Answer by Christian Liedtke for Hodge numbers of reduction mod $p$
As already pointed out, the Hodge numbers may go up under reduction modulo $p$. On the other hand, let me also point out that the situation can be controlled: 1.) For all $p$, where $\overline{X}_p$ is...
View ArticleAnswer by Daniel Litt for Hodge numbers of reduction mod $p$
Since people have addressed (2-4), I'll address (1). There is indeed a fast argument. Namely, you have a flat family of curves $X\to \operatorname{Spec}(\mathcal{O}_{K, p})$. You are interested in...
View ArticleAnswer by Donu Arapura for Hodge numbers of reduction mod $p$
For some explicit counterexamples to (2) and therefore also (4) again, see J. Suh,Plurigenera of general type surfaces in mixed characteristic, Compositio (2008).The only thing that you can say for...
View ArticleAnswer by user30035 for Hodge numbers of reduction mod $p$
This is only a comment but I'm a new user and can't make comments.For (4), the dimensions of the $H^i$ might not even be the same if $i=0$. let $X$ be an elliptic curve and choose two distinct points...
View ArticleHodge numbers of reduction mod $p$
Let $X$ be a projective variety defined over a number field $K$, and $p \in \textrm{Spec }\mathcal{O}_K$ a maximal ideal, so that reduction mod $p$ makes sense, and the resulting scheme (mod $p$)...
View Article