Spin/Pin-Structures and Real Enumerative Geometry

TL;DR

This monograph explores Spin and Pin structures from multiple perspectives, develops orientation techniques for real Cauchy-Riemann operators, and applies these to compare real curve counting invariants in enumerative geometry.

Contribution

It introduces an intrinsic perspective on Spin and Pin structures, establishes their equivalence, and applies these insights to real enumerative geometry, specifically comparing curve signs in Welschinger invariants.

Findings

Established equivalence of different perspectives on Spin/Pin structures.

Developed orientation methods for real Cauchy-Riemann operators.

Compared curve signs in Welschinger invariants using intrinsic Pin-structures.

Abstract

The present, partly expository, monograph consists of three parts. The first part treats Spin- and Pin-structures from three different perspectives and shows them to be suitably equivalent. It also introduces an intrinsic perspective on the relative Spin- and Pin-structures of Fukaya-Oh-Ohta-Ono and Solomon, establishes properties of these structures in both perspectives, and again shows them to be suitably equivalent. The second part uses the intrinsic perspective on (relative) Spin- and Pin-structures to detail constructions of orientations on the determinants of real Cauchy-Riemann operators and study their properties. The final part applies the results of the first two parts to the enumerative geometry of real curves and obtains an explicit comparison between the curve signs in the intrinsic definition of Welschinger and later Pin-structure dependent definitions. This comparison makes use of both the classical and instrinisc perspectives on Pin-structures and thus of the equivalence between them established in this monograph. The preface and the introductions to the three parts describe the present work in more detail.