# 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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