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Let $(M, \omega)$ denote a compact Kähler manifold. Since $d\omega =0$, $\omega$ represents a cohomology class in $H^2(M, \mathbb{R})$. Let $\rho$ denote the Ricci form of $M$, in local coordinates, we have $$\rho = \sqrt{-1} \ \text{Ric}_{i\overline{j}} dz^i \wedge d\overline{z}^j.$$

It is a well-known result that $\frac{1}{2\pi}\rho$ represents the first Chern class of $M$, i.e., $\left[ \frac{1}{2\pi} \rho \right] \in H^2(M, \mathbb{Z})$.

The Chern class may also be defined in a more intrinsic manner by means of the connecting homomorphism obtained from the exponential sequence of sheaves. This requires a discussion of divisors and the Picard group. This approach is presented in Chapter 1 of Griffiths and Harris.

I am currently writing some notes on the first Chern class and am at odds with how I want to define the first Chern class. I like the more intrinsic definition using the language of sheaves and line bundles, but feel that it is not clear why the image of this connecting homomorphism is represented by $\frac{1}{2\pi}\rho$.

The treatment in Griffiths and Harris is quite extensive and I feel that I am losing the forrest from the trees when looking at this treatment. Can anyone provide either some insight or references in which I may find a rather streamlined approach that introduces the first Chern class in an intrinsic manner, but also ends up proving that this must be represented by $\frac{1}{2\pi}$ times the Ricci form.

Please do not take this as any disrespect to the masterpiece that is Griffiths' and Harris' Principles of Algebraic Geometry.

Thanks in advance.

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    $\begingroup$ I hope the following comment helps a bit: I think the best definition of the first chern class for these types of questions is as the (unique) generator of the cohomology of $BS^1 \cong \mathbb{C}P^\infty$. Then you note that it lives in $H^2$ so it is actually the push-forward of the fundamental class of $\mathbb{C}P^1$. This shows that any relation between this definition and other definitions of the first chern class it is enough to prove it for $\mathbb{C}P^n$ (and in fact $\mathbb{C}P^1$). The rest follows from pulling back everything along the classifying map to some $\mathbb{C}P^n$. $\endgroup$ – Saal Hardali Jun 12 '18 at 12:45
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    $\begingroup$ look no further than the book of Huybrechts on complex geometry! $\endgroup$ – YangMills Jun 12 '18 at 20:14
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"Lectures on Kähler Geometry" by "Andrei Moroianu" computes the first Chern class of the canonical bundle in terms of the Ricci form in its Ch16.

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I recommend very much the book "Compact Manifolds with special holonomy" by Dominic Joyce and also, the lecture notes of Ballmann: http://people.mpim-bonn.mpg.de/hwbllmnn/archiv/kaehler0609.pdf

Moroianu class notes is also very pleasant!

One more reference is https://www.springer.com/la/book/9783540212904, by Huybrechts.

These are the references I studied on my master, I hope it helps.

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