On surfaces satisfying q=0, pg=0, c₁²=9

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 P² (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).