Birational cobordisms and factorization of birational maps

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 étale 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.

23 pages

Cited by in corpus (10)

• Orbifold Quantum Cohomology
• Functorial destackification of tame stacks with abelian stabilisers
• Quiver varieties, category O for rational Cherednik algebras, and Hecke algebras
• Ricci flow and birational surgery
• Small bandwidth ${\mathbb{C}}^{\ast }$-actions and birational geometry
• Geometric convergence of the Kähler-Ricci flow on complex surfaces of general type
• Birational geometry and derived categories
• Geometric Invariant Theory and Birational Geometry
• Toroidal varieties and the weak Factorization Theorem
• Toric non-equalized flips associated to ${\mathbb{C}}^{\ast }$-actions