  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.
