令$X$是一复$n$维Stein流形, $T(X)$和${{T}^{{*}}}({X})$分别表示$X$上切丛和余切丛, ${{T}_{z}}(X)$和${{T}_{z}}^{*}(X)$分别表示对于$z\in X$的纤维.用$s(z, \varsigma)$和${{s}^{*}}(z, \varsigma)$分别表示$T(X)$和${{T}^{{*}}}(\mbox{X})$的全纯截面和${C ^{1}}$截面.因为我们的兴趣在于函数或微分形式的积分表示, 所以我们可令$X$是更大Stein流形的相对紧集.

Henkin和Leiterer^[1]已获得如下两个重要结果:

(A) 若$X$是更大Stein流形的相对紧集, 则存在全纯映射$s$: $X\times X\to T(X)$和$X\times X$上全纯函数$\varphi$满足如下条件: (a)对于所有$z, \varsigma\in X$, $S(z, \varsigma)\in{{T}_{z}}(X)$; (b)对于每一固定的$z\in X$, $S(z, z)=0$和$s(z, \varsigma):X\to {{T}_{z}}(X)$在$\varsigma\in X$的邻域中是双全纯的; (c)对于所有$z\in X$, $\varphi(z, z)=1$; (d)若${{{\Bbb F}}_{s}}$是一解析层, 它是由${}_{X\times X}\mathfrak{M}$生成的, 则$\varphi\in{{{\Bbb F}}_{s}}((X\times X)\backslash\{(z, z):z\in X\})$; (e)存在一个$\kappa\ge 0$, 对于$T(X)$中每一范数$\left\|\cdot \right\|$我们有函数${{\varphi}^{\kappa}}{{\left\|\left.s\right\|\right.}^{-2}}$在$(X\times X)\backslash \{(z, z):z\in X\}$上是${C ^{2}}$的.

(B) 若$X$是更大Stein流形的相对紧集, 则对于每一全纯函数${{F}_{j}}$和$X\times X$的局部有限开复盖$\{{{V}_{j}}\}$, 存在全纯映射${{\alpha }^{*(j)}}:{{V}_{j}}\to {{T}^{*}}(X)$, 使得${{\alpha }^{*(j)}}(z, \varsigma )\in T{^{*}_{z}}(X)$, 且对于所有$(z, \varsigma )\in {{V}_{j}}$, 我们有

(1.1) $\begin{equation}\label{eq11} \varphi (z, \varsigma )({{F}_{j}}(\varsigma )-{{F}_{j}}(z))=\sum\limits_{r=1}^{n}{\alpha _{r}^{*(j)}}(z, \varsigma )s_{r}^{(j)}(z, \varsigma )\end{equation}$

以及对于$z\in D$和$\varsigma \in {{\delta }_{j}}=\{z\in \partial D:\left| {{F}_{j}}(z) \right|=1\}$, $j=1, 2, \cdots, N$, 还有

(1.2) $\begin{equation}\label{eq12}\sum\limits_{r=1}^{n}{\frac{\alpha _{r}^{*(j)}(z, \varsigma )}{\varphi (z, \varsigma )({{F}_{j}}(\varsigma )-{{F}_{j}}(z))}}s_{r}^{(j)}(z, \varsigma )=1.\end{equation}$

作者在${{\Bbb C}^{n}}$中对于一般有界域解析簇上已获得Koppelman-Leray型积分表示式^[2-3].本文我们研究在Stein流形的解析簇上如何建立微分形式的积分公式.首先, 使用不同的方法和技巧我们导出在Stein流形的两类有界域中对于复$n-m(0\le m<n)$维解析簇上微分形式的相应的积分表示式.其次, 我们得到在Stein流形的一般有界域中对于复$n-m(0\le m<n)$维解析簇上微分形式的统一的积分表示式.特别, 当$m=0$时本文所得公式正是Koppelman-Leray公式在Stein流形中的拓广.

我们还注意到:(a)对于Ⅱ -型有界域微分形式的积分表示至今还很少研究, 更谈不上研究两型有界域的统一积分表示式^[2-3]. (b)我们可进一步获得积分算子的一些估计式^[7-8] (为节省篇幅, 本文不予处理, 我们将另文给予讨论). (c)在Stein流形的解析簇上如何建立微分形式的积分公式至今仍缺少研究.其困难在于我们需证明本文的引理2.3, 本文填补了这个空缺.

${C ^{\infty }}$映射$\sigma :T(X)\to T^*(X)$在$T(X)$的每一纤维中, 确定一个范数

${{\left\| s(z, \left. \varsigma ) \right\| \right.}_{\sigma }}=\langle \sigma s(z, \varsigma ), s(z, \varsigma ){{\rangle}^{\frac{1}{2}}}\ge 0.$

选择$\{{X_{j}}\}$是从属于$\{{{U}_{j}}\}$的单位分解.对于$z\in {{U}_{j}}\subset X$, $s(z, \varsigma )\in {{T}_{z}}(x)$和${{\sigma }_{j}}s(z, \varsigma )\in T_{z}^{*}(X)$, 定义$\sigma s(z, \varsigma )=\sum\limits_{j}{{{X}_{j}}(z){{\sigma }_{j}}s(z, \varsigma )}$, 且记$\bar{s}(z, \varsigma )=\sigma s(z, \varsigma )$($\bar{s}(z, \varsigma )$用于替代${C ^{n}}$中的$\bar{\varsigma }-\bar{z}$).选择全纯截面${{s}_{1}}, {{s}_{2}}, \cdots, {{s}_{n}}\in {{T}_{z}}(X)$和${C ^{1}}$截面$s_{b, 1, \nu }^{*}, s_{b, 2, \nu }^{*}, \cdots, s_{b, n, \nu }^{*}\in T_{z}^{*}(X)$, $b=1, 2, \cdots, \beta$, $\nu =1, 2, \cdots, \alpha -l+1(1\le l\le \alpha \le k)$.

令${{J}_{\alpha }}=\{{{j}_{1}}, \cdots, {{j}_{\alpha }}\}$是$\{1, 2, \cdots, K\}$的有序子集, ${{j}_{1}}<\cdots <{{j}_{\alpha }}$.我们置

${{\Delta }_{K}}=\bigg\{\mu :({{\mu }_{0}}, {{\mu}_{1}}, \cdots, {{\mu }_{K}})\in {{\Bbb R}^{K+1}}, {{\mu }_{j}}\ge 0, j=0, 1, \cdots, K, \sum\limits_{j=0}^{K}{{{\mu }_{j}}}=1\bigg\}, $

${{\Delta }_{0{{J}_{\alpha -k+1}}}}=\{({{\mu }_{0}}, {{\mu }_{{{j}_{1}}}}, \cdots, {{\mu }_{{{j}_{\alpha-k+1}}}}):{{\mu }_{0}}+{{\mu }_{{{j}_{1}}}}+\cdots +{{\mu }_{{{j}_{\alpha -k+1}}}}=1, {{\mu }_{0}}, {{\mu }_{{{j}_{1}}}}, \cdots, {{\mu }_{{{j}_{\alpha -k\mbox{+}1}}}}\ge 0\}, $

$ {{\Delta }_{{{J}_{\alpha -k+1}}}}=\{({{\mu }_{{{j}_{1}}}}, \cdots, {{\mu }_{{{j}_{\alpha -k+1}}}}):{{\mu }_{{{j}_{1}}}}+\cdots +{{\mu }_{{{j}_{\alpha -k+1}}}}=1, {{\mu }_{{{j}_{1}}}}, \cdots, {{\mu }_{{{j}_{\alpha -k\mbox{+}1}}}}\ge 0\}, $

$ {{\Delta }_{0}}=\{{{\mu }_{0}}=1\}, $

为了简单起见, 我们用$({{\mu }_{1}}, {{\mu }_{2}}, \cdots, {{\mu }_{\alpha -k+1}})$替代$({{\mu }_{{{j}_{1}}}}, \cdots, {{\mu }_{{{j}_{\alpha -k+1}}}})$, ${{\Lambda }^{(\beta -1)}}$表示$\beta$维实空间${{\Bbb R}^{\beta }}$的$\beta-1$维链, ${{\Lambda }^{(0)}}=1$.

令$T={{({{T}_{1}}, {{T}_{2}}, \cdots, {{T}_{n}})}^{\tau }}$和${{T}_{p}}={{T}_{p}}(\zeta, z, \lambda, \mu )=\sum\limits_{\nu =1}^{\alpha -l+1}{{{\mu }_{\nu }}}T_{p}^{(\nu )}, T_{p}^{(\nu )}=t_{\beta, \nu }^{p}/{{M}_{\beta, \nu }}$, 其中$t_{\beta, \nu }^{p}=\sum\limits_{b=1}^{\beta }{{{\lambda }_{b}}}s_{b, p, \nu }^{*}(z, \varsigma )$, ${{M}_{\beta, \nu }}=\sum\limits_{p=1}^{n}{{{s}_{p}}(z, \varsigma )}t_{\beta, \nu }^{p}$.特别地,

当$l=k$时, $T=Q={{({{Q}_{1}}, \cdots, {{Q}_{n}})}^{\tau }}$, ${{Q}_{p}}={{Q}_{p}}(\varsigma, z, \lambda )=t_{\beta, 1}^{p}/{{M}_{\beta, 1}}, t_{\beta, 1}^{p}=\sum\limits_{b=1}^{\beta }{{{\lambda }_{b}}}s_{b, p, 1}^{*}(z, \varsigma )$, ${{M}_{\beta, 1}}=\sum\limits_{p=1}^{n}{{{s}_{p}}}(z, \varsigma )t_{\beta, 1}^{p}$;

当$\beta =1$时, $T={{\tilde{Q}}^{0}}={{({{\tilde{Q}}^{0}}_{1}, \cdots, {{\tilde{Q}}^{0}}_{n})}^{\tau }}$和${{\tilde{Q}}^{0}}_{p}={{\tilde{Q}}^{0}}_{p}(\zeta, z, \mu )=\sum\limits_{\nu =1}^{\alpha -l+1}{{{\mu }_{\nu }}}\tilde{Q}_{p}^{0(\nu )}, \tilde{Q}_{p}^{0(\nu )}=t_{1, \nu }^{p}/{{M}_{1, \nu }}$, 其中$t_{1, \nu }^{p}=s_{1, p, \nu }^{*}(z, \varsigma ), $${{M}_{1, \nu }}=\sum\limits_{p=1}^{n}{{{s}_{p}}}(z, \varsigma )s_{1, p, \nu }^{*}(z, \varsigma )$;

当$\beta =\mbox{1}$和$l=k$时, $T={{Q}^{0}}$$={{({{Q}^{0}}_{1}, \cdots, {{Q}^{0}}_{n})}^{\tau }}$和$Q_{p}^{0}=t_{1, 1}^{p}/{{M}_{1, 1}}$, 其中$t_{1, 1}^{p}=s_{1, p, 1}^{*}(z, \zeta )$, ${{M}_{1, 1}}=\sum\limits_{p=1}^{n}{{{s}_{p}}(z, \varsigma )s_{1, p, 1}^{*}}(z, \varsigma )$.这里以及本文剩余部分, $\tau$均表示矩阵转置.

若对于$\varsigma \in \partial D, z\in D$$, \lambda \in {{\Lambda }^{(\beta -1)}}$, ${{M}_{\beta, \nu }}(\varsigma, z, \lambda )\ne 0$和$\varphi (z, \varsigma )$在$D\times \partial D\subseteq D\times X$的邻域中是全纯函数, 函数${{\varphi }^{{{\kappa }^{*}}}}(z, \varsigma )/{{M}_{\beta, \nu }}$是${C ^{1}}$的, 其中${{\kappa }^{*}}$是一适当的非负整数, 则记${{M}_{\beta, \nu }}\in B$.

若全纯函数${{F}_{j}}(\varsigma, z)$可表示成

$\varphi (z, \varsigma ){{F}_{j}}(\varsigma, z)=\varphi (z, \varsigma )({{F}_{j}}(\varsigma )-{{F}_{j}}(z))=\sum\limits_{r=1}^{n}{\alpha _{r}^{*j}}(z, \varsigma )s_{r}^{(j)}(z, \varsigma ), $

且对任何固定的$z\in D$, 在$\varsigma \in \sigma _j\subset \partial D=\bigcup\limits_{j=1}^K\sigma _j$处, $\varphi (z, \varsigma ){{F}_{j}}(\varsigma, z)\ne 0$, 则记${{F}_{j}}(\varsigma, z)\in A$.本文我们总假设${{M}_{\beta, 1}}\in B$, ${{F}_{j}}(\varsigma, z)\in A$.

我们记${{h}_{{{j}_{\nu }}}}={{({{h}_{{{j}_{\nu }}1}}, \cdots, {{h}_{{{j}_{\nu }}n}})}^{\tau }}$, 其中${{h}_{{{j}_{\nu }}p}}=\alpha _{p}^{*{{j}_{\nu }}}(z, \varsigma )/\varphi (z, \varsigma ){{F}_{{{j}_{\nu }}}}(\zeta, z)$.令

(2.1) $ \begin{equation}\label{eq21} E_{l}^{[\rho ]}=({{h}_{{{j}_{1}}}}, \cdots, [{{h}_{{{j}_{\rho }}}}], \cdots, {{h}_{{{j}_{l}}}}), E_{l}^{{}}=({{h}_{{{j}_{1}}}}, \cdots, {{h}_{{{j}_{l}}}}). \end{equation}$

我们假设在Stein流形$X$中的有界域$D$的边界$\partial D$(这里以及本文剩余部分总假设$\partial D$是逐块光滑的)构成缝空间的链, 且可表示成

(2.2) $ \begin{equation}\label{eq22} \bar{D}\supset \partial D\equiv {{\bar{\sigma }}^{(1)}}\supset \cdots \supset {{\bar{\sigma }}^{(\theta )}}\supset{{\bar{\sigma }}^{(\theta +1)}}\supset \cdots \supset {{\bar{\sigma }}^{(\beta -1)}}\supset \bar{\sigma }_{{}}^{(\beta )}\mbox{, } \end{equation} $

其中${{\bar{\sigma }}^{(\theta )}}$是${{\sigma }^{(\theta )}}$的闭包$(\theta =1, 2, \cdots, \beta )$, 则Stein流形中的有界域将分成如下两类型：

(Ⅰ) ${{\sigma }^{(\theta )}}$的维数恰比${{\sigma }^{(\theta \mbox{+}1)}}$增加一维.这种有界域称为Ⅰ -型的.例如解析多面体$D$的边界构成缝空间的链, 且$\partial D$可表示成$\bar{D}\supset \partial D\equiv {{\bar{\sigma }}^{(1)}}\supset \cdots \supset {{\bar{\sigma }}^{(\theta )}}\supset {{\bar{\sigma }}^{(\theta +1)}}\supset$$\cdots \supset {{\bar{\sigma }}^{(\beta )}}$, 其中${{\sigma }^{(\theta )}}$和${{\sigma }^{(\theta +1)}}$分别是解析多面体$D$的实$2n-\theta$维和实$2n-\theta -1$维的棱边.

(Ⅱ) ${{\sigma }^{(\theta )}}$的维数至少比${{\sigma }^{(\theta \mbox{+}1)}}$增加一维.这种有界域称为Ⅱ -型的.例如典型域R的边界构成缝空间的链, 且缝空间的维数至少比其缝增加一维.

但它们可写为统一的形式:

(2.3) $ \begin{eqnarray}\label{eq23} \bar{D}\supset \partial D &\equiv& {{\bar{\sigma }}^{(1)}}\supset \cdots \supset {{\bar{\sigma }}^{(\theta )}}\supset {{\bar{\sigma }}^{(\theta +1)}}\supset \cdots \supset {{\bar{\sigma }}^{(\beta -1)}}\supset {{\bar{\sigma }}^{(\beta )}}\\ &\cong& \bar{\sigma }_{(1)}^{(\beta -1)}\supset \cdots \supset\bar{\sigma }_{(l)}^{(\beta -1}\supset \bar{\sigma }_{(l+1)}^{(\beta -1)}\cdots \supset \bar{\sigma }_{(k)}^{(\beta -1)}, \end{eqnarray}$

其中${{\bar{\sigma }}^{(\theta )}}$是${{\sigma }^{(\theta )}}$的闭包$(\theta =1, 2, \cdots, \beta )$, 且${{\sigma }^{(\theta )}}$的维数至少比${{\sigma }^{(\theta +1)}}$增加一维, 且${{\sigma }^{(\beta )}}$是$2n-\beta$维边缘链, 即存在$2n-\beta +1$维链${{B}_{0}}$, 使得$\partial {{B}_{0}}={{\varepsilon }_{0}}{{\sigma }^{(\beta )}}$, 其中${{\varepsilon }_{0}}$的值是由${{\sigma }^{(\beta )}}$的定向确定, 且通常我们可选择:当$\beta =1$时, ${{\varepsilon }_{0}}=1$; 当$\beta >1$时, ${{\varepsilon }_{0}}={{(-1)}^{n+1}}$; 而$\bar{\sigma }_{(l)}^{(\beta -1)}$是$\sigma _{(l)}^{(\beta -1)}$的闭包$(l=1, 2, \cdots, k)$, 且${{\sigma }^{(\theta )}}$的维数恰比${{\sigma }^{(\theta \mbox{+}1)}}$增加一维.置

$\sigma _{(l)}^{(\beta -1)}=\sum\limits_{1\le {{j}_{1}}<\cdots <{{j}_{l}}\le K}{\sigma _{{{j}_{1}}\cdots {{j}_{l}}}^{(\beta -1)}}=\sum\limits_{{{J}_{l}}}{\sigma _{{{J}_{l}}}^{(\beta -1)}}(K\ge n).$

若${{\Phi }_{1}}, \cdots, {{\Phi }_{m}}$是在$\bar{D}$的邻域${{U}_{{\bar{D}}}}$中的全纯函数, 则置

$Z({{\Phi }_{1}}, \cdots, {{\Phi }_{m}})=\{z\in {{U}_{{\bar{D}}}}:{{\Phi }_{1}}(z)=\cdots ={{\Phi }_{m}}(z)=0\}, m<n.$

我们假设$Z({{\Phi }_{1}}, \cdots, {{\Phi }_{m}})$横截相交于$\partial D$.置$\tilde{D}=Z({{\Phi }_{1}}, \cdots, {{\Phi }_{m}})\bigcap D$, 考虑

(2.4) $ \begin{eqnarray}\label{eq24} \bar{\tilde{D}}\supset \partial \tilde{D}&\equiv&{{\bar{\tilde{\sigma }}}^{(1)}}\supset \cdots \supset {{\bar{\tilde{\sigma }}}^{(\theta )}}\supset {{\bar{\tilde{\sigma }}}^{(\theta +1)}}\supset \cdots \supset {{\bar{\tilde{\sigma }}}^{(\beta -1)}}\supset {{\bar{\tilde{\sigma }}}^{(\beta )}}\\ &\cong &\bar{\tilde{\sigma }}_{(1)}^{(\beta -1)}\supset \cdots \supset\bar{\tilde{\sigma }}_{(l)}^{(\beta -1}\supset \bar{\tilde{\sigma }}_{(l+1)}^{(\beta -1)}\supset \cdots \supset \bar{\tilde{\sigma }}_{(k)}^{(\beta -1)}\mbox{, } \end{eqnarray} $

其中${{\tilde{\sigma }}^{(\theta )}}=Z({{\Phi }_{1}}, \cdots, {{\Phi }_{m}}) \bigcap {{\sigma }^{(\theta )}} $$(\theta =1, 2, \cdots, \beta )$和$\tilde{\sigma }_{(l)}^{\beta -1}=Z({{\Phi }_{1}}, \cdots, {{\Phi }_{m}})\bigcap \sigma _{(l)}^{(\beta -1)}$$(l=1, 2, \cdots, k)$.置$\tilde{\sigma }_{(l)}^{(\beta -1)}=\sum\limits_{1\le {{j}_{1}}<\cdots <{{j}_{l}}\le K}{\tilde{\sigma }_{{{j}_{1}}\cdots {{j}_{l}}}^{(\beta -1)}}=\sum\limits_{{{J}_{l}}}{\tilde{\sigma }_{{{J}_{l}}}^{(\beta -1)}}$.

由(B), 我们有

(2.5) $\begin{equation}\label{eq25}\varphi (z, \varsigma )({{\Phi }_{j}}(\varsigma )-{{\Phi }_{j}}(z))=\sum\limits_{r=1}^{n}{\beta _{r}^{*(j)}}(z, \varsigma )s_{r}^{(j)}(z, \varsigma )\mbox{, }\end{equation}$

和全纯映射${{\beta }^{*(j)}}(z, \varsigma ):{{V}_{j}}\to {{T}^{*}}(X)$, 其中$j=1, 2, \cdots, m$, 或

(2.6) $\begin{equation}\label{eq26}{{\Phi }_{j}}(\varsigma )-{{\Phi }_{j}}(z)=\sum\limits_{r=1}^{n}{\frac{\beta _{r}^{*(j)}(z, \varsigma )}{\varphi (z, \varsigma )}}s_{r}^{(j)}(z, \varsigma ). \end{equation}$

我们记${{H}_{jr}}(z, \varsigma )=\frac{\beta _{r}^{*(j)}(z, \varsigma )}{\varphi (z, \varsigma )}$.

令${{\Psi }_{j}}=({{H}_{1}}, \cdots, {{H}_{j}})$, 这里${{H}_{j}}={{({{H}_{j1}}, \cdots, {{H}_{jn}})}^{\tau }}$, $j=1, \cdots, m$.

$\left| \begin{array}{cc} 0 \\ {{H}_{11}} \\ \vdots \\ {{H}_{1n}} \\\end{array}\ \begin{array}{cc} \cdots \\ \cdots \\ \vdots \\ \cdots \\\end{array}\ \begin{array}{cc} 0 \\ {{H}_{m1}} \\ \vdots \\ {{H}_{mn}} \\\end{array}\ \begin{array}{cc} 1 \\ {{T}_{1}} \\ \vdots \\ {{T}_{n}} \\\end{array}\ \begin{array}{cc} 1 \\ {{h}_{{{j}_{1}}1}} \\ \vdots \\ {{h}_{{{j}_{1}}n}} \\\end{array}\ \begin{array}{cc} \cdots \\ \cdots \\ \vdots \\ \cdots \\\end{array}\ \begin{array}{cc} 1 \\ {{h}_{{{j}_{l}}1}} \\ \vdots \\ h{}_{{{j}_{l}}n} \\\end{array}\ \begin{array}{cc} 0 \\ {{\partial }_{\bar{\zeta }\bar{z}\lambda \mu }}{{T}_{1}} \\ \vdots \\ {{\partial }_{\bar{\zeta }\bar{z}\lambda \mu }}{{T}_{n}} \\\end{array}\ \begin{array}{cc} \cdots \\ \cdots \\ \vdots \\ \cdots \\\end{array}\ \begin{array}{cc} 0 \\ {{\partial }_{\bar{\zeta }\bar{z}\lambda \mu }}{{T}_{1}} \\ \vdots \\ {{\partial }_{\bar{\zeta }\bar{z}\lambda \mu }}{{T}_{n}} \\\end{array} \right|{{\varphi }^{\nu }}(z, \varsigma )=0, $

因为在$Z({{\Phi }_{1}}, \cdots, {{\Phi }_{m}})$和${{\Delta }_{{{J}_{\alpha -l+1}}}}$上,

其中${{\partial }_{\bar{\varsigma }\bar{z}\lambda \mu }}={{\partial }_{{\bar{\varsigma }}}}\mbox{+}{{\partial }_{{\bar{z}}}}\mbox{+}{{\mbox{d}}_{\lambda }}+{{d}_{\mu }}$, 我们有

(2.7) $\begin{eqnarray}\label{eq27}&&{{\varphi }^{\upsilon }}(z, \varsigma )\sum\limits_{\rho =1}^{l}{{{(-1)}^{l+\rho }}}{{\det }_{(m, l-1, 1, n-m-l)}}({{\Psi }_{m}}, E_{l}^{[\rho ]}, T, {{\partial }_{\bar{\zeta }\bar{z}\lambda \mu }}T)\\&=&{{\varphi }^{\upsilon }}(z, \varsigma ){{\det }_{(m, l, n-m-l)}}({{\Psi }_{m}}, E_{l}^{{}}, {{\partial }_{\bar{\zeta }\bar{z}\lambda \mu }}T) \mbox{, }\end{eqnarray}$

其中$\upsilon$取成适当大.特别地,

(2.8) $\begin{eqnarray}\label{eq28}&&{{\varphi }^{\upsilon }}(z, \varsigma )\sum\limits_{\rho =1}^{l}{{{(-1)}^{l+\rho }}}{{\det }_{(m, l-1, 1, n-m-l)}}({{\Psi }_{m}}, E_{l}^{[\rho ]}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\\&=&{{\varphi }^{\upsilon }}(z, \varsigma ){{\det }_{(m, l, n-m-l)}}({{\Psi }_{m}}, E_{l}^{{}}, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q), \end{eqnarray}$

(2.9) $\begin{eqnarray}\label{eq29}&&{{\varphi }^{\upsilon }}(z, \varsigma )\sum\limits_{\rho =1}^{l}{{{(-1)}^{l+\rho }}}{{\det }_{(m, l-1, 1, n-m-l)}}({{\Psi }_{m}}, E_{l}^{[\rho ]}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\\&=&{{\varphi }^{\upsilon }}(z, \varsigma ){{\det }_{(m, l, n-m-l)}}({{\Psi }_{m}}, {{E}_{l}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}}), \end{eqnarray}$

且当$l=1$时, 我们有

(2.10) $\begin{equation}\label{eq210}{{\varphi }^{\upsilon }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)={{\varphi }^{\upsilon }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, h_{{{j}_{1}}}^{{}}, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q), \end{equation}$

(2.11) $\begin{equation}\label{eq211}{{\varphi }^{\upsilon }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})={{\varphi }^{\upsilon }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, {{h}_{{{j}_{1}}}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}}).\end{equation}$

令

(2.12) $\begin{eqnarray}\label{eq212}&&B_{j-1}^{\Phi }(s(z, \varsigma ))\\&=&\frac{(n-j+1)!}{{{\left| \nabla _{j-1}^{\Phi }s(z, \varsigma ) \right|}^{2}}}\sum\limits_{1\le {{j}_{1}}<\cdots <{{j}_{j-1}}\le n}{{{(-1)}^{{{j}_{1}}+\cdots +{{j}_{j-1}}}}}\frac{\overline{\partial ({{\Phi }_{1}}, \cdots, {{\Phi }_{j-1}})}}{\partial ({{s}_{{{j}_{1}}}}, \cdots, {{s}_{{{j}_{j-1}}}})}\bigwedge_{l\ne {{j}_{1}}, \cdots, {{j}_{j-1}}}{\rm d}{{s}_{l}}(z, \varsigma ), \end{eqnarray}$

其中

(2.13) $\begin{equation}\label{eq213}{{\left| \nabla _{j-1}^{\Phi }(s(z, \varsigma )) \right|}^{2}}={{\sum\limits_{1\le {{j}_{1}}<\cdots <{{j}_{j-1}}\le n}{\left| \frac{\partial ({{\Phi }_{1}}, \cdots, {{\Phi }_{j-1}})}{\partial ({{s}_{{{j}_{1}}}}, \cdots, {{s}_{{{j}_{j-1}}}})} \right|}}^{2}}\ne 0, \zeta \in \partial \tilde{D}\mbox{, }\end{equation}$

$B_{0}^{\Phi }(s(z, \varsigma ))=n!\omega (s(z, \varsigma ))$, 这里$\omega (s(z, \varsigma ))={\rm d}{{s}_{1}}\wedge \cdots \wedge {\rm d}{{s}_{n}}$.

经计算得(参见文献[4-5])

(2.14) $\begin{equation}\label{eq214}B_{j-1}^{\Phi }(s(z, \varsigma ))=\tilde{e}(n-j+1)!{{\bigg(\frac{\partial ({{\Phi }_{1}}, \cdots, {{\Phi }_{j-1}})}{\partial ({{s}_{n-j+2}}, \cdots, {{s}_{n}})}\bigg)}^{-1}} \bigwedge_{i=1}^{n-j+1}{\rm d}{{s}_{i}}\mbox{, }\end{equation}$

其中$\tilde{e}={{(-1)}^{(j-1)(n-j+1)}}{{(-1)}^{j(j-1)/2}}$.且当$j>1$, 使用(2.14)式我们得

(2.15) $\begin{equation}\label{eq215}B_{j}^{\Phi }(s(z, \varsigma ))\wedge {\rm d}{{\Phi }_{j}}=\frac{{{(-1)}^{n+j-1}}}{n-j+1}B_{j-1}^{\Phi }(s(z, \varsigma )).\end{equation}$

当$j=1$时, 我们有

(2.16) $\begin{equation}\label{eq216}B_{1}^{\Phi }(s(z, \varsigma ))\wedge {\rm d}{{\Phi }_{1}}=\frac{{{(-1)}^{n}}}{n}B_{0}^{\Phi }(s(z, \varsigma ))={{(-1)}^{n}}(n-1)!{\rm d}{{s}_{1}}\wedge \cdots \wedge {\rm d}{{s}_{n}}.\end{equation}$

本文总假设$Z({{\Phi }_{1}}, \cdots, {{\Phi }_{j}})$横截$\partial D$和$\left| \nabla _{j}^{\Phi }(s(z, \varsigma )) \right|\ne 0$, 其中$1\le j\le m$.

引理 2.1^[3-4]  令${{D}_{j}}=Z({{\Phi }_{1}}, \cdots, {{\Phi }_{j}})\bigcap D, {{D}_{0}}=D, {{D}_{m}}=\tilde{D}$和${{M}_{1, 1}}\in B.$若$z\in \tilde{D}$, 则

(2.17) $\begin{eqnarray}\label{eq217}&&{{c}_{m}}\int_{\partial {{D}_{m}}}{{{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, {{Q}^{0}}, {{\partial }_{{\bar{\varsigma }}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \varsigma ))\\&=&\frac{1}{{{(2\pi{\rm i})}^{n}}}\int_{\partial D}{{{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(1, n-1)}}}({{Q}^{0}}, {{\partial }_{{\bar{\varsigma }}}}{{Q}^{0}})\wedge \omega (s(z, \varsigma )), \end{eqnarray}$

其中${{c}_{m}}={{(-1)}^{m(n+1)}}{{(-1)}^{m(m\mbox{+}1)/2}}/(n-m)!{{(2\pi {\rm i})}^{n-m}}$.

因为

${{(-1)}^{{{j}_{1}}+\cdots +{{j}_{n}}}}\bigwedge\limits^n_{{i=1}\atop {i\ne j_1, \cdots, j_m}}{\rm d}{{s}_{i}}=\delta \frac{\partial ({{\Phi }_{1}}, \cdots, {{\Phi }_{m}})}{\partial ({{s}_{{{j}_{1}}}}, \cdots, {{s}_{{{j}_{m}}}})}{{\bigg(\frac{\partial ({{\Phi }_{1}}, \cdots, {{\Phi }_{m}})}{\partial ({{s}_{n-m+1}}, \cdots, {{s}_{n}})}\bigg)}^{-1}} \bigwedge\limits_{i=1}^{n-m}{\rm d}{{s}_{l}}, $

其中$\delta = {{(-1)}^{m(n-m)}}{{(-1)}^{m(m+1)/2}}/(n-m)!{{(2\pi {\rm i})}^{n-m}}$, 我们有

(2.18) $\begin{eqnarray}\label{eq218}B_{m}^{\Phi }(s(z, \varsigma ))&=&\frac{(n-m)!\delta }{{{\left| \nabla _{m}^{\Phi }(s) \right|}^{2}}}\sum\limits_{{{j}_{1}}<\cdots <{{j}_{m}}}{\overline{\frac{\partial ({{\Phi }_{1}}, \cdots, {{\Phi }_{m}})}{\partial ({{s}_{{{j}_{1}}}}, \cdots, {{s}_{{{j}_{m}}}})}}\frac{\partial ({{\Phi }_{1}}, \cdots, {{\Phi }_{m}})}{\partial ({{s}_{{{j}_{1}}}}, \cdots {{s}_{{{j}_{m}}}})}}\\&&\times {{\bigg(\frac{\partial ({{\Phi }_{1}}, \cdots, {{\Phi }_{m}})}{\partial ({{s}_{n-m+1}}, \cdots, {{s}_{n}})}\bigg)}^{-1}} \bigwedge_{i=1}^{n-m} {\rm d}{{s}_{i}}\\ &=&\delta (n-m)!{{\bigg(\frac{\partial ({{\Phi }_{1}}, \cdots, {{\Phi }_{m}})} {\partial ({{s}_{n-m+1}}, \cdots, {{s}_{n}})}\bigg)}^{-1}} \bigwedge_{i=1}^{n-m}{\rm d}{{s}_{i}}.\end{eqnarray}$

令

(2.19) $\begin{equation}\label{eq219}U_{m}^{\Phi }(s(z, \varsigma ))=\delta (n-m)!{{\bigg(\frac{\partial ({{\Phi }_{1}}, \cdots, {{\Phi }_{m}})}{\partial ({{s}_{n-m+1}}, \cdots, {{s}_{n}})}\bigg)}^{-1}}\mbox{, }\end{equation}$

则

(2.20) $\begin{equation}\label{eq220}B_{m}^{\Phi }(s(z, \varsigma ))=U_{m}^{\Phi }(s(z, \varsigma )) \bigwedge_{i=1}^{n-m}{\rm d}{{s}_{i}}.\end{equation}$

由引理2.1我们可得:

引理 2.2  对于$\tilde{D}$中的任何固定点$z$, 令${{\tilde{U}}_{r}}=\{(z, \varsigma )\in {{U}_{\Delta }}\times {{U}_{\Delta }}:\left\| s(z, \varsigma ) \right\|<r\}$是足够小, 其中${{U}_{\Delta }}\subseteq X\times X$是$\Delta =\{(z, z):z\in \tilde{D}\subset X\}$的邻域, 则有

(2.21) $\begin{equation}\label{eq221}{{c}_{m}}\int_{\partial {{{\tilde{U}}}_{r}}}{{{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}\bigg({{\Psi }_{m}}, \frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}, {{\partial }_{{\bar{\varsigma }}}}\frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}\bigg)\wedge B_{m}^{\Phi }(s(z, \varsigma ))=1.\end{equation}$

证  设$\vartheta (\varsigma )$和$\theta (\varsigma )$分别是$\bar{s}(z, \varsigma )$和$s(z, \varsigma )$关于全纯图册$({{U}_{{{j}_{0}}}}, {{\pi }_{{{j}_{0}}}})$的表示, 和$z\in {{U}_{{{j}_{0}}}}$.

令

$\eta (\varsigma, z, \lambda )={{\varphi }^{\nu }}(z, \varsigma )\bigg[(1-\lambda )\frac{\left\langle \theta (\varsigma ), \vartheta (\varsigma ) \right\rangle \overline{\theta (\varsigma )}}{{{\left| \theta (\varsigma ) \right|}^{2}}}+\lambda \vartheta (\varsigma )\bigg], $

则通过计算我们得

(2.22) $\begin{eqnarray}\label{eq222}&&\frac{1}{{{(2\pi {\rm i})}^{n}}}\int_{\partial {{U}_{r}}}{{{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(1, n-1)}}}\bigg(\frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}, {{\partial }_{{\bar{\varsigma }}}}\frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}\bigg)\wedge \omega (s(z, \varsigma ))\\&=&\frac{1}{{{(2\pi {\rm i})}^{n}}}\int_{\partial {{U}_{r}}}{{{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(1, n-1)}}}\bigg(\frac{\vartheta (\varsigma )}{\left\langle \vartheta (\varsigma ), \theta (\varsigma ) \right\rangle }, {{\partial }_{{\bar{\varsigma }}}}\frac{\vartheta (\varsigma )}{\left\langle \vartheta (\varsigma ), \theta (\varsigma ) \right\rangle }\bigg)\wedge \omega (\theta (\varsigma ))\\&=&\frac{1}{{{(2\pi {\rm i})}^{n}}}\int_{\partial {{U}_{r}}}{{{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(1, n-1)}}}\bigg(\frac{\overline{\theta (\varsigma )}}{{{\left| \theta (\varsigma) \right|}^{2}}}, {{\partial }_{{\bar{\varsigma }}}}\frac{\overline{\theta (\varsigma )}} {{{\left| \theta (\varsigma) \right|}^{2}}}\bigg)\wedge \omega (\theta (\varsigma )), \end{eqnarray}$

其中当$z\in {{U}_{{{j}_{0}}}}$时, ${{\tilde{U}}_{r}}$被写成${{U}_{r}}$.

由上所述和Bochner-Martinelli公式得

(2.23) $\begin{eqnarray}\label{eq223}&&\frac{1}{{{(2\pi {\rm i})}^{n}}}\int_{\partial {{U}_{r}}}{{{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(1, n-1)}}}\bigg(\frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}, {{\partial }_{{\bar{\varsigma }}}}\frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}\bigg)\wedge \omega (s(z, \varsigma ))\\&=&\frac{1}{{{(2\pi {\rm i})}^{n}}}\int_{\partial \theta ({{U}_{r}})}{{{\varphi }^{\nu }}(z, \theta {}^{-1}(\xi )){{\det }_{(1, n-1)}}}\bigg(\frac{{\bar{\xi }}}{{{\left| \xi \right|}^{2}}}, \frac{{\rm d}\bar{\xi }}{{{\left| \xi \right|}^{2}}}\bigg)\wedge \omega (\xi )\\&=&{{\varphi }^{\nu }}[z, {{\theta }^{-1}}(0)]={{[\varphi (z, z)]}^{\nu }}=1. \end{eqnarray}$

且由(2.22)和(2.23)式我们有

(2.24) $\begin{eqnarray}\label{eq224}&&{{c}_{m}}\int_{\partial {{{\tilde{U}}}_{r}}}{{{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}\bigg({{\Psi }_{m}}, \frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}, {{\partial }_{{\bar{\varsigma }}}}\frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}\bigg)\wedge B_{m}^{\Phi }(s(z, \varsigma ))\\&=&\frac{1}{{{(2\pi {\rm i})}^{n}}}\int_{\partial {{U}_{r}}}{{{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(1, n-1)}}}\bigg(\frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}, {{\partial }_{{\bar{\varsigma }}}}\frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}\bigg)\wedge \omega (s(z, \varsigma ))\\&=&1. \end{eqnarray}$

引理2.2得证.

下面我们要证明下述的引理2.3.

若$\{({{U}_{j}}, {{\pi }_{j}})\}$是$X$的全纯图册, 且$\tilde{D}\subset D\subset \subset X$, 则由(A)(b)我们可找到$\alpha =\{(z, z):z\in \tilde{D}\}$的邻域${{U}_{\alpha }}\subseteq \tilde{D}\times \tilde{D}$, 使得映射$s(z, \varsigma )$对于每一固定的$z\in \tilde{D}$, 和$(z, \varsigma )\in {{U}_{\alpha }}$是双全纯的.我们考虑开集${{U}_{\varepsilon }}=\{(z, \varsigma )\in {{U}_{\alpha }}\times {{U}_{\alpha }}:\left\| s(z, \varsigma ) \right\|<\varepsilon \}$.

令$\theta (z, \varsigma )$和$\vartheta (z, \varsigma )$分别是$s(z, \varsigma )$和$\bar{s}(z, \varsigma )$关于$({{U}_{j}}, {{\pi }_{j}})$的表示.由(A)(b), 我们可找到这样的${{\varepsilon }_{0}}$, 使得:(a)映射$T$把$({{U}_{j}}\times {{U}_{j}})\bigcap {{U}_{{{\varepsilon }_{0}}}}$双全纯映为某开集${{V}_{{{\varepsilon }_{0}}}}\subseteq {C ^{n}}\times {C ^{n}}$. (b) $N:{{Y}_{{{\varepsilon }_{0}}}}\to {{U}_{j}}$是由$N({{\pi }_{j}}(z), \theta (z, \varsigma ))=N(\eta, \xi )=\varsigma$定义的全纯映射.我们可定义$T$的逆映射${{T}^{-1}}:{{V}_{{{\varepsilon }_{0}}}}\to ({{U}_{j}}\times {{U}_{j}})\bigcap {{U}_{{{\varepsilon }_{0}}}}$, 和对于所有$(\eta, \xi )\in {{Y}_{{{\varepsilon }_{0}}}}$我们有${{T}^{-1}}(\eta, \xi )=(z, \varsigma )=(\pi _{j}^{-1}(\eta ), N(\eta, \xi ))$和$\theta ({{T}^{-1}}(\eta, \xi ))=\theta (z, \varsigma )=\xi$, 以及对于所有$\eta \in {{\pi }_{j}}({{U}_{j}})$, 我们有${{T}^{-1}}(\eta, 0)=(\pi _{j}^{-1}(\eta ), \pi _{j}^{-1}(\eta ))$, 和当$\eta \in {{\pi }_{j}}({{U}_{j}})$, 则$\vartheta ({{T}^{-1}}(\eta, 0))=0$.

对于$0<\varepsilon \le {{\varepsilon }_{0}}$, 置${{\tilde{S}}_{\varepsilon }}=T(({{\tilde{U}}_{j}}\times {{U}_{j}})\bigcap \partial {{U}_{\varepsilon }})$, 其中${{\tilde{U}}_{j}}\subset \subset {{U}_{j}}$.我们可找到${{\varepsilon }_{1}}:0<{{\varepsilon }_{1}}\le {{\varepsilon }_{0}}$, 使得对于$0<\varepsilon \le {{\varepsilon }_{1}}$, ${{\tilde{S}}_{\varepsilon }}\subset \subset {{Y}_{{{\varepsilon }_{0}}}}$, 且存在$c<\infty$, 使得$\left| \theta (z, \varsigma ) \right|\le c\left\| s(z, \varsigma ) \right\|\le c\varepsilon$, 其中$(z, \varsigma )\in ({{\tilde{U}}_{j}}\times {{U}_{j}})\bigcap \partial {{U}_{\varepsilon }}$, 即对于$0<\varepsilon \le {{\varepsilon }_{1}}$, $\left| \xi \right|\le c\varepsilon$和$(\eta, \xi )\in {{\tilde{S}}_{\varepsilon }}$.

因为$\vartheta \circ {{T}^{-1}}$在${{Y}_{{{\varepsilon }_{0}}}}$上是${C ^{\infty }}$的, 由上所述存在$c<\infty$, 使得对于所有$0<\varepsilon \le {{\varepsilon }_{1}}$和$(\eta, \xi )\in {{\tilde{S}}_{\varepsilon }}$, 我们有^[1]

(2.25) $ \begin{equation}\label{eq225} \left\| {{\det }_{(1, n-1)}}(\vartheta, {{\partial }_{\bar{\eta }\bar{\xi }}}\vartheta )-{{\det }_{(1, n-1)}}(\vartheta, {{\partial }_{{\bar{\xi }}}}\vartheta ) \right\|\le c{{\varepsilon }^{2}}. \end{equation} $

令$(\pi _{j}^{-1})^*G$和${{N}^{*}}G$分别是由$\pi _{j}^{-1}$和$N$确定的$G$的拉回, 则

$(\pi _{j}^{-1})^*G(\eta )=\sum\limits_{\left| K \right|=q}{{{G}_{K}}}(\eta ){\rm d}{{\bar{\eta }}^{K}}, \;\;\; \eta \in {{\pi }_{j}}({{U}_{j}}), $

(2.26) $\begin{equation}\label{eq226} {{N}^{*}}G(\xi, \eta )=\sum\limits_{\left| K \right|\le q}{G({{\pi }_{j}}}(N(\eta, \xi ))){{\rm d}_{\eta \xi }}{{\overline{[{{\pi }_{j}}(N(\eta, \xi ))]}}^{K}}. \end{equation}$

对于$(\eta, \xi )\in {{Y}_{{{\varepsilon }_{0}}}}$, 我们定义

(2.27) $\begin{equation}\label{eq227}N_{\eta }^{*}G(\xi, \eta )=\sum\limits_{\left| K \right|\le q}{G({{\pi }_{j}}}(N(\eta, \xi ))){{\rm d}_{\eta }}{{\overline{[{{\pi }_{j}}(N(\eta, \xi ))]}}^{K}}. \end{equation}$

对于$0<\varepsilon \le {{\varepsilon }_{1}}$和$(\eta, \xi )\in {{\tilde{S}}_{\varepsilon }}$, 存在$c<\infty$使得

(2.28) $ \begin{equation}\label{eq228} \left| {{\pi }_{j}}(N(\eta, \xi ))-\eta \right|=\left| {{\pi }_{j}}(\varsigma )-{{\pi }_{j}}(z) \right|\le c\varepsilon, \left\| {{\rm d}_{\eta }}{{\pi }_{j}}(N(\eta, \xi ))-{\rm d}\eta \right\|\le c\varepsilon. \end{equation} $

因此

(2.29) $\begin{equation}\label{eq229}\lim\limits_{\varepsilon \to 0}\sup\limits_{(\eta, \xi )\in {{{\tilde{S}}}_{\varepsilon }}}\left\| N_{\eta }^{*}G(\xi, \eta )-(\pi _{j}^{-1})^*G(\eta ) \right\|=0, \end{equation}$

(2.30) $\begin{equation}\label{eq230}\left| \varphi ({{T}^{-1}}(\eta, \xi ))-\varphi ({{T}^{-1}}(\eta, 0)) \right|=\left| \varphi ({{T}^{-1}}(\eta, \xi ))-1 \right|\le c\varepsilon. \end{equation}$

令$v(z)$是具有紧支集在$\tilde{D}$中的${C ^{\infty }}$微分形式, $\{{{\chi }_{j}}\}$是从属于$\{{{U}_{j}}\}$的${{ C}^{\infty }}$单位分解, 且书${{v}_{j}}={{\chi }_{j}}v$和选择开集${{\tilde{U}}_{j}}\subset \subset {{U}_{j}}$.

令$({{T}^{-1}})^*$是${{T}^{-1}}$的拉回, 则

$\begin{eqnarray*}J&= :&{{c}_{m}}\int _{(z, \varsigma )\in ({{{\tilde{U}}}_{j}}\times {{U}_{j}})\bigcap \partial {{U}_{\varepsilon }}}{G(\varsigma )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}\\&&\times\bigg({{\Psi }_{m}}, \frac{\vartheta (z, \varsigma )}{\left\langle \left. \vartheta (z, \varsigma ), \theta (z, \varsigma ) \right\rangle \right.}, {{\partial }_{\bar{\varsigma }\bar{z}}}\frac{\vartheta (z, \varsigma )}{\left\langle \left. \vartheta (z, \varsigma ), \theta (z, \varsigma ) \right\rangle \right.}\bigg)\wedge B_{m}^{\Phi }(\theta (z, \varsigma )\wedge {{v}_{j}}(z)\\&=& {{c}_{m}}\int _{(\eta, \xi )\in {{{\tilde{S}}}_{\varepsilon }}}{{{N}^{*}}G(\eta, \xi )\wedge {{\varphi }^{\nu }}}({{T}^{-1}}(\eta, \xi )){{\det }_{(m, 1, n-m-1)}}\\&&\times\bigg({{\Psi }_{m}}, \frac{\vartheta ({{T}^{-1}}(\eta, \xi ))}{\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi )), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}, {{\partial }_{\bar{\eta }\bar{\xi }}}\frac{\vartheta ({{T}^{-1}}(\eta, \xi ))}{\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi )), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}\bigg)\\&& \wedge B_{m}^{\Phi }(\theta ({{T}^{-1}}(\eta, \xi )))\wedge (\pi _{j}^{-1})^*{{v}_{j}}(\eta )\\&=& {{c}_{m}}\int _{(\eta, \xi )\in {{{\tilde{S}}}_{\varepsilon }}}{{{N}^{*}}G(\eta, \xi )\wedge [{{\varphi }^{\nu }}}({{T}^{-1}}(\eta, \xi ))-1]{{\det }_{(m, 1, n-m-1)}} \\ &&\times \bigg({{\Psi }_{m}}, \frac{\vartheta ({{T}^{-1}}(\eta, \xi ))}{\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi )), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}, {{\partial }_{\bar{\eta }\bar{\xi }}}\frac{\vartheta ({{T}^{-1}}(\eta, \xi ))}{\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi )), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}\bigg) \\ && \wedge B_{m}^{\Phi }(\theta ({{T}^{-1}}(\eta, \xi )))\wedge (\pi _{j}^{-1})^*{{v}_{j}}(\eta )\\ &&+{{c}_{m}}\int _{(\eta, \xi )\in {{{\tilde{S}}}_{\varepsilon }}}{{{N}^{*}}G(\eta, \xi )\wedge\bigg [}{{\det }_{(m, 1, n-m-1)}}\bigg({{\Psi }_{m}}, \frac{\vartheta ({{T}^{-1}}(\eta, \xi ))}{\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi )), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}, \\ &&{{\partial }_{\bar{\eta }\bar{\xi }}}\frac{\vartheta ({{T}^{-1}}(\eta, \xi ))}{\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi )), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}\bigg) \\ && -{{\det }_{(m, 1, n-m-1)}}\bigg({{\Psi }_{m}}, \frac{\vartheta ({{T}^{-1}}(\eta, \xi ))}{\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi )), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}, \\ &&{{\partial }_{{\bar{\xi }}}}\frac{\vartheta ({{T}^{-1}}(\eta, \xi ))}{\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi )), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}\bigg)\bigg] \wedge B_{m}^{\Phi }(\theta ({{T}^{-1}}(\eta, \xi )))\wedge (\pi _{j}^{-1})^*{{v}_{j}}(\eta )\\ &&+{{c}_{m}}\int _{(\eta, \xi )\in {{{\tilde{S}}}_{\varepsilon }}}{{{N}^{*}}G(\eta, \xi ) \wedge }{{\det }_{(m, 1, n-m-1)}}\bigg({{\Psi }_{m}}, \frac{\vartheta ({{T}^{-1}}(\eta, \xi ))} {\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi )), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}, \\ &&{{\partial }_{{\bar{\xi }}}}\frac{\vartheta ({{T}^{-1}}(\eta, \xi ))}{\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi )), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}\bigg) \wedge B_{m}^{\Phi }(\theta ({{T}^{-1}}(\eta, \xi )))\wedge (\pi _{j}^{-1})^*{{v}_{j}}(\eta ).\end{eqnarray*}$

由(2.25)和(2.30)式得

$\begin{eqnarray*}\lim\limits_{\varepsilon \to 0}\, J&=&{{c}_{m}}\int _{(\eta, \xi )\in {{{\tilde{S}}}_{\varepsilon }}}{{{N}^{*}}G(\eta, \xi )\wedge }{{\det }_{(m, 1, n-m-1)}}\bigg({{\Psi }_{m}}, \frac{\vartheta ({{T}^{-1}}(\eta, \xi ))}{\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi) ), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}, \\&&{{\partial }_{{\bar{\xi }}}}\frac{\vartheta ({{T}^{-1}}(\eta, \xi ))}{\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi )), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}\bigg) \wedge B_{m}^{\Phi }(\theta ({{T}^{-1}}(\eta, \xi )))\wedge (\pi _{j}^{-1})^*{{v}_{j}}(\eta ).\end{eqnarray*}$

由于阶数的缘由, 我们有

$\begin{eqnarray*}\lim\limits_{\varepsilon \to 0}\, J&=&{{c}_{m}}\int _{(\eta, \xi )\in {{{\tilde{S}}}_{\varepsilon }}}{N_{\eta }^{*}G(\eta, \xi )\wedge }{{\det }_{(m, 1, n-m-1)}}\bigg({{\Psi }_{m}}, \frac{\vartheta ({{T}^{-1}}(\eta, \xi ))}{\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi )), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}, \\&&{{\partial }_{{\bar{\xi }}}}\frac{\vartheta ({{T}^{-1}}(\eta, \xi ))}{\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi )), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}\bigg)\wedge B_{m}^{\Phi }(\theta ({{T}^{-1}}(\eta, \xi )))\wedge (\pi _{j}^{-1})^*{{v}_{j}}(\eta ).\end{eqnarray*}$

由(2.29)式和引理2.2我们有

$\begin{eqnarray*}\lim\limits_{\varepsilon \to 0}\, J&=&{{c}_{m}}\int _{(\eta, \xi )\in {{{\tilde{S}}}_{\varepsilon }}}{{{({{\pi }^{-1}})}^{*}}G(\eta )\wedge }{{\det }_{(m, 1, n-m-1)}}\bigg({{\Psi }_{m}}, \frac{\vartheta ({{T}^{-1}}(\eta, \xi ))}{\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi )), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}, \\&&{{\partial }_{{\bar{\xi }}}}\frac{\vartheta ({{T}^{-1}}(\eta, \xi ))}{\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi )), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}\bigg)\wedge B_{m}^{\Phi }(\theta ({{T}^{-1}}(\eta, \xi )))\wedge (\pi _{j}^{-1})^*{{v}_{j}}(\eta )\\&=&{{c}_{m}}\int _{(\eta, \xi )\in {{{\tilde{S}}}_{\varepsilon }}}{{\rm det}_{(m, 1, n-m-1)}}\bigg({{\Psi }_{m}}, \frac{\vartheta ({{T}^{-1}}(\eta, \xi ))}{\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi )), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}, \\&&{{\partial }_{{\bar{\xi }}}}\frac{\vartheta ({{T}^{-1}}(\eta, \xi ))}{\left\langle \left. \vartheta ({{T}^{-1}}(\eta, \xi )), \theta ({{T}^{-1}}(\eta, \xi )) \right\rangle \right.}\bigg)B_{m}^{\Phi }({{T}^{-1}}(\eta, \xi ))\wedge {{(-1)}^{q}}{{(\pi _{j}^{-1})}^{*}}G(\eta )\wedge (\pi _{j}^{-1})^*{{v}_{j}}(\eta )\\&=&\int _{\eta \in {{\pi }_{j}}({{{\tilde{U}}}_{j}})}{(}-1{{)}^{q}}{{(\pi _{j}^{-1})}^{*}}G(\eta )\wedge (\pi _{j}^{-1})^*{{v}_{j}}(\eta ).\end{eqnarray*}$

从上面的论述, 我们已得如下引理2.3:

引理 2.3  对于在$\tilde{D}$中的任何固定的$z$, 则

(2.31) $\begin{eqnarray}\label{eq231}&&\lim\limits_{\varepsilon \to 0}{{c}_{m}}\int_{\partial {{{\tilde{U}}}_{\varepsilon }}}{G(\varsigma )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}\bigg({{\Psi }_{m}}, \frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}, {{\partial }_{\bar{\varsigma }\bar{z}}}\frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}\bigg)\\&&\wedge U_{m}^{\Phi }(s(z, \varsigma )) \bigwedge_{i=1}^{n-m}{\rm d}{{s}_{i}}={{(-1)}^{^{q}}}G(z), \end{eqnarray}$

其中$G(z)=\sum\limits_{\left| K \right|}{{{G}_{K}}}(z){\rm d}{{\bar{z}}_{K}}$是$(0, q) (q >0) $微分形式.

定理 3.1  若$D$是在Stein流形$X$中具有逐块光滑边界的有界域, 则对于在$\bar{D}$上的$(0, q)(q>0)$微分形式$G(z)$和$z\in \tilde{D}$, 在Stein流形$X$的复$n-m(0\le m<n)$维的解析簇中我们有拓广的Koppelman-Leray型公式

(3.1) $\begin{equation}\label{eq31} {(-1)^{q}}{G(z)=}{{{L}}_{\partial \tilde{D}}}G-({{R}_{\partial \tilde{D}}}+{{B}_{{\tilde{D}}}})\bar{\partial }G+\bar{\partial }({{R}_{\partial \tilde{D}}}+{{B}_{{\tilde{D}}}})G, \end{equation} $

其中

$({{B}_{{\tilde{D}}}}\Gamma )(z)=\\{{c}_{m}}\int_{{\tilde{D}}}{\Gamma (\varsigma) \wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}\bigg({{\Psi }_{m}}, \frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}, {{\partial }_{\bar{\varsigma }\bar{z}}}\frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}\bigg)\wedge B_{m}^{\Phi }(s(z, \varsigma )), ({{L}_{\partial \tilde{D}}}G)(z)\\={{c}_{m}}\int_{\partial \tilde{D}}{G(\varsigma )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, {{Q}^{0}}, {{\partial }_{\bar{\varsigma }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \varsigma )), ({{R}_{\partial \tilde{D}}}\Gamma )(z)\\ ={{c}_{m}}\int_{\partial \tilde{D}\times [0, 1]}{\Gamma (\varsigma )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, \eta, {{\partial }_{\bar{\varsigma }\bar{z}e}}\eta )\wedge B_{m}^{\Phi }(s(z, \varsigma )), $

这里$\Gamma (\varsigma )$为$\bar{\partial }G(\varsigma )$或$G(\varsigma )$, $\eta ={{({{\eta }_{1}}, \cdots, {{\eta }_{n}})}^{\tau }}$和${{\eta }_{p}}=e\frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}+(1-e)Q_{p}^{0}, \ e\in [0, 1], $$ p=1, \cdots, n. $

当$q=0$时, $G(z)$改写为可微函数$u(z)$, (3.1)式被写为拓广的Leray-Stokes型公式

$u(z)={{L}_{\partial \tilde{D}}}u-({{R}_{\partial \tilde{D}}}+{{B}_{{\tilde{D}}}})\bar{\partial }u.$

证  因为在$Z({{\Phi }_{1}}, \cdots, {{\Phi }_{m}})$上,

${{\partial }_{\bar{\varsigma }\bar{z}e}}[{{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, \eta, {{\partial }_{\bar{\varsigma }\bar{z}e}}\eta )]=0.$

因此我们有

(3.2) $\begin{eqnarray}\label{eq32} &&{{\partial }_{\bar{\varsigma }e}}{{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, \eta, {{\partial }_{\bar{\varsigma }\bar{z}e}}\eta )\\&=&-{{\partial }_{{\bar{z}}}}{{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, \eta, {{\partial }_{\bar{\varsigma }\bar{z}e}}\eta ), \\ &&{\rm d}[G(\varsigma )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, \eta, {{\partial }_{\bar{\varsigma }\bar{z}e}}\eta )\wedge B_{m}^{\Phi }(s(z, \varsigma ))]\\ &=&\bar{\partial }G(\varsigma )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, \eta, {{\partial }_{\bar{\varsigma }\bar{z}e}}\eta )\wedge B_{m}^{\Phi }(s(z, \varsigma ))\\& &-{{\partial }_{{\bar{z}}}}[G(\varsigma )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, \eta, {{\partial }_{\bar{\varsigma }\bar{z}e}}\eta )\wedge B_{m}^{\Phi }(s(z, \varsigma ))], \end{eqnarray}$

其中${\rm d}={{\partial }_{\varsigma }}+{{\partial }_{{\bar{\varsigma }}}}+{{d}_{e}}, {{\partial }_{\bar{\varsigma }\bar{z}e}}={{\partial }_{{\bar{\varsigma }}}}+{{\partial }_{{\bar{z}}}}+{{d}_{e}}, \varsigma \in \partial \tilde{D}$.

我们考虑积分域$\{e=1, \varsigma \in \tilde{D}-{{\tilde{S}}_{r}}\}\bigcup \{(\varsigma, e)\in \partial \tilde{D}\times [0, 1]\}$, 其边界是$\{e=0, \varsigma \in \partial \tilde{D}\}\bigcup \{e=1, \varsigma \in \partial {{\tilde{S}}_{r}}\}$.由Stokes公式我们有

(3.3) $ \begin{eqnarray}\label{eq33} &&{{c}_{m}}\int _{\partial \tilde{D}\times [0, 1]}{{}}\bar{\partial }G(\varsigma )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, \eta, {{\partial }_{\bar{\varsigma }\bar{z}e}}\eta )\wedge B_{m}^{\Phi }(s(z, \varsigma ))\nonumber\\& &-{{c}_{m}}{{\partial }_{{\bar{z}}}}\int _{\partial \tilde{D}\times [0, 1]}{G(\varsigma )}\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, \eta, {{\partial }_{\bar{\varsigma }\bar{z}e}}\eta )\wedge B_{m}^{\Phi }(s(z, \varsigma ))\nonumber\\ &&+{{c}_{m}}\int_{\tilde{D}-{{{\tilde{s}}}_{r}}}{\bar{\partial }G(\varsigma) \wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}\bigg({{\Psi }_{m}}, \frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}, {{\partial }_{\bar{\varsigma }\bar{z}}}\frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}\bigg)\wedge B_{m}^{\Phi }(s(z, \varsigma ))\nonumber\\& &-{{c}_{m}}{{\partial }_{{\bar{z}}}}\int_{\tilde{D}-{{{\tilde{s}}}_{r}}}{G(\varsigma) \wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}\bigg({{\Psi }_{m}}, \frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}, {{\partial }_{\bar{\varsigma }\bar{z}}}\frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}\bigg)\wedge B_{m}^{\Phi }(s(z, \varsigma ))\nonumber\\ &=&{{c}_{m}}\int_{\partial \tilde{D}}{G(\varsigma )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, {{Q}^{0}}, {{\partial }_{\bar{\varsigma }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \varsigma ))\\& &-{{c}_{m}}\int_{\partial {{{\tilde{s}}}_{r}}}{G(\varsigma) \wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}\bigg({{\Psi }_{m}}, \frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}, {{\partial }_{\bar{\varsigma }\bar{z}}}\frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}\bigg)\wedge B_{m}^{\Phi }(s(z, \varsigma )).\\\end{eqnarray} $

令$r\to 0$, 且在(3.3)式的两边取极限, 则由引理2.3我们得(3.1)式.

定理 3.2  若Stein流形$X$中有界域$D$的边界$\partial D$由(2.3)式定义, $\partial \tilde{D}$由(2.4)式定义(其中$\beta =1$), 则在$\bar{D}$上的$(0, q) (q >0)$微分形式$G(z)$和$z\in \tilde{D}$, 我们有Ⅰ -型有界域上复$n-m(0\le m<n)$维的解析簇的积分表示:

(a) 当$k>1$时, 我们有

(3.4) $\begin{eqnarray}\label{eq34}{{(-1)}^{q}}G(z)&=&{{c}_{m}}{{(-1)}^{(k-1)q}}\int _{\tilde{\sigma }_{(k)}^{(0)}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, , k, n-m-k)}}}({{\Psi }_{m}}, E_{k}^{{}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\\&&\wedge B_{m}^{\Phi }(s(z, \zeta ))-\sum\limits_{\theta =1}^{k-1}{{}}{{c}_{m}}{{(-1)}^{\theta q}}\int _{\tilde{\sigma }_{(\theta )}^{(0)}}{\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, \theta, 1, n-m-\theta -1)}}}\\&&(\Psi _m, E_\theta , Q^0\partial _{\bar{\zeta }\bar{z}}Q^0)\wedge B_m^\Phi (s(z, \zeta ))+\sum\limits_{\theta =1}^{k-1}c_m(-1)^{\theta q}\partial _{\bar{z}}\int _{\tilde{\sigma }_{(\theta )}^{(0)}}G(\zeta )\wedge \varphi ^\nu (z, \varsigma )\\&&{\det}_{(m, \theta, 1, n-m-\theta -1)}(\Psi _m, E_{\theta }, Q^0\partial_{\bar{\zeta }\bar{z}} Q^0)\wedge B_{m}^{\Phi }(s(z, \zeta ))+\tilde{L}_m(\varsigma, z).\end{eqnarray}$

(b) 当$k=1$时, 我们有

(3.5) $\begin{eqnarray}\label{eq35}{{(-1)}^{q}}G(z)&=&{{c}_{m}}\int _{\tilde{\sigma }_{(1)}^{(0)}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&&+{{\tilde{L}}_{m}}(\varsigma, z), \end{eqnarray}$

其中${{\tilde{L}}_{m}}(\varsigma, z)=-({{R}_{\partial \tilde{D}}}+{{B}_{{\tilde{D}}}})\bar{\partial }G +\bar{\partial }({{R}_{\partial \tilde{D}}}+{{B}_{{\tilde{D}}}})G$.

证  因为

$\begin{eqnarray*}&&{\rm d}[G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))]\\&=&\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&&+{{(-1)}^{q}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\partial }_{{\bar{\varsigma }}}}{{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&=&\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&&+{{(-1)}^{q}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, n-m-l)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta )), \\&&-{{(-1)}^{q}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\partial }_{{\bar{z}}}}{{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&=&\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&&+{{(-1)}^{q}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, n-m-l)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta )), \\&&-{{\partial }_{{\bar{z}}}}[G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))], \end{eqnarray*}$

则由上述和Stokes公式, 我们有

$\begin{eqnarray*}{{I}_{l}}:&=&{{c}_{m}}\int _{\tilde{\sigma }_{(l)}^{(0)}}{{}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, n-m-l)}}({{\Psi }_{m}}, E_{l}^{{}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&=&{{c}_{m}}\sum\limits_{{{j}_{1}}<\cdots <{{j}_{l}}}{{}}\int _{{{{\tilde{\sigma }}}^{(0)}}_{{{j}_{1}}\cdots {{j}_{l}}}}{G(\varsigma )\wedge }{{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, n-m-l)}}({{\Psi }_{m}}, E_{l}^{{}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&=&\frac{{{c}_{m}}}{l!}\sum\limits_{{{j}_{1}}, \cdots, {{j}_{l}}}{\int _{\tilde{\sigma }_{{{j}_{1}}\cdots {{j}_{l}}}^{(0)}}{G(\varsigma )\wedge }}{{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, n-m-l)}}({{\Psi }_{m}}, E_{l}^{{}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&=&\frac{{{c}_{m}}}{(l+1)!}\sum\limits_{\rho =1}^{l+1}{{}}\sum\limits_{{{j}_{1}}, \cdots, [{{j}_{\rho }}], \cdots, {{j}_{l+1}}}{\int _{\tilde{\sigma }_{{{j}_{1}}\cdots [{{j}_{\rho }}]\cdots {{j}_{l+1}}}^{(0)}}{G(\varsigma )\wedge }}{{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, n-m-l)}}\\&&({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \varsigma ))\end{eqnarray*}$

和

(3.6) $\begin{eqnarray}\label{eq36}{{(-1)}^{q}}{{I}_{l}}&=&\frac{{{c}_{m}}}{(l+1)!}\sum\limits_{\rho =1}^{l+1}{{}}\sum\limits_{{{j}_{1}}, \cdots, [{{j}_{\rho }}], \cdots, {{j}_{l+1}}}{\int _{\tilde{\sigma }_{{{j}_{1}}\cdots [{{j}_{\rho }}]\cdots {{j}_{l+1}}}^{(0)}}{{\rm d}[}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}\\&&({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \varsigma ))]\\&&-\frac{{{c}_{m}}}{(l+1)!}\sum\limits_{\rho =1}^{l+1}{{}}\sum\limits_{{{j}_{1}}, \cdots, [{{j}_{\rho }}], \cdots, {{j}_{l+1}}}{\int _{\tilde{\sigma }_{{{j}_{1}}\cdots [{{j}_{\rho }}]\cdots {{j}_{l+1}}}^{(0)}}{{\bar{\partial }}}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}\\&&({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \varsigma ))\\&&+\frac{{{c}_{m}}}{(l+1)!}\sum\limits_{\rho =1}^{l+1}{{}}\sum\limits_{{{j}_{1}}, \cdots, [{{j}_{\rho }}], \cdots, {{j}_{l+1}}}{{{\partial }_{{\bar{z}}}}\int _{\tilde{\sigma }_{{{j}_{1}}\cdots [{{j}_{\rho }}]\cdots {{j}_{l+1}}}^{(0)}}{{}}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}\\&&({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \varsigma ))\\&=&\frac{{{c}_{m}}}{(l+1)!}\sum\limits_{\rho =1}^{l+1}{{}}\sum\limits_{{{j}_{1}}, \cdots, [{{j}_{\rho }}], \cdots, {{j}_{l+1}}}{\sum\limits_{{{j}_{\rho }}}{{{(-1)}^{l+1+\rho }}}\int _{\tilde{\sigma }_{{{j}_{1}}\cdots {{j}_{l+1}}}^{(0)}}{G}}(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma )\\&&{{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, {{Q}^{0}}, {{\partial }_{\bar{\varsigma }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \varsigma ))\\&&-{{c}_{m}}\sum\limits_{{{j}_{1}}<\cdots <{{j}_{l}}}{\int _{\tilde{\sigma }_{{{j}_{1}}\cdots \cdots {{j}_{l}}}^{(0)}}{{\bar{\partial }}}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}\\&&({{\Psi }_{m}}, {{E}_{l}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \varsigma ))\\&&+{{c}_{m}}\sum\limits_{{{j}_{1}}<\cdots <{{j}_{l}}}{{{\partial }_{{\bar{z}}}}\int _{\tilde{\sigma }_{{{j}_{1}}\cdots {{j}_{l}}}^{(0)}}{G}}(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}\\&&({{\Psi }_{m}}, E_{l}^{{}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta )).\end{eqnarray}$

由(2.9)和(3.6)式我们得

(3.7) $\begin{eqnarray}\label{eq37}{{(-1)}^{q}}{{I}_{l}}&=&{{I}_{l+1}}-{{c}_{m}}\int _{\tilde{\sigma }_{(l)}^{(0)}}{{}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l}^{{}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))\nonumber\\&&+{{c}_{m}}{{\partial }_{{\bar{z}}}}\int _{\tilde{\sigma }_{(l)}^{(0)}}{{}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l}^{{}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta )), \nonumber \\&&l=1, 2, \cdots, k-1.\end{eqnarray}$

反复使用(3.7)式, 我们得

(3.8) $\begin{eqnarray}\label{eq38}{{I}_{1}}&=&{{(-1)}^{(k-1)q}}{{I}_{k}}-\sum\limits_{\theta =1}^{k-1}{{{(-1)}^{\theta q}}}{{c}_{m}}\int _{\tilde{\sigma }_{(\theta )}^{(0)}}{{\bar{\partial }}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, \theta, 1, n-m-\theta -1)}}\\&&({{\Psi }_{m}}, E_{\theta }^{{}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))+\sum\limits_{\theta =1}^{k-1}{{{(-1)}^{\theta q}}}{{c}_{m}}{{\partial }_{{\bar{z}}}}\int _{\tilde{\sigma }_{(\theta )}^{(0)}}{G}(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma )\\&&{{\det }_{(m, \theta, 1, n-m-\theta -1)}}({{\Psi }_{m}}, E_{\theta }^{{}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta )).\end{eqnarray}$

由(2.11)式我们有

(3.9) $\begin{eqnarray}\label{eq39}{{I}_{1}}&=&{{c}_{m}}\sum\limits_{{{j}_{1}}}{\int _{\tilde{\sigma }{{_{{{j}_{1}}}^{(0)}}_{{}}}}{G(\varsigma )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}}}({{\Psi }_{m}}, {{h}_{{{j}_{1}}}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&=&{{c}_{m}}\int _{\partial \tilde{D}}{G(\varsigma )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta )). \end{eqnarray}$

因此, 由(3.8), (3.9)和(3.1)式得(3.4)和(3.5)式.

定理 3.3  若Stein流形$X$中有界域$D$的边界$\partial D$由(2.3)式定义, $\partial \tilde{D}$由(2.4)式定义(其中$k=1$), 令${{\tau }_{m}}={{B}_{0}}\bigcap Z({{\Phi }_{1}}, \cdots, {{\Phi }_{m}})$, $\partial {{\tau }_{m}}={{\varepsilon }_{0}}{{\tilde{\sigma }}^{(\beta )}}$和$\tilde{C}=\partial ({{\Lambda }^{(\beta -1)}}\times {{\tau }_{m}})$.假设$\tilde{\Omega }$是具有以$\tilde{C}$和$\partial \tilde{D}$作为边缘的一个链, 则在$\bar{D}$上的$(0, q) (q >0)$微分形式$G(z)$和$z\in \tilde{D}$, 我们有Ⅱ -型有界域上复$n-m(0\le m<n)$维的解析簇的积分表示:

(3.10) $\begin{eqnarray}\label{eq310}{{(-1)}^{q}}G(z)&=&\varepsilon {{c}_{m}}\int _{{{\Lambda }^{(\beta -1)}}\times {{{\tilde{\sigma }}}^{(\beta )}}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\nonumber\\&&+{{c}_{m}}{{\partial }_{{\bar{z}}}}\int _{{\tilde{\Omega }}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&&-{{c}_{m}}\int _{{\tilde{\Omega }}}{\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&&+{{\tilde{L}}_{m}}(\varsigma, z), \end{eqnarray}$

其中$\varepsilon ={{(-1)}^{\beta -1}}{{\varepsilon }_{0}}, $${{\partial }_{\bar{\varsigma }\bar{z}\lambda }}\mbox{=}{{\partial }_{{\bar{\varsigma }}}}\mbox{+}{{\partial }_{{\bar{z}}}}\mbox{+}{{\mbox{d}}_{\lambda }}$, $Q={{({{Q}_{1}}, \cdots, {{Q}_{n}})}^{\tau }}$和${{Q}_{p}}=t_{\beta, 1}^{p}/{{M}_{\beta, 1}}(p=1, \cdots, n)$, 且如下条件成立

(3.11) $\begin{equation}\label{eq311}{\rm Rank}\frac{\partial (t_{\beta, 1}^{1}, \cdots, t_{\beta, 1}^{n})}{\partial ({{{\bar{\zeta }}}_{1}}, \cdots, {{{\bar{\varsigma }}}_{n}}, {{{\bar{z}}}_{1}}, \cdots, {{{\bar{z}}}_{n}})}\le n-m-\beta.\end{equation}$

证  令$\tilde{C}=\partial ({{\Lambda }^{(\beta -1)}}\times {{\tau }_{m}})$, 则$\tilde{C}=\partial {{\Lambda }^{(\beta -1)}}\times {{\tau }_{m}}+\varepsilon {{\Lambda }^{(\beta -1)}}\times {{\tilde{\sigma }}^{(\beta )}}$是实$2n-2m-1$维循环, 使用条件(3.11), 且经计算得

$\int _{\partial {{\Lambda }^{(\beta -1)}}\times {{\tau }_{m}}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\mbox{=}0, $

因此我们有

(3.12) $\begin{eqnarray}\label{eq312}&&\varepsilon {{c}_{m}}\int _{{{\Lambda }^{(\beta -1)}}\times {{{\tilde{\sigma }}}^{(\beta )}}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&=&{{c}_{m}}\int _{{\tilde{C}}}{G(\varsigma )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta )). \end{eqnarray}$

另一方面, 因为

${{\partial }_{\bar{\varsigma }\bar{z}\lambda }}[{{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\varsigma }\bar{z}\lambda }}Q)]={{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, n-m)}}({{\Psi }_{m}}, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)=0$

和

${{\partial }_{\bar{\varsigma }\lambda }}[{{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\varsigma }\bar{z}\lambda }}Q)=-{{\partial }_{{\bar{z}}}}[{{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)], $

在$Z({{\Phi }_{1}}, \cdots, {{\Phi }_{m}})$上有

(3.13) $\begin{eqnarray}\label{eq313}&&{\rm d}[G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))]\\&=&\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&&-{{\partial }_{{\bar{z}}}}[G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))]. \end{eqnarray}$

因$\tilde{C}$和$\partial \tilde{D}=\partial {{D}_{m}}$都是${{\Bbb R}^{\beta }}\times {C ^{n-m}}$上实$2n-2m-1$维循环, 因此由(3.13)式和Stokes公式得

(3.14) $ \begin{eqnarray}\label{eq314} &&\bigg(\int _{{\tilde{C}}}{-\int _{\partial \tilde{D}}{\bigg)}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\ &=&\int _{{\tilde{\Omega }}}{\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\& &-{{\partial }_{{\bar{z}}}}\int _{{\tilde{\Omega }}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta )). \end{eqnarray}$

从(3.13)和(3.14)式我们推得

(3.15) $\begin{eqnarray}\label{eq315}&&\varepsilon {{c}_{m}}\int _{{{\Lambda }^{(\beta -1)}}\times {{{\tilde{\sigma }}}^{(\beta )}}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&=&{{c}_{m}}\int_{\partial \tilde{D}}{G(\varsigma )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, {{Q}^{0}}, {{\partial }_{\bar{\varsigma }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \varsigma ))\\&&+{{c}_{m}}\int _{{\tilde{\Omega }}}{\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&&-{{c}_{m}}{{\partial }_{{\bar{z}}}}\int _{{\tilde{\Omega }}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta )).\end{eqnarray}$

由(3.15)和(3.1)式我们可得(3.10)式.

定理 3.4  若Stein流形$X$中有界域$D$的边界$\partial D$由(2.3)式定义, $\partial \tilde{D}$由(2.4)式定义, 则在$\bar{D}$上的$(0, q) (q >0)$微分形式$G(z)$和$z\in \tilde{D}$, 我们有Ⅰ -型有界域上复$n-m(0\le m<n)$维的解析簇的积分表示:

(a) 当$k>1$时, 我们有

(3.16) $\begin{eqnarray}\label{eq316}{{(-1)}^{q}}G(z)&=&\varepsilon {{\varepsilon }_{k}}{{c}_{m}}\int _{{{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{(k)}^{(\beta -1)}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, k, n-m-k)}}}({{\Psi }_{m}}, E_{k}^{{}}, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\\&&\wedge B_{m}^{\Phi }(s(z, \varsigma ))-\sum\limits_{\theta =1}^{k-1}(-1)^q\varepsilon \varepsilon _\theta c_m\int _{{\Lambda }^{(\beta -1)\times \tilde{\sigma }_{(\theta )}^{(\beta -1)}}} \bar{\partial }G(\zeta )\wedge \varphi ^\nu (z, \varsigma )\\&&{\det }_{(m, \theta, 1, n-m-\theta -1)}({{\Psi }_{m}}, E_{\theta }^{{}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \varsigma ))\\&&+\sum\limits_{\theta =1}^{k-1}{{{(-1)}^{q}}}\varepsilon {{\varepsilon }_{\theta }}{{c}_{m}}{{\partial }_{{\bar{z}}}}\int _{{{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{(\theta )}^{(\beta -1)}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, \theta, 1, n-m-\theta -1)}}}\\&&({{\Psi }_{m}}, E_{\theta }^{{}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \varsigma ))+{{{L}'}_{m}}(\varsigma, z), \end{eqnarray}$

(b) 当$k=1$时, 我们有

(3.17) $\begin{eqnarray}\label{eq317}{{(-1)}^{q}}G(z)&=&\varepsilon {{c}_{m}}\int _{{{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{(1)}^{(\beta -1)}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\\&&\wedge B_{m}^{\Phi }(s(z, \zeta ))+{{{L}'}_{m}}(\varsigma, z), \end{eqnarray}$

其中${{\varepsilon }_{l}}={{(-1)}^{(l-1)(q+\beta -1)}}\ (l=1, 2, \cdots, k;k>1)$, $Q={{({{Q}_{1}}, \cdots, {{Q}_{n}})}^{\tau }}$, 以及${{Q}_{p}}=t_{\beta, 1}^{p}/{{M}_{\beta, 1}}$$(p=1, \cdots, n)$, 且满足如下条件:

(3.18) $\begin{equation}\label{eq318}{\rm Rank}\frac{\partial (t_{\beta, 1}^{1}, \cdots, t_{\beta, 1}^{n})}{\partial ({{{\bar{\zeta }}}_{1}}, \cdots, {{{\bar{\varsigma }}}_{n}}, {{{\bar{z}}}_{1}}, \cdots, {{{\bar{z}}}_{n}})}\le n-m-\beta -k+1 \mbox{;}\end{equation}$

以及

(3.19) $\begin{eqnarray}\label{eq319}{{{L}'}_{m}}(\varsigma, z)&=&-{{c}_{m}}\int _{{\tilde{\Omega }}}{\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&&+{{c}_{m}}{{\partial }_{{\bar{z}}}}\int _{{\tilde{\Omega }}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&&+{{\tilde{L}}_{m}}(\varsigma, z). \end{eqnarray}$

证  因为

(3.20) $\begin{eqnarray}\label{eq320}&&\partial ({{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{{{j}_{1}}\cdots [{{j}_{\rho }}]\cdots {{j}_{l+1}}}^{(\beta -1)})\\&=&\partial {{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{{{j}_{1}}\cdots [{{j}_{\rho }}]\cdots {{j}_{l+1}}}^{(\beta -1)}+{{(-1)}^{\beta -1}}{{\Lambda }^{(\beta -1)}}\times \sum\limits_{{{j}_{\rho }}}{{{(-1)}^{l+1+\rho }}}\tilde{\sigma }_{{{j}_{1}}\cdots {{j}_{l+1}}}^{(\beta -1)}\end{eqnarray}$

和

(3.21) $\begin{eqnarray}\label{eq321}&&{\rm d}[G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))]\\&=&{{(-1)}^{q}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\partial }_{\bar{\varsigma }\lambda }}{{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&&+\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \varsigma ))\\&=&{{(-1)}^{q}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, n-m-l)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \varsigma ))\\&&-{{(-1)}^{q}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\partial }_{{\bar{z}}}}{{\det }_{(m, l, 1.n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&&+\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \varsigma ))\\&=&{{(-1)}^{q}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, n-m-l)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \varsigma ))\\&&-{{\partial }_{{\bar{z}}}}[G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))]\\&&+\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \varsigma )), \end{eqnarray}$

其中${\rm d}={{\partial }_{\zeta }}+{{\partial }_{{\bar{\zeta }}}}+{{\rm d}_{\lambda }}$, ${{\partial }_{\bar{\varsigma }\bar{z}\lambda }}={{\partial }_{{\bar{\varsigma }}}} +{{\partial }_{{\bar{z}}}}+{{\rm d}_{\lambda }}={{\partial }_{\bar{\varsigma }\lambda }}+{{\partial }_{{\bar{z}}}}$.则由(3.20)式, (3.21)式和Stokes公式我们有

$\begin{eqnarray*}{{\tilde{I}}_{l}}:&=&{{c}_{m}}\int _{{{\Lambda }^{(\beta -1)}}\times {{{\tilde{\sigma }}}^{(\beta -1)}}_{(l)}}{G}(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, n-m-l)}}({{\Psi }_{m}}, E_{l}^{{}}, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&=&{{c}_{m}}\sum\limits_{{{j}_{1}}<\cdots <{{j}_{l}}}{{}}\int _{{{\Lambda }^{(\beta -1)}}\times {{{\tilde{\sigma }}}^{(\beta -1)}}_{{{j}_{1}}\cdots {{j}_{l}}}}{G}(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, n-m-l)}}({{\Psi }_{m}}, E_{l}^{{}}, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&=&\frac{{{c}_{m}}}{l!}\sum\limits_{{{j}_{1}}, \cdots, {{j}_{l}}}{\int _{{{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{{{j}_{1}}\cdots {{j}_{l}}}^{(\beta -1)}}{G}}(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, n-m-l)}}({{\Psi }_{m}}, E_{l}^{{}}, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&=&\frac{{{c}_{m}}}{(l+1)!}\sum\limits_{\rho =1}^{l+1}{{}}\sum\limits_{{{j}_{1}}, \cdots, [{{j}_{\rho }}], \cdots, {{j}_{l+1}}}{\int _{{{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{{{j}_{1}}\cdots [{{j}_{\rho }}]\cdots {{j}_{l+1}}}^{(\beta -1)}}{G}}(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, n-m-l)}}\\&&({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\end{eqnarray*}$

和

(3.22) $\begin{eqnarray}\label{eq322}{{(-1)}^{q}}{{\tilde{I}}_{l}}&=&\frac{{{c}_{m}}}{(l+1)!}\sum\limits_{\rho =1}^{l+1}{{}}\sum\limits_{{{j}_{1}}, \cdots, [{{j}_{\rho }}], \cdots, {{j}_{l+1}}}{\int _{{{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{{{j}_{1}}\cdots [{{j}_{\rho }}]\cdots {{j}_{l+1}}}^{(\beta -1)}}}\\&&{\rm d}[G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, Q, {{\partial }_{\bar{\varsigma }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))]\\&&-\frac{{{c}_{m}}}{(l+1)!}\sum\limits_{\rho =1}^{l+1}{{}}\sum\limits_{{{j}_{1}}, \cdots, [{{j}_{\rho }}], \cdots, {{j}_{l+1}}}{\int _{{{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{{{j}_{1}}\cdots [{{j}_{\rho }}]\cdots {{j}_{l+1}}}^{(\beta -1)}}{\bar{\partial }G}}(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma )\\&&{{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, Q, {{\partial }_{\bar{\varsigma }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))]\\&&+\frac{{{c}_{m}}}{(l+1)!}\sum\limits_{\rho =1}^{l+1}{{}}\sum\limits_{{{j}_{1}}, \cdots, [{{j}_{\rho }}], \cdots, {{j}_{l+1}}}{{{\partial }_{{\bar{z}}}}\int _{{{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{{{j}_{1}}\cdots [{{j}_{\rho }}]\cdots {{j}_{l+1}}}^{(\beta -1)}}{G}}(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma )\\&&{{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, Q, {{\partial }_{\bar{\varsigma }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))]\\&=&\frac{{{c}_{m}}}{(l+1)!}\sum\limits_{\rho =1}^{l+1}{{}}\sum\limits_{{{j}_{1}}, \cdots, [{{j}_{\rho }}], \cdots, {{j}_{l+1}}}\bigg(\int _{\partial {{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{{{j}_{1}}\cdots [{{j}_{\rho }}]\cdots {{j}_{l+1}}}^{(\beta -1)}}\\&&+{{(-1)}^{\beta -1}}\sum\limits_{{{j}_{\rho }}}{{{(-1)}^{l+1+\rho }}}\int _{{{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{{{j}_{1}}\cdots {{j}_{l+1}}}^{(\beta -1)}}\bigg)\\&&G(\varsigma )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&&-{{c}_{m}}\sum\limits_{{{j}_{1}}<\cdots <{{j}_{l}}}{\int _{{{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{{{j}_{1}}\cdots {{j}_{l}}}^{(\beta -1)}}{{\bar{\partial }}}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}\\&&({{\Psi }_{m}}, E_{l}^{{}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&&+{{c}_{m}}\sum\limits_{{{j}_{1}}<\cdots <{{j}_{l}}}{{{\partial }_{{\bar{z}}}}\int _{{{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{{{j}_{1}}\cdots {{j}_{l}}}^{(\beta -1)}}{G}}(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}\\&&({{\Psi }_{m}}, E_{l}^{{}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta )), \end{eqnarray}$

使用条件(3.18), 且经计算我们可得

(3.23) $\begin{equation}\label{eq323}\int _{\partial {{\Lambda }^{(\beta -1)}}\times\tilde{\sigma }_{{{j}_{1}}\cdots [{{j}_{\rho }}]\cdots {{j}_{l+1}}}^{(\beta -1)}}{G(\zeta )\wedge }{{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}({{\Psi }_{m}}, E_{l+1}^{[\rho ]}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))=0.\end{equation}$

于是由(3.22), (3.23)和(2.8)式得

(3.24) $\begin{eqnarray}\label{eq324}{{(-1)}^{q}}{{\tilde{I}}_{l}}&=&{{(-1)}^{\beta -1}}{{\tilde{I}}_{l+1}}-{{c}_{m}}\sum\limits_{{{j}_{1}}<\cdots <{{j}_{l}}}{\int _{{{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{{{j}_{1}}\cdots {{j}_{l}}}^{(\beta -1)}}{{\bar{\partial }}}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}\\&&({{\Psi }_{m}}, E_{l}^{{}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&&+{{c}_{m}}\sum\limits_{{{j}_{1}}<\cdots <{{j}_{l}}}{{{\partial }_{{\bar{z}}}}\int _{{{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{{{j}_{1}}\cdots {{j}_{l}}}^{(\beta -1)}}{G}}(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}\\&&({{\Psi }_{m}}, E_{l}^{{}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta )), \end{eqnarray}$

其中$l=1, 2, \cdots, k-1$.

反复使用(3.24)式得

(3.25) $\begin{eqnarray}\label{eq325}\varepsilon {{\tilde{I}}_{1}}&=&\varepsilon {{\varepsilon }_{k}}{{\tilde{I}}_{k}}-\sum\limits_{l=1}^{k-1}{{{(-1)}^{q}}}\varepsilon {{\varepsilon }_{l}}{{c}_{m}}\int _{{{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{(l)}^{(\beta -1)}}{{\bar{\partial }}}G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}\\&&({{\Psi }_{m}}, {{E}_{l}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\\&&+\sum\limits_{l=1}^{k-1}{{{(-1)}^{q}}}\varepsilon {{\varepsilon }_{l}}{{c}_{m}}{{\partial }_{{\bar{z}}}}\int _{{{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{(l)}^{(\beta -1)}}{G}(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, l, 1, n-m-l-1)}}\\&&({{\Psi }_{m}}, {{E}_{l}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta )), \end{eqnarray}$

由(2.10)式我们有

(3.26) $\begin{eqnarray}\label{eq326}\varepsilon {{\tilde{I}}_{1}}&=&\varepsilon {{c}_{m}}\sum\limits_{{{j}_{1}}}{{}}\int _{{{\Lambda }^{(\beta -1)}}\times {{{\tilde{\sigma }}}^{(\beta -1)}}_{{{j}_{1}}}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, {{h}_{{{j}_{1}}}}, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta ))\nonumber\\&=&\varepsilon {{c}_{m}}\int _{{{\Lambda }^{(\beta -1)}}\times {{{\tilde{\sigma }}}^{(\beta )}}_{{}}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge B_{m}^{\Phi }(s(z, \zeta )). \end{eqnarray}$

从(3.25), (3.26)和(3.10)式得(3.16)和(3.17)式.

注 3.1  当$\beta =1$时, 我们有$Q={{Q}^{0}}, $${{\partial }_{\bar{\varsigma }\bar{z}\lambda }}Q={{\partial }_{\bar{\varsigma }\bar{z}}}{{Q}^{0}}, {{\tilde{\sigma }}^{(1)}}=\tilde{\sigma }_{(1)}^{(0)}=\partial \tilde{D}$, 且由(3.15)式得

$ \begin{eqnarray*} &&{{c}_{m}}\int _{{{{\tilde{\sigma }}}^{(1)}}}{G(\zeta ){{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))\\ &=&{{c}_{m}}\int_{\partial \tilde{D}}{G(\varsigma ){{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, {{Q}^{0}}, {{\partial }_{\bar{\varsigma }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \varsigma ))\\ &&-{{c}_{m}}{{\partial }_{{\bar{z}}}}\int _{{\tilde{\Omega }}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))\\ &&+{{c}_{m}}\int _{{\tilde{\Omega }}}{\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta )), \end{eqnarray*} $

这意味着

$\begin{eqnarray*} & &-{{c}_{m}}{{\partial }_{{\bar{z}}}}\int _{{\tilde{\Omega }}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))\\ &&{{c}_{m}}\int _{{\tilde{\Omega }}}{\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}({{\Psi }_{m}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge B_{m}^{\Phi }(s(z, \zeta ))=0 \end{eqnarray*} $

和${{{L}'}_{m}}(\varsigma, z)=$${{\tilde{L}}_{m}}(\varsigma, z)$.因此(3.16)和(3.17)式分别转化为(3.4)和(3.5)式.

注 3.2  当$k=1$时, 我们有$\tilde{\sigma }_{(1)}^{(\beta -1)}={{\tilde{\sigma }}^{(\beta )}}$, 且(3.17)式转化为(3.10)式.由注3.1和注3.2断言(3.16)和(3.17)式是(3.1), (3.4), (3.5)和(3.10)式的统一的积分表示式.

注 3.3  当$m=0$时, 则$B_{m}^{\Phi }(s(z, \zeta ))$$=B_{0}^{\Phi }(s(z, \zeta ))=n!\omega (s(z, \zeta ))$, ${{c}_{0}}=1/n!{{(2\pi {\rm i})}^{n}}$, $\tilde{\sigma }_{\alpha }^{\beta -1}\mbox{=}\sigma _{\alpha }^{\beta -1}$, $\tilde{\Omega }=\Omega $和$\partial \tilde{D}=\partial D$, 它是由(2.3)式定义, 且(3.18)式可写为

(3.27) $ \begin{equation}\label{eq327} {\rm Rank}\frac{\partial (t_{\beta, 1}^{1}, \cdots, t_{\beta, 1}^{n})}{\partial ({{{\bar{\zeta }}}_{1}}, \cdots, {{{\bar{\varsigma }}}_{n}}, {{{\bar{z}}}_{1}}, \cdots, {{{\bar{z}}}_{n}})}\le n-\beta -k+1. \end{equation} $

因此(3.16)和(3.17)式转化为如下的Koppelman-Leray公式的推广:

对于$k>1$,

(3.28) $\begin{eqnarray}\label{eq328}{{(-1)}^{q}}G(z)&=&\frac{\varepsilon {{\varepsilon }_{k}}}{{{(2\pi {\rm i})}^{n}}}\int _{{{\Lambda }^{(\beta -1)}}\times \sigma _{(k)}^{(\beta -1)}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(k, n-k)}}}(E_{k}^{{}}, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge {{\omega }_{\varsigma }}(s(z, \varsigma ))\nonumber\\&&-\sum\limits_{\theta =1}^{k-1}{{{(-1)}^{q}}}\frac{\varepsilon {{\varepsilon }_{\theta }}}{{{(2\pi {\rm i})}^{n}}}\int _{{{\Lambda }^{(\beta -1)}}\times \sigma _{(\theta )}^{(\beta -1)}}{\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(\theta, 1, n-\theta -1)}}}\nonumber\\&&(E_{\theta }^{{}}, Q, {{\partial }_{\bar{\zeta }\tilde{z}\lambda }}Q)\wedge {{\omega }_{\varsigma }}(s(z, \varsigma ))\nonumber\\&&+\sum\limits_{\theta =1}^{k-1}{{{(-1)}^{q}}}\frac{\varepsilon {{\varepsilon }_{\theta }}}{{{(2\pi {\rm i})}^{n}}}{{\partial }_{{\bar{z}}}}\int _{{{\Lambda }^{(\beta -1)}}\times \sigma _{(\theta )}^{(\beta -1)}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(\theta, 1, n-\theta -1)}}}\nonumber\\&&(E_{\theta }^{{}}, Q, {{\partial }_{\bar{\zeta }\tilde{z}\lambda }}Q)\wedge {{\omega }_{\varsigma }}(s(z, \varsigma ))+{L}'(\varsigma, z), \end{eqnarray}$

对于$k=1, $

(3.29) $\begin{eqnarray}\label{eq329}{{(-1)}^{q}}G(z)&=&\frac{\varepsilon }{{{(2\pi {\rm i})}^{n}}}\int _{{{\Lambda }^{(\beta -1)}}\times \sigma _{(1)}^{(\beta -1)}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(1, n-1)}}}(Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge \omega (s(z, \varsigma ))\\&&+{L}'(\varsigma, z), \end{eqnarray}$

其中

(3.30) $\begin{eqnarray}\label{eq330}{L}'(\varsigma, z)&=&-\frac{1}{{{(2\pi {\rm i})}^{n}}}\int _{\Omega }{\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(1, n-1)}}}(Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge \omega (s(z, \zeta ))\\&&+\frac{1}{{{(2\pi {\rm i})}^{n}}}{{\partial }_{{\bar{z}}}}\int _{\Omega }{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(1, n-1)}}}(Q, {{\partial }_{\bar{\zeta }\bar{z}\lambda }}Q)\wedge \omega (s(z, \zeta ))+{{\tilde{L}}_{{}}}(\varsigma, z), \end{eqnarray}$

(3.31) $\begin{equation}\label{eq331}\tilde{L}(\varsigma, z)=-({{R}_{\partial D}}+{{B}_{D}})\bar{\partial }G+\bar{\partial }({{R}_{\partial D}}+{{B}_{D}})G, \end{equation}$

其中

$({{B}_{D}}\Gamma )(z)=\frac{1}{{{(2\pi {\rm i})}^{n}}}\int_{D}{\Gamma (\varsigma \wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(1, n-1)}}}\bigg(\frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}, {{\partial }_{\bar{\varsigma }\bar{z}}}\frac{\bar{s}(z, \varsigma )}{\left\| s(z, \varsigma ) \right\|_{\sigma }^{2}}\bigg)\wedge \omega (s(z, \varsigma )), $

$({{R}_{\partial D}}\Gamma )(z)=\frac{1}{{{(2\pi {\rm i})}^{n}}}\int_{\partial D\times [0, 1]}{\Gamma (\varsigma \wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(1, n-1)}}}(\eta, {{\partial }_{\bar{\varsigma }\bar{z}e}}\eta )\wedge \omega (s(z, \varsigma )), $

这里$\Gamma (\varsigma )$为$\bar{\partial }G(\varsigma )$或$G(\varsigma )$.

特别, 当$\beta =1$时, 则我们有如下Ⅰ -型有界域的微分形式的积分表示:

对于$k>1, $

(3.32) $\begin{eqnarray}\label{eq332}{{(-1)}^{q}}G(z)&=&\frac{{{(-1)}^{(k-1)q}}}{{{(2\pi {\rm i})}^{n}}}\int _{\sigma _{(k)}^{(0)}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(k, n-k)}}}(E_{k}^{{}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge {{\omega }_{\varsigma }}(s(z, \varsigma ))\\&&-\sum\limits_{\theta =1}^{k-1}{{}}\frac{{{(-1)}^{\theta q}}}{{{(2\pi {\rm i})}^{n}}}\int _{\sigma _{(\theta )}^{(0)}}{\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(\theta, 1, n-\theta -1)}}}(E_{\theta }^{{}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge \omega (s(z, \varsigma ))\\&&+\sum\limits_{\theta =1}^{k-1}{{}}\frac{{{(-1)}^{\theta q}}}{{{(2\pi {\rm i})}^{n}}}{{\partial }_{{\bar{z}}}}\int _{\sigma _{(\theta )}^{(0)}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(\theta, 1, n-\theta -1)}}}(E_{\theta }^{{}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge \omega (s(z, \varsigma ))\\&&+\tilde{L}(\varsigma, z), \end{eqnarray}$

对于$k=1, $

(3.33) $\begin{equation}\label{eq333}{{(-1)}^{q}}G(z)=\frac{1}{{{(2\pi {\rm i})}^{n}}}\int _{\sigma _{(1)}^{(0)}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(1, n-1)}}}({{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge {{\omega }_{\varsigma }}(s(z, \varsigma ))+\tilde{L}(\varsigma, z).\end{equation}$

当$q=0$ (即可微函数)的情形, 我们还有相应于(3.32)和(3.33)式的公式.例如, 我们假设解析多面体是由${{Z}_{j}}(z)$$(j=1, 2, \cdots, K\ge n)$定义, $\partial D$表示为$\sum\limits_{j}{{{\sigma }_{j}}}$.在(3.32)式的相应公式中, 我们选择$k=n$和在$\varsigma \in {{\sigma }_{j}}, z\in D$处

$\varphi (z, \varsigma )[Z_j(\varsigma )-Z_j(z)]=\sum\limits_{p=1}^{n}{\alpha _{p}^{*(j)}}(z, \varsigma )s_{p}^{(j)}(z, \varsigma ).$

令${{h}_{{{j}_{\nu}p}}}=\frac{\alpha _{p}^{*(j_\nu)}(z, \varsigma )}{\varphi (z, \varsigma )[{{Z}_{j_\nu}}(\varsigma )-{{Z}_{j_\nu}}(z)]}, p, \nu=1, 2, \cdots, n$, 则由(3.32)式的相应公式得

$\begin{eqnarray*} u(z)&=&\frac{1}{{{(2\pi {\rm i})}^{n}}}\int _{\sigma _{(n)}^{(0)}}{u(\varsigma ) {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(n)}}}({{h}_{{{j}_{1}}}}, \cdots, {{h}_{{{j}_{n}}}})\wedge \omega (s(z, \zeta ))\\& &-\frac{1}{{{(2\pi {\rm i})}^{n}}}\sum\limits_{\theta =1}^{n-1}{{}}\int _{\sigma _{(\theta )}^{(0)}}{\bar{\partial }u(\varsigma )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(\theta, 1, n-\theta -1)}}}(E_{\theta }^{{}}, {{Q}^{0}}, {{\partial }_{{\bar{\zeta }}}}{{Q}^{0}})\wedge \omega (s(z, \zeta ))\\ & &-({{R}_{\partial \tilde{D}}}+{{B}_{{\tilde{D}}}})\bar{\partial }u. \end{eqnarray*}$

注 3.4  当$X={C ^{n}}$时, 我们可得${C ^{n}}$中相应于本文的积分公式(参见文献[2-3]).

从(2.7), (3.10), (3.16)和(3.17)式, 由计算可得如下定理.

定理 3.5  若Stein流形$X$中有界域$D$的边界$\partial D$由(2.3)式定义, $\partial \tilde{D}$由(2.4)式定义, 则对于$\bar{D}$上的$(0, q) (q >0)$微分形式$G(z)$和$z\in \tilde{D}$, 我们有如下公式:

(3.34) $ \begin{eqnarray}\label{eq334} {{(-1)}^{q}}G(z)&=&{{c}_{m}}\sum\limits_{\alpha =1}^{\gamma }{\varepsilon }\int _{{{\Lambda }^{(\beta -1)}}\times \tilde{\sigma }_{\alpha}^{(\beta -1)}\times {{\Delta }_{{{J}_{\alpha }}}}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(m, 1, n-m-1)}}}\\& &({{\Psi }_{m}}, T, {{\partial }_{\bar{\zeta }\bar{z}\lambda \mu }}T)\wedge B_{m}^{\Phi }(s(z, \zeta ))+{{{L}''}_{m}}(\varsigma, z), \end{eqnarray} $

其中$T={{({{T}_{1}}, \cdots, {{T}_{n}})}^{\tau }}$, 这里$T_{p}^{(\nu )}=t_{\beta, \nu }^{p}/{{M}_{\beta, \nu }}(p=1, \cdots, n)$, 以及如下条件被满足

(3.35) $\begin{equation}\label{eq335} {\rm Rank}\frac{\partial (t_{\beta, \nu }^{1}, \cdots, t_{\beta, \nu }^{n})}{\partial ({{{\bar{\zeta }}}_{1}}, \cdots, {{{\bar{\varsigma }}}_{n}}, {{{\bar{z}}}_{1}}, \cdots, {{{\bar{z}}}_{n}})}\le n-m-\beta -\alpha +1, 1\le \gamma \le k\le n-m. \end{equation} $

注 3.5  对于$q=0$ (即可微函数)的情形, 使用上述同样的方法, 我们也可得类似于(3.4), (3.5), (3.10), (3.16), (3.17)式.所以对于$q=0$和$m=0$, 我们也可得相应得积分公式.