In this chapter we deal with another important class of varieties — the nilpotent orbit closures of the adjoint action of a simple algebraic group on its Lie algebra. These varieties play an important role in representation theory. All such orbit closures have desingularizations which are total spaces of vector bundles over homogeneous spaces. We describe the applications of the geometric method. The vector bundles involved in the construction of these desingularizations are more complicated than in the case of determinantal varieties. The explicit formula for the terms of complexes F(ℒ)• is not known in general. Still, one can prove some interesting results.

The first two sections of the chapter are devoted to the nilpotent orbit closures for the general linear group.

In section 8.1 we describe the desingularizations of these orbit closures explicitly. We apply theorems from chapter 5 to prove that all orbit closures are normal, are Gorenstein, and have rational singularities. We also describe the combinatorial way of estimating the terms of the complexes F• in this case.

This method is then used in section 8.2 to describe the generators of the defining ideals of nilpotent orbit closures.

In section 8.3 we treat the case of general simple groups.