Preface to the second edition Preface to the first edition Introduction Part I. General Theory: 1. Preliminaries 2. Families of sheaves 3. The Grauert-Mullich Theorem 4. Moduli spaces Part II. Sheaves on Surfaces: 5. Construction methods 6. Moduli spaces on K3 surfaces 7. Restriction of sheaves to curves 8. Line bundles on the moduli space 9. Irreducibility and smoothness 10. Symplectic structures 11. Birational properties Glossary of notations References Index.

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The geometry of moduli spaces of sheaves

第12次 아시아ㆍ太平洋 矯政局長會議 參加記

13. 벼잎선충의 약제처리 방법별 방제 효과

To any finite group Γ⊂Sp(V) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ of the algebra ℂ[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of ℙ
 r
 , where r=number of conjugacy classes of symplectic reflections in $Γ$. The algebra Hκ, called a symplectic reflection algebra, is related to the coordinate ring of a Poisson deformation of the quotient singularity V/Γ. This leads to a symplectic analogue of McKay correspondence, which is most complete in case of wreath-products. If Γ is the Weyl group of a root system in a vector space ? and V=?⊕?*, then the algebras Hκ are certain ‘rational’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let $Γ=S

 n
 , the Weyl group of ?=??
 n
 . We construct a 1-parameter deformation of the Harish-Chandra homomorphism from ?(?)?, the algebra of invariant polynomial differential operators on ??
 n
 , to the algebra of S

 n
 -invariant differential operators with rational coefficients on the space ℂ
 n
 of diagonal matrices. The second order Laplacian on ? goes, under the deformed homomorphism, to the Calogero-Moser differential operator on ℂ
 n
 , with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: ?(?)? ↠ spherical subalgebra in Hκ, where Hκ is the symplectic reflection algebra associated to the group Γ=S

 n
 . This way, the deformed Harish-Chandra homomorphism becomes nothing but a description of the spherical subalgebra in terms of ‘quantum’ Hamiltonian reduction. In the ‘classical’ limit κ→∞, our construction gives an isomorphism between the spherical subalgebra in H∞ and the coordinate ring of the Calogero-Moser space. We prove that all simple H∞-modules have dimension n!, and are parametrised by points of the Calogero-Moser space. The family of these modules forms a distinguished vector bundle on the Calogero-Moser space, whose fibers carry the regular representation of S

 n
 . Moreover, we prove that the algebra $H∞ is isomorphic to the endomorphism algebra of that vector bundle.

Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism

We briefly review the formal picture in which a Calabi-Yau $n$-fold is the complex analogue of an oriented real $n$-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a Calabi-Yau 3-fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of \cite{LT}, \cite{BF} in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in $\Pee^3$, and Donaldson-- and Gromov-Witten-- like invariants of Fano 3-folds. It also allows us to define the holomorphic Casson invariant of a Calabi-Yau 3-fold $X$, prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general $K3$ fibration $X$, enabling us to compute the invariant for some ranks and Chern classes, and equate it to Gromov-Witten invariants of the ``Mukai-dual'' 3-fold for others. As an example the invariant is shown to distinguish Gross' diffeomorphic 3-folds. Finally the Mukai-dual 3-fold is shown to be Calabi-Yau and its cohomology is related to that of $X$.

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A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on $K3$ fibrations

We give an algorithmic description of the action of the Chern classes of tautological bundles on the cohomology of Hilbert schemes of points on a smooth surface within the framework of Nakajima's oscillator algebra. This leads to an identification of the cohomology ring of Hilb
 n
 (A2) with a ring of explicitly given differential operators on a Fock space. We end with the computation of the top Segre classes of tautological bundles associated to line bundles on Hilb
 n
 up to n=7, extending computations of Severi, LeBarz, Tikhomirov and Troshina and give a conjecture for the generating series.

Chern classes of tautological sheaves on Hilbert schemes of points on surfaces

Let S be a smooth projective surface and S [n] the Hilbert scheme of zerodimensional subschemes of S of length n. We proof that the class of S [n] in the complex cobordism ring depends only on the class of the surface itself. Moreover, we compute the cohomology and holomorphic Euler characteristics of certain tautological sheaves on S [n] and prove results on the general structure of certain integrals over polynomials in Chern classes of tautological sheaves. Let S be a smooth projective surface over the field of complex numbers. For a nonnegative integer n let S [n] denote the Hilbert scheme parameterizing zerodimensional subschemes of length n. By a well-known result of Fogarty [10] the scheme S [n] is smooth and projective of dimension 2n, and is irreducible if S is irreducible. Let = U ⊗ Q be the complex cobordism ring with rational coefficients. Milnor [20] showed that is a polynomial ring freely generated by the cobordism classes [CPi] for i ∈ N. For a smooth and projective complex surface we define

On the Cobordism Class of the Hilbert Scheme of a Surface

We give an algorithmic description of the action of the Chern classes of tautological bundles on the cohomology of Hilbert schemes of points on surfaces within the framework of Nakajima's oscillator algebra. This leads to an identification of the cohomology ring of Hilbert schemes of the affine plane with a ring of differential operators on a Fock space. We end with the computation of the top Segre classes of tautological bundles associated to line bundles on Hilb^n up to n=7, and give a conjecture for the generating series.

Chern Classes of Tautological Sheaves on Hilbert Schemes

Moduli spaces of semistable sheaves on a K3 or abelian surface with respect to a general ample divisor are shown to be locally factorial, with the exception of symmetric products of a K3 or abelian surface and the class of moduli spaces found by O’Grady. Consequently, since singular moduli space that do not belong to these exceptional cases have singularities in codimension ≥4 they do no admit projective symplectic resolutions.

Manfred Lehn

Papers

The geometry of moduli spaces of sheaves

Chern classes of tautological sheaves on Hilbert schemes of points on surfaces

On the Cobordism Class of the Hilbert Scheme of a Surface

Chern Classes of Tautological Sheaves on Hilbert Schemes

Singular symplectic moduli spaces