Timeline for Holomorphic line bundles on arbitrary simplicial toric varieties as restrictions
Current License: CC BY-SA 3.0
8 events
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| Jan 23, 2018 at 6:39 | comment | added | Mtheorist | @JasonStarr I think the toric manifolds described in the textbook are not all projective. If you look at pages 367, 368, and 372, noncompact Calabi-Yaus such as $\mathcal{O}(-N)\rightarrow \mathbb{C}P^{N-1}$, $\mathcal{O}(-1)\oplus \mathcal{O}(-1)\rightarrow \mathbb{C}P^{1}$ and ALE spaces are also included, and I don't see how they can be immersed in compact projective spaces. | |
| Jan 22, 2018 at 15:52 | comment | added | Mtheorist | @JasonStarr That was very helpful. I noticed you used the external tensor product '$\boxtimes$' here. Does its definition agree with the one in ncatlab.org/nlab/show/external+tensor+product ? | |
| Jan 22, 2018 at 15:28 | comment | added | Jason Starr | 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. | |
| Jan 22, 2018 at 15:24 | comment | added | Mtheorist | @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 $\iota:X\rightarrow \mathbb{P}^{n_1}\times \ldots \times \mathbb{P}^{n_m}$? | |
| Jan 22, 2018 at 14:56 | comment | added | Jason Starr | 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. | |
| Jan 22, 2018 at 13:12 | comment | added | Mtheorist | 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 . | |
| Jan 22, 2018 at 13:01 | comment | added | Jason Starr | 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. | |
| Jan 22, 2018 at 12:48 | history | asked | Mtheorist | CC BY-SA 3.0 |