Does the small étale topos of a smooth proper variety over a perfect field of positive characteristic determine its Hodge numbers? We consider it as a Grothendieck topos over the étale topos of the field.



Does the étale topos determine the Hodge numbers?
No. See Proposition 2.14 in *[Canonical models of surfaces of general type in positive characteristic](http://www.numdam.org/item/PMIHES_1988__67__97_0/)*


