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The definition and basic properties of nef locally free sheaves appear for instance in the second volume of Lazarsfeld's book "Positivity in Algebraic Geometry" (beginning of chapter 6).

However, I am in a situation where some of the sheaves I deal with are not locally free, but only coherent; so I would like to know whether there is a well-behaved notion of nefness for coherent sheaves. The only mention of this that I found is at the end of section 1 of Kodaira Dimension of Subvarieties by Peternell-Schneider-Sommese, but it is just the definition with no references and no discussion of basic properties.

So my question is: is there a reference that gives an analogue of Theorem 6.2.12 in Lazarsfeld for nef coherent sheaves? (The results I'm mostly interested in are: (a) quotient of nef is nef; (b) pullback of nef is nef; and (c) extension of nef by nef is nef.)

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Definition A coherent sheaf $\mathcal{F}$ on an algebraic variety $X$ is nef if the following condition holds: For every irreducible curve $C\subset X$, the line bundle $O(1)$ is nef on $\mathbb{P}(\mathcal{F}|_C)$.

I haven't seen this definition of a nef coherent sheaf either, but I think most of the properties you mention just follow formally from the properties of ordinary nefness. Here is a proof for a) and b):

a) A quotient of a nef sheaf is nef. Let $C$ be as in the definition. If $F\to E\to 0$ is a surjection, this restricts to a surjection $F|_C\to E|_C\to 0$ and hence gives an embedding $\mathbb{P}(\mathcal{F}|_C)\hookrightarrow \mathbb{P}(\mathcal{F}|_C)$ such that $O(1)$ on $\mathbb{P}(\mathcal{F}|_C)$ restricts to $O(1)$ on $\mathbb{P}(\mathcal{E}|_C)$. Since the restriction of a nef line bundle is nef, $E$ is also nef.

b) A pullback of nef is nef. Similarily, if $f:X\to Y$ is finite, then $f$ restricts to a finite map $f:C'=f^{-1}C\to C$ and hence there is a finite map $F:\mathbb{P}(f^*\mathcal{F}|_C)\to \mathbb{P}(\mathcal{F}|_C)$ such that $O(1)=F^*O(1)$. Now the claim just follows from the corresponding statement for line bundles.

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