A family of indecomposable rank-$n$ vector bundles on $\mathbb P^n\times\mathbb P^n$ in positive characteristics

TL;DR

This work constructs a family of indecomposable rank-$n$ vector bundles on $X=\mathbb{P}^n\times\mathbb{P}^n$ over a field of positive characteristic by taking kernels of canonical Frobenius-based maps to twists of a divisor $\mathcal{A}\in |L+h|$. The bundles are built as kernels of $p_h^*F^{a*}Q_h\to \mathcal{O}_{k\mathcal{A}}(qL)$ with $q=p^a$ and $1\le k\le q$, yielding indecomposable objects not arising as pullbacks from a factor, and the authors develop a detailed theory of their Chern classes, cohomology, and restriction behavior. They establish strong nondegeneracy results, deduce vanishing and global generation properties of twists, analyze jumping lines, and prove an A-symmetry relating the two factors under factor interchange. The construction blends Frobenius techniques with monad-like descriptions and opens a path toward moduli interpretations via divisors in $|L+h|$, enriching the landscape of indecomposable vector bundles in positive characteristic.

Abstract

We construct a sequence of rank-$n$ indecomposable vector bundles on $\mathbb P^n\times\mathbb P^n$ for every $n\geq 2$ and in every positive characteristic that are not pullbacks via any map $\mathbb P^n\times\mathbb P^n\to \mathbb P^{m} $

Theorems & Definitions (14)

• Theorem
• Lemma 1
• proof
• Example 2
• Corollary 3
• proof
• Lemma 5
• Remark 6
• Corollary 7
• proof