Asymptotically Z-stable bundles over projective surfaces

TL;DR

This paper develops a method to construct new examples of asymptotically Z-stable bundles over projective surfaces, linking stability to solutions of deformed Hermitian--Yang--Mills equations in the large-volume limit.

Contribution

It introduces a technique for constructing rank 3 asymptotically Z-stable bundles as extensions, and presents an analogue of the Hoppe criterion for rank 2 bundles.

Findings

Constructed new strictly a.Z-stable bundles over various surfaces.

Linked a.Z-stability to solutions of deformed Hermitian--Yang--Mills equations.

Provided an analogue of the Hoppe criterion for rank 2 bundles.

Abstract

We study the existence of asymptotically $Z$-stable (a.Z stable) bundles over polycyclic surfaces. Our choice of polynomial central charge is related to the existence of solutions of the deformed Hermitian--Yang--Mills equations, with vanishing $B$-field, in the large-volume limit. The main result is a technique to construct rank $3$, strictly a.Z-stable bundles as extensions of a line bundle by a $\mu$-stable bundle of rank $2$. In particular, this leads to new examples of strictly a.Z-stable bundles over $\mathbb{P}^2$, the product $\mathbb{P}^1\times \mathbb{P}^1$, and the blow-up $\mathrm{Bl}_q\mathbb{P}^2$. We also present an analogue of the Hoppe criterion for the a.Z-stability of vector bundles of rank $2$, which may be of independent interest.