Residues for flags of holomorphic foliations (Q2411354)

A 2-flag of foliations is a pair of foliations \((\mathcal{F}_1, \mathcal{F}_2)\) such that the leaves of \(\mathcal{F}_1\) are contained in the leaves of \(\mathcal{F}_2\). The authors consider 2-flags formed by singular holomorphic foliations on a complex manifold \(M\). In this work, a Baum-Bott type residue theorem for flags of holomorphic foliations is shown. Some relations between the residues of the flag and the residues of their corresponding foliations are studied. A Nash residue for flags is defined, and a partial answer to the Baum-Bott type rationality conjecture in this context is given. Several examples are provided. The present Baum-Bott theorem for 2-flags detects some characteric classes that the Bott vanishing theorem for foliations does not detect. See also [the second author et al., Int. J. Math. 24, No. 11, Article ID 1350093, 12 p. (2013; Zbl 1296.32011); \textit{R. S. Mol}, An. Acad. Bras. Ciênc. 83, No. 3, 775--786 (2011; Zbl 1262.32021)].