/sci/ - Science & Math

Let [math]X[/math] be a smooth projective variety over [math]\mathbb{C}[/math] cut out by polynomials with integral coefficients.

Let [math]{{\bar X}_p}[/math] be the corresponding variety over the algebraic closure of [math]{\mathbb{F}_p}[/math] defined by taking the polynomials mod p.

We know the weil cohomology theories of the two varieties are related by [math]H_{et}^i\left( {{{\bar X}_p},{\mathbb{Q}_\ell }} \right) \otimes \mathbb{C} \cong {H^i}\left( {X,\mathbb{C}} \right)[/math].

I was wondering if we could find relations between other cohomology groups of these varieties, particularly cohomology in the tangent sheaf.

The reasoning behind this is I want to see if there are any interesting relation between [math]{T_{{{\operatorname{Def} }_X}}}[/math] and [math]{T_{{{\operatorname{Def} }_{{{\bar X}_p}}}}}[/math].