Algebraic Morse theory and the weak factorization theorem

Abstract

We develop a Morse-like theory for complex algebraic varieties. In this theory a Morse function is replaced by a ℂ*-action. The critical points of the Morse function correspond to connected fixed point components. “Passing through the fixed points” induces some simple birational transformations called blow-ups, blow-downs and flips which are analogous to spherical modifications.

In classical Morse theory by means of a Morse function we can decompose the manifold into elementary pieces – “handles”. In algebraic Morse theory we decompose a birational map between two smooth complex algebraic varieties into a sequence of blow-ups and blow-downs with smooth centers.