DHYM connections on higher rank holomorphic vector bundles over ${\mathbb{P}}(T_{{\mathbb{P}^{2}}})$

TL;DR

This paper constructs explicit examples of deformed Hermitian Yang-Mills (dHYM) connections on higher rank, slope-unstable holomorphic vector bundles over a Fano threefold, expanding the known cases and providing algebraic criteria for their existence.

Contribution

It presents the first explicit non-trivial dHYM connection on higher rank unstable bundles and offers algebraic conditions for dHYM connections on sums of line bundles over rational homogeneous varieties.

Findings

Constructed explicit dHYM connections on higher rank unstable bundles.

Provided algebraic criteria for dHYM existence on line bundle sums.

Discovered new examples of dHYM connections on vector bundles.

Abstract

We construct the first explicit non-trivial example of deformed Hermitian Yang-Mills (dHYM) connection on a higher rank slope-unstable holomorphic vector bundle over a Fano threefold. Additionally, we provide a sufficient algebraic condition in terms of central charges for the existence of dHYM connections on Whitney sum of holomorphic line bundles over rational homogeneous varieties. As a consequence, we obtain several new examples of dHYM connections on higher rank holomorphic vector bundles.