Classification of Smooth Minimal Kähler Fourfolds Without Effective Divisors and Surfaces

Pisya Vikash

Classification of Smooth Minimal Kähler Fourfolds Without Effective Divisors and Surfaces

Algebraic Geometry 2026-07-05 v1

Authors: Pisya Vikash

Abstract

We prove that if $X$ is a compact K\"ahler fourfold with pseudo--effective canonical bundle and no subvarieties of codimension one or two, then $K_X$ is a torsion line bundle. By the Beauville--Bogomolov decomposition theorem, it follows that $X$ is either a quotient of a complex torus or an irreducible holomorphic symplectic manifold.

Keywords

Kähler fourfolds pseudo-effective canonical bundle torsion line bundle Beauville-Bogomolov decomposition irreducible holomorphic symplectic manifold

Cite

@article{arxiv.2607.04536,
  title  = {Classification of Smooth Minimal Kähler Fourfolds Without Effective Divisors and Surfaces},
  author = {Pisya Vikash},
  journal= {arXiv preprint arXiv:2607.04536},
  year   = {2026}
}