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.

  • 2
    $\begingroup$ 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
  • 4
    $\begingroup$ look no further than the book of Huybrechts on complex geometry! $\endgroup$ – YangMills Jun 12 '18 at 20:14

"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.

| cite | improve this answer | |

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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.