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I have seen both definitions and this is getting me more and more confused.

Are Chern classes dual to the degeneracy cycles of smooth sections or holomorphic?

They can't be the same thing, can they?

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    $\begingroup$ I believe you mean smooth sections rather than dual sections. $\endgroup$
    – Michael Albanese
    Commented Jul 15, 2015 at 16:47
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    $\begingroup$ Chern classes of a smooth complex vector bundle on a smooth manifold can be described in terms of cycles where generic sections vanish or are linearly dependent. In particular in the special case of a holomorphic bundle (on a complex manifold) this is true, if there are enough sections. $\endgroup$
    – Tom Goodwillie
    Commented Jul 15, 2015 at 17:28

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