Stable toric sheaves. I : Chern classes

Abstract

We reduce Hartshorne's conjecture on indecomposable rank 2 vector bundles in the semistable case to the non-existence of smoothable rank 2 semistable toric sheaves on the projective space. We then study Chern classes for rank 2 torsion-free toric sheaves. For the reflexive ones, we derive a simple formula for the Chern polynomial, and in the general torsion-free case we introduce an iterative construction method based on elementary injections, allowing us to prescribe Chern classes. This yields infinite families of explicit examples on P4 and P5, and establishes existence on Pn for all n greater than 3, with Chern classes satisfying all known constraints arising from locally freeness and indecomposability. We also provide simple obstructions for smoothability.

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