How do I compute the compact cohomology of a hypersurface?
For example, let $f$ be a Newton polynomial of a polytope in $\mathbb{R}^n$ and let $X = (f=0)$
inside $(\mathbb{C}^\\*)^n$ (maybe there is some dependency on the coefficients of $f\;$?). Can you tell me anything about $H^\*\_c(X)$? Perhaps I should know better, but I don't.
Thanks!



How do I compute the compact cohomology of a hypersurface?
The classic reference is Danilov-Khovanskii's "Newton polyhedra and an algorithm for calculating Hodge-Deligne numbers". There is subsequent work by Cox, Batyrev, Malvyutov, etc. but they are mainly concerned with more general toric ambient spaces; if you want a hypersurface in the torus then this original paper should have all you need. 


