A necessary condition for liftings of positive characteristic varieties with finite fundamental groups

Abstract: In this paper, we introduce a necessary condition for the existence of characteristic zero liftings of certain smooth, proper varieties in positive characteristic, using \'etale homotopy theory and Wall's finiteness obstruction. For a variety with finite \'etale fundamental group pi, we define a notion of mod-l finite dominatedness based on the F_l-chain complex of the universal cover of its l-profinite \'etale homotopy type. We prove that such a variety can be lifted to characteristic zero only if this complex is quasi-isomorphic to a bounded complex of finitely generated projective F_l[pi]$-modules. This extends classical finiteness results in topology to the l-adic and algebro-geometric setting.