TITLE: Does the étale topos determine the Hodge numbers?
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: No. See Proposition 2.14 in Canonical models of surfaces of general type in positive characteristic