Birational geometry of symplectic resolutions of nilpotent orbits Yoshinori

Let G be a complex simple Lie group and let g be its Lie algebra. Then G has the adjoint action on g. The orbit Ox of a nilpotent element x ∈ g is called a nilpotent orbit. A nilpotent orbit Ox admits a non-degenerate closed 2-form ω called the Kostant-Kirillov symplectic form. The closure Ōx of Ox then becomes a symplectic singularity. In other words, the 2-form ω extends to a holomorphic 2-form on a resolution of Ōx. A resolution of Ōx is called a symplectic resolution if this extended form is everywhere non-degenerate on the resolution. A typical symplectic resolution of Ōx is obtained as the Springer resolution T (G/P ) → Ōx for a suitable parabolic subgroup P ⊂ G. Here T (G/P ) is the cotangent bundle of the homogenous space G/P . Spaltenstein [S] and Hesselink [He] obtained a necessary and sufficient condition for Ōx to have a Springer resolution when g is a classical simple Lie algebra. Moreover, [He] gave an explicit number of such parabolics P up to conjugacy class that give Springer resolutions of Ōx (cf. §2). Recently, Fu [Fu 1] has shown that every symplectic (projective) resolution is obtained as a Springer resolution. The following is one of main results of this paper.