The Universal Deformation Space of a Source-Compact Poisson Manifold

• Applicant: Professor Dr. Ioan Marcut
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 580538076

This project lies in the field of Poisson geometry, a branch of differential geometry that generalizes the classical Poisson bracket from Hamiltonian mechanics. It is part of a broader program studying how such geometric structures can be deformed. The set of all Poisson structures on a manifold, identifying isomorphic ones, forms the so-called Poisson moduli space. This space describes how one Poisson structure can be continuously transformed into another. In general, it is poorly understood: it is often infinite-dimensional, possibly locally disconnected, and general tools for its study are lacking. The project introduces and investigates a new class of Poisson manifolds – the so-called source-compact Poisson manifolds. They are defined via their Weinstein groupoid, an object from the integrability theory of Lie algebroids that plays a role in Poisson geometry similar to the fundamental groupoid in topology. “Source-compact” refers to a compactness property of this groupoid that ensures good geometric behavior even when the groupoid is not smooth. The goal of the project is to understand and classify deformations of such structures by constructing a universal deformation space – a larger Poisson manifold that captures all nearby structures at once. It is expected that this “completion” of the original manifold is itself rigid (i.e. admits no further deformations) and locally parametrizes the Poisson moduli space. Expected consequences include that source-compact Poisson manifolds form an open class among all Poisson structures, have unobstructed infinitesimal deformations, and exhibit singular foliations that remain stable under deformation. The project builds on the applicant’s earlier work on local normal forms and rigidity theorems in Poisson geometry. Analytically, it relies on the Nash–Moser inverse function theorem for Fréchet spaces, a central tool for treating infinite-dimensional geometric problems. In the complex-algebraic context, universal Poisson deformations appeared in the work of Namikawa. In the smooth case, a result by the applicant – the main inspiration for this project – identified such a deformation space for certain Poisson structures on spheres arising from Lie theory. Closely related are the recently introduced Poisson manifolds of compact type. Compared to them, source-compact Poisson manifolds satisfy a stronger compactness property but need not be integrable. While integrability is not stable under deformations, the central hypothesis of the project is that source-compactness is stable.

• DFG Programme: Research Grants