A Hopf index for isotropic sections of orthogonal bundles

TL;DR

This paper introduces an orthogonal analogue of the Hopf index for isotropic sections of complex orthogonal bundles, defining a square-root Euler class $\sqrt e(\underline E,s)$ via cosection localisation and presenting eight equivalent formulæ paralleling the classical case. A key feature is the refinement in rank $4$ (i.e., $n=2$) where $\sqrt e(\underline E,s)$ splits into a pair $(d_1,d_2)$ with $\sqrt e=d_1-d_2$, and concrete computations are given through homotopy, projective geometry on $Q\cong \mathbb P^1\times\mathbb P^1$, and a 2-periodic Clifford complex, including explicit examples. The paper also demonstrates that, in general, $\sqrt e(\underline E,s)$ need not equal the length of the zero set, and discusses deformation and localization phenomena in relation to cosection-localised virtual cycles and to DT$^4$ virtual cycles. An additional proof of OH5 via a projective compactification connects the index to Edidin–Graham computations, highlighting the interplay between topology, algebraic geometry, and derived/virtual techniques in higher-rank isotropic settings.

Abstract

The Hopf index equates the multiplicity of a zero of a section of a vector bundle with a winding number. We give eight analogues for isotropic sections of bundles with quadratic form. There are applications to cosection localised virtual cycles and to DT$^4$ virtual cycles.