[Paper Review] On the second exterior power of tangent bundles of Fano fourfolds with Picard number $rho(X)geqslant2$

Kazunori Yasutake|arXiv (Cornell University)|Dec 4, 2012

Algebraic Geometry and Number Theory16 references3 citations

TL;DR

This paper classifies Fano fourfolds with Picard number at least 2 for which the second exterior power of the tangent bundle is numerically effective (nef). Using techniques from birational geometry and the theory of Fano bundles, it proves that such fourfolds are either the blow-up of $\mathbb{P}^4$ at a point or have nef tangent bundle. The result extends Campana and Peternell's classification of Fano threefolds to fourfolds and confirms a conjectural criterion for nefness of $\Lambda^2\mathcal{T}_X$ via extremal rational curves.

ABSTRACT

In this paper, we classify Fano fourfolds whose the second exterior power of tangent bundles are numerically effective with Picard number greater than one.

Motivation & Objective

• To extend the classification of Fano threefolds with $\Lambda^2\mathcal{T}_X$ nef to Fano fourfolds.
• To investigate the structure of Fano fourfolds with $\rho(X)\geq2$ under the condition that $\Lambda^2\mathcal{T}_X$ is numerically effective.
• To verify whether the nefness of $\Lambda^2\mathcal{T}_X$ on extremal rational curves implies global nefness, as posed by Campana and Peternell.
• To determine the possible geometric types of such fourfolds using birational and vector bundle theory.

Proposed method

• Analyzes the nefness of $\Lambda^2\mathcal{T}_X$ via the canonical bundle and Chern classes, using the fact that $\det(\Lambda^2\mathcal{T}_X) = -\binom{3}{1}K_X = -3K_X$.
• Applies the theory of numerically flat vector bundles and H-semistability to show that if $\kappa(X) = 0$, then $X$ admits an étale cover isomorphic to an abelian variety.
• Uses the classification of Fano bundles on $\mathbb{P}^2$, $\mathbb{P}^3$, and $Q_3$ to analyze the structure of $X$ when it is a $\mathbb{P}^n$-bundle or a blow-up.
• Applies results on extremal contractions and the structure of Fano manifolds with nef tangent bundles to classify the possible cases.
• Employs the fact that $\Lambda^2\mathcal{T}_X$ is nef on all extremal rational curves if and only if it is globally nef, confirming a conjectural criterion.
• Reduces the problem to the case where $\mathcal{T}_X$ is nef or $X$ is a blow-up of $\mathbb{P}^4$ at a point, using the classification of Fano fourfolds with nef tangent bundles.

Experimental results

Research questions

• RQ1 Which Fano fourfolds with $\rho(X) \geq 2$ have $\Lambda^2\mathcal{T}_X$ numerically effective?
• RQ2 Does the nefness of $\Lambda^2\mathcal{T}_X$ on all extremal rational curves imply global nefness for Fano fourfolds?
• RQ3 What are the possible geometric structures of Fano fourfolds with $\rho(X) \geq 2$ and $\Lambda^2\mathcal{T}_X$ nef?
• RQ4 How does the classification of Fano bundles on $\mathbb{P}^3$ and $Q_3$ constrain the structure of such fourfolds?
• RQ5 Can the blow-up of $\mathbb{P}^4$ at a point be the only non-tangent-nef example among Fano fourfolds with $\rho(X) \geq 2$?

Key findings

• Fano fourfolds with $\rho(X) \geq 2$ and $\Lambda^2\mathcal{T}_X$ nef are either the blow-up of $\mathbb{P}^4$ at a point or have $\mathcal{T}_X$ nef.
• The nefness of $\Lambda^2\mathcal{T}_X$ on all extremal rational curves implies its global nefness, confirming a conjecture of Campana and Peternell for this class.
• If $\kappa(X) = 0$, then $X$ admits an étale cover isomorphic to an abelian variety, under the assumption that $\Lambda^2\mathcal{T}_X$ is nef.
• The only Fano fourfolds with $\rho(X) \geq 2$ and $\Lambda^2\mathcal{T}_X$ nef that are not covered by the tangent bundle being nef are the blow-ups of $\mathbb{P}^4$ at a point.
• The classification of Fano bundles on $\mathbb{P}^3$, $Q_3$, and $\mathbb{P}^2$ is used to rule out other possible structures for $X$.
• The result establishes a sharp dichotomy: either $X$ is a blow-up of $\mathbb{P}^4$ at a point or $\mathcal{T}_X$ is nef, with no other examples existing under the given conditions.