The sections of the holomorphic line bundle $\mathcal{O}(n)$ are acted on by the covariant derivative $$ d+nA, $$ where $A$ is the connection on the $U(1)$ bundle to which $\mathcal{O}(n)$ is associated.
My question is, are there holomorphic line bundles associated to $U(1)^M$ bundles, where $M$ is some integer? The sections of such a holomorphic line bundle would be acted on by the covariant derivative
$$ d+\sum_b^M n_b A_b, $$ where $n_b$ denotes the representation w.r.t. each $U(1)$ factor. It would be natural to denote such a line bundle as $\mathcal{O}(n_1,n_2,\ldots,n_b)$. Do such holomorphic line bundles exist in the mathematical literature? References would be highly appreciated.
Edit: To be more precise, I am interested in the holomorphic line bundles described in the previous paragraph defined over toric manifolds $X=(\mathbb{C}^N-\mathcal{P})/({\mathbb{C}^{\times}})^k$, especially the case where $k>1$.
(Here $\mathcal{P}$ denotes a subset of $\mathbb{C}^N$ fixed under a continuous subgroup of $({\mathbb{C}^{\times}})^k$.)
In particular, I would like to understand the generalization of the line bundles over $\mathbb{P}^1 \times \mathbb{P}^1$ which are denoted as $\mathcal{O}(m,n)$, mentioned in Line bundles and vector bundles on $\mathbb P^1 \times \mathbb P^1$. There, it is explained in the comments that $\mathcal{O}(m,n)= p_1^*\mathcal{O}(m)\otimes p_2^*\mathcal{O}(n)$, where $p_1$ and $p_2$ denote the projections onto the two factors.
