Determination of Baum–Bott residues of higher codimensional foliations

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.

Keywords

Baum–Bott residues, localization, holomorphic foliations, characteristic classes

2010 Mathematics Subject Classification

32A27, 32S65, 57R30

Full Text (PDF format)

Received 26 October 2017

Accepted 2 March 2018

Published 9 July 2019