dv^{I^{\mathrm{G}}}$$+0(t^{-n+l+1})$

5.

The proof of Main Theorem III (Complex Manifolds)

Let $M$ be a complex manifold with a Hermitian metric 9. Choose a local chart and a orthnormal
system as in the previous section. Using the Levi-Civita connection , we have the following
representation for differential and its adjoint acting on $A^{p,q}(M)$ :$\nabla$$d$$\theta$$d= \sum_{\dot{g}=1}^{n}e(\alpha^{\rho})\nabla_{Z_{j}}+\sum_{j=1}^{n}e(^{-}\dot{d})\nabla_{Z_{j}},$.

$\theta=-\sum_{j=1}^{n}t(Z_{j})\nabla_{Z_{f}}-\sum_{j=1}^{n}\iota(\overline{Z}_{j})\nabla_{Z_{f}}$and the torsion vanish. But
The connection is the Levi-Civita connection. So both
does not preserve type of vector fields, that is,for the complex structure . In this case,generally we have $R(z_{:}, Z_{j})\neq 0$ ,. For the representation of our operator , weandand give characterization of these connections.
introduce connection$\nabla$$\nabla g$$\nabla I\neq 0$$T$$\nabla$$I$$L$$R(\overline{Z}_{i},\overline{Z}_{j})\neq 0$$\nabla^{S}$$\nabla^{\tilde{S}}$Deflnition 5.1(i) Letbe the Hermitian connection of $M$ , that is, the unique connection which satisfies thefollouring $cond;tions_{j}$$\nabla^{\overline{S}}$$\nabla^{\overline{S}}g=0$,,$\nabla^{\overline{S}}I=0$$T^{\acute{S}}(V, W)=0$$(V\in T^{(1,0)}(M),$ $W\in T^{(0,1)}(M))$

.

170

Let

$\tilde{S}_{\alpha\beta}^{\gamma}(\alpha, \beta,\gamma\in\Lambda)$

be the following functions of this connection$\nabla_{Z_{\alpha}}^{\overline{S}}Z_{j}=\sum_{k=1}^{n}\tilde{S}_{\alpha j}^{k}Z_{k}$(2) Let$\nabla^{S}$,$\nabla^{\overline{s}_{j}}$$\nabla^{\overline{S}}z_{\mathrm{Y}=}z_{\alpha J}\sum_{k=1}^{n}\tilde{S}_{\alpha\overline{j}}^{\overline{k}}\overline{Z}_{k}$.

be the unigue connection which satisfies the following conditions;,$\nabla^{S}g=0$$\nabla^{S}I=0$,

and$g(W, T^{S}(U, V))+g(U,T^{S}(\overline{W}, V))=0$

for

$U,$

$V\in T^{(1,0)}(M),$ $W\in T^{(0,1)}(M)$.

Proposition 5.2 We have the following representation(5.1)$\overline{\partial}=\sum_{r=1}^{n}ajD_{r}$,$\overline{\partial}"=-\sum_{t=1}^{n}a_{\overline{r}}(D_{f}+\sum_{j=1}^{n}\overline{d_{\overline{j}\mathrm{r}}})$

,

where
$(\alpha\in\Lambda)$$D_{\alpha}=Z_{\alpha}- \sum_{j,k=1}^{n}c_{\alpha_{J}}^{k}\neg a_{j}^{*}a_{\overline{k}}-\sum_{j.k=1}^{n}\tilde{S}_{\alpha j}^{k}a_{j}^{*}a_{k}$wzth functions$c_{\alpha\beta}^{\gamma}(\alpha, \beta,\gamma\in\Lambda)$

defined by$\nabla_{Z_{\alpha}}Z_{\beta}=\sum_{\gamma\in\Lambda}c_{\alpha\beta}^{\gamma}Z_{\gamma}$.

The following proposition holds for the Kaehler form $\Phi(u,v)=g(Iu, v)$ .
Proposition 5.3$i \partial\overline{\partial}\Phi=\sum_{j,k,\ell,m=1}^{n}\omega_{\overline{\ell}\overline{m}jk}\overline{\omega}^{p}\wedge\overline{\omega}^{m}\wedge\dot{d}\wedge\overline{\omega}^{k}$

,

where
.

$\omega_{\overline{\ell}\overline{m}jk}=-\frac{1}{2}R_{\zeta\overline{m}jk^{+\frac{1}{2}\sum_{t=1}^{n},\frac{f}{p}}}\{_{\mathrm{C}\frac{f\overline}{m}j}c_{k\overline{\ell}}^{f}+c_{\ell k}^{r=}c_{j\overline{m}}^{f}-c_{\hslash k}^{f}c_{j\overline{\ell}}^{r}-\mathrm{c}_{j}c_{k\dagger\hslash}^{f}\}$

We have the following representation formula for
Kodaira formula, using (5.1).

Theorem 5.4 It holds on$A^{0},{}^{\mathrm{t}}(M)= \sum_{q=0}^{n}A^{0,q}(M)$$L$

on complex manifolds instead of Bochner-,$L= \overline{\partial}\overline{\partial}^{\mathrm{r}}+\overline{\partial}^{n}\overline{\partial}=-\frac{1}{2}\{\sum_{j=1}^{n}(\nabla_{Z_{j}}^{S}\nabla_{Z_{f}}^{S}+\nabla_{Z_{\dot{f}}}^{S}\nabla_{Z_{f}}^{S})-\nabla_{D}^{S}$$+ \sum_{\mathrm{j},k,r=1}^{n}e(^{-}\dot{d})\mathrm{t}(\overline{Z}_{k})g(R^{\overline{S}}(Z_{j}, Z_{k})Z_{f},\overline{Z}_{f})\}$$-, \sum_{pm,j,k=1}^{\hslash}\omega_{t\overline{m}jk}\overline{a}_{p}^{\mathrm{r}}\overline{a}_{m}^{*}\overline{a}_{j}\overline{a}_{k}-2\sum_{r,\ell,k=1}^{n}\omega_{rTtk}\overline{a}_{\ell}^{l}\overline{a}_{k}$where$D= \sum_{\mathrm{r}=1}^{n}\{\nabla_{Z_{\mathrm{r}}}^{S}Z_{r}+\nabla_{Z_{f}}^{S}\overline{Z}_{r}+T^{S}(Z_{r}, Z_{r})\}$

.

,

171

Remark 5.5 If $M$ is a Kaehler manifold, then we have the following equations;$\partial\overline{\partial}\Phi=0,$$\nabla^{S}=\nabla^{\overline{S}}=\nabla,$$T^{S}=T^{\overline{S}}=0$.

In the case, we can construct the fundamental solution foron complex
manifolds as a pseudo-diffeatial operator, using the above Theorem 5.4 instead of (3.1). By
the similar argument we obtain the assertion(l) of The Main theorem III. But in this oee, thesupsertrace has the ingularity with respect to as. So, in this case we may say that “ a local
version of Reimann-Roch theorem does not hold.”$\partial\overline{\partial}\Phi\neq 0$$(_{Tt}^{\theta}+L)$

$s$

$t$

$tarrow \mathrm{O}$On the other hand, in the caseintroducing a curvature transformation
which corresponds a new connection, we obtain the assertion (2) of the Main
Theorem III by (5.1) and the above theorem. Here A is defined as follows:$\partial\overline{\partial}\Phi=0$$R^{M}(Z_{j}, Z_{k})$$\nabla^{M}=2\nabla-\nabla^{S}$(5.2)$( \Lambda)_{pq}=,\sum_{pm=1}^{n}g(R^{M}(Z_{p},Z_{m})X_{q},X_{p})\overline{\omega}^{\ell}\wedge\omega^{m}$,$Z_{j}= \frac{1}{\sqrt{2}}(X_{j}-iX_{n+j}),Z_{j}=\frac{1}{\sqrt{2}}(X_{j}+iX_{n+j}\rangle$.

This assertion coincides with the result which is proved in J.M.Bismut[2] by the probabilistic
method.

References[1] N.Berline, E.Getzler and M.Vergne, Heat Kernels and Dirac Operators. Springer-Verlag, 1992.
[2] J.M.Bismut, A local index theorem for non K\"auer manifolds, Math.Ann. 284 (1989), 681-699.
[3] H.l.Cycon,R.G.

$\mathrm{R}\mathrm{o}\alpha \mathrm{e},\mathrm{W}.\mathrm{K}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{c}\mathrm{h}$in Physics, Springer,1987.

and B.Simon, Schr\"odinger operators, Texts and Monographs[4] P.G\"unther and R.Schimming, Curvature and spectrum of compact Riemannian manifolds,

J.Diff.Geom. 12 (1977), 599-618.

[5] E.Getzler, The locd Atiyah-Stnger index theorem, Critical phenomena, radom systems, gauge
theories, K.Sterwalder and R.Stora, . Les Houches, Sessin XLIII, (1984), 967-974, North$\alpha \mathrm{k}$Holland.

[6] E.Getzler, A short proof of the local Atiyah-Singer index Theorem., Topology 25 (1986), 111-

117.

[7] P.B.Gilkey, Curvature and the eigenvalues of the Laplacian for geometrical eUiptic complexes,
Ph.D.Dissertation, Harvard University, 1972.

[8] P.B.Gilkey, Invariance Theory, the Heat Equation, and the $Atiyah- s$;lish or Perish, Inc.,1984.

Inder Theorem, Pub-

[9] C.Iwasaki and N.Iwasaki, Parametrtx for a Degenerate Parabolic Equation and its Application to the Asymptotic Behavior of Spectrd Functions for Stationary Problems,
Publ.Res.Inst. Math. Sci. 17 (1981), 557-655.
[10] C.Iwasaki, A proof of the Gauss-Bonnet-Chm Theorem by the symbol calculus of pseudodiffeerentid opemtors, Japanese J.Math.21 (1995), 235-285.

172[11] C.Iwasaki, Symbolic cdculus for construction of the fundamnetal solution
for a degerate equation and a local version of Riemann Roch theorem, Geometry,Analysis and Applications, 2000,

83-92, World Scientic[12] C.Iwasaki, Symbolic construction of the fandamnetal solution and alocd inda, 数 U- 析研究所講究録 Vol. 1412(2005), r 超局所解析の展望 (Recent Rends in Microlocal Analysis)」, $67- 79$.
[13] C.Iwasaki, Symbolic cdculus of Pseudodifferentid operators and curvatureof manifolds, toappear in Modem ?kends in Pseudo-Differential Operators.
[14] S.Kobayashi-K.Nomizu, Foundations ofDifferentid Geometry, I,II, John Wiley&Sons, 1963.
[15] T.Kotake, An andytic proof of the classicd Riemann-Roch theorem, Global Analyis,

Proc.Symp.Pure Math. XVI Providence, 1970.

[16] S.Murakami, Manifolds, Kyoritsusshuppan, 1969 (in Japanese).
[17] V.K.Patodi, An andytic proof of Riemann-Roch-Hirzebruch theorem
for Kaehler manifold,
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$\mathrm{G}\infty \mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}$2167 Shosha Himeji Hyogo 671-2201 Japanemail adress iwasaki@sci.u-hyogo.ac.jp