Toroidal varieties and the weak factorization theorem

We develop the theory of stratified toroidal varieties, which gives, together with the theory of birational cobordisms 73], a proof of the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blow-ups and blow-downs with smooth centers. Dedicated to Professor Andrzej Biaynicki-Birula