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.