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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.

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

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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.

I should say that you may also want to look at the first two sections, but I think the specific explanation you want (along with quite a few references) is in Section 3.

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  • $\begingroup$ Thank you very much for the reference! $\endgroup$ Apr 3, 2021 at 22:58

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