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Working paper

Acyclicity of non-linearizable line bundles on fake projective planes

arxiv.org. math. Cornell University, 2016. No. 1602.06107.
Galkin S., Karzhemanov I., Shinder E.
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 (assuming it exists) must be acyclic. In the present note we give a new short proof of this statement and show acyclicity of some other line bundles on those fake projective planes with at least 9 automorphisms. Similarly to our earlier work we employ simple representation theory for non-abelian finite groups. The novelty stems from the idea 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. Our argument also exploits J. Rogawski's vanishing theorem and the linearization of an auxiliary line bundle.