2025/08/17 by Kirti Joshi, Joshi, Kirti
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2508.12339
openalex publication_date 2025/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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).