On the continuation of analytic sets
Hirotaka Fujimoto · Journal of the Mathematical Society of Japan · 1966
\S 1. Introduction.1.It is well known as Hartogs-Osgood's theorem that for a relatively compact domain $D$ in $C^{n}(n\geqq 2)$ with the connected boundary $\partial D$ every holo- morphic function in a connected neighborhood of $\partial D$ is continuable to $D$ .In [21], Rothstein gave an analogous continuation theorem of analytic sets in H. FU J I M OT $0$ $ v\tau$ is $*$ -strongly $(r+s)$ -convex at $p$ .Indeed, we can take a nowhere degenerate holomorphic mapping $\varphi^{\prime}$ of a neighborhood $U$ of $p$ into a domain $D^{\prime}$ in $C^{n\prime}$ and a strongly $(r+s)$ -convex function $v^{\prime}$ on $D^{\prime}$ with $v=v^{\prime}\varphi^{\prime}$ on $U^{\prime}$ as follows.By Definition 2.5, $v$ is represented as $ v=\tilde{v}\varphi$ for a suitable nowhere degenerate holomorphic mapping $\varphi$ of a neighborhood $U$ of $p$ into a domain $D$ in $C^{n}$ and a strongly s-convex function $\tilde{v}$ on $D$ .Since $\dim_{p}(\varphi\tau)^{-1}(\varphi\tau)(p)=\dim_{p}\tau^{-1}\tau(p)=r$ , we can find $r$ holomorphic functions $\varphi_{n+1}^{\prime},$ $\cdots$ , $\varphi_{n+r}^{\prime}$ in a neighborhood $U^{\prime}$ of $p(U^{\prime}\subset U)$ such that the mapping $\varphi^{\prime}=\varphi\tau\times$ $(\varphi_{n+1}^{\prime}, \cdots , \varphi_{n+r}^{\prime})$ of $U^{\prime}$ into $C^{n+r}$ is nowhere degenerate on $U^{\prime}$ .Now we define the canonical extension $y/of\tilde{v}$ putting $v^{\prime}(p_{1}, p_{2})=\tilde{v}(p_{1})$ for any $(p_{1}, p_{2})\in D\times C^{n}$ , which is strongly $(r+s)$ -convex on $D^{\prime}=D\times C^{n}$ .These $\varphi^{\prime},$ $v^{\prime}$ and $D^{\prime}$ have the desired properties.PROPOSITION 2.7 (Maximum Principle).$A*$ -strongly s-convex function $v$ on a complex space $X$ cannot take its maximum at any interior point $p$ of $X$ with $\dim_{p}X\geqq s$ .PROOF.Assume that $X$ is of dimension at least $s$ at $p$ .According to Definition 2.5, we take a nowhere degenerate holomorphic mapping $\varphi$ of a neighborhood $U$ of $p$ into $D$ in $C^{n}$ and a strongly s-convex function $\tilde{v}$ on $D$ with $ v=\tilde{v}\varphi$ .Making $U$ and $D$ sufficiently small, we may assume that $\varphi$ maps $U$ properly onto an analytic set in $D$ (cf.Remmert [17]).By the assumption, we have $\dim_{p}X=\dim_{\varphi(p)}\varphi(U)\geqq s$ .On the other hand, $\tilde{v}$ is strongly pluri- subharmonic on $L=\{z_{1}=\ldots=z_{s-1}=0\}$ for suitable local coordinates $z_{1},$ $\cdots$ , $z_{n}$ $(\varphi(p)=(O))$ and hence on the analytic subset $\varphi(U)\cap L$ .Since $\dim_{\varphi(p)}\varphi(U)\cap L$ $\geqq s-(s-1)=1$ and $\tilde{v}$ is not constant on any irreducible component of $\varphi(U)\cap L$ , $\tilde{v}$