Morelli-W\l odarczyk cobordism and examples of rooftop flips

Abstract

We introduce the notion of rooftop flip, namely a small modification among normal projective varieties which is modeled by a smooth projective variety of Picard number 2 admitting two projective bundle structures. Examples include the Atiyah flop and the Mukai flop, modeled respectively by $\mathbb{P}^1\times \mathbb{P}^1$ and by $\mathbb{P}\left(T_{\mathbb{P}^2}\right)$. Using the Morelli-Wlodarczyk cobordism, we prove that any smooth projective variety of Picard number 1, endowed with a $\mathbb{C}^*$-action with only two fixed point components, induces a rooftop flip.