Mori dream bonds and C ∗ ${\mathbb {C}}^*$ -actions

Abstract

We construct a correspondence between Mori dream regions arising from small modifications of normal projective varieties and $C^{\ast }$-actions on polarized pairs which are bordisms. Moreover, we show that the Mori dream regions constructed in this way admit a chamber decomposition on which the models are the geometric quotients of the $C^{\ast }$-action. In addition, we construct, from a given $C^{\ast }$-action on a polarized pair for which there exist at least two admissible geometric quotients, a $C^{\ast }$-equivariantly birational $C^{\ast }$-variety, whose induced action is a bordism, called the pruning of the variety.