Compact Kähler three-folds with nef anti-canonical bundle

Abstract

In this paper, we prove that a non-projective compact Kähler three-fold with nef anti-canonical bundle is, up to a finite étale cover, one of the following: a manifold with vanishing first Chern class; the product of a K3 surface and the projective line; or a projective space bundle over a $2$-dimensional torus. This result extends Cao-Höring's structure theorem for projective manifolds to compact Kähler manifolds in dimension $3$. For the proof, we investigate the Minimal Model Program for compact Kähler three-folds with nef anti-canonical bundles by using the positivity of direct image sheaves, $\mathbb{Q}$-conic bundles, and orbifold vector bundles.

The final version (v3). 39 pages. The term '3-folds' in the title has been changed to ''three-folds'' to align with the publication version. An erroneous argument in Case 1 of Subsection 4.3 has been replaced with a correct alternative proof. To appear in Math. Ann. 391, 1253-1289 (2025)

References in corpus (1)

• On the minimal model program for projective varieties with pseudo-effective tangent sheaf