Local and global applications of the Minimal Model Program for co-rank one foliations on threefolds

Calum Spicer, Roberto Svaldi

Published 2019-08-14Version 1

We show that the Minimal Model Program for co-rank one foliations on threefolds terminates by proving foliation flips terminate. Moreover, we recover a full suite of powerful results on the birational structure of co-rank one foliations on threefolds (Connectedness theorem, Inversion of adjunction, Non-vanishing theorem), despite the well-known failure of the foliated analogue of the vanishing results which are (classically) used to prove these statements in the Minimal Model Program in the case of log pairs. We then turn to several applications of the MMP to the analysis of local and global properties of foliations on threefolds. In particular, we study foliation singularities proving the existence of first integrals for isolated canonical foliation singularities (an extension of Malgrange's theorem) and derive a complete classification of terminal foliated threefolds singularities. We show the existence of separatrices for log canonical singularities. We also prove some hyperbolicity properties of foliations, showing that the failure of the canonical bundle to be nef implies the existence of entire holomorphic curves contained in the open strata of a natural stratification of the singular locus of the foliation.