A Poisson--Poincar\'e--Dulac for Poisson Connections

TL;DR

This paper establishes a Poisson--Poincaré--Dulac theorem for logarithmic Poisson-flat connections on holomorphic Poisson manifolds, providing a normal form classification under nonresonance conditions and linking it to a Poisson Riemann--Hilbert correspondence.

Contribution

It introduces a Poisson Poincaré--Dulac theorem for logarithmic flat connections, defining a normal form and a Poisson Riemann--Hilbert correspondence in this setting.

Findings

Normal form classification of logarithmic Poisson-flat connections

Construction of a twisted leafwise fundamental groupoid

Explicit examples illustrating the theory

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 Poincar\'e--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.