Infinite-dimensional complex projective varieties

Abstract

Let $V_i$, $0\le i \le s$, $s \ge 1$, be complex Banach spaces. Assume $V_i$ infinite-dimensional for $1\le i \le s$, $V_0$ finite-dimensional and $V_0 \ne 0$. Let $\pi : {\bf {P}}(V_0)\times \dots \times {\bf {P}}(V_s) \to {\bf {P}}(V_0)$ be the projection. Let $X$ be a closed analytic subset of finite codimension of ${\bf {P}}(V_0)\times \dots \times {\bf {P}}(V_s)$. Here we prove that $\pi (X)$ is a closed analytic subset of ${\bf {P}}(V_0)$ with the following universal property. For every finite-dimensional reduced analytic space $Y$ and every holomorphic map $f: X \to Y$ there is a unique holomorphic map $g: \pi (X) \to Y$ such that $f = g\circ (\pi \vert X)$.