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Cohomological splitting conditions of vector bundles on $\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_s}$

Damian Maingi

TL;DR

The paper extends regularity and vector-bundle splitting criteria from biprojective to multiprojective spaces $\mathbb P^{n_1}\times\cdots\times\mathbb P^{n_s}$. By defining $(p_1,\dots,p_s)$-regularity and employing the Künneth formula, it develops aCM- and regularity-based vanishing criteria that imply that rank $r$ bundles split as sums of twisted line bundles $\mathcal O(t_i,\dots,t_i)$ under suitable hypotheses. When $Reg(E)=0$, the results further identify specific summands, including line bundles tied to standard basis directions and twisted wedge bundles $\mathcal O\boxtimes\Omega^{a}_{\mathbb P^{n_i}}(a+1)$, giving concrete splitting decompositions. These findings generalize Ballico–Malaspina’s work to higher-dimensional multiprojective products and provide a framework for cohomological splitting criteria in this broader setting.

Abstract

In this paper we extend the results of Ballico and Malaspina on regularity and splitting conditions on multiprojective spaces $\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_s}$.

Cohomological splitting conditions of vector bundles on $\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_s}$

TL;DR

The paper extends regularity and vector-bundle splitting criteria from biprojective to multiprojective spaces . By defining -regularity and employing the Künneth formula, it develops aCM- and regularity-based vanishing criteria that imply that rank bundles split as sums of twisted line bundles under suitable hypotheses. When , the results further identify specific summands, including line bundles tied to standard basis directions and twisted wedge bundles , giving concrete splitting decompositions. These findings generalize Ballico–Malaspina’s work to higher-dimensional multiprojective products and provide a framework for cohomological splitting criteria in this broader setting.

Abstract

In this paper we extend the results of Ballico and Malaspina on regularity and splitting conditions on multiprojective spaces .
Paper Structure (4 sections, 9 theorems, 37 equations)

This paper contains 4 sections, 9 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Splitting Criteria for Vector Bundles on Multiprojective spaces
  3. Main Results
  4. Acknowledgment

Key Result

Theorem 1.2

Let $X$ and $Y$ be projective varieties over a field $k$. Let $\mathscr{F}$ and $\mathscr{G}$ be coherent sheaves on $X$ and $Y$ respectively. Let $\mathscr{F}\boxtimes\mathscr{G}$ denote $p_1^*(\mathscr{F})\otimes p_2^*(\mathscr{G})$ then ${H^m(X\times Y,\mathscr{F}\boxtimes\mathscr{G}) \cong \bigo

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 1.2: Künneth formula
  • Lemma 1.3
  • Remark 1.4
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3: Theorem 1.3 3
  • Lemma 2.4: Theorem 1.4 3
  • Lemma 2.5: Theorem 3.5, 3
  • Theorem 3.1
  • ...and 5 more