Is there literature about chern classes of vector bundles on DM-stacks? I had a look at a lot of different papers about intersection theory on stacks and related stuff and this seems to be known, but I couldn't find a good reference on this topic. The strongest statement I could find was "there are chern classes in the Chow-cohomology satisfying all the usual properties".

A particular thing I am interested in, is the projection formula, i.e. given a proper morphism of DM-stacks $p:M\longrightarrow N$, a vectorbundle $E$ on $N$ and a cycle $\alpha\in A_*(M)_\mathbb{Q}$ is it true that $p_*(c_i(p^*E)\cap\alpha)=c_i(E)\cap p_*\alpha$? I guess this is one of the usual properties, but does it hold for any proper morphism, or only representable ones?

What I did was trying to check the definition of the Chow-cohomology from Vitsoli's paper on Intersection theory on stacks for chern classes, but I got already stuck at the compatibility with the Gysin homomorphism, because of my poor knowledge about normal bundles and stacks... so I gave up trying to prove it myself.

Can anyone help me out with a hint to more detailed literature, or some hints how the proof of the projection formula works? I would also be happy with some precise statement under which circumstances the projection formula holds. Thank you in advance.


1 Answer 1


The formula holds for arbitrary proper morphisms. If $X \to Y$ is a proper morphism of DM stacks, where $X$ has finite inertia (the hypotheses in my paper are more stringent, but the theory has been refined since then), there exists a finite map $V \to X$, where $V$ is an algebraic space, and the pushforward from the rational Chow group of $V$ to that of $X$ is surjective. This allows to reduce the non-representable case to the representable one.

About the more general question, I am reasonably certain that the construction of Chern classes in Fulton's book goes through in the formalism in my old paper without any major problem, but it's been a while since I wrote it, so it's hard to be completely positive.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.