Horrocks correspondence on arithmetically Cohen–Macaulay varieties

Malaspina, Francesco; Rao, Aroor

doi:10.2140/ant.2015.9.981

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Horrocks correspondence on arithmetically Cohen–Macaulay varieties

Francesco Malaspina and A. Prabhakar Rao

Vol. 9 (2015), No. 4, 981–1003

DOI: 10.2140/ant.2015.9.981

Abstract

We describe a vector bundle $ℰ$ on a smooth $n$ -dimensional arithmetically Cohen–Macaulay variety in terms of its cohomological invariants $H_{*}^{i} (ℰ)$ , $1 \leq i \leq n - 1$ , and certain graded modules of “socle elements” built from $ℰ$ . In this way we give a generalization of the Horrocks correspondence. We prove existence theorems, where we construct vector bundles from these invariants, and uniqueness theorems, where we show that these data determine a bundle up to isomorphism. The cases of the quadric hypersurface in $ℙ^{n + 1}$ and the Veronese surface in $ℙ^{5}$ are considered in more detail.

Keywords

vector bundles, cohomology modules, Horrocks correspondence, smooth ACM varieties

Mathematical Subject Classification 2010

Primary: 14F05

Secondary: 14J60

Milestones

Received: 6 October 2014

Revised: 23 February 2015

Accepted: 7 April 2015

Published: 30 May 2015

Authors

Francesco Malaspina
Dipartimento di Scienze Matematiche Politecnico di Torino Corso Duca degli Abruzzi 24 I-10129 Torino Italy

A. Prabhakar Rao
Department of Mathematics University of Missouri – St. Louis Saint Louis, MO 63121 United States

Vol. 9, No. 4, 2015