A Poisson--Poincaré--Dulac for Poisson Connections
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866908835366567936 |
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| author | Corrêa, Maurício Peña, Miguel Rodríguez |
| author_facet | Corrêa, Maurício Peña, Miguel Rodríguez |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14132 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Poisson--Poincaré--Dulac for Poisson Connections Corrêa, Maurício Peña, Miguel Rodríguez Algebraic Geometry Complex Variables Differential Geometry Symplectic Geometry 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. |
| title | A Poisson--Poincaré--Dulac for Poisson Connections |
| topic | Algebraic Geometry Complex Variables Differential Geometry Symplectic Geometry |
| url | https://arxiv.org/abs/2602.14132 |