A characterization of nilpotent varieties of complex semisimple Lie algebras

A normal complex algebraic variety X is called a symplectic variety (cf. [Be]) if its regular locus Xreg admits a holomorphic symplectic 2-form ω such that it extends to a holomorphic 2-form on a resolution f : X̃ → X. Affine symplectic varieties are constructed in various ways such as nilpotent orbit closures of a semisimple complex Lie algebra (cf. [CM]), Slodowy slices to nilpotent orbits (cf. [Sl]) or symplectic reductions of holomorphic symplectic manifolds with Hamiltonian actions. Usually these examples show up with C∗-actions. In this lecture we shall characterizes the nilpotent variety of a complex semisimple Lie algebra among affine symplectic varieties from a view point of algebraic geometry. Let g be a complex semisimple Lie algebra and let N be the nilpotent variety of g. It is well known that N is an affine normal variety and its regular locus admits a holomorphic symplectic 2-form ωKK called the KostantKirilliv 2-form. Then (N,ωKK) is an affine symplectic variety in our sense. Moreover, the scalar multiplication determines a C∗-action on g and it induces a C∗-action also on N . The Kostant-Kirillov 2-form ωKK has weight 1 with respect to this C∗-action. The adjoint group G acts on g and let g//G be the GIT quotient of the G-action. Namely g := SpecC[g], where C[g] is the G-invariant ring of the coordinate ring C[g] of g. By a theorem of Chevalley, C[g] is isomorphic to a polynomial ring C[f1, ..., fr] generated by algebraically independent G-invariant homogeneous polynomials fi. Here r coincides with the rank of g. Let χ : g → g//G = C be the adjoint quotient map. Then N = χ−1(0). In particular, N is a complete intersection of r homogeneous polynomials in the affine space g.