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I would like to know if there is a notion of normal bundle to a quotient map. In one specific case, let $G$ be a finite group acting on an algebraic variety $V$. Denote by the map $\pi:V \rightarrow V/G$ the quotient map. Is it possible to define the normal bundle to $\pi$ having good properties. For example, I would like to get the projection formula. If $V/G$ is a substack of $X$ of co-dimension $r,$ do we get a vector bundle of rank $r$ on $V$ as the normal bundle to the composition $V \rightarrow V/G \rightarrow X$? I am interested in the case where $V/G$ is a Deligne-Mumford stack.

I would like to know the conditions under which the equality $$i^*(i_*(x))=x.c_r(N)$$ for $x\in A^*(V)$ holds.

Here, $N$ is the desired normal bundle. What is the condition under which the Chern classes of this bundle exist?

The question may not be stated correctly and clearly. I apologize about it.

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  • $\begingroup$ Are you looking at the quotient " $V/G$ " as a GIT quotient V//G or as a Deligne-Mumford stack [V/G] ? $\endgroup$
    – Qfwfq
    Commented Apr 19, 2010 at 14:13

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For any map $f:X\to Y$ of smooth stacks, you can define a virtual normal bundle $N_{X/Y}$ as $[f^*T_Y]-[T_X]$. That would be an element of the K-ring $K^*(X)$.

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    $\begingroup$ I like your name. $\endgroup$
    – Dan Petersen
    Commented Apr 19, 2010 at 20:43

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