Complex surfaces 6: Kodaira dimensions 1 and 2
In this lecture we continue the overview of complex projective urfaces by discussing those of Kodaira dimensions 1 and 2. The surfaces of dimension 1 are all elliptic surfaces with a map onto a curve whose fibers are mostly elliptic curves. We describe Kodaira's classification of the possible fibers,
and mention an analogy with elliptic curves over Z. For curves of Kodaira dimension 2 we give a few examples and describe the possible values of the Chern numbers.
(Correction: in the video I got the history of the BMY inequality slightly wrong: I should have said that Van de Ven proved the BMY inequality with a constant of 8, Bogomolov with a constant of 4, and Miyaoka and Yau with a constant of 3.)