Equivariant Enumerative Geometry and Donaldson-Thomas Theory

Abstract

Chapter 1. Introduction 1.1 A bit of history 1.2 Donaldson-Thomas theory 1.3 Pandharipande-Thomas theory of local curves 1.4 Higher rank K-theoretic Donaldson-Thomas theory of points 1.5 Donaldson-Thomas theory of Calabi-Yau 4-folds 1.6 Nested Quot schemes 1.7 Organization of the thesis 1.8 Cross-reference 1.9 Conventions I Donaldson-Thomas theory in dimension three Chapter 2. Localization in Equivariant Enumerative Geometry 2.1 Une promenade equivariante 2.2 Une promenade virtuelle Chapter 3. Double nested Hilbert schemes and the local PT theory of curves 3.1 Introduction 3.2 Double nested Hilbert schemes 3.3 Moduli space of stable pairs 3.4 K-theory class of the perfect obstruction theory 3.5 Universality 3.6 The leading term 3.7 Toric computations 3.8 Summing the theory up 3.9 K-theoretic refinement 3.10 Appendix: The combinatorial identities Chapter 4. Higher rank K-theoretic Donaldson-Thomas theory of points 4.1 Introduction 4.2 The local Quot scheme: critical and equivariant structure 4.3 Invariants attached to the local Quot scheme 4.4 Higher rank vertex on the local Quot scheme 4.5 The higher rank K-theoretic DT partition function 8 CONTENTS 4.6 The higher rank cohomological DT partition function 4.7 Elliptic Donaldson-Thomas invariants 4.8 Higher rank DT invariants of compact toric 3-folds II Donaldson-Thomas theory in dimension four Chapter 5. Donaldson-Thomas theory of Calabi-Yau 4-folds 5.1 Taking square roots 5.2 Oh-Thomas virtual structures Chapter 6. K-theoretic DT/PT for toric Calabi-Yau 4-folds 6.1 Introduction 6.2 K-theoretic vertex formalism 6.3 Limits of K-theoretic conjecture 6.4 Hilbert schemes of points 6.5 Local resolved conifold 6.6 Appendix: Combinatorial sign rule Chapter 7. Stable pair invariants of local Calabi-Yau 4-folds 7.1 Introduction 7.2 Moduli spaces 7.3 Invariants 7.4 Appendix: Tables III Nested Quot schemes Chapter 8. On the motive of the nested Quot scheme of points on a curve 8.1 Introduction 8.2 Nested Quot schemes of points 8.3 Tangent space and smoothness in the case of curves 8.4 Bia lynicki-Birula decomposition 8.5 The motive of the nested Quot scheme on a curve 8.6 Appendix: Motivic exponentials Chapter 9. A note on the smoothness of the nested Quot scheme of points 9.1 Introduction 9.2 Properties of the moduli space 9.3 Proof of the theorem Samenvatting Acknowledgements Curriculum vitae Bibliography