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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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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? – Richard Montgomery Nov 27 '10 at 3:42
I've added the arxiv ag tag, if that's ok with the OP. – babubba Nov 27 '10 at 23:19
up vote 12 down vote accepted

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.

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Thanks. This looks super-duper. Now looking forward to understanding it all. – Eric Zaslow Nov 27 '10 at 14:25
For other readers, here a link to the article:… – Eric Zaslow Nov 27 '10 at 15:57

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