# 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...')

• I don't know the answer, but it should make sense, as we can define line bundles on stacks and we can define projective stacks, so presumably we can have very ample line bundles on stacks, in particular. Also, there's some intersection theory on stacks, but again, I don't know much. Mar 16 '10 at 0:14
• Yes, I know one can define some intersection theory, but I'm not sure that copying the usual definitions is the right thing to do. For example, should we define ample bundles by Serre vanishing or by Nakai-Moishezon's criterion, or something else? In the scheme case, they are of course equivalent, but I don't know for stacks. Mar 16 '10 at 0:35
• Here is a related question: mathoverflow.net/questions/204701/… May 11 '15 at 20:45