Inequalities and enumerative formulas for flags of Pfaff systems

In this work, we study inequalities and enumerative formulas for flags of Pfaff Systems on [Formula: see text]. More specifically, we establish a bound for the number of independent twisted [Formula: see text]-forms that leave invariant a one-dimensional holomorphic foliation and deduce inequalities that relate the degrees in the flags, which can be interpreted as a version of the Poincaré problem for flags. Moreover, by restricting to a flag of specific holomorphic foliations/distributions, we obtain inequalities involving the degrees. As a consequence, we obtain stability results for the tangent sheaf of some rank two holomorphic foliations/distributions.