On the complement of nef divisors on projective manifolds

Abstract

Let $X'$ be a complex projective manifold, $\dim X'>1$, $Z$ a connected analytic subset of codimension one which is the support of a nef effective Cartier divisor $D$ on $X'$, $X:=X'\setminus Z$. 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.