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

TL;DR

This paper studies surfaces with specific invariants, showing they are Hodge-Witt and ordinary in positive characteristic, characterizing the projective plane via fundamental groups, and demonstrating the existence of fake projective planes in positive characteristics.

Contribution

It provides new characterizations of surfaces with given invariants, including a characteristic-free criterion for the projective plane and results on the existence of fake projective planes in positive characteristic.

Findings

Surfaces with given invariants are Hodge-Witt and often ordinary in positive characteristic.

Frobenius split surfaces with these invariants are isomorphic to the projective plane.

Fake projective planes have good ordinary reduction at almost all primes and exist in positive characteristic.

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).