Let $K$ be an algebraically closed field of characteristic $0$. Let $P$ be a proposition on a non-singular projective variety over $K$ which is stated in the language of algebraic geometry. Suppose $P$ holds when $K = \mathbb{C}$. Does $P$ hold on any such $K$?
Remark We may take as $P$, for example, the Kodaira vanishing theorem.