A Hopf index for isotropic sections of orthogonal bundles

TL;DR

This paper introduces eight analogues of the Hopf index for isotropic sections of orthogonal bundles, with applications to virtual cycles in algebraic geometry.

Contribution

It extends the classical Hopf index concept to isotropic sections of orthogonal bundles, providing new tools for virtual cycle computations.

Findings

Eight analogues of the Hopf index are established.

Applications to cosection localized virtual cycles.

Implications for DT^4 virtual cycles.

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.