Tautological vector bundles

Conversely, starting from this description, Kronheimer constructed resolutions of the simple singularities [49] as follows. In order to explain the hyper-Kahler structures, we use the description given in Theorem 2.1 rather than that in Theorem 1.14. Let M = Hom(l/, Q 0 F) Hom(VF, V) Hom(F, W). Then its F-hxed point is 

The hermitian metric and the quaternion module structure on M descends to Mp. In particular, M is a hyper-Kahler manifold. There is a natural action on M of a Lie group Ur(F) = rifcU(Ffc). This action preserves the hyper-Kahler structure. The corresponding hyper-Kahler moment map is p o o where i is the inclusion M C M, /r is the hyper-Kahler moment map for U(F)-action on M, and p is the orthogonal projection to 0 u Vk) in u(F). We denote this hyper-Kahler moment map also by p = (/ri, /T2, / s)- This increases the flexibility of the choice of parameters. Take = (Co Cn Cn) ( = 1) 2, 3) such that (I is a scalar matrix in u(14)- Then we can consider a hyper-Kahler quotient 

This is the Kronheimer s construction. Since the hyper-Kahler manifolds which are obtained as the F-fixed component of the Hilbert scheme (or its deformation) satisfy (o = 

Many properties of simple singularities can be studied by using the description given in previous sections. Since these have been discussed in other papers (e.g. see [49, 50, 66]), we shall explain only briefly. 

Here the superscript indicates the factor from which the tautological homomorphism is pulled back. For example, B is a pull-back of Bi by pi, hence is a section of Hom(pjV,PiV). 

---

Now we restrict everything to the T-fixed component X. We restrict the tautological vector bundle V to A, which is still denoted by V. It has a structure of a T-module and decomposes into irreducibles ... 

Moreover, (4.11) gives an interesting interpretation of the McKay correspondence. Irreducible summands Vk of the tautological vector bundle V have following properties ... 

---