On surfaces satisfying $q=0,p_g=0,c_1^2=9$

Abstract

I consider the class of surfaces $X$ over algebraically closed fields with numerical invariants given in the title. In characteristic zero, this class contains fake projective planes which were introduced by David Mumford. I prove that in characteristic $p>0$ such surfaces are Hodge-Witt and also ordinary under additional assumptions. In particular, fake projective planes are Hodge-Witt (Theorem 3.1). I show that if $X$ is Frobenius split then $X\simeq \mathbb{P}^2$ (Theorem 4.1). I also establish a characteristic free characterization of the projective plane using the Nori fundamental group scheme (Theorem 5.1). Finally, I show that any fake projective plane over a number field has good ordinary reduction at all but finitely many primes and in particular fake projective planes exist in positive characteristics (Theorem 6.1).