# 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.