Morelli-Włodarczyk 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 P 1 × P 1 and by P (T P 2 ). Using the Morelli-W lodarczyk cobordism, we prove that any smooth projective variety of Picard number 1, endowed with a C *action with only two fixed point components, induces a rooftop flip.