---
OA_type: closed access
_id: '20615'
abstract:
- lang: eng
  text: "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.\r\n\r\nThis 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.\r\n\r\nPart 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."
article_processing_charge: No
author:
- first_name: Xujia
  full_name: Chen, Xujia
  id: 968ad14a-fd86-11ee-a420-ea29715511a3
  last_name: Chen
- first_name: Aleksey
  full_name: Zinger, Aleksey
  last_name: Zinger
citation:
  ama: Chen X, Zinger A. <i>Spin/Pin-Structures and Real Enumerative Geometry</i>.
    World Scientific Publishing; 2024. doi:<a href="https://doi.org/10.1142/13476">10.1142/13476</a>
  apa: Chen, X., &#38; Zinger, A. (2024). <i>Spin/Pin-structures and real enumerative
    geometry</i>. World Scientific Publishing. <a href="https://doi.org/10.1142/13476">https://doi.org/10.1142/13476</a>
  chicago: Chen, Xujia, and Aleksey Zinger. <i>Spin/Pin-Structures and Real Enumerative
    Geometry</i>. World Scientific Publishing, 2024. <a href="https://doi.org/10.1142/13476">https://doi.org/10.1142/13476</a>.
  ieee: X. Chen and A. Zinger, <i>Spin/Pin-structures and real enumerative geometry</i>.
    World Scientific Publishing, 2024.
  ista: Chen X, Zinger A. 2024. Spin/Pin-structures and real enumerative geometry,
    World Scientific Publishing,p.
  mla: Chen, Xujia, and Aleksey Zinger. <i>Spin/Pin-Structures and Real Enumerative
    Geometry</i>. World Scientific Publishing, 2024, doi:<a href="https://doi.org/10.1142/13476">10.1142/13476</a>.
  short: X. Chen, A. Zinger, Spin/Pin-Structures and Real Enumerative Geometry, World
    Scientific Publishing, 2024.
date_created: 2025-11-10T08:40:10Z
date_published: 2024-01-01T00:00:00Z
date_updated: 2025-11-10T15:28:49Z
day: '01'
doi: 10.1142/13476
extern: '1'
language:
- iso: eng
month: '01'
oa_version: None
publication_identifier:
  eisbn:
  - '9789811278556'
  isbn:
  - '9789811278532'
publication_status: published
publisher: World Scientific Publishing
quality_controlled: '1'
scopus_import: '1'
status: public
title: Spin/Pin-structures and real enumerative geometry
type: book
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2024'
...