Global nilpotent variety is Lagrangian

Let (M,ω) be a smooth symplectic algebraic variety. A (possibly singular) algebraic subvariety Y ⊂ M is said to be isotropic, resp. Lagrangian, if the tangent space, TyY , at any regular point y ∈ Y is an isotropic, resp. Lagrangian, vector subspace in the symplectic vector space TyM (we always assume Y to be reduced, but not necessarily irreducuble). The following characterisation of isotropic subvarieties proved, e.g. in [CG, Prop. 1.3.30] will be used later : Y ⊂ M is isotropic if and only if for any smooth locally closed subvariety W ⊂ Y (here W is possibly contained in the singular locus of Y ), we have ω| W = 0. An advantage of this characterisation is that it allows to extend the notion of ‘being isotropic’ from algebraic subvarieties to semi-algebraic constructible subsets. Thus, we call a semi-algebraic constructible subset Y ⊂ M isotropic if ω| W = 0 for any smooth locally closed algebraic variety W ⊂ Y .