Stable toric sheaves. I : Chern classes

We study rank 2 torus-equivariant torsion-free sheaves on the complex projective space. For reflexive sheaves 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 ${\mathbb{P}}^4$ and ${\mathbb{P}}^5$, and establishes existence on ${\mathbb{P}}^n$ for all $n\ge 3$, with Chern classes satisfying all known constraints arising from locally freeness and indecomposability. We also provide simple obstructions for smoothability.