Small bandwidth $\mathbb{C}^\ast$-actions and birational geometry

Dealing with Oda and Hironaka conjectures about factorization of birational maps, in 1990's Morelli and Włodarczyk showed that birational maps can be related to varieties with $\mathbb{C}^\ast$-action via algebraic cobordims construction. We note that conversely, varieties with $\mathbb{C}^\ast$- action yield interesting birational transformations. This topic is a part of a joint project with Occhetta, Romano, and Sola Conde about varieties with contact structure.