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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...')

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  • $\begingroup$ 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. $\endgroup$ Mar 16 '10 at 0:14
  • $\begingroup$ 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. $\endgroup$ Mar 16 '10 at 0:35
  • $\begingroup$ Here is a related question: mathoverflow.net/questions/204701/… $\endgroup$ May 11 '15 at 20:45
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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

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  • $\begingroup$ I can not comment Andrea's comment up there, so I do it here: you might be able to combine Charlie's comment and my answer to see, that Nakai-Moishezon criterion works for stacks with projective coarse moduli space (though I don't have the time and energy to work out the details...). $\endgroup$ Mar 16 '10 at 0:46
  • $\begingroup$ Their speculation that some smooth stack mey be hiding behind the singularities of the minimal model program seems particularly interesting. Has this line been pursued any further? $\endgroup$ Mar 16 '10 at 11:02
  • $\begingroup$ @Zsolt: Why is your first sentence true? I know this result only for tame DM-stacks. $\endgroup$ Aug 12 '13 at 8:19
  • $\begingroup$ @Lennart: A reference is Kresch--Vistoli "On coverings of DM-stacks and surjectivity of the Brauer map" Lemma 2. $\endgroup$ May 8 '15 at 16:04
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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.

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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.

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