Donaldson type invariants for algebraic surfaces : transition of moduli stacks / Takuro Mochizuki.

Format:

Book

Author/Creator:

Mochizuki, Takuro, 1972-

Series:

Lecture notes in mathematics (Springer-Verlag) ; 1972.

Lecture notes in mathematics, 0075-8434 ; 1972

Language:

English

Subjects (All):

Surfaces, Algebraic.

Invariants.

Moduli theory.

Physical Description:

xxiii, 383 pages ; 24 cm.

Place of Publication:

Berlin : Springer, [2009]

Summary:

We are defining and studying an algebro-geometric analogue of Donaldson invariants by using moduli spaces of semistable sheaves with arbitrary ranks on a polarized projective surface. We are interested in relations among the invariants, which are natural generalizations of the "wall-crossing formula" and the "Witten conjecture" for classical Donaldson invariants. Our goal is to obtain a weaker version of these relations, by systematically using the intrinsic smoothness of moduli spaces. According to the recent excellent work of L. Goettsche, H. Nakajima and K. Yoshioka, the wall-crossing formula for Donaldson invariants of projective surfaces can be deduced from such a weaker result in the rank two case!

Contents:

1 Introduction 1

1.1 Problems 1

1.2 Construction of Invariants 3

1.2.1 Orientation and Reduced L-Bradlow Pairs 3

1.2.2 Virtual Fundamental Classes 5

1.3 Transition Formulas in a Simple Case 6

1.4 Rank Two Case 9

1.4.1 Dependence on Polarizations 9

1.4.2 Reduction to Integrals over Hilbert Schemes 11

1.5 Higher Rank Case 12

1.5.1 pg > 0 12

1.5.2 Transition Formula in the Case pg = 0 13

1.5.3 Weak Intersection Rounding Formula 15

1.6 Master Space 16

1.6.1 Master Space in the Case that a 2-Stability Condition is Satisfied 17

1.6.2 Gm-Localization Method (I) 19

1.6.3 Enhanced Master Space 20

1.6.4 Gm-Localization Method (II) 22

2 Preliminaries 25

2.1 Some Convention 25

2.1.1 Product and Projection 25

2.1.2 Vector Bundles 26

2.1.3 Coherent Sheaves on a Product 26

2.1.4 Quotient Stacks 26

2.1.5 Signature in Complexes 26

2.1.6 Filtrations and Complexes on a Curve 27

2.1.7 Virtual Vector Bundle 28

2.1.8 Compatible Diagrams 29

2.2 Geometric Invariant Theory 29

2.2.1 GIT Quotient and Algebraic Stacks 29

2.2.2 Mumford-Hilbert Criterion and Some Elementary Examples 30

2.3 Cotangent Complex 32

2.3.1 Basic Facts 32

2.3.2 Quotient Stacks 34

2.3.3 Some More Examples 39

2.4 Obstruction Theory 44

2.4.1 Definition and Fundamental Theorems 44

2.4.2 Easy Example 46

2.4.3 Locally Free Subsheaves 48

2.4.4 Filtrations of a Vector Bundle on a Curve 51

2.5 Equivariant Complexes on Deligne-Mumford Stacks with GIT Construction 55

2.5.1 Locally Free Resolution 55

2.5.2 Equivariant Representative 56

2.6 Elementary Remarks on Some Extremal Sets 57

2.6.1 Preparation for a Proof of Proposition 4.3.3 57

2.6.2 Preparation for a Proof of Proposition 4.4.4 59

2.7 Twist of Line Bundles 61

2.7.1 Construction 61

2.7.2 Weight of the Induced Action 61

3 Parabolic L-Bradlow Pairs 63

3.1 Sheaves with Some Structure and their Moduli Stacks 64

3.1.1 Orientation 64

3.1.2 Parabolic Structure 65

3.1.3 L-Bradlow Pairs and Reduced L-Bradlow Pairs 66

3.1.4 Type and Moduli Stacks 68

3.1.5 Tautological Line Bundle and Relations Among Some Moduli Stacks 69

3.2 Hilbert Polynomials 71

3.2.1 Hilbert Polynomials of Coherent Sheaves 71

3.2.2 Hilbert Polynomials of Parabolic Sheaves 71

3.2.3 Hilbert Polynomial for Parabolic L-Bradlow Pairs 72

3.2.4 Hilbert Polynomials Associated to a Type 72

3.3 Semistability 73

3.3.1 Semistability Conditions and the Associated Moduli Stacks 73

3.3.2 Harder-Narasimhan Filtration and Partial Jordan-Holder Filtration 76

3.3.3 (d, l)-Semistability 77

3.4 Some Boundedness 78

3.4.1 Fundamental Theorems 78

3.4.2 Boundedness of Semistable L-Bradlow Pairs 79

3.4.3 Boundedness of Yokogawa Family 80

3.5 1-Stability Condition and 2-Stability Condition 84

3.5.1 Parabolic Sheaf 84

3.5.2 Parabolic L-Bradlow Pair 85

3.5.3 Parabolic L-Bradlow Pair 88

3.6 Quot Schemes 89

3.6.1 Preliminary 89

3.6.2 Quotient Sheaves 89

3.6.3 Quotient Quasi-Parabolic Sheaves and Maruyama-Yokogawa Construction 90

3.6.4 Quotient L-Bradlow Pair 91

3.6.5 Quotient Reduced L-Bradlow Pair 93

3.6.6 Quotient Reduced L-Bradlow Pair 94

3.6.7 Oriented Objects 94

3.6.8 Quotient Stacks 95

4 Geometric Invariant Theory and Enhanced Master Space 97

4.1 Semistability Condition and Mumford-Hilbert Criterion 98

4.1.1 Statement 98

4.1.2 Mumford-Hilbert Criterion 101

4.1.3 A Lemma 102

4.1.4 Proof of the Claim 1 in Proposition 4.1.2 103

4.1.5 Proof of the Claim 2 of Proposition 4.1.2 104

4.1.6 Complement 105

4.2 Perturbation of Semistability Condition 106

4.2.1 Preliminary 106

4.2.2 d+-Semistability and d

Semistability 107

4.2.3 (d, l)-Semistability 109

4.3 Enhanced Master Space 113

4.3.1 Construction 113

4.3.2 Proof of Lemma 4.3.4 114

4.3.3 Proof of Lemma 4.3.5 116

4.3.4 Proof of Lemma 4.3.6, Step 1 117

4.3.5 Proof of Lemma 4.3.6, Step 2 118

4.3.6 Proof of Lemma 4.3.6, Step 3 119

4.4 Fixed Point Set of Torus Action on Enhanced Master Space 120

4.4.1 Preliminary 120

4.4.2 Statements 122

4.4.3 Step 1 122

4.4.4 Step 2 124

4.4.5 Step 3 125

4.4.6 Step 4 126

4.4.7 End of the Proof of Proposition 4.4.4 127

4.5 Enhanced Master Space in Oriented Case 128

4.5.1 Construction 128

4.5.2 Obvious Fixed Point Sets 129

4.5.3 Fixed Point Sets Associated to Decomposition Types 130

4.5.4 Statement 130

4.5.5 Ambient Stack 131

4.5.6 Fixed Point Set of the Ambient Space 131

4.5.7 Proof of Proposition 4.5.3 132

4.6 Decomposition of Exceptional Fixed Point Sets 134

4.6.1 Statement 134

4.6.2 Preliminary 136

4.6.3 Construction of the Stack S 136

4.6.4 Universal Sheaf 138

4.7 Simpler Cases 138

4.7.1 Case in Which a 2-Stability Condition is Satisfied 138

4.7.2 Oriented Reduced L-Bradlow Pairs 140

5 Obstruction Theories of Moduli Stacks and Master Spaces 145

5.1 Deformation of Torsion-Free Sheaves 146

5.1.1 Construction of a Basic Complex 146

5.1.2 The Trace-Free Part and the Diagonal Part 148

5.1.3 Preparation for Master space 150

5.1.4 Basic Complex on the Moduli Stack M (m, y) 151

5.1.5 Line Bundles 153

5.2 Relative Obstruction Theory for Orientations 154

5.2.1 Construction of a Complex 154

5.2.2 Relative Obstruction Property 155

5.3 Relative Obstruction Theory for L-Sections 156

5.3.1 Construction of a Complex 156

5.3.2 Relative Obstruction Property 158

5.3.3 Preparation for Obstruction Theory of Master Space 160

5.3.4 Preparation for Proposition 6.2.1 161

5.3.5 Another Construction 162

5.4 Relative Obstruction Theory for Reduced L-Sections 164

5.4.1 Construction of a Complex 164

5.4.2 Relative Obstruction Property 166

5.4.3 Preparation for Obstruction Theory of Master Space 168

5.4.4 Preparation for Proposition 6.2.1 170

5.5 Relative Obstruction Theory for Parabolic Structures 170

5.5.1 Construction of a Complex 170

5.5.2 Relative Obstruction Property 173

5.5.3 Decomposition into the Trace-Free Part and the Diagonal Part 173

5.6 Obstruction Theory for Moduli Stacks of Stable Objects 174

5.6.1 Relative Complexes 174

5.6.2 Construction of Complexes and Morphisms 175

5.6.3 Obstruction Theories of Quot Schemes and Moduli Stacks 176

5.6.4 Obstruction Theories of Moduli Stacks of Stable Objects 177

5.7 Obstruction Theory for Enhanced Master Spaces and Related Stacks 178

5.7.1 Enhanced Master Space 178

5.7.2 Substack M* 179

5.7.3 Moduli Stack M (m, y, [L]) 181

5.7.4 Moduli Stack M (m, y, L) 184

5.7.5 Moduli Stack Ms (y, a*, +) 184

5.7.6 Case in Which a 2-Stability Condition is Satisfied 185

5.7.7 Oriented Reduced L-Bradlow Pairs 186

5.8 Moduli Theoretic Obstruction Theory of Fixed Point Set 187

5.8.1 Statement 187

5.8.2 Moduli Stack of Split Objects with Orientations 188

5.8.3 Embedding into Moduli Stack of Non-Split Objects 190

5.8.4 Some Compatibility 194

5.8.5 Deformation 196

5.8.6 Proof of Proposition 5.8.1 198

5.8.7 Case in Which a 2-Stability Condition is Satisfied 199

5.8.8 Oriented Reduced L-Bradlow Pairs 200

5.9 Equivariant Obstruction Theory of Master Space 201

5.9.1 Statements 201

5.9.2 Gm-Equivariant Lift of Ob (M) and ob (M) 204

5.9.3 Comparison of Gm-Equivariant Structures of T-1LM 205

5.9.4 Gm-Equivariant structure of Ob(M*) 207

5.9.5 Proof of Proposition 5.9.2 208

5.9.6 Proof of Proposition 5.9.3 208

5.9.7 Case in Which a 2-Stability Condition is Satisfied 209

5.9.8 Oriented Reduced L-Bradlow Pairs 210

6 Virtual Fundamental Classes 213

6.1 Perfectness of Obstruction Theories 213

6.1.1 Moduli Stacks of Semistable Objects 213

6.1.2 Master Spaces and Some Related Stacks 214

6.1.3 Vanishing of Some Cohomology Groups 216

6.1.4 Proof of

the Propositions in Subsection 6.1.1 219

6.1.5 Expected Dimension 220

6.2 Comparison of Oriented Reduced Case and Unoriented Unreduced Case 221

6.2.1 Statements 221

6.2.2 Proof of Proposition 6.2.1 223

6.3 Rank One Case 224

6.3.1 Moduli of L-Abelian Pairs 224

6.3.2 Involutivity and Relation with Seiberg-Witten Invariants 227

6.3.3 Parabolic Hilbert Schemes 228

6.3.4 Splitting into Moduli of Abelian Pairs and Parabolic Hilbert Schemes 231

6.3.5 Morphism to Moduli of Abelian Pairs 232

6.3.6 Morphism to Parabolic Hilbert Scheme 237

6.3.7 Mixed Case 239

6.3.8 Proof of Proposition 6.3.8 240

6.4 Bradlow Perturbation 242

6.4.1 Statements 242

6.4.2 Construction of B 244

6.4.3 Compatibility of the Obstruction Theories 244

6.4.4 Ambient Smooth Stack 248

6.5 Comparison with Full Flag Bundles 250

6.6 Parabolic Perturbation 252

6.6.1 Statement 252

6.6.2 Construction of a Stack B with an Obstruction Theory 254

6.6.3 Compatibility of the Obstruction Theories 257

6.6.4 Smooth Ambient Stack 260

6.6.5 Proof of Proposition 6.6.1 261

7 Invariants 263

7.1 Preliminary 263

7.1.1 Ring Rl 264

7.1.2 Homomorphisms 266

7.1.3 Evaluation 267

7.1.4 Equivariant Case 271

7.1.5 Ring RCH 273

7.1.6 Equivariant Euler Class 274

7.1.7 Twist by Line Bundle 275

7.2 Transition Formulas in Simpler Cases 276

7.2.1 Basic Case 276

7.2.2 Twist with the Euler Class of the Relative Tangent Bundle 279

7.2.3 Oriented Reduced L-Bradlow Pairs 282

7.3 Invariants 286

7.3.1 Construction 286

7.3.2 Easy Property 289

7.3.3 Integrals over Mss (y, a*, +) 290

7.3.4 Another Expression 290

7.3.5 Deformation Invariance 291

7.4 Rank Two Case 293

7.4.1 Reduction to the Sum of Integrals over the Products of Hilbert Schemes 293

7.4.2 Dependence on Polarizations 296

7.4.3 Proof of Theorem 7.4.3 (I) 298

7.4.4 Proof of Theorem 7.4.3 (II) 300

7.5 Higher Rank Case (pg > 0) 303

7.5.1 Transition Formula in the Case pg > 0 303

7.5.2 Reduction to the Sum of Integrals over the Products of Hilbert Schemes 306

7.5.3 Independence from Polarizations in the Case pg > 0 309

7.6 Transition Formula (pg = 0) 311

7.6.1 Statement 311

7.6.2 Step 1 313

7.6.3 Step 2 315

7.6.4 Step 3 318

7.7 Weak Wall Crossing Formula 319

7.7.1 Statement 319

7.7.2 Proof of Theorem 7.7.1 321

7.7.3 Weak Wall Crossing Formula in the Rank 3 Case 323

7.7.4 Weak Intersection Rounding Formula in the Rank 3 Case 325

7.7.5 Transition for a Critical Parabolic Weight 330

7.8 Weak Intersection Rounding Formula 331

7.8.1 Preliminary 331

7.8.2 Statement 335

7.8.3 Preparation from Combinatorics 335

7.8.4 Proof of Theorem 7.8.2 337.

Notes:

Includes bibliographical references (pages 341-345) and index.

ISBN:

9783540939122

3540939121

OCLC:

297148418