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A Poisson--Poincaré--Dulac for Poisson Connections

Maurício Corrêa, Miguel Rodríguez Peña

Abstract

We study Poisson-flat connections with logarithmic poles along a simple normal crossings divisor on a holomorphic Poisson manifold, where flatness is required only along the symplectic foliation. After identifying the relevant logarithmic cotangent Poisson Lie algebroid, we define an Euler--Poisson principal part and a residue theory adapted to the canonical logarithmic Hamiltonian generators. Under a precise nonresonance hypothesis, we establish a Poisson Poincaré--Dulac theorem: any logarithmic Poisson-flat connection with prescribed principal part is holomorphically gauge equivalent to a pure Euler--Poisson normal form with constant commuting residues, and this normal form is unique up to Casimir-valued gauge transformations lying in the common centralizer of the residues. To encode both leafwise transport and boundary winding, we construct a twisted leafwise fundamental groupoid via the real oriented blow-up and a 2-pushout that adjoins canonical tangential meridians. In the nonresonant regime, the normal form yields a meridional character determined by the residues, and hence a logarithmic Riemann--Hilbert correspondence in the Poisson setting for this groupoid. Finally, we illustrate the theory with rank-two Poisson modules (Poisson triples) and an explicit family of examples.

A Poisson--Poincaré--Dulac for Poisson Connections

Abstract

We study Poisson-flat connections with logarithmic poles along a simple normal crossings divisor on a holomorphic Poisson manifold, where flatness is required only along the symplectic foliation. After identifying the relevant logarithmic cotangent Poisson Lie algebroid, we define an Euler--Poisson principal part and a residue theory adapted to the canonical logarithmic Hamiltonian generators. Under a precise nonresonance hypothesis, we establish a Poisson Poincaré--Dulac theorem: any logarithmic Poisson-flat connection with prescribed principal part is holomorphically gauge equivalent to a pure Euler--Poisson normal form with constant commuting residues, and this normal form is unique up to Casimir-valued gauge transformations lying in the common centralizer of the residues. To encode both leafwise transport and boundary winding, we construct a twisted leafwise fundamental groupoid via the real oriented blow-up and a 2-pushout that adjoins canonical tangential meridians. In the nonresonant regime, the normal form yields a meridional character determined by the residues, and hence a logarithmic Riemann--Hilbert correspondence in the Poisson setting for this groupoid. Finally, we illustrate the theory with rank-two Poisson modules (Poisson triples) and an explicit family of examples.
Paper Structure (22 sections, 13 theorems, 167 equations, 4 figures)

This paper contains 22 sections, 13 theorems, 167 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Geometric and Poisson background
  3. Holomorphic Poisson manifolds
  4. Divisors and logarithmic geometry
  5. Standing hypotheses
  6. Local SNC charts and logarithmic Hamiltonians
  7. Logarithmic Poisson connections
  8. Definition and basic properties
  9. Residues and Euler--Poisson principal parts
  10. Euler--Poisson principal parts and residues
  11. Leafwise meridians and monodromy
  12. Non-resonance and the Poisson--Poincaré--Dulac normal form
  13. Joint generalized eigenspaces and non-resonance
  14. Poisson--Poincaré--Dulac elimination in the non-resonant case
  15. Uniqueness up to Casimir--centralizer gauge
  16. ...and 7 more sections

Key Result

Lemma 3.1

Assume (H3). Then the Koszul bracket $[-,-]_\sigma$ preserves logarithmic forms: Equivalently, $A_\sigma(\log D):=\Omega_X^1(\log D)$ is a holomorphic Lie algebroid with anchor $\rho=\sigma^\#:\Omega_X^1(\log D)\to T_X(-\log D)$.

Figures (4)

  • Figure 1: If two gauges send $\Theta$ to the same Euler--Poisson normal form, their ratio stabilizes the normal form. Theorem \ref{['thm:uniq']} identifies this stabilizer as Casimir-valued and contained in the simultaneous centralizer of the residue tuple.
  • Figure 2: Schematic local model for the oriented blow-up when $r=2$: the corner variables $(\rho_1,\rho_2)\in[0,\varepsilon)^2$ encode approach to $D_1\cup D_2$; each boundary face carries the canonical angular circle; over the corner stratum the angular fiber is $(S^1)^2$. The full local chart is $[0,\varepsilon)^2\times (S^1)^2\times \mathbf D^{2(n-2)}$.
  • Figure 3: The twisted leafwise groupoid is obtained by adjoining canonical tangential meridians as boundary isotropy and gluing along the collar.
  • Figure 4: Tangential completion as a pushout: on the collar we adjoin meridional labels in $\mathbb{Z}^3$, and on the overlap we force the label to vanish.

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 3.1: Logarithmic cotangent is a Lie subalgebroid
  • proof
  • Definition 3.2: Logarithmic Poisson connection
  • Remark 3.3: Gauge transformation
  • Definition 3.4: Poisson curvature / flatness
  • ...and 37 more