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This paper is devoted to coregular submanifolds in Poisson geometry. We showthat their local Poisson saturation is an embedded Poisson submanifold, and wegive a normal form for this Poisson submanifold around the coregularsubmanifold. This result recovers the normal form around Poisson transversals,and it yields Poisson versions of some normal form/rigidity results aroundconstant rank submanifolds in symplectic geometry. As an application, we provea uniqueness result concerning coisotropic embeddings of Dirac manifolds inPoisson manifolds. We also show how our results generalize to the setting ofcoregular submanifolds in Dirac geometry.

The Poisson saturation of coregular submanifolds