Determination of Baum–Bott residues of higher codimensional foliations

Abstract

Let F be a singular holomorphic foliation, of codimension k, on a complex compact manifold such that its singular set has codimension ≥ k + 1. In this work we determinate Baum-Bott residues for 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 F . Also, we show that Cenkl's algorithm for non-expected dimensional singularities holds dropping the Cenkl's regularity assumption.

Lehmann-Suwa residues of codimension one holomorphic foliations and applications

“…Using a recently result of Corrêa-Lourenço [13] we can verify that the above computations are correct. In fact, as in [13,…”

“…In [7], Brunella and Perrone determine the Baum-Bott residue [3] of a codimension one holomorphic foliation along a singular component of codimension two via integration over a 3-sphere of a certain 3-form (see for instance Section 5). In general, the determination of Baum-Bott residues (in terms of the Grothendieck residues) of singular holomorphic foliations of arbitrary codimension have been obtained by Corrêa and Lourenço [13]. In Section 6, we will prove (see for instance Theorem 6.2) that the Baum-Bott and Lehmann-Suwa residues are related when the codimension one foliation F is a simple almost Liouvillian foliation (see for instance Definition 6.1).…”

“…Definition 6.1. We say that the germ F is an almost Liouvillian foliation at 0 ∈ C n if there exist a germ of closed meromorphic 1-form γ 0 and a germ of holomorphic 1-form γ 1 at 0 ∈ C n such that (13) dω = (γ 0 + γ 1 ) ∧ ω.…”

Analytic varieties invariant by foliations and Pfaff systems

“…An explicit calculation of the residues is difficult in general, see [5,29,9,19,32,38,10,39]. If the foliation F has dimension one with isolated singularities, Baum and Bott in [6] show that residues can be expressed in terms of a Grothendieck residue, i.e, for each x ∈ Sing(F ) we have…”

Localization formulas on complex supermanifolds

“…Localization theories and their techniques have been very useful tools in the study of several areas of mathematics and theoretical physics, for instance in the supersymmetric gauge theories [18], symplectic geometry [10], equivariant cohomology [1,3], singular holomorphic foliations [2,5,9], Morita-Futaki invariants [8,11,12], Superconnections and the Index Theorem [4]. We refer to [18,27] for more details about applications of localization techniques in supersymmetric quantum field theories.…”