I have developed a keen interest in understanding the Calabi-Yau manifold. I have been following "Lectures on Kähler Geometry" by Andrei Moroianu and several online resources. However, it seems that I have become lost in the details, particularly when trying to comprehend the computation of the Chern class. It has been rather unintuitive for me thus far.
Proposition 16.1. To every complex vector bundle $E$ over a smooth manifold $M$ one can associate a cohomology class $c_1(E) \in H^2(M, \mathbb{Z})$ called the first Chern class of $E$ satisfying the following axioms:
(1) Naturality: For every smooth map $f: M \rightarrow N$ and complex vector bundle $E$ over $N$, one has $f^*\left(c_1(E)\right)=c_1\left(f^* E\right)$, where the left term denotes the pull-back in cohomology and $f^* E$ is the pull-back bundle defined by $f^* E_x=E_{f(x)}, \forall x \in M$. (2) Whitney sum formula: For every bundles $E, F$ over $M$ one has $c_1(E \oplus F)=c_1(E)+c_1(F)$, where $E \oplus F$ is the Whitney sum defined as the pull-back of the bundle $E \times F \rightarrow M \times M$ by the diagonal inclusion of $M$ in $M \times M$.
(3) Normalization: The first Chern class of the tautological bundle of $\mathbb{C P}^1$ is equal to -1 in $H^2\left(\mathbb{C P}^1, \mathbb{Z}\right) \simeq \mathbb{Z}$, which means that the integral over $\mathbb{C P}^1$ of any representative of this class equals -1 .
Another one,
Let $E$ be a complex vector bundle over a manifold $M$, and let $F=dA+A\land A$ be the curvature two-form of a connection $A$ on $E$. We define the total chern class $c(E)=\det(1+\frac{i}{2\pi}E)=1+c_1(E)+\cdots$ where $c_k(E)\in H^{2k}(M,\mathbb R)$.
To overcome this challenge, I have decided to narrow my focus and study a single particular example in order to gain a more comprehensive understanding and motivation (like study only that part which is needed in this process otherwise it's seemed I will never complete the goal). Specifically, I am planning to work with the complex torus. My objective is to demonstrate that the complex torus possesses Kähler metrics with a vanishing first Chern class.
Unfortunately, I have been unable to find a single resource that provides a detailed explanation of these concepts for a complex torus. I would greatly appreciate it if anyone could suggest a resource that covers these aspects thoroughly. Thank you in advance. TIA
If you are happy to make use of Chern--Weil theory, then one can proceed as follows: A complex torus is a quotient of $\mathbf{C}^n$ by a lattice $\Lambda$ of maximal rank. The Euclidean metric $\omega_{\delta}$ on $\mathbf{C}^n$ is flat, Kähler, and invariant under translations. Hence, $\mathbf{T}^n = \mathbf{C}^n/ \Lambda$ admits a flat Kähler metric; in particular, its Ricci curvature vanishes identically $\text{Ric}(\omega_{\delta}) =0$.
The first Chern class $c_1(X) : = c_1(-K_X)$ is represented by $1/2\pi$ times the Ricci curvature of a Kähler metric. Hence, $$c_1(\mathbf{T}^n) = c_1(-K_{\mathbf{T}^n}) = \frac{1}{2\pi} \{ \text{Ric}(\omega_{\delta}) \} = 0.$$