All Questions

Let $S$ be a connected smooth complex-analytic space. Let $\phi:X\to S$ be a proper holomorphic submersion with connected fibers. Can the fibers of $\phi$ have infinitely many distinct Hodge diamonds? ...

Let $X$ be an open (connected) Riemann surface of finite Euler characteristic. And $K_1 \subset \cdots K_n \subset$ be an sequence of closures of bounded open subsets with smooth boundary of $X.$ ...