Suppose $E$ is a rank $r$ vector bundle over a projective variety $X$, denote the zero section by $i\colon X\to E$. Given an effective cycle $a\in A_{k+r}(E)$, the Gysin pullback gives us a class $i^![a]\in A_k(E)$. Can this class always be represented by an effective cycle on $X$?
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2$\begingroup$ Since Gysin pullback for the zero section is, by definition, the inverse of the flat pullback isomorphism, you can phrase your question in terms of whether flat pullback of a non-effective cycle may be effective. Indeed it may: for a rank $1$ vector bundle $E\to X$ whose first Chern class is nonzero, nonetheless the image of the zero section is an effective cycle on $E$ that is rationally equivalent to the pullback of the first Chern class. $\endgroup$– Jason StarrCommented Oct 12, 2015 at 15:19
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$\begingroup$ "nonzero" --> "non-effective". Sorry for the typo. $\endgroup$– Jason StarrCommented Oct 12, 2015 at 16:17
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1$\begingroup$ Don't you want to end up in $A_k(X)$? If I'm not misreading, then consider $E = \mathcal O(-1)$ over $X = \mathbb P^1$, and $a=[$the zero section$]$. $\endgroup$– Allen KnutsonCommented Oct 12, 2015 at 21:58
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