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Indices and residues: from Poincaré-Hopf to Baum-Bott, and Marco Brunella

Maurício Corrêa, José Seade

TL;DR

This expository work connects invariants of vector fields and holomorphic foliations on singular spaces, weaving together ICIS indices (GSV, homological, virtual, radial) with Baum–Bott residues for singular foliations. It establishes precise algebraic and topological formulations: GSV equals the Milnor-fiber PH index, the homological index provides algebraic expressions that relate to GSV via Tjurina numbers, and the virtual index localizes top Chern data through Verdier specialization. In the foliations setting, Baum–Bott residues localize characteristic classes along singular sets, with 1-dimensional foliations on surfaces yielding concrete links to Milnor numbers and Camacho–Sad indices, and nonnegativity of GSV indices furnishing obstructions to the Poincaré problem. The text then extends these ideas to higher dimensions and logarithmic settings, presenting Aleksandrov’s logarithmic index and log Baum–Bott residues, and highlighting adjunction-type relations and global residue theorems that unify disparate strands of singularity theory and foliation theory. Overall, the article clarifies the deep interplay between local singularity invariants and global geometric/topological data across complex analytic and holomorphic foliations contexts, via residues, specialization, and index theory.

Abstract

In this expository article, we study and discuss invariants of vector fields and holomorphic foliations that intertwine the theories of complex analytic singular varieties and singular holomorphic foliations on complex manifolds: two different settings with many points in common.

Indices and residues: from Poincaré-Hopf to Baum-Bott, and Marco Brunella

TL;DR

This expository work connects invariants of vector fields and holomorphic foliations on singular spaces, weaving together ICIS indices (GSV, homological, virtual, radial) with Baum–Bott residues for singular foliations. It establishes precise algebraic and topological formulations: GSV equals the Milnor-fiber PH index, the homological index provides algebraic expressions that relate to GSV via Tjurina numbers, and the virtual index localizes top Chern data through Verdier specialization. In the foliations setting, Baum–Bott residues localize characteristic classes along singular sets, with 1-dimensional foliations on surfaces yielding concrete links to Milnor numbers and Camacho–Sad indices, and nonnegativity of GSV indices furnishing obstructions to the Poincaré problem. The text then extends these ideas to higher dimensions and logarithmic settings, presenting Aleksandrov’s logarithmic index and log Baum–Bott residues, and highlighting adjunction-type relations and global residue theorems that unify disparate strands of singularity theory and foliation theory. Overall, the article clarifies the deep interplay between local singularity invariants and global geometric/topological data across complex analytic and holomorphic foliations contexts, via residues, specialization, and index theory.

Abstract

In this expository article, we study and discuss invariants of vector fields and holomorphic foliations that intertwine the theories of complex analytic singular varieties and singular holomorphic foliations on complex manifolds: two different settings with many points in common.

Paper Structure

This paper contains 15 sections, 32 theorems, 138 equations, 6 figures.

Table of Contents

  1. Indices of vector fields on isolated complete intersection singularities
  2. The GSV index
  3. The homological index
  4. The virtual index
  5. Radial index and the Milnor number
  6. Singular Holomorphic foliations, Baum-Bott residues and indices
  7. Singular holomorphic foliations and Pfaffian systems
  8. Baum--Bott residues
  9. Indices for 1-dimensional holomorphic foliations on surfaces
  10. Baum-Bott-Brunella indices
  11. The GSV index for 1-dimensional holomorphic foliations
  12. Indices in higher dimensions and the Poincaré problem
  13. GSV-indices for vector fields as residues
  14. Log Baum--Bott residues and Aleksandrov's logarithmic index
  15. Final remarks

Key Result

proposition 1

The $\mathrm{GSV}$ index has the following properties: (1) The $\mathrm{GSV}$ index of $v$ at $0$ equals the total Poincaré-Hopf index of $v$ in the Milnor fiber: (2) If $v$ is everywhere transverse to ${\bf K}$, then where $n$ is the complex dimension of $V$ and $\mu$ is the Milnor number.

Figures (6)

  • Figure 1: The GSV equals the PH index in a Milnor fiber.
  • Figure 2: The virtual index is the localization at $0$ of the PH index in a Milnor fiber.
  • Figure 3: The radial index
  • Figure 4:
  • Figure 5:
  • ...and 1 more figures

Theorems & Definitions (55)

  • definition 1
  • remark 1
  • definition 2
  • proposition 1
  • definition 3
  • theorem 1
  • definition 4
  • theorem 2
  • theorem 3
  • definition 5
  • ...and 45 more