In general, the equivalence of the stability and the solvability of an equation is an important problem in geometry.
In this talk, we introduce the J-equation on holomorphic vector bundles over compact Kahler manifolds, as an extension
of the line bundle case and the Hermitian-Einstein equation over Riemann surfaces. We investigate some fundamental
properties as well as examples. In particular, we give algebraic obstructions called the (asymptotic) J-stability in terms of
subbundles on compact Kahler 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.