Compact K\"ahler 3-manifolds without non-trivial subvarieties

TL;DR

This paper proves that compact K"ahler 3-manifolds with no non-trivial subvarieties are tori, advancing the classification of such manifolds and confirming a special case of a broader conjecture.

Contribution

It establishes that such manifolds are tori, using advanced techniques like the Brunella pseudo-effectivity theorem and a version of the hard Lefschetz theorem.

Findings

Any compact K"ahler 3-manifold with no non-trivial subvarieties is a torus.

Normal compact K"ahler 3-folds with terminal singularities and nef canonical bundle are cyclic quotients of simple tori.

Progress towards the bimeromorphic classification of compact K"ahler 3-folds.

Abstract

We prove that any compact K\"ahler 3-dimensional manifold which has no non-trivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of 'simple manifolds', central in the bimeromorphic classification of compact K\"ahler manifolds. The proof follows from the Brunella pseudo-effectivity theorem, combined with fundamental results of Siu and of the second author on the Lelong numbers of closed positive (1,1)-currents, and with a version of the hard Lefschetz theorem for pseudo-effective line bundles, due to Takegoshi and Demailly-Peternell-Schneider. In a similar vein, we show that a normal compact and K\"ahler 3-dimensional analytic space with terminal singularities and nef canonical bundle is a cyclic quotient of a simple non-projective torus if it carries no effective divisor. This is a crucial step to complete the bimeromorphic classification of compact K\"ahler 3-folds