Compact Kähler three-folds with nef anti-canonical bundle

TL;DR

We address the problem of classifying non-projective compact Kähler threefolds with nef $-K_X$ by extending Cao–Höring’s structure theorem to the Kähler setting. The approach combines a Minimal Model Program for Kähler threefolds, the positivity of direct image sheaves, and the use of $\mathbb{Q}$-conic/conic bundles and orbifold vector bundles to obtain locally constant fibrations after finite covers. The main result shows that, up to a finite étale cover, such $X$ must be Calabi–Yau type ($c_1(X)=0$), the product $K3\times\mathbb{P}^1$, or a $\mathbb{P}^1$‑bundle over a 2‑dimensional complex torus with numerically flat rank‑2 bundle; equivalently, the nefness of $-K_X$ imposes strong rigidity in dimension three. This provides a comprehensive structure theorem for nef anti-canonical geometry on compact Kähler threefolds and introduces techniques (orbifold bundles, positivity of direct images) that may extend to broader non-projective settings.

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.

Theorems & Definitions (71)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1: BEG
• Lemma 2.2
• proof : Proof of Lemma \ref{['support-lemma']}
• Definition 2.3
• Proposition 2.4: cf. CH19, CCM21
• Remark 2.5
• proof