Inequalities and enumerative formulas for flags of Pfaff systems

In this work, we study inequalities and enumerative formulas for flags of Pfaff systems on ${\mathbb{P}}_{\mathbb{C}}^n$. More specifically, we find the number of independent Pfaff systems that leave invariant a one-dimensional holomorphic foliation and deduce inequalities relating the degrees in the flags, which can be interpreted as the Poincaré problem for flags. Moreover, restricting to a flag of specific holomorphic foliations/distributions, we obtain inequalities involving the degrees. As a consequence, we prove stability results for the tangent sheaf of some rank two holomorphic foliations/distributions.