This is a general question that I have. Let $X$ be a projective variety over an algebraically closed field $k$. Let $L$ be an ample line bundle over $X$. Let $F$ be a vector bundle on $X$. We say that $F$ is $L$-semistable if for any coherent subsheaf $0\neq E\subset F$ of strictly lesser rank, $\mu\_L(E)\leq\mu\_L(F)$. Here $\mu\_L(F)=\frac{c\_1(F).L^{n-1}}{r}$, where $r=$ rank$(F)$ and $n=dim\ X$. 


My question is, why do we need $L$ to be ample? The definitions and basic properties seem to go through without ampleness. Even the fact that semistability is an open condition does not seem to depend on the fact that $L$ is ample, and existence of Harder-Narasimhan filtration/Jordan Holder filtration too seems to hold without ampleness. Why can't we just take $L$ to be a globally generated or nef and big line bundle for example.


$\textbf{Edit:}$


If $L$ is a big and nef bundle, it seems like the following will go through still,


a) Definition and basic properties of like there is no nontrivial homomorphism of semi stable sheaves from one with higher slope to one with lesser slope etc.


b) Existence and uniqueness of Harder-Narasimhan and Jordan-Holder filtrations


c) stability/semistability is an open condition.


Am I mistaken. Is there some subtle/obvious point where we need ampleness for this?



Why do we need ampleness in the definition of stability/semistability
You are right, the condition that $L$ be ample can be weakened. In fact, on an $n$-dimensional normal projective variety $X$ one can measure (semi-)stability with respect to an arbitrary *movable* curve class $\alpha \in N\_1(X)\_{\mathbb R}$. A numerical curve class $\alpha$ is said to be movable if $D \cdot \alpha \ge 0$ for all effective divisors $D$ on $X$. Then most of the basic properties you mentioned continue to hold.


The classical case ($L$ ample) is recovered by taking $\alpha$ to be the class of a complete intersection curve $C = H\_1 \cap \cdots \cap H\_{n-1}$, where the $H\_i$ are general members of some very ample linear system $|mL|$, $m$ sufficiently large.


This (along with several applications) is worked out in the paper "[Movable curves and semistable sheaves](http://arxiv.org/abs/1408.4308)" by Greb, Kebekus, and Peternell.


You might also be interested in the survey "[Moduli of vector bundles on higher-dimensional base manifolds - Construction and Variation](http://arxiv.org/abs/1503.00319)" by Greb, Ross, and Toma, where this machinery is applied.


