Exceptional collection of objects on some fake projective planes

Abstract

The purpose of the article is to explain a new method to establish the existence of an exceptional collection of length three for a fake projective plane M with non-trivial automorphism group, related to a conjecture of Galkin-Katzarkov-Mellit-Shinder in 2015. Our method shows that 30 fake projective planes support such a sequence, most of which are new. In particular, this provides many new H-phantom categories.