I'm currently reading Min Ru's book Nevanlinna theory and its relation to Diophantine approximation and there's a bit that I'm confused. In Theorem A2.3.1, also known as the First Main theorem of Nevanlinna theory, it writes
Let $X$ be a complex projective variety. Let $f: \mathbb{C} \to X$ be holomorphic, $L \to X$ be a Hermitian line bundle, $s \in H^0(X,L)$ with $D = [s=0]$. Assume that $s \circ f \not\equiv 0$, then $$T_{f,L}(r) = m_f(r,D)+N_f(r,D)+O(1).$$
The functions $m_f$ and $N_f$ are the proximity and counting functions of $f$ respectively, and they are both defined with respect to a line bundle. Here $D$ is a very ample Cartier divisor, also viewed as a line bundle. What this theorem says is that the height $T_{f,L}(r)$ of $f$ with respect to the line bundle $L$ can be defined in terms of the proximity and counting functions of $f$ with respect to this $D$. I'm new to this and so I'm unsure about the following:
- What does $D = [s=0]$ mean here?
- I know that $s$ is a section $X \rightarrow L$. But if $s = 0$, how can we have $s \circ f \not\equiv 0$?