Weak Fano bundles of rank $2$ over hyperquadrics $Q^n$ of dimension $n \ge 5$

Technical Summary: Weak Fano Bundles of Rank 2 over Hyperquadrics $Q_n$ of Dimension $n \ge 5$

Problem Statement
The paper addresses the classification of rank 2 weak Fano bundles over smooth quadric hypersurfaces $Q_n$ defined over an algebraically closed field of characteristic zero. While rank 2 Fano bundles (where the projectivization is a Fano variety) have been classified on various spaces including $Q_n$ for arbitrary dimensions, the classification of weak Fano bundles (where the projectivization is a weak Fano variety, i.e., $-K_X$ is nef and big) on quadric hypersurfaces $Q_n$ with $n \ge 3$ remained open. Specifically, this work targets the case where the dimension of the quadric is $n \ge 5$.

Methodology
The author adopts a strategy analogous to the classification of Fano bundles on $Q_n$ found in [1], but adapts the arguments to the weaker conditions imposed by weak Fano bundles. The proof relies on three main technical pillars:

1. Global Generation of Twisted Bundles: A central step is proving that for a normalized weak Fano bundle $E$ of rank 2 on $Q_n$ ($n \ge 5$), the twisted bundle $E(n-1)$ is globally generated. This is established by analyzing the anticanonical divisor $-K_{\mathbb{P}(E)}$ of the projectivization. The author divides the proof into cases based on the parity of $n$ and the value of the first Chern class $c_1(E)$ (normalized to 0 or -1). By utilizing Kleiman's criterion and vanishing theorems (specifically Theorem 2.6 and 2.7 regarding nef and big divisors), the author demonstrates that $E(n-1)$ is $(n-1)$-regular with respect to the ample line bundle $\mathcal{O}_{Q_n}(1)$, which implies global generation via Castelnuovo-Mumford regularity.

2. Splitting Criteria: The paper utilizes established splitting criteria for vector bundles on $Q_n$. Specifically, Proposition 2.8 provides a condition under which an unstable, globally generated rank 2 bundle splits into a direct sum of line bundles. This criterion involves an inequality relating the second Chern class $c_2$ and the first Chern class $c_1$ relative to the dimension $n$.

3. Numerical Analysis and Vanishing: The proof distinguishes between two numerical cases for the normalized bundle $E$:

   • Case 1 ($c_1(E)^2 \ge 4c_2(E)$): Here, the bundle is unstable. The author shows that the twisted bundle $F = E(n-1)$ satisfies the numerical inequality required by the splitting criterion (Proposition 2.8), forcing $E$ to split.
   • Case 2 ($c_1(E)^2 < 4c_2(E)$): Here, the bundle is stable. The author employs a contradiction argument for $n \ge 12$. By combining the global generation of $F$ with bounds derived from the geometry of the zero locus of sections and the inequality $c_1(F)^2 < 4c_2(F)$, a numerical contradiction is reached using trigonometric estimates (Proposition 2.9). For the remaining dimensions $5 \le n \le 11$, the author performs detailed calculations involving Chern classes, Segre classes, and the Euler characteristic $\chi$ restricted to lower-dimensional quadrics (specifically $Q_5$). These calculations eliminate all possibilities except for the specific case of the Cayley bundle on $Q_5$.

Key Contributions and Results
The primary result is Theorem 1.1, which classifies rank 2 weak Fano bundles on $Q_n$ for $n \ge 5$:

Let $E$ be a weak Fano bundle of rank 2 over $Q_n$ where $n \ge 5$. Then $E$ is either a direct sum of line bundles or, up to a twist by a line bundle, the Cayley bundle on $Q_5$.

The classification proceeds as follows:

• If $c_1(E)^2 \ge 4c_2(E)$, the bundle splits.
• If $c_1(E)^2 < 4c_2(E)$, no such bundles exist for $n \ge 12$.
• For $5 \le n \le 11$, if $c_1(E)^2 < 4c_2(E)$, the only solution is $n=5$ and $E$ being a twist of the Cayley bundle (defined as a stable rank 2 bundle with $(c_1, c_2) = (-1, 1)$).

Significance and Claims
The paper claims to complete the classification of rank 2 weak Fano bundles on higher-dimensional quadrics, filling a gap left by previous classifications of Fano bundles and weak Fano bundles on other varieties (such as projective spaces and del Pezzo 3-folds). The author notes that while the basic idea follows the approach used for Fano bundles in [1], the calculations for weak Fano bundles are "more complicated" and require "more discussion," particularly in handling the vanishing theorems and the specific numerical constraints on the Chern classes.

The work also briefly acknowledges in Remark 4.6 that the case for $n=4$ presents additional complexities, particularly when $c_1(E) = -1$, suggesting that the classification for $n=4$ is not fully resolved by the methods presented here. The paper does not propose future applications or experimental proposals, focusing strictly on the algebraic geometric classification.