[Paper Review] Projective manifolds whose tangent bundle contains a strictly nef subsheaf

Jie Liu, Wenhao Ou+1 more|arXiv (Cornell University)|Apr 18, 2020

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes that a projective manifold with a locally free strictly nef subsheaf in its tangent bundle is isomorphic to a projective bundle over a hyperbolic manifold. If the fundamental group is virtually abelian, the manifold is isomorphic to a projective space, extending results on positivity in algebraic geometry and characterizing manifolds with strictly nef tangent subbundles via fibrations and hyperbolicity.

ABSTRACT

Suppose that $X$ is a projective manifold whose tangent bundle $T_X$ contains a locally free strictly nef subsheaf. We prove that $X$ is isomorphic to a projective bundle over a hyperbolic manifold. Moreover, if the fundamental group $π_1(X)$ is virtually abelian, then $X$ is isomorphic to a projective space.

Motivation & Objective

• To understand the geometric structure of projective manifolds whose tangent bundle contains a strictly nef subsheaf.
• To generalize classical results on ample subsheaves of the tangent bundle (e.g., Andreatta-Wi{\ss}niewski) to the strictly nef setting.
• To investigate whether the existence of a strictly nef subsheaf in the tangent bundle still forces the manifold to be a projective space or a projective bundle.
• To classify the possible structures of such manifolds using tools from positivity theory, MRC fibrations, and projective geometry.
• To provide a characterization of projective spaces via the strictly nef condition on the tangent bundle when the fundamental group is virtually abelian.

Proposed method

• Utilize the theory of numerically projectively flat vector bundles and their positivity properties.
• Apply the MRC (Maximal Rationally Connected) fibration to analyze the structure of the manifold and reduce the problem to base manifolds.
• Use the projectivization of vector bundles to relate the geometry of the tangent sheaf to the tautological line bundle on projective bundles.
• Analyze the degeneration of rational curves in the fibers to deduce hyperbolicity of the base.
• Employ transcendental methods, including the study of Hermitian flat and strictly nef vector bundles over curves.
• Leverage known results on strictly nef tangent bundles (e.g., LOY19) and extend them to the case of subsheaves.

Experimental results

Research questions

• RQ1 What geometric structure arises when a projective manifold has a locally free strictly nef subsheaf in its tangent bundle?
• RQ2 Can the existence of a strictly nef subsheaf in the tangent bundle still force the manifold to be a projective space, as in the ample case?
• RQ3 How does the fundamental group influence the structure of such manifolds, particularly when it is virtually abelian?
• RQ4 What are the two possible structures for the strictly nef subsheaf in the tangent bundle, and how do they differ from the ample case?
• RQ5 To what extent does strict nefness, despite not being closed under exterior products, still control the global geometry of the manifold?

Key findings

• A projective manifold $X$ with a locally free strictly nef subsheaf $\mathscr{F}$ in its tangent bundle $T_X$ admits a $\mathbb{P}^d$-bundle structure $\varphi: X \to T$ for some $d \geq r = \operatorname{rk}(\mathscr{F})$.
• The base manifold $T$ is hyperbolic, meaning every holomorphic map from $\mathbb{C}$ to $T$ is constant.
• There are exactly two possible structures for the strictly nef subsheaf $\mathscr{F}$: either $\mathscr{F} \cong T_{X/T}$ and $X$ is a flat projective bundle over $T$, or $\mathscr{F}$ is numerically projectively flat with fibers isomorphic to $\mathcal{O}_{\mathbb{P}^d}(1)^{\oplus r}$.
• If the fundamental group $\pi_1(X)$ is virtually abelian, then $X$ is isomorphic to $\mathbb{P}^n$, extending the result of LOY19 to the strictly nef case.
• The result generalizes Theorem 1.2 (Andreatta-Wi{\ss}niewski) to the strictly nef setting, showing that such manifolds are not necessarily projective spaces unless additional topological conditions are imposed.
• The paper constructs examples showing that the strictly nef condition is not sufficient to force $X \cong \mathbb{P}^n$ without further assumptions, highlighting the necessity of the fundamental group condition.