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3 added 12 characters in body

See this question.

Also, read Milnor-Stasheff or Hatcher's book "Vector Bundles and K-Theory". In particular, Milnor-Stasheff and Hatcher prove that there is a unique "theory of Chern classes" for complex vector bundles over topological spaces satisfying axioms totally analogous to the C0, C1, C2, C3. Milnor-Stasheff constructs Chern classes using Steenrod squaresthe Thom isomorphism theorem; Hatcher constructs them using the Leray-Hirsch theorem. Hatcher's construction (in topology) is essentially the same as Grothendieck's construction (in algebraic geometry). I think if you study the two constructions (Hatcher's and Grothendieck's) carefully, their equivalence should follow fairly easily. I did this once a while ago.

I don't think you need any GAGA theorem. I think you just need the fact that there is an analytification functor.

Appendix C of Milnor-Stasheff then proves the equivalence with the Chern-Weil theory.

See this question.

Also, read Milnor-Stasheff or Hatcher's book "Vector Bundles and K-Theory". In particular, Milnor-Stasheff and Hatcher prove that there is a unique "theory of Chern classes" for complex vector bundles over topological spaces satisfying axioms totally analogous to the C0, C1, C2, C3. Milnor-Stasheff constructs Chern classes using Steenrod squares; Hatcher constructs them using the Leray-Hirsch theorem. Hatcher's construction (in topology) is essentially the same as Grothendieck's construction (in algebraic geometry). AnywayI think if you study the two constructions (Hatcher's and Grothendieck's) carefully, so this proves their equivalence of Grothendieck's definition of Chern classes in algebraic geometry with Chern classes in topologyshould follow fairly easily. I did this once a while ago.

I don't think you need any GAGA theorem. I think you just need the fact that there is an analytification functor.

Appendix C of Milnor-Stasheff then proves the equivalence with the Chern-Weil theory.

1

See this question.

Also, read Milnor-Stasheff or Hatcher's book "Vector Bundles and K-Theory". In particular, Milnor-Stasheff and Hatcher prove that there is a unique "theory of Chern classes" for complex vector bundles over topological spaces satisfying axioms totally analogous to the C0, C1, C2, C3. Milnor-Stasheff constructs Chern classes using Steenrod squares; Hatcher constructs them using the Leray-Hirsch theorem. Hatcher's construction (in topology) is essentially the same as Grothendieck's construction (in algebraic geometry). Anyway, so this proves equivalence of Grothendieck's definition of Chern classes in algebraic geometry with Chern classes in topology.

Appendix C of Milnor-Stasheff then proves the equivalence with the Chern-Weil theory.