For complex affine Poisson algebras, the Poisson Dixmier-Moeglin equivalence holds exactly when |C|-separability of Poisson prime ideals forces finite separability, and exactly when all symplectic cores are locally closed.
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Poisson Dixmier-Moeglin equivalence from a topological point of view
For complex affine Poisson algebras, the Poisson Dixmier-Moeglin equivalence holds exactly when |C|-separability of Poisson prime ideals forces finite separability, and exactly when all symplectic cores are locally closed.