Hodge numbers rule out good reduction

Question

A theorem of Fontaine says that if a geometrically connected smooth proper variety $X$ over $mathbb{Q}$ has good reduction everywhere then $h^{i, j}(X)=0$ for $ineq j$, $i+jleq 3$.
This means that the variety's Hodge numbers can "disqualify" it from having good reduction everywhere. For example, there do exist varieties whose non-zero Hodge numbers are $h^{0, 0}=h^{1, 0}=h^{0, 1}=h^{1, 1}=1$ but none of them have good reduction everywhere (in this case it was known before Fontaine but it is just an example).
Can this happen locally? We consider the geometrically connected smooth proper varieties over $mathbb{Q}_p$ whose Hodge numbers are equal to some fixed values. Can they all have bad reduction (assuming at least one such variety exists)?

R. van Dobben de Bruyn · Answer

Because the inverse Hodge problem is a very difficult question, I think it's unlikely that there will be an answer for this question for any given set of Hodge numbers. For most given Hodge diamonds, we don't even know whether it can be realised by any smooth projective variety (or Kähler manifold); and when it can, we know little about the geometry of these varieties.
However, Kotschick and Schreieder proved that the Hodge ring of varieties in characteristic $0$ is generated by $mathbf P^1$, $mathbf P^2$, and any elliptic curve $E$. Since we can choose these with good reduction, this implies that there are no linear relations (like $h^{i,j}(X) = 0$ for a given $(i,j) in {0,ldots,n}^2$) among the Hodge numbers of smooth projective varieties with good reduction.

Hodge numbers rule out good reduction

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