2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78399Building on the work of the fourth author in math.AG/9904074, we prove 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 blowings up and blowings down with smooth centers. Such a factorization exists which is functorial with respect to absolute isomorphisms, and compatible with a normal crossings divisor. The same holds for algebraic and analytic spaces. Another proof of the main theorem by the fourth author appeared in math.AG/9904076.The previous versions of this paper used Morelli's algorithm for strong factorization of toric birational maps, in which a gap was recently found by K. Karu. In this version we rely only on weak factorization of toric birational maps (due to Wlodarczyk and Morelli). The results remain the sameAlgebraic GeometryComplex VariablesDifferential Geometry14E05;14L30;14M25;58E05;57R90text