# Holomorphic line bundles associated to multiple U(1) groups, defined over toric manifolds

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.

• You want a line bundle over what space exactly? Over projective spaces all line bundles are of the form $\mathcal{O}(n)$ for some $n$. – Denis Nardin Nov 6 '17 at 8:41
• I am interested in line bundles over toric manifolds $X=(\mathbb{C}^N-\mathcal{P})/({\mathbb{C}^{\times}})^k$, especially the case where $k>1$. I would appreciate more knowledge about the $k=1$ case as well. – Mtheorist Nov 6 '17 at 8:53
• I don't really know much about toric manifolds, but as far as I can tell you are basically asking if there are interesting group homomorphisms $U(1)\to U(1)^M$ (since a line bundle is literally the same amount of data as an $U(1)$-principal bundle) – Denis Nardin Nov 6 '17 at 11:49
• I am trying 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 mathoverflow.net/questions/120081/…. It is explained in the comments over there that $\mathcal{O}(m,n)= p_1^*\mathcal{O}(m)\otimes p_2^*\mathcal{O}(m)$. – Mtheorist Nov 6 '17 at 12:47