$J$-equation on holomorphic vector bundles

We introduce the $J$-equation on holomorphic vector bundles over compact K\"ahler manifolds and investigate some fundamental properties as well as examples of them. In particular, we give algebraic obstructions called the (asymptotic) $J$-stability in terms of subbundles on compact K\"ahler surfaces, and a numerical criterion on vortex bundles via dimensional reduction. Also, we discuss an application for the vector bundle version of the deformed Hermitian-Yang-Mills equation in the small volume regime.
Comment: 52 pages, made some corrections, Theorem 1.4 (3) is extended for sufficiently smooth vector bundles