Hartogs type extension theorem for the complement of effective and numerically effective divisors

In these notes we generalize the Ohsawa’s results on the Hartogs extension phenomenon in the complement of effective divisors in Kähler manifolds with semipositive non-flat normal bundle. Namely, we prove that the Hartogs extension phenomenon occurs in the complement of effective and nef divisors with connected supports in Kähler manifolds. We use homological algebra methods instead of a construction of the $(n-1)$-convex exhaustion function. Also, the Demailly-Peternell vanishing theorem is a crucial argument for us. Moreover, we obtain geometric characterizations of the Hartogs phenomenon for the complement of basepoint-free divisors.