Torification and Factorization of Birational Maps

TL;DR

This paper proves the weak factorization conjecture for birational maps over algebraically closed fields of characteristic zero, showing they can be decomposed into blowings up and down with smooth centers.

Contribution

It establishes a functorial and compatible factorization of birational maps into blowings up and down, confirming a key conjecture in algebraic geometry.

Findings

Proves the weak factorization conjecture in characteristic zero.

Shows the factorization is functorial and compatible with normal crossings divisors.

Extends the result to algebraic and analytic spaces.

Abstract

Building 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.