Hyper-Kahler quotients

In this section, we shall see that the quotient in Theorem 3.23 is, in fact, a hyper-Kahler quotient. Let us review on the hyper-Kahler structure and hyper-Kahler quotients briefly. The interested reader should read Hitchin s book [38]. First, we recall the definition of Kahler manifolds. 

Definition 3.25. Let X be a 2n-dimensional manifold. A Kahler structure of X is a pair of a Riemmnian metric g and an almost complex structure / which satisfies the following conditions  

It is well-known (see e.g. [48, Chapter IX, Theorem 4.3]) that the above conditions (3.25.2), (3.25.3) (under (3.25.1)) are equivalent to requiring that I is parallel with respect to the Levi-Civita connection of g, i.e. V/ = 0. This is also equivalent to the condition  

The hyper-Kahler structure is a quaternionic version of the Kahler structure. However, there is no good definition of the integrability (i.e. the existence of local charts) for the almost hyper-complex structure. Hence we generalize the second equivalent dehnition explained in above. 

Definition 3.26. Let X be a 4n-dimensional manifold. A hyper-Kahler structure of X consists of a Riemmanian metric g and a triple of almost complex structures /, J, K which satisfy the following conditions  

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We call this quotient space a hyper-Kahler quotient. 

Example 3.40. The moduli space of instantons on a 4-dimensional hyper-Kahler manifold X can be considered as a hyper-Kahler quotient. Let us take a smooth vector bundle E over X with a Hermitian metric. Let us denote the space of metric connections on E by A. Its tangent space at A G M can be identified with... 

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... 

Suppose hrst that S = C. In this case there exists a natural Kahler metric on (T C)[ 1 since it is obtained by the hyper-Kahler quotient construction. In the notation of the ADHM description, we may take /([(Hi, H2, )]) = 52 P As shown in Proposition 5.13, the critical value is given by f S C) = ... 

When X = C2, X can be identified with the set of GLJl(C)-orbits of (Hi, B2, i) where Bi, B2 are commuting n x n-matrices and i is a cyclic vector (Theorem 1.14). Many properties of (C2) are derived from this description. In Chapter 3, we shall regard the description as a geometric invariant theory quotient and a hyper-Kahler quotient. This description is very similar to the definition of quiver varieties which were studied in [62]. 

The purpose of this chapter is to construct a hyper-Kahler metric on the Hilbert scheme (C2) of n points on C2. This will be accomplished by identifying (C2) with a hyper-Kahler quotient (see Theorem 3.23). 

P.B. Kronheimer, The construction of ALE spaces as a hyper-Kahler quotients, J. Differential Geom. 29 (1989) 665-683. 

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