Asymptotically Z-stable bundles over projective surfaces

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 $\hat{I}¼$-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.

13 pages