dg.differential geometry – Chern class cohomology coefficients complex / real / integral?


I read Chern classes from Kobayashi and Nomizu.

Given a vector bundle $ pi: E rightarrow M $ with fiber $ mathbb {C} ^ r $ and group $ GL (r, mathbb {C}) $ They connect for everyone $ k leq r $ a cohomology class of $ M $ and call it that $ k $Chern class. It looks like Cohomology is $ H ^ * (M, mathbb {C}) $ and not $ H ^ * (M, mathbb {R}) $, Can someone explain what happens here?

given $ pi: E rightarrow M $, To let $ p: P rightarrow M $ be the associated $ GL (r, mathbb {C}) $ bundle up. We have
$$ text {det} left ( lambda I- frac {1} {2 pi sqrt {-1}} X right) = sum_ {k = 0} ^ rf_k (X) lambda ^ {rk} $$ to the $ X in mathfrak {gl} (r, mathbb {C}) $,
Here $ f_k: mathfrak {gl} (r, mathbb {C}) rightarrow mathbb {C} $ are $ GL (r, mathbb {C}) $ immutable degree $ k $ homogeneous poly. on $ mathfrak {gl} (r, mathbb {C}) $, These $ f_k $ can be seen as $ GL (r, mathbb {C}) $ invariant symmetric multlinear map
$$ underbrace { mathfrak {gl} (r, mathbb {C}) times cdots times mathfrak {gl} (r, mathbb {C})} _ {k- text {times}} rightarrow mathbb {C} $$
to give an element of $ I _ { mathbb {C}} (G) $,

After fixing a connection to the main package $ P (M, G) $There is a complex version of Weil Homomorphism $ I _ { mathbb {C}} (G) rightarrow H ^ * (M, mathbb {C}) $, These $ f_k in I ^ k _ { mathbb {C}} (G) $ gives an element $ c_k $ in the $ H ^ {2k} (M, mathbb {C}) $, But they write $ c_k in H ^ {2k} (M, mathbb {R}) $,

What is missing here?

Does that mean $ c_k in H ^ {2k} (M, mathbb {C}) $ is picture of an element in $ H ^ {2k} (M, mathbb {R}) $ under a map $ H ^ {2k} (M, mathbb {R}) rightarrow H ^ {2k} (M, mathbb {C}) $? One that trigger $ mathbb {R} rightarrow mathbb {C} $ defined as $ a mapsto a + i 0 $?

EDIT: The book "Calculus to Kohomology" by Ib Madsen and Jxrgen Tornehave says in a note $ 18.12 (Page $ 189) The

definition $ 18.3 (from the Chern class) gives cohomology classes $ H ^ * (M, mathbb {C}) $but actually all classes lies in real cohomology, This results from (previous result).

There was no clear explanation (for me) for this comment.

EDIT: User Jessica L (last seen 7 years ago) said

Classes of chunks can be defined by topological means (see Milnor's book on characteristic classes), which yields elements $ c_k (V) in H ^ {2k} (M; mathbb {Z}) $, The normalization in the Chern-Weil theory is chosen so that the corresponding elements of de Rham are cohomology groups $ H ^ {2k} (M; mathbb {R}) $ align with the integral elements and integrate to integers

I think that answer and my question are related. Therefore any reference (containing more details) is welcome.