Positivity in stack geometry I was wondering how much of the theory say of Lazarsfeld books can be carried to algebraic stacks (if this has been done).
Do we have a sensible notion of an ample (big, nef) line bundle? Of an ample vector bundle? How many of the usual results carry over? Do we have multiplier ideal sheaves? Are the usual vanishing theorems valid in this settings? And so on.
I hope the question even makes sense. I am complete beginner with stacks, so it may even turn out that the relevant objects cannot be defined and the question is too naive.
EDIT: It seems from the answers below that there is some tentative notion of positivity for line bundles. But I'd be more interested in knowing whether something has been done for vector bundles, or multiplier ideal sheaves, and whether vanishing theorems other than Kodaira are known to hold in this context.
Another interesting question (but here I'm really wildly speculating) would be if there are notions of plurisubharmonic functions over (complex) differentiable stacks, and associated analytic multiplier ideals.
Please note that a negative answer would be of equal interest to me ('no, we don't know yet how to generalize these objects...')
 A: For a DM stack (of finite type) every line bundle has some power, which is the pullback of some line bundle on the coarse moduli space (assuming that our stack has a coarse moduli space). You can define ample, big and nef this way, I think. Then for example Kodaira vanishing holds, as far as I know. Here is an article in this subject: http://math.berkeley.edu/~molsson/KVrev2.pdf
A: Whatever you mean by ample line bundle on a DM (say) stack, you cannot of course require that some power of the bundle embeds the stack in projective space.  You could ask that some power of the line bundle embeds the stack into a weighted projective stack, but this imposes restrictions on the kinds of stacks you will be talking about.  This is studied in a preprint by Abramovich and Hassett where they call such stacks cyclotomic.  If you define ampleness in terms of some other positivity (like Kleiman's criterion, Nakai-Moishezon, etc), then you will have many of the same theorems as in the case of varieties - because more or less this positivity will just be "pulled back" from the coarse moduli space.  So the answer depends on the situation you are in and the kinds of properties in which you are interested.
A: Perhaps this   paper of Andrew Kresch will be helpful. He discusses the notion of a quasi-projective DM  stack at length and gives several characterizations of this type of quasi-projectivity.
