On the complement of nef divisors on projective manifolds

Abstract

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

Did corrections in sections 2.3 and 4. Added section 5. Rewrite section 1