Corollary 6.5.9BFB355F9DDF9

Proof. Let $k=\dim Y$ and $n=\dim X$ near $a$. By the definition of a complex submanifold (definition 5.2.3), there are holomorphic coordinates $w=(w',w'')\in \mathbb C^ k\times \mathbb C^{n-k}$ centered at $a$ in which $Y=\{ w''=0\} $; by proposition 5.9.3, $p$ is plurisubharmonic near $0$ in these coordinates. On a compact neighborhood of $a$, the densities of smooth positive volume forms with respect to coordinate Lebesgue measure are bounded above and below by positive constants. Hence we may test integrability with Lebesgue measure $\mathrm dV_{2k}$ on $Y=\mathbb C^ k\times \{ 0\} $ and $\mathrm dV_{2n}$ in the coordinates $w$.

Choose a ball $B$ around $0$ so small that $p$ is plurisubharmonic on a neighborhood of $\overline B$, that $f$ is defined on $B\cap Y$, and that $\int _{B\cap Y}|f|^2\mathrm e^{-p}\, \mathrm dV_{2k}<\infty $. The ball $B$ is a bounded pseudoconvex domain, by corollary 6.3.5 applied to the exhaustion function $|w|^2$ of $\mathbb C^ n$, and $B\cap Y$ is its intersection with the linear subspace $\{ w''=0\} $ of codimension $n-k$. By corollary 6.5.8 there is $F\in \mathcal O(B)$ with $F|_{B\cap Y}=f$ and $\int _ B|F|^2\mathrm e^{-p}\, \mathrm dV_{2n}<\infty $. (If $k=0$, the integral over $Y$ is the weighted value $|f(a)|^2\mathrm e^{-p(a)}$; if $p|_ Y$ is identically $-\infty $ near $a$, then $f=0$ and $F=0$, in accordance with the conventions of theorem 6.5.4.) $\square$