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

TL;DR

This paper extends the Hartogs extension theorem to the complement of effective and nef divisors in K"ahler manifolds, using homological algebra and vanishing theorems, broadening the understanding of extension phenomena in complex geometry.

Contribution

It generalizes Hartogs extension results to nef divisors with connected supports in K"ahler manifolds using algebraic methods and vanishing theorems, avoiding convex exhaustion constructions.

Findings

Hartogs extension holds for complements of nef divisors in K"ahler manifolds.

Homological algebra methods can replace convex exhaustion in extension proofs.

Geometric characterizations are provided for the Hartogs phenomenon in specific divisor cases.

Abstract

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.