Stable toric sheaves. I : Chern classes
Carl Tipler
TL;DR
The paper develops an inductive, toric-framework to construct rank-$2$ torus-equivariant torsion-free sheaves on $ ext{P}^n$ with prescribed Chern data. Central tools are Perling's reflexive-resolution and a novel factorization into saturated elementary injections, which yields explicit control of Chern polynomials via log-expansions. The author proves obstructions to smoothability in several codimension regimes, then uses the method to produce infinite families of stable, indecomposable examples on $ ext{P}^4$ and $ ext{P}^5$, and a general existence result for all $n\ge 3$ with prescribed Chern classes satisfying known constraints. These constructions illuminate Hartshorne's conjecture by linking deformations of torus-equivariant torsion-free sheaves to the existence of indecomposable rank-$2$ bundles, and they demonstrate the power of toric methods to generate new explicit examples beyond Horrocks–Mumford. The work also lays a foundation for broader prescription problems in higher rank and dimensions via the explicit Chern-class calculus for elementary injections.
Abstract
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 $n\geq 3$, with Chern classes satisfying all known constraints arising from locally freeness and indecomposability. We also provide simple obstructions for smoothability.