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Moduli Spaces

Book Description

Moduli theory is the study of how objects, typically in algebraic geometry but sometimes in other areas of mathematics, vary in families and is fundamental to an understanding of the objects themselves. First formalised in the 1960s, it represents a significant topic of modern mathematical research with strong connections to many areas of mathematics (including geometry, topology and number theory) and other disciplines such as theoretical physics. This book, which arose from a programme at the Isaac Newton Institute in Cambridge, is an ideal way for graduate students and more experienced researchers to become acquainted with the wealth of ideas and problems in moduli theory and related areas. The reader will find articles on both fundamental material and cutting-edge research topics, such as: algebraic stacks; BPS states and the P = W conjecture; stability conditions; derived differential geometry; and counting curves in algebraic varieties, all written by leading experts.

Table of Contents

  1. Cover
  2. Series Page
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface
  7. List of contributors
  8. 1. Introduction to algebraic stacks
    1. Introduction
    2. 1.1 Topological stacks: triangles
      1. 1.1.1 Families and their symmetry groupoids
      2. 1.1.2 Continuous families
      3. 1.1.3 Classification
      4. 1.1.4 Scalene triangles
      5. 1.1.5 Isosceles triangles
      6. 1.1.6 Equilateral triangles
      7. 1.1.7 Oriented triangles
      8. 1.1.8 Stacks
      9. 1.1.9 Versal families
      10. 1.1.10 Degenerate triangles
      11. 1.1.11 Change of versal family
      12. 1.1.12 Weierstrass compactification
    3. 1.2 Formalism
      1. 1.2.1 Objects in continuous families: categories fibered in groupoids
      2. 1.2.2 Families characterized locally: prestacks
      3. 1.2.3 Families which can be glued: stacks
      4. 1.2.4 Topological stacks
      5. 1.2.5 Deligne–Mumford topological stacks
      6. 1.2.6 Lattices up to homothety
      7. 1.2.7 Fundamental groups of topological stacks
    4. 1.3 Algebraic stacks
      1. 1.3.1 Groupoid fibrations
      2. 1.3.2 Prestacks
      3. 1.3.3 Algebraic stacks
      4. 1.3.4 The coarse moduli space
      5. 1.3.5 Bundles on stacks
      6. 1.3.6 Stacky curves: the Riemann–Roch theorem
  9. 2. BPS states and the P = W conjecture
    1. 2.1 Introduction
    2. 2.2 Hausel–Rodriguez-Villegas formula and P = W
      1. 2.2.1 Hausel–Rodriguez-Villegas formula
      2. 2.2.2 Hitchin system and P = W
    3. 2.3 Refined stable pair invariants of local curves
      1. 2.3.1 TQFT formalism
      2. 2.3.2 Refined invariants from instanton sums
    4. 2.4 HRV formula as a refined GV expansion
  10. 3. Representations of surface groups and Higgs bundles
    1. 3.1 Introduction
    2. 3.2 Lecture 1: Character varieties for surface groups and harmonic maps
      1. 3.2.1 Surface group representations and character varieties
      2. 3.2.2 Review of connections and curvature in principal bundles
      3. 3.2.3 Surface group representations and flat bundles
      4. 3.2.4 Flat bundles and gauge equivalence
      5. 3.2.5 Harmonic metrics in flat bundles
      6. 3.2.6 The Corlette–Donaldson theorem
    3. 3.3 Lecture 2: G-Higgs bundles and the Hitchin–Kobayashi correspondence
      1. 3.3.1 Lie theoretic preliminaries
      2. 3.3.2 The Hitchin equations
      3. 3.3.3 G-Higgs bundles, stability and the Hitchin–Kobayashi correspondence
      4. 3.3.4 The Hitchin map
      5. 3.3.5 The moduli space of SU(p, q)- Higgs bundles
    4. 3.4 Lecture 3: Morse–Bott theory of the moduli space of G-Higgs bundles
      1. 3.4.1 Simple and infinitesimally simple G-Higgs bundles
      2. 3.4.2 Deformation theory of G-Higgs bundles
      3. 3.4.3 The C[sup(*)]-action and topology of moduli spaces
      4. 3.4.4 Calculation of Morse indices
      5. 3.4.5 The moduli space of Sp(2n, R)-Higgs bundles
  11. 4. Introduction to stability conditions
    1. 4.1 Torsion theories and t-structures
      1. 4.1.1 μ-stability on curves (and surfaces): recollections
      2. 4.1.2 Torsion theories in abelian categories
      3. 4.1.3 t-structures on triangulated categories
      4. 4.1.4 Torsion theories versus t-structures
    2. 4.2 Stability conditions: definition and examples
      1. 4.2.1 Slicings
      2. 4.2.2 Stability conditions
      3. 4.2.3 Aut(D)-action and GL[sup(+)](2, R)-action
      4. 4.2.4 Stability conditions on curves
    3. 4.3 Stability conditions on surfaces
      1. 4.3.1 Classification of hearts
      2. 4.3.2 Construction of hearts
    4. 4.4 The topological space of stability conditions
      1. 4.4.1 Topology of Slice(D)
      2. 4.4.2 Topology of Stab(D)
      3. 4.4.3 Main result
    5. 4.5 Stability conditions on K3 surfaces
      1. 4.5.1 Main theorem and conjecture
      2. 4.5.2 Autoequivalences
      3. 4.5.3 Building up Stab(X)[sup(º)]
      4. 4.5.4 Moduli space rephrasing
    6. 4.6 Further results
      1. 4.6.1 Non-compact cases
      2. 4.6.2 Compact cases
  12. 5. An introduction to d-manifolds and derived differential geometry
    1. 5.1 Introduction
    2. 5.2 C[sup(∞)]-rings and C[sup(∞)]-schemes
      1. 5.2.1 C[sup(∞)]-rings
      2. 5.2.2 C[sup(∞)]-schemes
      3. 5.2.3 Modules over C[sup(∞)]-rings, and cotangent modules
      4. 5.2.4 Quasicoherent sheaves on C[sup(∞)]-schemes
    3. 5.3 The 2-category of d-spaces
      1. 5.3.1 The definition of d-spaces
      2. 5.3.2 Gluing d-spaces by equivalences
      3. 5.3.3 Fibre products in dSpa
    4. 5.4 The 2-category of d-manifolds
      1. 5.4.1 The definition of d-manifolds
      2. 5.4.2 ‘Standard model’ d-manifolds, 1- and 2-morphisms
      3. 5.4.3 The 2-category of virtual vector bundles
      4. 5.4.4 Equivalences in dMan, and gluing by equivalences
      5. 5.4.5 Submersions, immersions and embeddings
      6. 5.4.6 D-transversality and fibre products
      7. 5.4.7 Embedding d-manifolds into manifolds
      8. 5.4.8 Orientations on d-manifolds
      9. 5.4.9 D-manifolds with boundary and corners, d-orbifolds
      10. 5.4.10 D-manifold bordism, and virtual cycles
      11. 5.4.11 Relation to other classes of spaces in mathematics
    5. A Basics of 2-categories
  13. 6. 13/2 ways of counting curves
    1. 0 Introduction
    2. ½ Naive counting of curves
    3. 1½ Gromov–Witten theory
    4. 2½ Gopakumar–Vafa / BPS invariants
    5. 3½ Donaldson–Thomas theory
    6. 4½ Stable pairs
    7. 5½ Stable unramified maps
    8. 6½ Stable quotients
    9. Appendix: Virtual classes