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.



The Hodge  numbers of a covering
The answer goes back to Esnault and Viehweg:
$$H^q(Y,\Omega\_Y^p)= H^q(X,\Omega\_X^p)\oplus H^q(X,\Omega\_X^p(\log Z)\otimes L^{-1})$$
where $L$ is the anti-invariant part of the direct image of $O\_Y$ to $X$ under the
natural $\mathbb{Z}/2$ action.


