On the complement of nef divisors on projective manifolds

Abstract: Let $X'$ be a complex projective manifold, $\dim X'>1$, $Z$ be a connected analytic subset of codimension one that is the support of a nef effective Cartier divisor $D$ on $X'$, $X:=X'\setminus Z$. We prove that the following assertions are equivalent: $X$ is not Hartogs; $H^{1}{c}(X,\mathcal{O}{X})$ is $\infty$-dimensional; $D$ is abundant of Iitaka dimension one; $X$ is a proper fibration over an affine curve; there exists a compact analytic set of codimension one of $X$ that is not removable.