# Some questions concerning reflexive and saturated sheaves

Disclaimer: I'm a Ph.D. in complex differential geometry, so algebraic geometry is not my strongest area, but it is an area of great interest to me.

Work over $$\mathbb{C}$$, and let $$(X,\mathcal{O}_X)$$ be a smooth variety. Here are some definitions:

1. Declare an $$\mathcal{O}_X$$-submodule $$\mathcal{F} \subset \mathcal{T}_X$$ to be saturated if the quotient $$\mathcal{T}_X / \mathcal{F}$$ is torsion-free.

2. A coherent sheaf $$\mathcal{E}$$ is said to be reflexive if it is isomorphic to its double dual via the canonical map.

Fact: A torsion-free sheaf $$\mathcal{S}$$ is reflexive if and only if there is a locally free sheaf $$\mathcal{V}$$ such that $$\mathcal{S} \subset \mathcal{V}$$ and $$\mathcal{V}/\mathcal{S}$$ is torsion-free.

Q1. Do algebraic geometers refer only to $$\mathcal{O}_X$$-submodules of the tangent sheaf as saturated, or can they be defined more generally (for any sheaf)?

Q2. The fact claimed above elucidates a relationship between reflexive and saturated sheaves. How closely related are these notions, and how do algebraic geometers think about these objects intuitively (geometrically)?

Q3. Can reflexive sheaves be non-coherent? And would one say that a coherent sheaf is reflexive if $$\mathcal{E}$$ is isomorphic to $$\mathcal{E}^{\ast \ast}$$ but the isomorphism is not given by the canonical map?

I apologise in advance if any of these questions are too elementary, and please let me know if further clarification is required :)

Update: I have found instances (for example, these notes of Campana: https://mast.queensu.ca/~mikeroth/proceedings/Campana-Survey-Special-Manifolds.pdf ) which discuss saturated subsheaves of $$\Omega_X^1$$, which partially addresses the first question.

• This might be better suited for Math Overflow, but welcome! I am also quite bad with algebraic geometry so I can't help here. Mar 30, 2021 at 20:58

I've never worked with reflexive sheaves in any meaningful way, but Section 3 of these notes by Karl Schwede (https://www.math.utah.edu/~schwede/Notes/GeneralizedDivisors.pdf) explain how on a normal scheme (basically one whose singular locus has codimension at least $$2$$), the usual correspondence between Cartier divisors and line bundles/invertible sheaves can be extended to a correspondence between Weil divisors and reflexive sheaves given via the same $$\mathcal O_X(D)$$ construction (as a subsheaf of the sheaf of rational functions on $$X$$). As one would hope, Cartier divisors can be characterized among Weil divisors as those $$D$$ for which the reflexive sheaf $$\mathcal O_X(D)$$ is invertible.