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In https://arxiv.org/pdf/hep-th/0005247.pdf, on page 60 and 61, it is mentioned that the connection of $\mathcal{O}(-n)$ over a (simplicial) toric variety of the form $$ (\mathbb{C}^N \backslash U)/(\mathbb{C}^*) $$ is $$ A= -n{i\over 2}{\displaystyle~\sum \!{}_{i=1}^N (\overline{\phi}_i{d} \phi_i-{d}\overline{\phi}_i \phi_i)~\over \displaystyle \sum \!{}_{i=1}^NQ_i|\phi_i|^2}, $$ in terms of complex homogeneous coordinates $(\phi_i,\overline{\phi}_i)$ which obey the equivalence relation

$$ (\phi_1,\dots,\phi_N)\sim (\lambda^{Q_1} \phi_1, \ldots, \lambda^{Q_N} \phi_N), \\ $$ with the weights $Q_i\in \mathbb Z$ and $\lambda\in \mathbb{C}^* = \mathbb{C}-\{0\}$.

My question is, how does the connection generalize to the case of a general (simplicial) toric variety $$ (\mathbb{C}^N \backslash U)/(\mathbb{C}^*)^m, $$ which contains a set of homogeneous coordinates $(\phi_i,\overline{\phi}_i)$ equipped with a number $m$ of equivalence relations $$ (\phi_1,\dots,\phi_N)\sim (\lambda_r^{Q^{(r)}_1} \phi_1, \ldots, \lambda_r^{Q^{(r)}_N} \phi_N), \\ $$ for $r=1,\dots, m$ with the weights $Q^{(r)}_i\in \mathbb Z$ and $\lambda_r\in \mathbb{C}^* = \mathbb{C}-\{0\}$?

Since the Picard group for a general simplicial toric variety is $\mathbb{Z}^{N-m}$, an arbitrary line bundle on such a variety is of the form $\mathcal{O}(k_1,k_2,\ldots,k_{N-m})$, so I believe the connection should be expressible in terms of the integers $k_1,k_2,\ldots,k_{N-m}$.

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