Classification of Smooth Minimal Kähler Fourfolds Without Effective Divisors and Surfaces

Abstract

We prove that if \(X\) is a compact Kähler 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.