Determination of Baum-Bott residues of higher codimensional foliations

TL;DR

This paper extends the computation of Baum-Bott residues for higher codimensional singular holomorphic foliations on complex manifolds, removing previous genericity and regularity assumptions, and expresses residues via Grothendieck residues.

Contribution

It generalizes Baum-Bott residue formulas by removing genericity and regularity hypotheses, providing new expressions in terms of Grothendieck residues for higher codimensional foliations.

Findings

Residues expressed via Grothendieck residues on transversal discs.

Removal of Baum-Bott's generic hypothesis.

Validation of Cenkl's algorithm without regularity assumptions.

Abstract

Let $\mathscr{F}$ be a singular holomorphic foliation, of codimension $k$, on a complex compact manifold such that its singular set has codimension $\geq k+1$. In this work we determinate Baum-Bott residues for $\mathscr{F}$ with respect to homogeneous symmetric polynomials of degree $k+1$. We drop the Baum-Bott's generic hypothesis and we show that the residues can be expressed in terms of the Grothendieck residue of an one-dimensional foliation on a $(k+1)$-dimensional disc transversal to a $(k+1)$-codimensional component of the singular set of $\mathscr{F}$. Also, we show that Cenkl's algorithm for non-expected dimensional singularities holds dropping the Cenkl's regularity assumption.