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In the question titled "Line bundles and vector bundles on $\mathbb P^1 \times \mathbb P^1$" it was explained how any holomorphic line bundle on $\mathbb P^1 \times \mathbb P^1$ is of the form $\mathcal O(m,n) = p_1^*(\mathcal O(m)) \otimes p_2^*(\mathcal O(n))$, where $p_1$ and $p_2$ denote the projections on the two factors.

In addition, in "Monad Bundles in Heterotic String Compactifications", it is explained in Section 3.1 that for a complete intersection Calabi-Yau (CICY) (defined as an intersection of zero loci of polynomials in an ambient space, which is a product of $m$ projective spaces $\mathbb{P}^{n_i}$), denoted $X$, its holomorphic line bundles $\mathcal{O}_X(k^1,\ldots,k^m)$ can be obtained by restricting $\mathcal{O}_{\mathbb{P}^{n_1}}(k^1)\otimes\ldots\otimes\mathcal{O}_{\mathbb{P}^{n_m}}(k^m)$ to $X$.

I would like to know how to generalize the above to holomorphic line bundles on an arbitrary simplicial toric variety. Is it possible to show that a holomorphic line bundle on a simplicial toric variety $$ X=(\mathbb{C}^N \backslash U)/(\mathbb{C}^*)^m, $$ denoted $\mathcal{O}_X(k^1,\ldots,k^m)$, is equal to the restriction of a holomorphic line bundle of the form $$ \mathcal O_{X_1}(k^1) \otimes \mathcal O_{X_2}(k^2)\otimes\dots\otimes \mathcal O_{X_m}(k^m) $$ to $X$, where $X$ can be defined as an intersection of zero loci of polynomials in the ambient space $X_1\times X_2\ldots \times X_m$? All the $X_i$ here are spaces which have Picard number equal to 1.

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  • $\begingroup$ I do not quite understand your question. Are you asking whether the Picard group of a projective, toric variety with only finite quotient singularities has a $\mathbb{Z}$-basis of globally generated invertible sheaves? That is true simply because the Picard group is a finitely generated, free Abelian group and the ample cone is open. $\endgroup$
    – Jason Starr
    Commented Jan 22, 2018 at 13:01
  • $\begingroup$ I am not interested solely in projective toric varieties, I am interested in any simplicial toric variety. Such a variety can be defined as in equation 15.81 of claymath.org/library/monographs/cmim01c.pdf . $\endgroup$
    – Mtheorist
    Commented Jan 22, 2018 at 13:12
  • $\begingroup$ The footnote on that very page of the textbook repeats the well-known characterization of projective, toric varieties with only finite quotient singularities as the Geometric Invariant Theory quotients for the actions of certain connected multiplicative subgroup schemes of the coordinate torus with its standard action on complex affine space. Thus, your toric manifolds are projective, toric varieties. $\endgroup$
    – Jason Starr
    Commented Jan 22, 2018 at 14:56
  • $\begingroup$ @JasonStarr Thank you for pointing that out. Would I then be correct in saying that for the projective, toric varieties I am interested in, $\mathcal{O}_X(k^1,\ldots,k^m)=\iota^*(\mathcal{O}_{\mathbb{P}^{n_1}}(k^1)\otimes\ldots\otimes\mathcal{O}_{\mathbb{P}^{n_m}}(k^m))$, where $\iota$ is the inclusion map $\iota:X\rightarrow \mathbb{P}^{n_1}\times \ldots \times \mathbb{P}^{n_m}$? $\endgroup$
    – Mtheorist
    Commented Jan 22, 2018 at 15:24
  • $\begingroup$ For a projective, toric variety, there exists a closed immersion $\iota:X\hookrightarrow \mathbb{P}^{n_1}\times \dots \times \mathbb{P}^{n_m}$ such that every holomorphic invertible sheaf on $X$ is of the form $\iota^*(\mathcal{O}_{\mathbb{P}^{n_1}}(k_1)\boxtimes \dots \boxtimes \mathcal{O}_{\mathbb{P}^{n_m}}(k_m))$ for a unique ordered $m$-tuple of integers $(k_1,\dots,k_m)$. In fact, that is true for every projective variety whose Picard group is a finitely generated, free Abelian group. $\endgroup$
    – Jason Starr
    Commented Jan 22, 2018 at 15:28

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