Birational cobordisms and factorization of birational maps

TL;DR

This paper introduces a Morse-like theory to decompose birational maps of smooth projective varieties into elementary steps, laying the groundwork for proving the weak factorization conjecture in characteristic zero.

Contribution

It develops a new Morse-like theoretical framework for decomposing birational maps into elementary steps, facilitating the proof of the weak factorization conjecture.

Findings

Decomposition of birational maps into elementary steps using K^*-actions.

Foundation for proving the weak factorization conjecture in characteristic zero.

Connection of the theory with geometric invariant theory.

Abstract

In this paper we develop a Morse-like theory in order to decompose birational maps and morphisms of smooth projective varieties defined over a field of characteristic zero into more elementary steps which are locally \'etale isomorphic to equivariant flips, blow-ups and blow-downs of toric varieties. A crucial role in the considerations is played by K^*-actions where K is the base field. This paper serves as a basis for proving the weak factorization conjecture on factorization of birational maps in characteristic zero into blow-ups and blow-downs. This is carried out in two subsequent papers, one by the author (Combinatorial structures on toroidal varieties: a proof of the weak Factorization Theorem) and one joint with Abramovich, Karu and Matsuki (Torification and factorization of birational maps). In the last paper, the ideas of the present paper are discussed using geometric invariant theory.