A vanishing theorem on fake projective planes with enough automorphisms

Abstract

For every fake projective plane $X$ with automorphism group of order 21, we prove that $H^i(X,2L)=0$ for all $i$ and for every ample line bundle $L$ with $L^2=1$. 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 $K$. As an immediate consequence, there are exceptional sequences of length 3 on such fake projective planes.

arXiv comment17 pages

References in corpus (1)

• Exceptional collections on 2-adically uniformised fake projective planes

Cited by in corpus (3)

• Cycles, derived categories, and rationality
• Quasiphantom categories on a family of surfaces isogenous to a higher product
• Examples of surfaces with canonical maps of maximal degree