JongHae Keum, Fake Projective Planes I

Fake projective planes (abbreviated as FPPs) are 2-dimensional complex manifolds with the same Betti numbers as the projective plane, but not isomorphic to it.

FPPs can be uniformized by a complex 2-ball. In other words, they are ball quotients having the minimum possible Betti numbers.

The existence of such a surface was first proved by Mumford in 1979, via 2-adic uniformization.

Not always algebraic varieties are described via polynomial equations: sometimes they are constructed via uniformization: this means, as quotients of certain domains in a complex vector space, called bounded symmetric domains, via the action of discontinuous groups. Then general theorems (as Kodaira’s) imply the algebraicity of these quotient complex manifolds. The problem concerning the algebro-geometrical properties of such varieties constructed via uniformization and especially the description of their projective embeddings (and the corresponding polynomial equations) lies at the crossroads of several allied fields: the theory of arithmetic groups and division algebras, complex algebraic and differential geometry, linear systems, use of group symmetries, and topological and homological tools in the study of quotient spaces. Of particular importance are the so-called ball quotients, especially in dimension 2, since they yield the surfaces with the maximal possible canonical volume K² for a fixed value of the geometric genus p_g.

In the first lecture I will introduce basic properties of FPPs and their position in the classification theory of algebraic surfaces.

In the second I will discuss recent progress on them, such as their derived categories, bicanonical maps and their equations.