## The Extensional Koppelman-Leray Type Integral Formulas in the Analytic Varieties of Stein Manifolds

Chen Shujin,

Abstract

In this paper, we study how to establish integral formulas for differential forms in the analytic varieties of Stein manifolds. Firstly using different method and technique we derive the corresponding integral representation formulas of differential forms for the complex n-m(0 ≤ m < n) dimensional analytic varieties in two types of bounded domains of Stein manifolds. Secondly we obtain the unified integral representation formulas of differential forms for the complex n-m dimensional analytic varieties in the general bounded domains of Stein manifolds, i.e. Koppelman-Leray type integral formulas for the complex n-m dimensional analytic varieties in the bounded domains of Stein manifolds. In particular, when m=0, the formulas obtained in this paper are the extension of Koppelman-Leray formula in the Stein manifolds.

Keywords： Stein manifold ; Analytic varieties ; Unified formula ; Extension ; Differential form ; Integral formula

Chen Shujin. The Extensional Koppelman-Leray Type Integral Formulas in the Analytic Varieties of Stein Manifolds. Acta Mathematica Scientia[J], 2020, 40(6): 1492-1510 doi:

## 1 引言

$X$是一复$n$维Stein流形, $T(X)$${{T}^{{*}}}({X})分别表示X上切丛和余切丛, {{T}_{z}}(X)$${{T}_{z}}^{*}(X)$分别表示对于$z\in X$的纤维.用$s(z, \varsigma)$${{s}^{*}}(z, \varsigma)分别表示T(X)$${{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)$$X\times X上全纯函数\varphi满足如下条件: (a)对于所有z, \varsigma\in X, S(z, \varsigma)\in{{T}_{z}}(X); (b)对于每一固定的z\in X, S(z, z)=0$$s(z, \varsigma):X\to {{T}_{z}}(X)$$\varsigma\in X的邻域中是双全纯的; (c)对于所有z\in X, \varphi(z, z)=1; (d)若{{{\Bbb F}}_{s}}是一解析层, 它是由{}_{X\times X}\mathfrak{M}生成的, 则\varphi\in{{{\Bbb F}}_{s}}((X\times X)\backslash\{(z, z):z\in X\}); (e)存在一个\kappa\ge 0, 对于T(X)中每一范数\left\|\cdot \right\|我们有函数{{\varphi}^{\kappa}}{{\left\|\left.s\right\|\right.}^{-2}}$$(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}}$$X\times X的局部有限开复盖\{{{V}_{j}}\}, 存在全纯映射{{\alpha }^{*(j)}}:{{V}_{j}}\to {{T}^{*}}(X), 使得{{\alpha }^{*(j)}}(z, \varsigma )\in T{^{*}_{z}}(X), 且对于所有(z, \varsigma )\in {{V}_{j}}, 我们有 $$\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 )$$ 以及对于z\in D$$\varsigma \in {{\delta }_{j}}=\{z\in \partial D:\left| {{F}_{j}}(z) \right|=1\}$, $j=1, 2, \cdots, N$, 还有

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

## 2 一些引理

${C ^{\infty }}$映射$\sigma :T(X)\to T^*(X)$$T(X)的每一纤维中, 确定一个范数 选择\{{X_{j}}\}是从属于\{{{U}_{j}}\}的单位分解.对于z\in {{U}_{j}}\subset X, s(z, \varsigma )\in {{T}_{z}}(x)$${{\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)$${C ^{1}}截面s_{b, 1, \nu }^{*}, s_{b, 2, \nu }^{*}, \cdots, s_{b, n, \nu }^{*}\in T_{z}^{*}(X), b=1, 2, \cdots, \beta, \nu =1, 2, \cdots, \alpha -l+1(1\le l\le \alpha \le k). {{J}_{\alpha }}=\{{{j}_{1}}, \cdots, {{j}_{\alpha }}\}$$\{1, 2, \cdots, K\}$的有序子集, ${{j}_{1}}<\cdots <{{j}_{\alpha }}$.我们置

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

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

$\beta =\mbox{1}$$l=k时, T={{Q}^{0}}$$={{({{Q}^{0}}_{1}, \cdots, {{Q}^{0}}_{n})}^{\tau }}$$Q_{p}^{0}=t_{1, 1}^{p}/{{M}_{1, 1}}, 其中t_{1, 1}^{p}=s_{1, p, 1}^{*}(z, \zeta ), {{M}_{1, 1}}=\sum\limits_{p=1}^{n}{{{s}_{p}}(z, \varsigma )s_{1, p, 1}^{*}}(z, \varsigma ).这里以及本文剩余部分, \tau均表示矩阵转置. 若对于\varsigma \in \partial D, z\in D$$, \lambda \in {{\Lambda }^{(\beta -1)}}$, ${{M}_{\beta, \nu }}(\varsigma, z, \lambda )\ne 0$$\varphi (z, \varsigma )$$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 }}$${C ^{1}}的, 其中{{\kappa }^{*}}是一适当的非负整数, 则记{{M}_{\beta, \nu }}\in B. 若全纯函数{{F}_{j}}(\varsigma, z)可表示成 且对任何固定的z\in D, 在\varsigma \in \sigma _j\subset \partial D=\bigcup\limits_{j=1}^K\sigma _j处, \varphi (z, \varsigma ){{F}_{j}}(\varsigma, z)\ne 0, 则记{{F}_{j}}(\varsigma, z)\in A.本文我们总假设{{M}_{\beta, 1}}\in B, {{F}_{j}}(\varsigma, z)\in A. 我们记{{h}_{{{j}_{\nu }}}}={{({{h}_{{{j}_{\nu }}1}}, \cdots, {{h}_{{{j}_{\nu }}n}})}^{\tau }}, 其中{{h}_{{{j}_{\nu }}p}}=\alpha _{p}^{*{{j}_{\nu }}}(z, \varsigma )/\varphi (z, \varsigma ){{F}_{{{j}_{\nu }}}}(\zeta, z). $$\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}}}}).$$ 我们假设在Stein流形X中的有界域D的边界\partial D(这里以及本文剩余部分总假设\partial D是逐块光滑的)构成缝空间的链, 且可表示成 $$\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{, }$$ 其中{{\bar{\sigma }}^{(\theta )}}$${{\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$$\cdots \supset {{\bar{\sigma }}^{(\beta )}}, 其中{{\sigma }^{(\theta )}}$${{\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.

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

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

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

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

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

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

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

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

$\theta (z, \varsigma )$$\vartheta (z, \varsigma )分别是s(z, \varsigma )$$\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$$({{U}_{j}}\times {{U}_{j}})\bigcap {{U}_{{{\varepsilon }_{0}}}}双全纯映为某开集{{V}_{{{\varepsilon }_{0}}}}\subseteq {C ^{n}}\times {C ^{n}}. (b) N:{{Y}_{{{\varepsilon }_{0}}}}\to {{U}_{j}}是由N({{\pi }_{j}}(z), \theta (z, \varsigma ))=N(\eta, \xi )=\varsigma定义的全纯映射.我们可定义T的逆映射{{T}^{-1}}:{{V}_{{{\varepsilon }_{0}}}}\to ({{U}_{j}}\times {{U}_{j}})\bigcap {{U}_{{{\varepsilon }_{0}}}}, 和对于所有(\eta, \xi )\in {{Y}_{{{\varepsilon }_{0}}}}我们有{{T}^{-1}}(\eta, \xi )=(z, \varsigma )=(\pi _{j}^{-1}(\eta ), N(\eta, \xi ))$$\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$.

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

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

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

$v(z)$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}}\}$${{ C}^{\infty }}单位分解, 且书{{v}_{j}}={{\chi }_{j}}v和选择开集{{\tilde{U}}_{j}}\subset \subset {{U}_{j}}. ({{T}^{-1}})^*$${{T}^{-1}}$的拉回, 则

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

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

$q=0$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

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

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

$\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}$

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

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

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

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

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

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

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

$\tilde{C}$$\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 \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 \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)$$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

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

(b) 当$k=1$时, 我们有

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

${{{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 $$\partial \tilde{D}=\partial D, 它是由(2.3)式定义, 且(3.18)式可写为 $$\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.$$ 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, \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, \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 \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} $$\label{eq331}\tilde{L}(\varsigma, z)=-({{R}_{\partial D}}+{{B}_{D}})\bar{\partial }G+\bar{\partial }({{R}_{\partial D}}+{{B}_{D}})G,$$ 其中 这里\Gamma (\varsigma )$$\bar{\partial }G(\varsigma )$$G(\varsigma ). 特别, 当\beta =1时, 则我们有如下Ⅰ -型有界域的微分形式的积分表示: 对于k>1, \begin{eqnarray}\label{eq332}{{(-1)}^{q}}G(z)&=&\frac{{{(-1)}^{(k-1)q}}}{{{(2\pi {\rm i})}^{n}}}\int _{\sigma _{(k)}^{(0)}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(k, n-k)}}}(E_{k}^{{}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge {{\omega }_{\varsigma }}(s(z, \varsigma ))\\&&-\sum\limits_{\theta =1}^{k-1}{{}}\frac{{{(-1)}^{\theta q}}}{{{(2\pi {\rm i})}^{n}}}\int _{\sigma _{(\theta )}^{(0)}}{\bar{\partial }G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(\theta, 1, n-\theta -1)}}}(E_{\theta }^{{}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge \omega (s(z, \varsigma ))\\&&+\sum\limits_{\theta =1}^{k-1}{{}}\frac{{{(-1)}^{\theta q}}}{{{(2\pi {\rm i})}^{n}}}{{\partial }_{{\bar{z}}}}\int _{\sigma _{(\theta )}^{(0)}}{G(\zeta )\wedge {{\varphi }^{\nu }}(z, \varsigma ){{\det }_{(\theta, 1, n-\theta -1)}}}(E_{\theta }^{{}}, {{Q}^{0}}, {{\partial }_{\bar{\zeta }\bar{z}}}{{Q}^{0}})\wedge \omega (s(z, \varsigma ))\\&&+\tilde{L}(\varsigma, z), \end{eqnarray} 对于k=1, $$\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).$$ q=0 (即可微函数)的情形, 我们还有相应于(3.32)和(3.33)式的公式.例如, 我们假设解析多面体是由{{Z}_{j}}(z)$$(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$\varsigma \in {{\sigma }_{j}}, z\in D$

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

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Henkin G , Leiterer J . Theory of Functions on Complex Manifolds. Boston: Basel, 1984

Chen S J .

General integral formulas of differential forms on the analytic varieties

Chin Quart J Math, 2017, 32 (4): 355- 370

Chen S J .

The Leray-Stokes type integral representation formulas on the analytic varieties

Chin Quart J Math, 2018, 33 (4): 395- 416

Hatiafratis T E .

Integral representation formulas on analytic varieties

Pacific J Math, 1986, 123: 71- 91

Chen S J .

General integral representation of holomorphic functions on the analytic subvariety

Publ RIMS Kyoto Univ, 1993, 29: 511- 533

Lu Q K .

On Cauchy-Fantappie formula

Acta Math Sinica, 1966, 16 (3): 344- 363

Range R M .

An integral kernal for weakly pseudoconex domains

Math Ann, 2013, 356 (2): 793- 808

Range R M .

A pointwise a-priori estimate for the Neumann problem on weakly pseudoconvex domains

Pacific J Math, 2015, 275 (2): 409- 432

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