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Splitting of supervector bundles on projective superspaces

Charles Almeida, Ugo Bruzzo

TL;DR

The paper extends Horrocks-type splitting criteria to supervector bundles on the projective superspace $\mathbb{P}^{n|m}$. By leveraging the split and projected structure of $\mathbb{P}^{n|m}$ and reducing to the bosonic part, the authors show that arithmetically Cohen-Macaulay supervector bundles with vanishing intermediate cohomology split as a direct sum of even and odd line bundles when $n\ge2$ and $m\ge1$. The proof hinges on lifting a splitting from the bosonic reduction via obstruction vanishing in $H^{1}$, made possible by the fundamental exact sequence and projection-formula techniques. The result provides a concrete classification in the supergeometry setting and clarifies limitations by exhibiting non-splitting examples in low-dimensional cases, while highlighting the dependence on the split structure and the challenges for general superschemes like supergrassmannians.

Abstract

We provide a splitting criterion for supervector bundles over the projective superspaces $\mathbb{P}^{n|m}$. More precisely, we prove that a rank $p|q$ supervector bundle on $\mathbb{P}^{n|m}$ with vanishing intermediate cohomology is isomorphic to the direct sum of even and odd line bundles, provided that $n \geq 2$. For $n=1$ we provide an example of a supervector bundle that cannot be written as a sum of line bundles.

Splitting of supervector bundles on projective superspaces

TL;DR

The paper extends Horrocks-type splitting criteria to supervector bundles on the projective superspace . By leveraging the split and projected structure of and reducing to the bosonic part, the authors show that arithmetically Cohen-Macaulay supervector bundles with vanishing intermediate cohomology split as a direct sum of even and odd line bundles when and . The proof hinges on lifting a splitting from the bosonic reduction via obstruction vanishing in , made possible by the fundamental exact sequence and projection-formula techniques. The result provides a concrete classification in the supergeometry setting and clarifies limitations by exhibiting non-splitting examples in low-dimensional cases, while highlighting the dependence on the split structure and the challenges for general superschemes like supergrassmannians.

Abstract

We provide a splitting criterion for supervector bundles over the projective superspaces . More precisely, we prove that a rank supervector bundle on with vanishing intermediate cohomology is isomorphic to the direct sum of even and odd line bundles, provided that . For we provide an example of a supervector bundle that cannot be written as a sum of line bundles.
Paper Structure (6 sections, 8 theorems, 50 equations)

This paper contains 6 sections, 8 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preparatory results
  3. Superschemes.
  4. Projected and split superschemes.
  5. Projective superschemes.
  6. Splitting criteria for supervector bundles on superprojective spaces

Key Result

Theorem 1

Let $\mathcal{E}$ be a rank $p|q$ supervector bundle on $\mathbb{P}^{n|m}$, with $n \geq 2$ and $m \geq 1,$ such that $H^{i}_*(\mathbb{P}^{n}, \mathcal{E}) = 0$, for $1 \leq i \leq n-1$. Then there exists two sequences of integers $a_1 \geq a_2 \geq \ldots, \geq a_p$, and $b_1 \geq b_2 \geq \ldots,

Theorems & Definitions (20)

  • Theorem 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 5
  • proof
  • Remark 6
  • Definition 7
  • Lemma 8
  • proof
  • ...and 10 more