Symplectic resolutions for nilpotent orbits

In this paper, firstly we calculate Picard groups of a nilpotent orbit �� in a classical complex simple Lie algebra and discuss the properties of being ℚ-factorial and factorial for the normalization ��tilde; of the closure of ��. Then we consider the problem of symplectic resolutions for ��tilde;. Our main theorem says that for any nilpotent orbit �� in a semi-simple complex Lie algebra, equipped with the Kostant-Kirillov symplectic form ω, if for a resolution π:Z��tilde;, the 2-form π*(ω) defined on π−1(��) extends to a symplectic 2-form on Z, then Z is isomorphic to the cotangent bundle T *(G/P) of a projective homogeneous space, and π is the collapsing of the zero section. It proves a conjecture of Cho-Miyaoka-Shepherd-Barron in this special case. Using this theorem, we determine all varieties ��tilde; which admit such a resolution.