Poincare polynomial of

Let us apply the above result to our situation. Recall that we have two different descriptions of The first one, given in Theorem 1.14, is simple, but does not give 

This induces the action on In terms of the ADHM data, this action is given by 

Note that we are using the description 3.23, otherwise 5q, P are not well-defined. 

For generic e (0 e C 1), the critical points of / coincide with the fixed points of T -action. Moreover f —oo,c]) is compact, since f c together with the equation [Bi, B ] + [B2, 53] -f iO) = dx implies a bound on Bi p+ B2 p-f i p, Therefore we can apply Morse theory to our situation. 

we shall identify the fixed point set. By definition, [(Bi, S2, 0] is a fixed 

Let us apply the above result to our situation. Recall that we have two different descriptions of (C2). The first one, given in Theorem 1.14, is simple, but does not give a Kahler metric. The second one, given in Theorem 3.23, defines a Kahler metric, but is complex because of the equation Hi = y/ ldx- We use both descriptions properly according to situations. 

we shall identify the fixed point set. By definition, [(Bi, B2, )] 6 (C2) is a fixed point if and only if there exists a homomorphism A T2 — U(V ) satisfying the following conditions  

From the conditions (5.3), the only components of Bi, B2 and i which might survive are 

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The Poincare polynomial of a symmetric product, Proc. Camb. Phil. Soc. 58 (1962), 563-568. 

In this chapter we shall calculate the Poincare polynomial of This was hrst accom-... 

In this chapter, we shall prove the following formula for Poincare polynomial of Hilbert scheme of n-points on a quasi-projective nonsingular surface X ... 

Theorem 6.1. The generating function of the Poincare polynomials of the Hilbert scheme parameterizing n-points in X, is given by... 

In this chapter we shall calculate the Poincare polynomial of (C2). This was first accomplished by Ellingsrud and Strpmme [14]. They have used the Bialynicki-Birula decomposition associated with the natural torus action on (C2), and then compute the Poincare polynomial using the Weil conjecture. Our approach is essentially the same, but we use Morse theory instead of the Weil conjecture. 

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