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Given a perfect obstruction theory $E^\bullet$ over a space $X$, we know that if $X$ is smooth, that the virtual fundamental class $[X, E^\bullet]$ is given by

$$[X, E^\bullet] = c_{top}\big((E^{-1})^\vee\big)$$

Suppose now that we can write $X = A \times B$ where $B$ is smooth. Let $p : X \to A$ be the projection. Is there something that we can say about $p_*[X,E^\bullet]$? In particular, if $A$ were smooth then we could compute $p_*[X, E^\bullet]$ by integration along the fibre, $B$.

If $A$ is not smooth, then can we still obtain $p_*[X, E^\bullet]$ by integrating the top chern class of a relative obstruction bundle of some kind?

What if $A$ is 0-dimensional?

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Did you check the final section of Behrend-Fantechi? They define "relative obstruction theories" and the associated virtual classes. Also, in Behrend's follow-up paper on Gromov-Witten invariants, he uses the relative version to verify the Kontsevich-Manin axioms (so that might also be a place to check). –  Jason Starr Mar 28 '12 at 20:24
    
... I did not check that. I looked a lot at the comparison lemma, but I somehow missed that section. I'll go look now. –  Simon Rose Mar 28 '12 at 21:00
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