SPIN/PIN-STRUCTURES AND REAL ENUMERATIVE GEOMETRY - Chen Xujia

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Résumé :
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 Cech cohomology perspective on fiber bundles and Lie group covering spaces--

Sommaire:
Preface; About the Authors; Spin- and Pin-Structures: Main Results and Examples of Part I; The Lie Groups Spin(n) and Pin?(n); Proof of Theorem 1.4(1): Classical Perspective; Proof of Theorem 1.4(1): Trivializations Perspectives; Equivalence of Definitions 1.1-1.3; Relative Spin- and Pin-Structures; Orientations for Real CR-Operators: Main Results and Applications of Part II; Base Cases; Intermediate Cases; Orientations for Twisted Determinants; Real Enumerative Geometry: Pin-Structures and Immersions; Counts of Rational Curves on Surfaces; Counts of Stable Real Rational Maps; Counts of Real Rational Curves vs. Maps; Appendices: Cech Cohomology; Lie Group Covers; Bibliography; Index of Terms; Index of Notation;