Contents

Preface ix

Introduction 1

Chapter 1. Hilbert scheme of points 5

1.1. General Results on the Hilbert scheme 5

1.2. Hilbert scheme of points on the plane 7

1.3. Hilbert scheme of points on a surface 12

1.4. Symplectic structure 13

1.5. The Douady space 15

Chapter 2. Framed moduli space of torsion free sheaves on P2 17

2.1. Monad 18

2.2. Rank 1 case 24

Chapter 3. Hyper-K¨ ahler metric on

(C2)[n]

29

3.1. Geometric invariant theory and the moment map 29

3.2. Hyper-K¨ ahler quotients 37

Chapter 4. Resolution of simple singularities 47

4.1. General Statement 47

4.2. Dynkin diagrams 49

4.3. A geometric realization of the McKay correspondence 52

Chapter 5. Poincar´e polynomials of the Hilbert schemes (1) 59

5.1. Perfectness of the Morse function arising from the moment map 59

5.2. Poincar´ e polynomial of

(C2)[n]

63

Chapter 6. Poincar´ e polynomials of Hilbert schemes (2) 73

6.1. Results on intersection cohomology 73

6.2. Proof of the formula 75

Chapter 7. Hilbert scheme on the cotangent bundle of a Riemann surface 79

7.1. Morse theory on holomorphic symplectic manifolds 79

7.2. Hilbert scheme of T ∗Σ

7.3. Analogy with the moduli space of Higgs bundles 85

Chapter 8. Homology group of the Hilbert schemes and the Heisenberg algebra 89

8.1. Heisenberg algebra and Clifford algebra 89

8.2. Correspondences 91

8.3. Main construction 93

8.4. Proof of Theorem 8.13 96

vii

Preface ix

Introduction 1

Chapter 1. Hilbert scheme of points 5

1.1. General Results on the Hilbert scheme 5

1.2. Hilbert scheme of points on the plane 7

1.3. Hilbert scheme of points on a surface 12

1.4. Symplectic structure 13

1.5. The Douady space 15

Chapter 2. Framed moduli space of torsion free sheaves on P2 17

2.1. Monad 18

2.2. Rank 1 case 24

Chapter 3. Hyper-K¨ ahler metric on

(C2)[n]

29

3.1. Geometric invariant theory and the moment map 29

3.2. Hyper-K¨ ahler quotients 37

Chapter 4. Resolution of simple singularities 47

4.1. General Statement 47

4.2. Dynkin diagrams 49

4.3. A geometric realization of the McKay correspondence 52

Chapter 5. Poincar´e polynomials of the Hilbert schemes (1) 59

5.1. Perfectness of the Morse function arising from the moment map 59

5.2. Poincar´ e polynomial of

(C2)[n]

63

Chapter 6. Poincar´ e polynomials of Hilbert schemes (2) 73

6.1. Results on intersection cohomology 73

6.2. Proof of the formula 75

Chapter 7. Hilbert scheme on the cotangent bundle of a Riemann surface 79

7.1. Morse theory on holomorphic symplectic manifolds 79

7.2. Hilbert scheme of T ∗Σ

7.3. Analogy with the moduli space of Higgs bundles 85

Chapter 8. Homology group of the Hilbert schemes and the Heisenberg algebra 89

8.1. Heisenberg algebra and Clifford algebra 89

8.2. Correspondences 91

8.3. Main construction 93

8.4. Proof of Theorem 8.13 96

vii