FACTORIZATION OF BIRATIONAL MAPS (after J. Włodarczyk)

Lecture given on the Mathematische Arbeitstagung by Dan Abramovich on June 23, 1999

This lecture is centered around the following theorem: Theorem 1 (Weak Factorization). Let ϕ : X_{1} −−→ X_{2} be a birational map between two nonsingular complex projective varieties X_{1} and X_{2}. Then ϕ can be factorized into a sequence of blowings up and blowings down with smooth centers. In other words, there exist nonsingular complex projective varieties V_{i} and a diagram

W_{1}  W_{n−1}
↙  ↘  ↙ ⋯ ↘  ↙  ↘
V_{1}  V_{2}  V_{n−1}  V_{n}
||  ||
X_{1}  X_{2}

where W_{i} → V_{i} and W_{i} → V_{i+1} are blowings up with nonsingular centers.

Much of the work leading to this theorem is due to Jarosław Włodarczyk, who also completed one of two currently known proofs of the theorems. The other proof is given in a joint paper of Kalle Karu, Kenji Matsuki, Jarosław Włodarczyk, and myself.

A more general version of this theorem is true - the reader is referred to the latter paper for a statement.

After giving a general overview of the results leading to the theorem, the lecture focuses on Włodarczyk’s birational cobordisms, which are the crucial tools behind the two proofs.

Given a birational map ϕ : X_{1} −−→ X_{2}, a birational cobordism B_{ϕ}(X_{1}, X_{2}) is a variety of dimension dim(X_{1}) + 1 with an action of the multiplicative group ℂ^{∗}. It is analogous to the usual cobordism B(M_{1}, M_{2}) between differentiable manifolds M_{1} and M_{2} given by a Morse function f. In the differential setting one can construct an action of the additive real group ℝ, where the “time” t ∈ ℝ acts as a diffeomorphism induced by integrating the vector field grad(f); hence the multiplicative group (ℝ_{>0}, ×) = exp(ℝ, +) acts as well. The critical points of f are precisely the fixed points of the action of the multiplicative group, and the homotopy type of fibers of f changes when we pass through these critical points. Analogously, in the algebraic setting “passing through” the fixed points of the ℂ^{∗}- action induces a birational transformation. Looking at the action on the tangent space at each fixed point, we obtain a locally toric description of the transformation. This already gives a factorization of ϕ into locally toric birational maps among varieties with locally toric structures.

Department of Mathematics, Boston University 111 Cummington Street Boston, MA 02139 U.S.A.

Email: abrmovic@math.bu.edu
Web site: http://math.bu.edu/people/abrmovic