On the complement of nef divisors on projective manifolds

Let $X^{\mathrm{\prime }}$ be a complex projective manifold, $\dim X^{\mathrm{\prime }}>1$, $Z$ a connected analytic subset of codimension one which is the support of a nef effective Cartier divisor $D$ on $X^{\mathrm{\prime }}$, $X:=X^{\mathrm{\prime }}\setminus Z$. Let $\kappa (D)$ be the Iitaka dimension of $D$. We prove that $X$ is not Hartogs if and only if $D$ is abundant and $\kappa (D)=1$. In particular, $X$ is not Hartogs if and only if $X$ is a proper fibration over an affine curve.