A note on symplectic singularities

Proof of Corollary 1. Assume that X has only terminal singularities. Then Codim(Σ ⊂ X) ≥ 3. By Theorem, Σ has no codimension 3 irreducible components. Therefore Codim(Σ ⊂ X) ≥ 4. Conversely, assume Codim(Σ ⊂ X) ≥ 4. Take a resolution f : Y → X of X and denote by {Ei} the f -exceptional divisors. Since X has canonical singularities, we have KY = f KX +ΣaiEi with non-negative integers ai. One can prove that ai are actually all positive by the same argument as §1 . We can define a symplectic variety Z similarly as a compact normal Kaehler space Z whose regular locus V admits an everywhere non-degenerate holomorphic closed 2-form which extends to a regular 2-form on an arbitrary resolution Z̃ of Z. Then the following result is also a corollary of our theorem.