There is a still unsettled question raised by M. Brunella and A. Lins Neto asking whether any codimension-one holomorphic foliation \({G}\) on \(\mathbb{P}^n\) is either the pull-back of a holomorphic foliation \({F}\) on \(\mathbb{P}^2\) by a rational map or admits a transverse projective structure with poles on some invariant hypersurface. In the first case, one has a foliation whose leaves are foliated by the levels of the rational map. This makes it interesting to study flags of foliations – loosely speaking, collections of foliations of increasing dimension and whose tangent sheaves form a sequence under inclusion.
The work being reviewed studies the relation between the degrees of the different distributions (hence also foliations) belonging to flags of holomorphic distributions on \(\mathbb{P}^n\). Apart from the main results, it contains a couple of illustrative examples.
The behaviour shown in the paper can be roughly summarised saying that the degree of the distributions in a flag decreases with the dimension of the distributions.
We state one of the main results to give an idea of the contents:
{Theorem. } Let \(\mathcal{F}:=({F}, {G})\) be a flag of reduced holomorphic foliations on \(\mathbb{P}^n\), \(n\geq 3\). If \(\dim({F}) = \dim({G})-1\) and \(\text{Sing}({G})\) has a Baum-Kupka component \(K\), then \[ \deg({G}) \leq \deg({F}). \] Notice that the print edition contains a typo in the statement of Corollary 1.4, where the inequalities are in the wrong direction.