Fake Projective Planes: Properties and Recent Progress

Explore the intricacies of fake projective planes (FPPs) in this 42-minute conference talk by JongHae Keum from the Korea Institute for Advanced Study. Delve into the definition and characteristics of FPPs, including their Betti numbers, Chern numbers, and canonical class. Discover how FPPs are uniformized by the unit complex 2-ball and their connection to co-compact arithmetic subgroups of PU(2, 1). Learn about Mumford's pioneering work in proving the existence of FPPs through 2-adic uniformization. Examine the intersection of various mathematical fields involved in studying FPPs, including arithmetic groups, division algebras, complex algebraic and differential geometry, group symmetries, and topological and homological tools. Gain insights into the significance of ball quotients, particularly in dimension 2, and their relation to surfaces with maximal canonical volume. Conclude with an overview of recent advancements in FPP research, covering derived categories, bicanonical maps, and equations.