Local and global applications of the Minimal Model Program for co-rank 1 foliations on threefolds

Abstract

We provide several applications of the minimal model program to the local and global study of co-rank 1 foliations on threefolds. Locally, we prove a singular variant of Malgrange's theorem, a classification of terminal foliation singularities and the existence of separatrices for log canonical singularities. Globally, we prove termination of flips, a connectedness theorem on log canonical centres, a non-vanishing theorem and some hyperbolicity properties of foliations.

On the connectedness principle and dual complexes for generalized pairs

“…Despite its simplicity, this statement has many powerful applications: for example, inversion of adjunction (see [23,Theorem 17.6], [21], and [15] for a more recent and general version) or the fact that log canonical singularities are Du-Bois (see [26]), or yet again, the study of the geometry and boundedness of varieties of Fano-type and complements (see [5,18,22]). Perhaps more surprisingly, the connectedness principle has also been used to study hyperbolicity questions related to the positivity of log pairs and even foliations (see [35,36]). We work with the following setup: we consider log pairs (𝑋, 𝐵) together with a contraction 𝑓 : 𝑋 → 𝑆, such that −(𝐾 𝑋 + 𝐵) 𝑓 -nef.…”

Toward classification of codimension 1 foliations on threefolds of general type

“…A result of McQuillan [37, Corollary I.2.2] says that a germ of a normal $$\operatorname{\mathbb {Q}}$$‐Gorenstein surface with a terminal foliation is isomorphic to a finite cyclic quotient of a smooth surface germ with a regular foliation. For more details on terminal foliation singularities on threefolds, see [45, section 5].…”

“…Recently, in a series of works [19, 44, 45], the MMP for codimension 1 foliations on threefolds has been established. Moreover, some structure theorems for codimension 1 foliations are known in higher dimensions, for example, in the case $$K_{\operatorname{\mathcal {F}}} \equiv 0$$ [22, 34] or $$-K_{\operatorname{\mathcal {F}}}$$ ample [1–3].…”

Closed meromorphic 1-forms

“…Recently, Spicer and Svaldi [61], established the existence of separatrices for germs of codimension one foliations with log canonical singularities on (C 3 , 0). We refer to their work for the definition of log canonical singularities for foliations.…”