Exceptional collection of objects on some fake projective planes

TL;DR

This paper introduces a new method to establish the existence of exceptional collections on certain fake projective planes, revealing many new examples and contributing to the understanding of their derived categories.

Contribution

The paper presents a novel approach to prove the existence of exceptional collections of length three on fake projective planes with automorphisms, expanding known examples significantly.

Findings

30 fake projective planes support such exceptional collections

Most of these are newly identified examples

The work introduces many new H-phantom categories

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.