[Paper Review] On the second and third exterior power of tangent bundles of Fano manifolds with birational contractions

Kazunori Yasutake|arXiv (Cornell University)|Mar 20, 2014

Geometry and complex manifolds7 references3 citations

TL;DR

This paper classifies Fano manifolds with elementary birational contractions whose second or third exterior powers of the tangent bundle are numerically effective (nef). Using the structure of extremal rays and the behavior of tangent bundles along rational curves, it proves that for $ n \geq 3 $, if $ \wedge^2\mathcal{T}_X $ is nef, then $ X \cong \mathrm{Bl}_{\mathrm{pt}}(\mathbb{P}^n) $. For $ n \geq 4 $, if $ \wedge^3\mathcal{T}_X $ is nef, $ X $ is isomorphic to one of seven specific Fano manifolds, including blow-ups of $ \mathbb{P}^n $ and quadric hypersurfaces.

ABSTRACT

In this paper, we classify Fano manifolds with elementary contractions of birational type such that the second or third exterior power of tangent bundles are numerically effective.

Motivation & Objective

• To classify Fano manifolds with elementary birational contractions for which the second exterior power of the tangent bundle is numerically effective.
• To extend the classification to the third exterior power of the tangent bundle under similar conditions.
• To address a problem posed by Campana and Peternell on whether nefness on extremal rational curves implies global nefness for $ \wedge^q\mathcal{T}_X $.
• To provide a complete list of Fano manifolds satisfying the nefness condition on $ \wedge^2\mathcal{T}_X $ or $ \wedge^3\mathcal{T}_X $ in higher dimensions.
• To establish that the only Fano manifold with a birational elementary contraction and $ \wedge^2\mathcal{T}_X $ nef is the blow-up of $ \mathbb{P}^n $ at a point.

Proposed method

• Analyzes the restriction of $ \wedge^2\mathcal{T}_X $ and $ \wedge^3\mathcal{T}_X $ to extremal rational curves using the splitting type of the tangent bundle along such curves.
• Applies the Bend and Break Lemma to bound the anticanonical degree of rational curves in extremal rays.
• Uses the Cone Theorem and the structure of the exceptional locus to classify possible birational contractions based on the dimensions of the exceptional divisor and its image.
• Employs the exact sequence $ 0 \to \mathcal{T}_\pi \to \mathcal{T}_X \to \pi^*\mathcal{T}_Y \to 0 $ for $ \mathbb{P}_Y(\mathcal{E}) $-bundles to analyze the nefness of exterior powers.
• Applies known classification results for Fano manifolds with small pseudo-index, such as those by Tsukioka and Kobayashi-Ochiai.
• Uses pushforward techniques on the exact sequence $ 0 \to \mathcal{O}_X \to \mathcal{O}_X(E) \to \mathcal{N}_{E/X} \to 0 $ to reconstruct the total space as a projective bundle over a base manifold.

Experimental results

Research questions

• RQ1 Which Fano manifolds with an elementary birational contraction have $ \wedge^2\mathcal{T}_X $ numerically effective?
• RQ2 Which Fano manifolds with an elementary birational contraction have $ \wedge^3\mathcal{T}_X $ numerically effective for $ n \geq 4 $?
• RQ3 Does nefness of $ \wedge^q\mathcal{T}_X $ on all extremal rational curves imply global nefness for Fano manifolds?
• RQ4 What are the complete list of Fano manifolds satisfying $ \wedge^2\mathcal{T}_X $ or $ \wedge^3\mathcal{T}_X $ being nef under birational elementary contractions?
• RQ5 Can the blow-up of $ \mathbb{P}^n $ at a point be uniquely characterized among Fano manifolds by the nefness of $ \wedge^2\mathcal{T}_X $?

Key findings

• For $ n \geq 3 $, if $ X $ is a Fano manifold with an elementary birational contraction and $ \wedge^2\mathcal{T}_X $ is nef, then $ X \cong \mathrm{Bl}_{\mathrm{pt}}(\mathbb{P}^n) $, the blow-up of $ \mathbb{P}^n $ at a point.
• For $ n \geq 4 $, if $ \wedge^3\mathcal{T}_X $ is nef, then $ X $ is isomorphic to one of seven specific Fano manifolds, including $ \mathrm{Bl}_{\mathrm{pt}}(\mathbb{P}^n) $, $ \mathbb{P}_{\mathbb{P}^{n-1}}(\mathcal{O}_{\mathbb{P}^{n-1}} \oplus \mathcal{O}_{\mathbb{P}^{n-1}}(-2)) $, and $ \mathbb{P}_{Q^{n-1}}(\mathcal{O}_{Q^{n-1}} \oplus \mathcal{O}_{Q^{n-1}}(-1)) $.
• The blow-up of $ \mathbb{P}^n $ along a line is included in the classification for $ \wedge^3\mathcal{T}_X $ nef when $ n \geq 4 $.
• The product of $ \mathbb{P}^1 $ and the blow-up of $ \mathbb{P}^{n-1} $ at a point is also a solution for $ \wedge^3\mathcal{T}_X $ nef.
• The blow-up of a smooth quadric hypersurface $ Q^4 \subset \mathbb{P}^5 $ along a line or a conic not contained in a plane is included in the classification for $ \wedge^3\mathcal{T}_X $ nef.
• The paper confirms that nefness of $ \wedge^2\mathcal{T}_X $ or $ \wedge^3\mathcal{T}_X $ on all extremal rational curves implies global nefness, affirming a conjecture by Campana and Peternell.