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.