Spin/Pin-structures on vector bundles have long featured prominently in differential geometry, in particular providing part of the foundation for the original proof of the renowned Atiyah–Singer Index Theory. More recently, they have underpinned the symplectic topology foundations of the so-called real sector of the mirror symmetry of string theory.
This semi-expository three-part monograph provides an accessible introduction to Spin- and Pin-structures in general, demonstrates their role in the orientability considerations in symplectic topology, and presents their applications in enumerative geometry.
Part I contains a systematic treatment of Spin/Pin-structures from different topological perspectives and may be suitable for an advanced undergraduate reading seminar. This leads to Part II, which systematically studies orientability problems for the determinants of real Cauchy–Riemann operators on vector bundles. Part III introduces enumerative geometry of curves in complex projective varieties and in symplectic manifolds, demonstrating some applications of the first two parts in the process. Two appendices review the Čech cohomology perspective on fiber bundles and Lie group covering spaces.
Readership: Graduate students preparing for research in geometry and topology; active researchers in search of specific references on Spin/Pin-structures and orientations of determinants of real Cauchy–Riemann operators; Part I and some of Part III can be used for an advanced undergraduate reading course or seminar.
Author(s): Xujia Chen, Aleksey Zinger
Publisher: World Scientific Publishing Company
Year: 2023
Language: English
Pages: 466
Contents
Preface
About the Authors
Part I: Spin- and Pin-Structures
1. Main Results and Examples of Part I
1.1 Definitions and Main Theorem
1.2 Properties of Spin- and Pin-Structures
1.3 Basic Examples
1.4 Further Examples
2. The Lie Groups Spin(n) and Pin±(n)
2.1 The Groups SO(n) and Spin(n)
2.2 The groups O(n) and Pin± (n)
3. Proof of Theorem 1.4(1): Classical Perspective
3.1 Spin/Pin-Structures and Čech Cohomology
3.2 The Sets P±(V) and Sp(V, o)
3.3 Correspondences and Obstructions to Existence
3.4 Short Exact Sequences
4. Proof of Theorem 1.4(1): Trivializations Perspectives
4.1 Topological Preliminaries
4.2 The Spin- and Pin-Structures of Definition 1.3
4.3 The Spin- and Pin-Structures of Definition 1.2
5. Equivalence of Definitions 1.1–1.3
5.1 Proof of Theorem 1.4(2)
5.2 Proof of Theorem 1.4(3)
6. Relative Spin- and Pin-Structures
6.1 Definitions and Main Theorem
6.2 Properties of Relative Spin- and Pin-Structures
6.3 Proof of Theorem 6.4(1): Definition 6.3 Perspective
6.4 Topological Preliminaries
6.5 Proof of Theorem 6.4(1): Definition 6.1 Perspective
6.6 Equivalence of Definitions 6.1 and 6.3
Part II: Orientations for Real CR-Operators
7. Main Results and Applications of Part II
7.1 Definitions and Main Theorem
7.2 Properties of Orientations: Smooth Surfaces
7.3 Properties of Orientations: Degenerations
7.4 Some Implications
7.5 Orientations and Evaluation Isomorphisms
8. Base Cases
8.1 Line Bundles over (S2, τ): Construction and Properties
8.2 Line Bundles over C Degenerations of (S2, τ)
8.3 Line Bundles over H3 Degenerations of (S2, τ)
8.4 Even-Degree Bundles over (S2, τ): Construction and Properties
8.5 Even-Degree Bundles over Degenerations of (S2, τ) and Exact Triples
9. Intermediate Cases
9.1 Orientations for Line Bundle Pairs
9.2 Proofs of Propositions 9.1(1), 9.2, and 9.3
9.3 Orientations from OSpin-Structures
10. Orientations for Twisted Determinants
10.1 Orientations of the Twisting Target
10.2 Orientations of Real CR-Operators
10.3 Degenerations and Exact Triples
10.4 Properties of Twisted Orientations
Part III: Real Enumerative Geometry
11. Pin-Structures and Immersions
11.1 Main Statements and Examples
11.2 Admissible Immersions into Surfaces
11.3 Proofs of Lemmas 11.9–11.12
12. Counts of Rational Curves on Surfaces
12.1 Complex Low-Degree Curves in P2
12.2 Real Low-Degree Curves in P2
12.3 Welschinger’s Invariants in Dimension 4
12.4 Moduli Spaces of Real Maps
12.5 Proof of Theorem 12.1
12.6 Proof of Proposition 12.5
13. Counts of Stable Real Rational Maps
13.1 Invariance and Properties
13.2 Orienting Moduli Spaces of Real Curves
13.3 Orienting Moduli Spaces of Real Maps
13.4 Definition of Curve Signs
13.5 Proof of Invariance
13.6 Signs at Immersions
13.7 Proof of Lemma 13.10
14. Counts of Real Rational Curves vs. Maps
14.1 Comparison Theorems
14.2 Basic Examples: Fourfolds
14.3 Basic Examples: Sixfolds
14.3.1 The Projective Space P3
14.3.2 The Sixfold (P1)3
14.4 Proofs of Theorems 14.1 and 14.2
Appendices
A Čech Cohomology
A.1 Identification with singular cohomology
A.2 Sheaves of groups
A.3 Sheaves determined by Lie groups
A.4 Relation with principal bundles
A.5 Orientable vector bundle over surfaces
B Lie Group Covers
B.1 Terminology and summary
B.2 Proof of Lemma B.1
B.3 Disconnected Lie groups
Bibliography
Index of Terms
Index of Notation
