Manifolds with two projective bundle structures

Abstract

In this paper we classify varieties of Picard number two having two projective bundle structures of any relative dimension, under the assumption that these structures are mutually uniform. As an application we prove the Campana–Peternell conjecture for varieties of Picard number one admitting C ∗ \mathbb {C}^* -actions of a certain kind.

Morelli-Włodarczyk cobordism and examples of rooftop flips

“…Except for a nonhomogeneous sporadic example appearing in [15] (see also [8, §2]), in all the known examples Y is also a rational homogeneous variety. In [12,Remark 3.3] there is a complete classification of rational homogeneous varieties with Picard number 2 admitting two projective bundle structures. The corresponding smooth drums X constructed from a rational homogeneous variety Y are well-known in literature: they are one of the horospherical varieties classified by Pasquier, see [16, Theorem 0.1].…”

“…Smooth projective varieties with two different projective bundle structures have been already studied in the literature (see for instance [12]): in particular, in [13,Lemma 4.4] the authors have constructed a correspondence between them and smooth projective varieties X of Picard number 1 admitting a C * -action having only two fixed point components. Motivated by the example of Atiyah flop and its connection with a variety admitting two projective bundle structures, we introduce the notion of rooftop flip.…”