For a variety [Formula: see text], a big [Formula: see text]-divisor [Formula: see text] and a closed connected subgroup [Formula: see text] we define a [Formula: see text]-invariant version of the [Formula: see text]-threshold. We prove that for a Fano variety [Formula: see text] and a connected subgroup [Formula: see text] this invariant characterizes [Formula: see text]-equivariant uniform [Formula: see text]-stability. We also use this invariant to investigate [Formula: see text]-equivariant [Formula: see text]-stability of some Fano varieties with large groups of symmetries, including spherical Fano varieties. We also consider the case of [Formula: see text] being a finite group.

Let X be a non-uniruled compact Kähler space of dimension 3. We show that the group of bimeromorphic automorphisms of X is Jordan. More generally, the same result holds for any compact Kähler space admitting a quasi-minimal model.

We aim to classify codimension 1 foliations F with canonical singularities and ν(K F ) < 3 on threefolds of general type. We prove a classification result for foliations satisfying these conditions and having non-trivial algebraic part. We also describe purely transcendental foliations F with the canonical class K F being not big on manifolds of general type in any dimension, assuming that F is non-singular in codimension 2.

Let X be a complex projective variety. Suppose that the group of birational automorphisms of X contains finite subgroups isomorphic to (Z/N i Z) r for r fixed and N i arbitrarily large. We show that r does not exceed 2 dim(X). We also show that the same result holds for groups of bimeromorphic automorphisms of compact Kähler threefolds.

The aim of this paper is to classify codimension 1 foliations ℱ with canonical singularities and 𝜈(𝐾 ℱ ) < 3 on threefolds of general type. I prove a classification result for foliations satisfying these conditions and having nontrivial algebraic part. We also describe purely transcendental foliations ℱ with the canonical class 𝐾 ℱ being not big on manifolds of general type in any dimension, assuming that ℱ is nonsingular in codimension 2.