The purpose of the talk is to
explain a joint work with Vincent Koziarz and
Donald Cartwright on the geometry of a complex two ball quotient constructed
earlier by Cartwright and Steger. The surface has the smallest possible Euler
number 3 among surfaces of general type but is a not a fake projective plane.
Basic geometric properties of the surface had not been understood, such
as the genus of the Albanese fibration. We would
answer some questions in this direction and use the example to study several
problems in the geometry of algebraic surfaces and complex ball quotients. |