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In derived category, there is a slogan, "cohomology is bad, chain complex is good". In the theory of characteristic classes, we could associate a vector bundle to cohomology classes of the base space.

Can we lift up (naturally) these cohomology classes to an actual complex?

I am particularly interested in Chern class and vector bundle over an algebraic variety over complex numbers.

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    $\begingroup$ You can realize Chern classes as obstruction classes (obstructions to sections of the associated complex frame bundle), which themselves have a direct geometric definition as obstruction cocycles. Actually explicitly specifying these obstruction cocyles is more difficult. $\endgroup$ – Tobias Shin Jan 28 at 20:24
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    $\begingroup$ Maybe I'm misunderstanding something but in the slogan it's cohomology groups that are bad. Cohomology classes are fine (in fact they can be interpreted as maps between complexes) $\endgroup$ – Denis Nardin Jan 28 at 20:25
  • $\begingroup$ @DenisNardin You're right, I misinterpret the slogan a little bit. What I want is like Tobias said, lifting up the cohomology group to a chain complex, e.g. de Rham or simplicial, and then looking for particular nice representatives of characteristic classes in that complex. $\endgroup$ – userabc Jan 28 at 20:40
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    $\begingroup$ @userabc That's kind of the opposite of what the slogan is suggesting to do, but in Chern-Weil theory you construct explicit representatives for the Chern classes in the deRham complex, and that has some applications. $\endgroup$ – Denis Nardin Jan 28 at 20:42
  • $\begingroup$ As Tobias mentions, obstruction cocycles is the formulation that's natural at the cochain complex level. But depending on exactly what you want, you might not like this formalism -- it requires working rather explicitly with the tangent bundle. Perhaps tell us what kind of thing you are looking for. $\endgroup$ – Ryan Budney Jan 28 at 21:17

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