Spin(7)-instantons, stable bundles and the Bogomolov inequality for complex 4-tori

TL;DR

This paper introduces a Spin-rotation technique using gauge theory on Spin(7)-manifolds to transform holomorphic structures on complex 4-tori, leading to new examples and a refined Bogomolov inequality for stable bundles.

Contribution

It develops a novel Spin-rotation method to generate new holomorphic structures on complex 4-tori and derives a stronger Bogomolov inequality for stable bundles.

Findings

Constructed non-trivial examples of Spin-rotation transformations.

Established a Bogomolov type inequality stronger than the classical version.

Provided restrictions on the existence of stable bundles on 4-dimensional abelian varieties.

Abstract

Using gauge theory for Spin(7)-manifolds of dimension 8, we develop a procedure, called Spin-rotation, which transforms a (stable) holomorphic structure on a vector bundle over a complex torus of dimension 4 into a new holomorphic structure over a different complex torus. We show non-trivial examples of this procedure by rotating a decomposable Weil abelian variety into a non-decomposable one. As a byproduct, we obtain a Bogomolov type inequality, which gives restrictions for the existence of stable bundles on an abelian variety of dimension 4, and show examples in which this is stronger than the usual Bogomolov inequality.