A vanishing theorem on fake projective planes with enough automorphisms

Abstract

For every fake projective plane X X with automorphism group of order 21, we prove that H i ( X , 2 L ) = 0 H^i(X, 2L)=0 for all i i and for every ample line bundle L L with L 2 = 1 L^2=1 . For every fake projective plane with automorphism group of order 9, we prove the same vanishing for every cubic root (and its twist by a 2-torsion) of the canonical bundle K K . As an immediate consequence, there are exceptional sequences of length 3 on such fake projective planes.