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Vector bundles without intermediate cohomology and the trichotomy result

Giorgio Ottaviani

Abstract

Horrocks proved in 1964 that vector bundles on $P^n$ 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.

Vector bundles without intermediate cohomology and the trichotomy result

Abstract

Horrocks proved in 1964 that vector bundles on 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.
Paper Structure (11 sections, 25 theorems, 22 equations)

This paper contains 11 sections, 25 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. The Horrocks criterion, the spinor bundles, classical roots on ACM bundles
  3. The Horrocks criterion on ${\mathbb P}^n$ and its classical roots
  4. Spinor bundles and the splitting criterion on quadrics
  5. Improvements of Horrocks criterion for small rank
  6. Grassmannians and beyond, the appearance of many ACM bundles
  7. The approach with the derived category
  8. The Grothendieck groups and a bit of K-theory
  9. The derived category of ${\mathbb P}^n$ and $Q_n$
  10. Cohomological characterizations via derived category
  11. The trichotomy for ACM varieties

Key Result

Theorem 1.1

Horrocks criterion (1964)Hor Let $E$ be a vector bundle on ${\mathbb P}^n({\mathbb C})$. $E=\oplus_{i=1}^r{\mathcal{O}}(a_i)$ for some integers $a_i$$\Longleftrightarrow$$H^i(E(t))=0,\quad 1\le i\le n-1, \forall t\in{\mathbb Z}$.

Theorems & Definitions (30)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3: G. Gherardelli
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7: Evans-Griffith criterion (1981)
  • Theorem 1.8: Kumar-Peterson-Rao (2003)
  • Theorem 1.9: Ancona-Peternell-Wisniewski 1994, Malaspina 2009
  • Theorem 1.11: Arrondo-Costa
  • ...and 20 more