I am considering the following Calabi-Yau three-fold $X$ defined as a hypersurface in $\mathbb{P}^4$ by the vanishing of a holomorphic equation \begin{equation} f(z)=\{z_0^5+z_1^5+z_2^5+z_3^5+z_4^5=0\} \subset \mathbb{P}^4. \end{equation} In the following expression (a Fubini-Study-like ansatz) \begin{equation} K=\frac{1}{k\pi}\log s_{\alpha} h^{\alpha \overline{\beta}}\overline{s}_{\overline{\beta}} \end{equation}
I am only focusing on the $s_{\alpha}$.
They are sections of $\mathscr{O}_X(k)$ which means homogeneous functions of the coordinates $z_i$ of degree $k$ modulo $f=0$. In the particular case of the Fermat quintic, the $\{s_{\alpha}\}$ are chosen to be monomials of degree $k$ on $X$. For example: \begin{equation} z_iz_j, \: \: \: (0 \leq i \leq j \leq 4) \: \: \: (k=2) \\ z_iz_jz_k, \: \: \: (0 \leq i \leq j \leq k \leq 4) \: \: \: (k=3) \end{equation}
That is what I don’t understand : how such monomials can be related to the defining equation $f$ of the quintic hypersurface $X$ where only sum of single degree $5$ complex coordinates appear ?
For $d \geq 0$, I know that the sections of $\mathscr{O}_{\mathbb{P}^n}(d)$ are the homogeneous polynomials of degree $d$ in $n+1$ variables, but what does it mean for those sections to be « defined on the Fermat quintic $X$ » ?
I am learning basic theory of complex manifolds not as a professional, only math-enthusiast, so perhaps my question is very naive. Many thanks for any help.