Algebraicity of a proper holomorphic map

Question

Let $f:X\to Y$ be a proper holomorphic map between complex analytic spaces. Suppose that $Y\subset \Bbb{P}^n$ is a closed analytic subset. By Chow's theorem, $Y$ is a projective algebraic variety and $f_*O_X$ is a coherent algebraic sheaf (Grauert direct image theorem+GAGA). Is $X$ algebraic? Does $f$ comes from a morphism of algebraic varieties?