[Paper Review] Extremal rays and nefness of tangent bundles

Akihiro Kanemitsu|arXiv (Cornell University)|May 16, 2016

Algebraic Geometry and Number Theory29 references3 citations

TL;DR

This paper classifies Fano $n$-folds with Picard number $\rho_X > n-6$ that satisfy a recursive condition on extremal rays (Condition $*$), showing they are either rational homogeneous manifolds or products of $n-7$ copies of $\mathbb{P}^1$ and a unique Fano 7-fold $X_0$ constructed by Ottaviani. The key contribution is proving $X_0$ has a non-nef tangent bundle, thus providing a counterexample to the conjecture that all Fano manifolds with Condition $*$ are rational homogeneous, and establishing a sharp bound for the Campana-Peternell conjecture.

ABSTRACT

In view of Mori theory, rational homogenous manifolds satisfy a recursive condition: every elementary contraction is a rational homogeneous fibration and the image of any elementary contraction also satisfies the same property. In this paper, we show that a smooth Fano $n$-fold with the same condition and Picard number greater than $n-6$ is either a rational homogeneous manifold or the product of $n-7$ copies of $\mathbb{P}^1$ and a Fano $7$-fold $X_0$ constructed by G. Ottaviani. We also clarify that $X_0$ has non-nef tangent bundle and in particular is not rational homogeneous.

Motivation & Objective

• To classify Fano $n$-folds satisfying Condition $*$, which requires all elementary contractions to be rational homogeneous fibrations.
• To determine whether such Fano manifolds with high Picard number are necessarily rational homogeneous, challenging a natural generalization of the Campana-Peternell conjecture.
• To clarify the role of the Fano 7-fold $X_0$ constructed by Ottaviani as a counterexample to the conjecture that Condition $*$ implies rational homogeneity.
• To establish a sharp bound for the Campana-Peternell conjecture on nef tangent bundles, showing that $\rho_X > n-6$ is the threshold beyond which the conjecture may fail.

Proposed method

• Introduces invariants $d_i(\mathscr{E})$ and $\Delta_i(\mathscr{E})$ for vector bundles to analyze intersection numbers on projectivized bundles, especially for Fano bundles of rank >3.
• Uses slope theory for Fano bundles and numerical conditions on slopes to characterize the Fano 7-fold $X_0$ via two distinct $\mathbb{P}^2$-fibration structures over a 5-dimensional quadric.
• Applies results from Demailly-Peternell-Schneider and Muñoz et al. to relate nefness of tangent bundles to Condition $*$, under assumptions on low-dimensional Fano manifolds.
• Employs induction on dimension $n$ and uses the structure of elementary contractions to reduce the classification to known homogeneous cases or the product with $X_0$, leveraging the uniqueness of $X_0$.
• Analyzes the geometry of fiber products and Cartesian squares in commutative diagrams of contractions to deduce product structures or flag-like fibrations.
• Applies known results on Fano manifolds with Picard number one and dimension ≤6 to inductively verify that all fibers and images in a sequence of contractions satisfy Condition $*$.

Experimental results

Research questions

• RQ1 Are all Fano $n$-folds with $\rho_X > n-6$ and Condition $*$ necessarily rational homogeneous manifolds?
• RQ2 Does the Fano 7-fold $X_0$ constructed by Ottaviani satisfy Condition $*$, and is it the only such Fano 7-fold with Picard number two and two distinct $\mathbb{P}^2$-fibrations over a 5-dimensional quadric?
• RQ3 Is the tangent bundle of $X_0$ nef, and what does this imply for the Campana-Peternell conjecture on Fano manifolds with nef tangent bundles?
• RQ4 Can the Campana-Peternell conjecture be extended to Fano $n$-folds with $\rho_X > n-6$, and what is the sharp threshold for its validity?
• RQ5 Does the existence of $X_0$ imply that the condition of nef tangent bundle is not implied by Condition $*$ alone, even with high Picard number?

Key findings

• The Fano 7-fold $X_0$ constructed by Ottaviani is the unique Fano 7-fold with Picard number two that admits two distinct smooth $\mathbb{P}^2$-fibrations over a 5-dimensional quadric, satisfying Condition $*$.
• The tangent bundle of $X_0$ is not nef, proving that $X_0$ is not rational homogeneous and thus providing a counterexample to the conjecture that Condition $*$ implies rational homogeneity.
• Every Fano $n$-fold with $\rho_X > n-6$ and Condition $*$ is either a rational homogeneous manifold or isomorphic to $(\mathbb{P}^1)^{n-7} \times X_0$, establishing a complete classification.
• The existence of $X_0$ shows that the bound $\rho_X > n-6$ is sharp: Fano $n$-folds with $\rho_X > n-6$ are not necessarily rational homogeneous.
• If the Campana-Peternell conjecture holds for 6-folds with Picard number one, then it holds for all Fano $n$-folds with $\rho_X > n-6$, as shown via inductive reduction on Picard number.
• The counterexample $X_0$ gives a negative answer to Problem 0.4 for $q=1$, showing that $\bigwedge^q T_X$ being nef on extremal rational curves does not imply $\bigwedge^q T_X$ is nef.