I will discuss a program, joint with K. O’Grady, to investigate the birational geometry of locally symmetric varieties of

K3 type (similar considerations apply to the case of ball quotients).

The motivation for our study is the search for geometric compactifications for the moduli of polarized K3 surfaces. Namely, as a consequence of Torelli theorem, the moduli of polarized K3 surfaces (with canonical singularities) can be identified to a locally symmetric variety D/\Gamma. As such, there are natural `arithmetic’ 

compactifications, e.g. the Baily-Borel (BB) compactification. 

Unfortunately, the BB compactification has obscure geometric meaning. 

Consequently, it is natural to compare it with more geometric compactifications, such as those given by GIT. I will explain that there is a natural continuous interpolation between BB and GIT compactifications, and that there is a rich geometric and arithmetic structure behind this picture. The case of quartic K3s will be discussed in detail.

(Related work includes the Hassett-Keel program on the moduli of curves, and the work of Looijenga on semi-toric compactifications for locally symmetric varieties of K3 type)