Log Geometry and the Moduli Space of Toric Varieties

In [2], Alexeev constructed a proper moduli space of polarized toric varieties. However, in addition to the main component containing the toric varieties, there were additional irreducible components which were tricky to eliminate in a canonical way. Later, Olsson [1] showed how adding a log structure to the toric varieties under consideration effectively restricted the problem enough to single out the primary irreducible component. The goal of these notes is to give a broad overview of Olsson’s work in [1]. We’ll look at some examples which illustrate the fundamental ideas and go through the broad strokes of some of the proofs. In particular, we’ll try to illustrate the importance of the log structure and the role they play in putting sufficiently strong conditions on our moduli spaces.