The infinitesimal multiplicities and orientations of the blow-up set of the Seiberg-Witten equation with multiple spinors

TL;DR

This paper develops a geometric framework for analyzing the blow-up set of the Seiberg-Witten equations with multiple spinors, linking its structure to topological invariants and providing measure bounds.

Contribution

It introduces multiplicities and orientations for tangent cones of the blow-up set, connecting the set's geometry to homology and characteristic classes.

Findings

Blow-up set determines a homology class equal to the Poincaré dual of the first Chern class.

Established a lower bound for the Hausdorff measure of the blow-up set.

Constructed tangent cone multiplicities and orientations for the blow-up set.

Abstract

I construct multiplicies and orientations of tangent cones to any blow-up set $Z$ for the Seiberg-Witten equation with multiple spinors. This is used to prove that $Z$ determines a homology class, which is shown to be equal to the Poincar\'{e} dual of the first Chern class of the determinant line bundle. I also obtain a lower bound for the 1-dimensional Hausdorff measure of $Z$.