Existence of Rank Two Vector Bundles on Higher Dimensional Toric Varieties

TL;DR

This paper constructs indecomposable rank two vector bundles on certain Fano toric varieties, providing counterexamples to Hartshorne's conjecture in higher dimensions, but not on projective spaces.

Contribution

It introduces a new class of Fano toric varieties with indecomposable rank two bundles, expanding understanding of vector bundle structures beyond projective spaces.

Findings

Constructed indecomposable rank two bundles on specific Fano toric varieties.

Counterexamples to Hartshorne's conjecture in certain higher-dimensional cases.

Shows the conjecture remains open for projective spaces P^n.

Abstract

In the mid 70's, Hartshorne conjectured that, for all n > 7, any rank 2 vector bundles on P^n is a direct sum of line bundles. This conjecture remains still open. In this paper, we construct indecomposable rank two vector bundles on a large class of Fano toric varieties. Unfortunately, this class does not contain P^n