Geometric Invariant Theory, Holomorphic Vector Bundles and the Harder-Narasimhan Filtration (SpringerBriefs in Mathematics)

This book introduces key topics on Geometric Invariant Theory, a technique to obtaining quotients in algebraic geometry with a good set of properties, through various examples. It starts from the classical Hilbert classification of binary forms, advancing to the construction of the moduli space of semistable holomorphic vector bundles, and to Hitchin’s theory on Higgs bundles. The relationship between the notion of stability between algebraic, differential and symplectic geometry settings is also covered. Unstable objects in moduli problems -- a result of the construction of moduli spaces -- get specific attention in this work. The notion of the Harder-Narasimhan filtration as a tool to handle them, and its relationship with GIT quotients, provide instigating new calculations in several problems. Applications include a survey of research results on correspondences between Harder-Narasimhan filtrations with the GIT picture and stratifications of the moduli space of Higgs bundles. Graduate students and researchers who want to approach Geometric Invariant Theory in moduli constructions can greatly benefit from this reading, whose key prerequisites are general courses on algebraic geometry and differential geometry. Preface Contents List of Symbols 1 Introduction 2 Preliminaries 2.1 Algebraic Varieties and Groups 2.1.1 Algebraic Varieties 2.1.2 Group Actions 2.2 Sheaf Theory and Schemes 2.2.1 Sheaves and Cohomology 2.2.2 Schemes 2.3 Holomorphic Vector Bundles 2.3.1 Vector Bundles 2.3.2 Line Bundles 2.3.3 Divisors 3 Geometric Invariant Theory 3.1 Quotients and the Notion of Stability 3.2 Hilbert–Mumford Criterion 3.3 Symplectic Stability 3.4 Examples 3.5 Maximal Unstability 4 Moduli Space of Vector Bundles 4.1 GIT Construction of the Moduli Space 4.2 Harder-Narasimhan Filtration 4.3 Other Constructions of the Moduli Space of Vector Bundles 4.3.1 Analytical Construction of the Moduli Space of Vector Bundles 4.3.2 Moduli Space of Representations of the Fundamental Group 4.4 Moduli Space of Higgs Bundles 4.4.1 Hitchin's Construction 4.4.2 Higher Rank and Dimensional Higgs Bundles 5 Unstability Correspondence 5.1 Correspondence for Vector Bundles 5.1.1 Main Correspondence: Holomorphic Vector Bundles 5.1.2 Other Correspondences for Augmented Bundles 5.2 Quiver Representations 5.3 (G,h)-Constellations 6 Stratifications on the Moduli Space of Higgs Bundles 6.1 Shatz Stratification 6.2 C-Action and Białynicki-Birula Stratification 6.3 Stratifications in Rank Three 6.3.1 Sketch of the Proof of Theorem 6.1 6.3.2 Relationship Between Shatz and Biłynicki-Birula Stratifications for Rank Three Higgs Bundles 6.4 Homotopy Groups References Index