Let $X$ be a Kahler manifold and $Z\subset X$ be a smooth hypersurface. How to compute the Hodge rhomb of the double covering $Y\to X$ ramified over $Z$? (And what I have to know? Would the map $H^*(X)\to H^*(Z)$ be enough?)

P.S. I tried the Gysin sequence, but it looks like there are many loose ends.

Let $X$ be a Kahler manifold and $Z\subset X$ be a smooth hypersurface. How to compute the Hodge diamond of the double covering $Y\to X$ ramified over $Z$? (And what I have to know? Would the map $H^*(X)\to H^*(Z)$ be enough?)

P.S. I tried the Gysin sequence, but it looks like there are many loose ends.