Dolbeault 复形

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定义 · 8.3 复微分形式

称为Dolbeault 复形(Dolbeault complex)。注意 ${\overline{\partial }}^2=0$。由 $\overline{\partial }$-闭的 $(r,s)$-形式（即满足 $\overline{\partial }\omega =0$ 的 $\omega \in {\Omega }^{r,s}(M)$）构成的集合称为 $(r,s)$-上闭链(cocycle)，记作 $Z_{\overline{\partial }}^{r,s}(M)$。由 $\overline{\partial }$-恰当的 $(r,s)$-形式（即满足对某个 $\eta \in {\Omega }^{r,s-1}(M)$ 有 $\omega =\overline{\partial }\eta$ 的 $\omega \in {\Omega }^{r,s}(M)$）构成的集合称为 $(r,s)$-上边界链(coboundary)，记作 $B_{\overline{\partial }}^{r,s}(M)$。复向量空间

$$H_{\partial }^{r,s}(M)\equiv Z_{\partial }^{r,s}(M)/B_{\partial }^{r,s}(M)\text{\hspace{1em}(8.46)}$$

英文原文

is called the Dolbeault complex. Note that ${\overline{\partial }}^2=0$ . The set of $\overline{\partial }$ -closed $(r,s)$ -forms (those $\omega \in {\Omega }^{r,s}(M)$ such that $\overline{\partial }\omega =0$ ) is called the $(r,s)$ -cocycle and is denoted by $Z_{\overline{\partial }}^{r,s}(M)$ . The set of $\overline{\partial }$ -exact $(r,s)$ -forms (those $\omega \in {\Omega }^{r,s}(M)$ such that $\omega =\overline{\partial }\eta$ for some $\eta \in {\Omega }^{r,s-1}(M)$ ) is called the $(r,s)$ -coboundary and is denoted by $B_{\overline{\partial }}^{r,s}(M)$ . The complex vector space

$$H_{\partial }^{r,s}(M)\equiv Z_{\partial }^{r,s}(M)/B_{\partial }^{r,s}(M)\text{\hspace{1em}(8.46)}$$

出处：8.3 复微分形式 · node a-d376218733575606c9bd70ea · 源节点 SHA-256 23a52818c6f1ff3a…