Classification of Smooth Minimal Kähler Fourfolds Without Effective Divisors and Surfaces

Abstract

We prove the following two statements: (1) 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; (2) If $X$ is a compact Kähler fourfold with no subvarieties of codimension one or two, then $X$ is not uniruled. Using the Beauville-Bogomolov decomposition and a characterization of Uniruled manifolds by Wenhao Ou, we get that if $X$ is a compact Kähler fourfold with no subvarieties of codimension one or two, then either $X$ is a torus or an irreducible holomorphic symplectic manifold.

arXiv commentImproved main theorem. Comments are welcome