Moscow Mathematical Journal

Volume 23, Issue 1, January–March 2023 pp. 97–111.

On Automorphic Forms of Small Weight for Fake Projective Planes

Authors: Sergey Galkin (1), Ilya Karzhemanov (2), and Evgeny Shinder (3)
Author institution:(1) PUC-Rio, Departamento de Matemática, Rua Marquês de São Vicente 225, Gávea, Rio de Janeiro;
HSE University;
Independent University of Moscow
(2) Laboratory of AGHA, Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141701, Russia
(3) School of Mathematics and Statistics, University of Sheffield, The Hicks Building, Hounsfield Road, Sheffield, S3 7RH, United Kingdom;
HSE University

Summary:

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, 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.

2020 Math. Subj. Class. 14J29, 32N15, 14F06.

Keywords: Fake projective planes, automorphic forms, exceptional collections.

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