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!
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$\begingroup$ Are you working over $\mathbb{C}$ or $\mathbb{R}$? That is: do you want your hypersurface in $\mathbb{C}^n$ or $\mathbb{R}^n$ or someplace else? Your notation makes this unclear? $\endgroup$ – Richard Montgomery Nov 27 '10 at 3:42

$\begingroup$ I've added the arxiv ag tag, if that's ok with the OP. $\endgroup$ – babubba Nov 27 '10 at 23:19
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The classic reference is DanilovKhovanskii's "Newton polyhedra and an algorithm for calculating HodgeDeligne 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.

$\begingroup$ Thanks. This looks superduper. Now looking forward to understanding it all. $\endgroup$ – Eric Zaslow Nov 27 '10 at 14:25

$\begingroup$ For other readers, here a link to the article: iopscience.iop.org/00255726/29/2/A02/pdf/… $\endgroup$ – Eric Zaslow Nov 27 '10 at 15:57