Let $X$ be a smooth variety, $L$ a line bundle on $X$ and $G$ a reductive group actin on $X$ with a linearization of the action to $L$. Say we are over the complex numbers.
Both the concept of GIT stability and the Hilbert-Mumford weight are defined for any $L$ as above. However, the Hilbert-Mumford Criterion is proven just when $L$ is ample. I would like to know if there are any extensions to the non-ample case. (E.g. an easy case should be when $L$ is big and the point is in the open set where $L$ is ample).
References and counterexamples are welcome!
