Given an uncountable cardinal κ with $\kappa ={\kappa }^{<\kappa }$ and $2^{\kappa }$ regular, we show that there is a forcing that preserves cofinalities less than or equal to $2^{\kappa }$ and forces the existence of a well-order of H(κ⁺) that is definable over ⟨H(κ⁺),∈⟩ by a Σ₁-formula with parameters. This shows that, in contrast to the case "κ = ω", the existence of a locally definable well-order of H(κ⁺) of low complexity is consistent with failures of the GCH at κ. We also show that the forcing mentioned above introduces a Bernstein subset...