Rank-two weak Fano bundles on a four-dimensional quadric hypersurface $Q^4$

Technical Summary: Rank-Two Weak Fano Bundles on a Four-Dimensional Quadric Hypersurface

Problem Statement
This paper addresses the classification of rank 2 weak Fano bundles on a smooth four-dimensional quadric hypersurface, denoted $Q_4$. A vector bundle $E$ on a smooth projective variety $X$ is defined as a weak Fano bundle if its projectivization $\mathbb{P}(E)$ is a weak Fano variety, meaning the anticanonical divisor $-K_{\mathbb{P}(E)}$ is nef and big.

While the classification of rank 2 Fano bundles (where $-K_{\mathbb{P}(E)}$ is ample) on quadric hypersurfaces of dimension $n \geq 4$ was completed by Ancona, Peternell, and Wiśniewski [APW94], and the weak Fano case for dimensions $n \geq 5$ was recently settled by the author [Tak26], the specific case of $Q_4$ remained open. The difficulty in dimension 4 arises from the structure of the Chow group $A_2(Q_4) \simeq \mathbb{Z}\alpha \oplus \mathbb{Z}\beta$, where $\alpha$ and $\beta$ represent the classes of the two distinct families of planes on $Q_4$. Unlike higher dimensions where the second Chern class $c_2(E)$ has a single component, on $Q_4$ it possesses two independent components, $c_2(E) = a\alpha + b\beta$. This increases the number of numerical cases to consider and necessitates more refined geometric arguments to exclude non-existent bundles.

Methodology
The author employs a combination of numerical restrictions, vanishing theorems, and geometric restrictions to planes and lines on $Q_4$. The methodology proceeds as follows:

1. Normalization and Numerical Bounds: The bundle $E$ is normalized by twisting with a line bundle such that $c_1(E) \in \{0, -1\}$. Using the weak Fano condition, the author establishes that the divisor $\xi + \frac{4-c_1}{2}H$ on $\mathbb{P}(E)$ is nef and big (where $\xi$ is the tautological divisor and $H$ is the pullback of the hyperplane class). Applying numerical inequalities from [APW94] to this divisor yields upper and lower bounds for the components $a$ and $b$ of $c_2(E)$.
2. Riemann-Roch Reduction: The Riemann-Roch theorem is applied to reduce the infinite set of possible integer pairs $(a, b)$ to a finite list of candidates for both $c_1 = 0$ and $c_1 = -1$.
3. Exclusion via Restriction and Vanishing: The remaining candidates are eliminated using several techniques:
   • Splitting Criteria: Utilizing criteria from [APW94] and [MP97] to determine when a bundle must split into a direct sum of line bundles based on the existence of global sections and the sign of Chern classes.
   • Restriction to Planes: Restricting the bundle to $\alpha$-planes and $\beta$-planes (isomorphic to $\mathbb{P}^2$). The author utilizes Fritzsche's classification [Fri83] of linear-uniform bundles on quadrics. If the restriction to every plane in a family is uniform, the bundle on $Q_4$ is constrained to be either split or a spinor bundle.
   • Vanishing Theorems: Applying Schneider's vanishing theorem (as recalled in [LN18]) and Kodaira-type vanishing results for nef and big bundles to derive contradictions for specific Chern class configurations.
   • Intersection Theory: Computing intersection numbers on $\mathbb{P}(E)$ to show that certain numerical cases would violate the nefness of the anticanonical divisor.

Key Contributions and Results
The main result, Theorem 1.1, provides a complete classification of rank 2 weak Fano bundles on $Q_4$ up to twisting by a line bundle. The theorem states that any such bundle $E$ falls into one of the following three categories:

1. Split Bundles: $E$ is a direct sum of two line bundles.
2. Spinor Bundles: $E$ is one of the two spinor bundles on $Q_4$.
3. Stable Bundles: $E$ is an $H_{Q_4}$-stable bundle with $c_1(E) = -H_{Q_4}$ and $c_2(E) = \alpha + \beta$. These correspond to the bundles constructed in [APW94, Example 2.2] via Serre's construction from two disjoint planes.

The paper systematically eliminates all other numerical possibilities derived in Section 3. Notably, the case $c_1=0$ is shown to yield only split bundles (Corollary 5.8), while the case $c_1=-1$ yields split bundles, the two spinor bundles, and the specific stable bundles with $c_2=(1,1)$ (Corollary 6.10).

Significance
The paper claims to complete the classification of rank 2 weak Fano bundles on smooth quadric hypersurfaces. While the Fano case was known and the weak Fano case for $n \geq 5$ was recently resolved, the four-dimensional case presented unique challenges due to the rank of the second Chow group. The significance lies in resolving these additional numerical cases by combining standard positivity arguments with the specific geometry of planes on $Q_4$. The result confirms that, despite the increased complexity of the numerical constraints in dimension 4, the classification remains structurally similar to the higher-dimensional cases: the bundles are either split, spinor, or the specific stable bundles previously identified in the Fano context. The work extends the understanding of weak Fano bundles on Fano varieties of Picard number one, complementing previous classifications on projective spaces and del Pezzo threefolds.