Hartogs type extension theorem for the complement of effective and numerically effective divisors
S. V. Feklistov
TL;DR
This work generalizes Ohsawa's Hartogs extension results to complements of effective and nef divisors with connected support on compact Kähler manifolds, using a homological-algebra framework rather than $(n-1)$-convex exhaustion. A key achievement is a cohomological criterion that equates the Hartogs phenomenon on $X\setminus Z$ with a surjectivity condition on restriction maps, enabling applications to $\mathcal{O}$ and $\mathcal{O}(-D)$. The main theorem shows that if $D$ is effective, nef, and $[D]^{2}\neq 0$ in $H^{2,2}(X,\mathbb{R})$, then $X\setminus Z$ exhibits Hartogs, leveraging the Demailly-Peternell vanishing theorem. Additional results give geometric characterizations for basepoint-free divisors and provide toric-geometry examples illustrating the criterion. Overall, the paper connects cohomological vanishing, divisor positivity, and neighborhood geometry to explain when holomorphic extension phenomena occur in complements of divisors.
Abstract
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.