Let (M,ω) be a Kaehler manifold, an holomorphic fiber bundle E is Hermite-Einstein with potential ϕ∈Λ1(M)⊗End(E) if there are a hermitian metric h over E, and a Chern connection ∇ such that:
Λ(F(∇)+d∇ϕ)=λId
with F(∇), the curvature of the Chern connection and Λ, the contraction with ω, λ is a constant.
Have we an Hermite-Einstein metric for any potential ϕ?