  In these notes we generalize the Ohsawa's results on the Hartogs extension
phenomenon in the complement of effective divisors in K\"ahler 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 K\"ahler 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.
