Hilbert scheme of points on the plane

The symplectic form on the moduli space A4r,ci,c2 a stable sheaf on X, then the tangent space of A r,ci,c2 Ext ( , ), and the symplectic structure is dehned by 

In the last part, we use the fact that X is a K3 surface or an abelian surface. For the proof of the non-degeneracy and the closedness, we refer to [56]. In Chapter 3, we shall show that the framed moduli space of torsion free sheaves on has a holomorphic symplectic form. 

The relation between the two theorems is as follows. Let be a rank I torsion free sheaf on X, then its double dual is locally free. There exists a natural inclusion 8 and the cokernel dehnes an element in If we associate 8 with 8 8 /8)  

In Chapter 3, we shall explain the hyper-Kahler structure, which is closely related with the holomorphic symplectic structure. Actually, a hyper-Kahler manifold has a holomorphic symplectic structure as we shall explain in Chapter 3. The converse is also true if X is compact. 

Proposition 1.12. Let X be a eompaet Kdhler manifold whieh admits a holomorphic sympleetie strueture. Then X has a hyper-Kahler metrie. 

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On a cell decomposition of the Hilbert scheme of points in the plane, Invent. Math 91 (1988), 365-370. 

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