# Degree of a line bundle corresponding to an effective divisor is positive

Let $M$ be a compact Kähler manifold with Kähler form $\omega$. For any line bundle $\mathcal{L}$ on $M$, we define the $\omega$-degree of $\mathcal{L}$ to be $deg(\mathcal{L}) :=\int_{M}c_{1}(\mathcal{L})\wedge \omega ^{n-1}$.

How does one show that if $D$ is an effective cartier divisor on $M$ and $\mathcal{O}(D)$ the corresponding line bundle, then $\deg \mathcal{O}(D) \geq 0$? The book I'm reading (Kobayashi's Differential geometry on complex vector bundles) states that the integral above actually becomes $\int_{D} \omega^{n-1}$ and thus concludes from there, but I can't figure out why it's true. Thanks in advance!

Note that $c_1(\mathcal{O}(D)) \in H^2(M; \mathbb{Z})$ is Poincaré dual to $[D] \in H_{2n-2}(M; \mathbb{Z})$; that is, $[D] = [M]\cap c_1(\mathcal{O}(D))$. Therefore,
• Can you please explain why $c_1 \mathcal{O}(D)$ is Poincare dual to $[D]$? Also, can you suggest a good reference to learn about these? – yojusmath Dec 3 '17 at 17:50