Canonical extensions of manifolds with nef tangent bundle

TL;DR

The paper investigates when canonical extensions $Z_X$ of compact Kähler manifolds with nef tangent bundle are Stein, framing the question proposed by Greb–Wong. It develops a structure theory along locally constant Albanese fibrations, showing that under the weak Campana–Peternell conjecture the extension factors into pieces whose Steinness follows from known results on tori bases and Fano fibres, and then deduces the Steinness of $Z_X$ up to finite étale covers. A key result is the decomposition $Z_{X,[\omega_X]}\simeq Z_{\Omega^1_{X/T},[\omega_{X/T}]}\times_T Z_{\alpha^*\Omega^1_T,[\alpha^*\omega_T]}$ with a pushdown decomposition of $[\omega_X]$, enabling a fibre-bundle analysis and the use of Matsushima-type theorems to conclude Steinness in the nef-tangent setting under CP conjecture. The paper also treats the special case of ruled surfaces, proving that for $X=\mathbb{P}(\mathcal{E})\to C$ with $g(C)\ge2$ the canonical extension cannot be Stein, aligning with the broader conjectural picture and extending existing results for surfaces.

Abstract

To any compact Kähler manifold $(X, ω)$ one may associate a bundle of affine spaces $Z_X\rightarrow X$ called a \emph{canonical extension} of $X$. In this paper we prove that if the tangent bundle of $X$ is nef, then the total space $Z_X$ is a Stein manifold. This partially answers a question raised by Greb-Wong of whether these two properties are actually equivalent. We also complement some known results for surfaces in the converse direction.

Theorems & Definitions (27)

• Theorem 2
• Lemma 3
• Definition 1.4
• Conjecture 1.5
• Remark 1.6
• Theorem 2.1
• Conjecture 2.2
• Proposition 3.1
• Proof : of \ref{['splitting_canonical_extension_2']}
• Proposition 3.2