Let $s_1,\ldots, s_k$ be linearly independent global holomorphic sections of a holomorphic line bundle $E$ over a compact algebraic manifold $X$, with volume form $\Omega$.
For $m$ large, let $\{e_1,\ldots, e_p\}$ be a basis of $H^0(X,L^m)$, where $L$ is ample. Write $\|e\|^2 := |e_1|^2 + \cdots + |e_p|^2$ and $\|s\|^2 := |s_1|^2 + \cdots + |s_k|^2$. If $t$ is a complex linear combination $\sum_{i,j} \lambda_{ij} e_is_j$, then $\int_X \frac{|t|^2}{\|e\|^2 \|s\|^2} \Omega$ is finite, being bounded above by $\sum_{i,j} |\lambda_{ij}|^2 vol_\Omega X$, up to some constant.
Conversely, for a global holomorphic section $t$ of $L^mE$ over $X$, does the finiteness of
$$\int_X \frac{|t|^2}{\|e\|^2 \|s\|^2} \Omega$$
imply that $t$ is a complex linear combination $\sum_{1\le i\le p,1\le j\le k} \lambda_{ij} e_is_j$?