Let $N$ be a positive integer. Let $f:X\to S=\mathrm{Spec}\:\mathbb{Z}[1/N]$ be a smooth projective morphism of relative dimension 2 such that $R^1f\_\*\mathcal{O}\_X$ and $R^2f\_\*\mathcal{O}\_X$ are both locally free $\mathcal{O}\_S$-modules. Can $\mathrm{dim}\_{k(s)}H^0\big(X\_s, \Omega^1\_{X\_{s}/k(s)}\big)$ and $\mathrm{dim}\_{k(s)}H^1\big(X\_s, \Omega^1\_{X\_{s}/k(s)}\big)$ depend on the closed point $s\in S$?



Can $h^{1, 0}$ and $h^{1, 1}$ jump for smooth projective surfaces over $\mathbb{Z}[1/N]$?
Yes. An Enriques surface with classical reduction at $p=2$ gives such an example. See Illusie "Complexe de de Rham-Witt et cohomologie cristalline" Prop. II 7.3.8(b), p. 658.


