On Automorphic Forms of Small Weight for Fake Projective Planes

Abstract

On the projective plane there is a unique cubic root of the canonical bundle and this root is acyclic. On fake projective planes such root exists and is unique if there are no 3-torsion divisors (and usually exists, but not unique, otherwise). Earlier we conjectured that any such cubic root must be acyclic. In the present note we give two short proofs of this statement and show acyclicity of some other line bundles on the fake projective planes with at least $9$ automorphisms. Similarly to our earlier work we employ simple representation theory for non-abelian finite groups. The first proof is based on the observation that if some line bundle is non-linearizable with respect to a finite abelian group, then it should be linearized by a finite, \emph{non-abelian}, Heisenberg group. For the second proof, we also demonstrate vanishing of odd Betti numbers for a class of abelian covers, and use linearization of an auxiliary line bundle as well.