Vector bundles without intermediate cohomology and the trichotomy result

2024/02/11 by Giorgio Ottaviani, Ottaviani, Giorgio · 2 citations
Mathematics · Physics and Astronomy · #14-02 #14F08 #14F17 #14M15 #14M17 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2402.07254

openalex publication_date 2024/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Horrocks proved in 1964 that vector bundles on Pⁿ without intermediate cohomology split as direct sum of line bundles. This result has been the starting point of a great research activity on other varieties, showing interesting connections with derived categories and other areas. We follow some paths into this fascinating story, which has classical roots. The story has a culmination with the trichotomy result (finite/tame/wild) for arithmetically Cohen Macaulay (ACM) varieties obtained by Faenzi and Pons-Llopis in 2021. This is an expanded version of the talk given at the conference "Homemade Algebraic Geometry" in July 2023 at Alcalá de Henares celebrating Enrique Arrondo's 60th birthday.

Cited by

• Rank 2 aCM and Ulrich bundles on Fano and Calabi--Yau double coverings of ℙ³
• Split distributions on Grassmann manifolds and smooth quadric hypersurfaces

Related

• An elementary proof that vector bundles on \mathbb P¹ split
• Linear and Steiner bundles on projective varieties
• Splitting of Vector Bundles on Toric Varieties
• Rank 3 arithmetically Cohen-Macaulay bundles on hypersurfaces
• Arithmetically Cohen-Macaulay Bundles on Threefold Hypersurfaces