[Paper Review] Rational curves and ampleness properties of the tangent bundle of algebraic varieties

Frédéric Campana, Thomas Peternell|arXiv (Cornell University)|Jul 8, 1996

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes deep connections between the ampleness properties of the tangent bundle of a projective algebraic manifold and the geometry of rational curves on it. It proves that if the tangent bundle admits an ample subbundle of rank at least n−2, then the manifold is isomorphic to projective space; further, rational connectivity implies the existence of a free, $T_X$-ample family of rational curves, linking positivity of the tangent bundle to global rational curve structure.

ABSTRACT

The purpose of this paper is to translate positivity properties of the tangent bundle (and the anti-canonical bundle) of an algebraic manifold into existence and movability properties of rational curves and to investigate the impact on the global geometry of the manifold $X$. Among the results we prove are these: \quad If $X$ is a projective manifold, and ${\cal E} \subset T_X$ is an ample locally free sheaf with $n-2\ge rk {\cal E}\ge n$, then $X \simeq \EP_n$. \quad Let $X$ be a projective manifold. If $X$ is rationally connected, then there exists a free $T_X$-ample family of (rational) curves. If $X$ admits a free $T_X$-ample family of curves, then $X$ is rationally generated.

Motivation & Objective

• To investigate how positivity properties of the tangent bundle and anti-canonical bundle influence the existence and movability of rational curves on algebraic manifolds.
• To determine the geometric consequences of ampleness conditions on subbundles of the tangent bundle.
• To clarify the relationship between rational connectedness and the existence of free, $T_X$-ample families of rational curves.
• To establish sufficient conditions under which a projective manifold must be isomorphic to projective space.

Proposed method

• Analyzes the ampleness of locally free sheaves $\mathcal{E} \subset T_X$ inside the tangent bundle of a projective manifold $X$.
• Applies techniques from algebraic geometry, particularly the theory of ample vector bundles and rational curves.
• Uses the concept of $T_X$-ampleness for families of curves to study their movability and covering properties.
• Employs the notion of rational connectedness and constructs free rational curves via deformation theory.
• Relies on results from the minimal model program and positivity theory in algebraic geometry.
• Applies the criterion that if $\mathcal{E} \subset T_X$ is ample with $n-2 \geq \text{rk}\, \mathcal{E} \geq n$, then $X \simeq \mathbb{P}^n$.

Experimental results

Research questions

• RQ1 Under what conditions on the tangent bundle does a projective manifold become isomorphic to projective space?
• RQ2 How does the existence of a free $T_X$-ample family of rational curves relate to the rational generation of a manifold?
• RQ3 What is the precise relationship between rational connectedness and the ampleness of the tangent bundle?
• RQ4 Can ampleness of a subbundle of the tangent bundle of rank $n-2$ or higher force the manifold to be $\mathbb{P}^n$?
• RQ5 What global geometric properties are implied by the existence of a $T_X$-ample family of rational curves?

Key findings

• If $X$ is a projective manifold and $\mathcal{E} \subset T_X$ is an ample locally free sheaf with $n-2 \geq \text{rk}\, \mathcal{E} \geq n$, then $X$ is isomorphic to $\mathbb{P}^n$.
• If $X$ is rationally connected, then there exists a free $T_X$-ample family of rational curves on $X$.
• If $X$ admits a free $T_X$-ample family of curves, then $X$ is rationally generated.
• The ampleness of the tangent bundle or its subbundles strongly constrains the global geometry of $X$, particularly in the case of high rank subbundles.
• The results establish a strong link between positivity of the tangent bundle and the presence of abundant, movable rational curves.
• The paper provides a characterization of projective space via ampleness conditions on subbundles of the tangent bundle.