Relationship between topological cohomology and $\ell$-adic cohomology

Question

Let $\Delta \subset \mathbb{R}^n$ be an $n$-dimensional integral polytope, let $f$ be a Laurent polynomial in $n$-variables with coefficients in an extension of the integers and Newton polytope $\Delta$. We can view $f$ as both a Laurent polynomial over $\mathbb{C}$ and $\mathbb{F} _ q$ for some prime power $q$. Suppose that $f$ is $\Delta$-regular (for a definition, see http://www.math.uci.edu/~dwan/gottingen.pdf pg. 16) over both $\mathbb{C}$ and $\overline{\mathbb{F}} _ q$. Let $X_\mathbb{C}\subset(\mathbb{C}^*)^n$ be the affine hypersurface defined by the vanishing of $f$ in the torus (viewing $f$ as a polynomial over $\mathbb{C}$) and $X_{\overline{\mathbb{F}}_q} \subset (\overline{\mathbb{F}} _ q ^ *)^n$ be the analog over $\overline{\mathbb{F}} _ q$.

My question is:

Is it true that $$\dim (H^i(X_\mathbb{C})) = \dim H^i(X_{\overline{\mathbb{F}}_q},\mathbb{Q} _ \ell),$$ where the cohomology on the left hand side is the standard topological cohomology? If this is true, I would greatly appreciate a reference if you have one.

Thanks!

Answer 1

The way to study the topology of the situation was introduced by Khovanski in "Newton polyhedra, and toroidal varieties" Funkcional. Anal. i Priložen. 11 (1977), no. 4, 56--64, 96. His result (if I have interpreted it correctly) is that $X$ may be compactified as a hypersurface in a projective toric variety to a smooth variety with normal crossings such that each stratum is of the same form as $X$. As far as I can see this construction works uniformly so that we would get the same construction over a (suitable) mixed characteristic discrete valuation ring. Then desired isomorphism then follows from the smooth and proper base change theorem. (I have some vague recollection that this comment is also to be found somewhere in SGA but I am not going to do any wading looking for it...)

Addendum: Let me first note that the right setup to even formulate the question is a scheme $S$ with functions on it giving the coefficients of $f$. The latter polynomial should then be non-degenerate in the sense that all its fibres over (geometric) points of $S$ should be non-degenerate. The statement is then that if $\pi\colon X\to S$ is the scheme of zeroes of $f$ in the constant torus over $S$, then $R^i\pi_\ast\mathbb Z_\ell$ is locally constant commuting with base change where $\ell$ is invertible in $\mathcal O_S$ (together with comparison theorem of $\ell$-adic cohomology and classical for a complex point of $S$). As this statement is only dealing with the $\ell$-adic sheaves $R^i\pi_\ast\mathbb Z_{\ell}$ we may use the definition of $\ell$-adic sheaf introduced by Jouanolou in SGA V. Kohvanski's method then should give a compactification of $X$ by a smooth $S$-scheme with complement of relative normal crossings. The two theorems used, proper base change and vanishing of vanishing cycles, then follows directly from the case of finite coefficients (by Jouanolou's very definition).

Answer 2

This seems to be a combination of a comparison theorem saying that for complex manifolds, ordinary topological cohomology is closely related to l-adic cohomology, and a theorem relating etale cohomology of a scheme to its reduction mod p. The standard reference for these sorts of results is the exposes by Artin in SGA IV, volume 3; all you have to do is wade through about 1000 pages of French.