By Hiraku Nakajima

The Hilbert scheme $X^{[n]}$ of a floor $X$ describes collections of $n$ (not unavoidably targeted) issues on $X$. extra accurately, it's the moduli area for $0$-dimensional subschemes of $X$ of size $n$. lately it used to be discovered that Hilbert schemes initially studied in algebraic geometry are heavily with regards to a number of branches of arithmetic, akin to singularities, symplectic geometry, illustration theory-even theoretical physics. The dialogue within the ebook displays this option of Hilbert schemes. for instance, a building of the illustration of the countless dimensional Heisenberg algebra (i.e., Fock area) is gifted. This illustration has been studied broadly within the literature in reference to affine Lie algebras, conformal box conception, and so forth. despite the fact that, the development offered during this quantity is totally specified and offers the unexplored hyperlink among geometry and illustration conception. The e-book deals a pleasant survey of present advancements during this quickly turning out to be topic. it's appropriate as a textual content on the complex graduate point

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**Example text**

B1 −z1 b = ( −(B2 −z2 ) B1 −z1 i ) ⊕ a = B2 −z2 j W ⊗ OC2 The next lemma completes the proof of the theorem. 7. 1 is given, and deﬁne endomorphisms a, b as above. Then (1) ker a = 0. (2) b is surjective if and only if there exists no proper subspace S V such that Bα (S) ⊂ S, (α = 1, 2) and im i ⊂ S. Proof. It is easy to see that a is injective and b surjective on ∞ . Hence it is suﬃcient to prove the lemma on C2 . On the ﬁber at z = (z1 , z2 ) ∈ C2 , a and b induce the homomorphisms σ τ z z V −→ V ⊕ V ⊕ W −→ V, ⎛ ⎞ B1 − z1 σz = ⎝B2 − z2 ⎠ , τz = −(B2 − z2 ) B1 − z1 j (1) If σz is not injective, there exists v = 0 ∈ V such that ⎧ ⎪ ⎨B1 v = z1 v B2 v = z2 v .

Then √ ωC (Iv, w) = g(JIv, w) + −1g(KIv, w) √ √ = −1(g(Jv, w) + −1g(Kv, w)) √ = −1ωC (v, w). This means that ωC is of type (2, 0). It is clear that dωC = 0 and ωC are nondegenerate. Hence ωC is a holomorphic symplectic form. (3) One of the advantages of the hyper-K¨ahler structure is that one can identify two diﬀerent complex manifolds with one hyper-K¨ ahler manifold. Namely, a hyper-K¨ ahler manifold (X, g, I, J, K) gives two complex manifolds (X, I) and (X, J), which are not isomorphic in general.

24 2. FRAMED MODULI SPACE OF TORSION FREE SHEAVES ON P2 Conversely, suppose there exists z ∈ C2 such that τz is not surjective, then t τz is not injective. Hence there exists φ = 0 ∈ ker t τz . If we take S = ker φ V , then we have Bα (S) ⊂ S and im i ⊂ S. 2. Rank 1 case As remarked before, we can identify the framed moduli space M(1, n) of rank [n] 1 torsion free sheaves with (C2 ) . 9. 1. 8. Assume r = 1. 1 is given. Then (1) j = 0. 9. We need the following elementary lemma. 9. 1, but not necessarily (ii).