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Tensor-generating line bundles on toric varieties

  • Daigo Ito
  • Michael R. Zeng
  • Xiangru Zeng
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Abstract

A line bundle is ⊗-generating if its powers generate the derived category of perfect complexes. Refining the derived Nakai--Moishezon criterion of Ito--Olander, we characterize ⊗-generating line bundles on proper toric varieties by a combinatorial criterion on the fan, leading to the notion of ⊗-generating fans characterizing ⊗-generating canonical bundles. Furthermore, we prove an analogue of Kleiman's criterion of ampleness for ⊗-generation. As applications, we provide several examples including: (i) smooth proper fourfolds with no ⊗-generating line bundle, by showing that products of two or more positive-dimensional proper toric varieties admit a ⊗-generating line bundle only when they are projective, (ii) families of smooth non-projective proper threefolds with ⊗-generating canonical bundles. Finally, we study subvarieties obstructing ⊗-generation of canonical bundles and implications to autoequivalences of derived categories.

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Cite

BibTeXRIS
BibTeX
@misc{ito2026tensorgenerating,
  title = {{Tensor-generating line bundles on toric varieties}},
  author = {Daigo Ito and Michael R. Zeng and Xiangru Zeng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22279},
  eprint = {2610.22279},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22279},
}
RIS
TY  - UNPB
TI  - Tensor-generating line bundles on toric varieties
AU  - Daigo Ito
AU  - Michael R. Zeng
AU  - Xiangru Zeng
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.22279
UR  - https://arxiv.org/abs/2610.22279
AB  - A line bundle is ⊗-generating if its powers generate the derived category of perfect complexes. Refining the derived Nakai--Moishezon criterion of Ito--Olander, we characterize ⊗-generating line bundles on proper toric varieties by a combinatorial criterion on the fan, leading to the notion of ⊗-generating fans characterizing ⊗-generating canonical bundles. Furthermore, we prove an analogue of Kleiman's criterion of ampleness for ⊗-generation. As applications, we provide several examples including: (i) smooth proper fourfolds with no ⊗-generating line bundle, by showing that products of two or more positive-dimensional proper toric varieties admit a ⊗-generating line bundle only when they are projective, (ii) families of smooth non-projective proper threefolds with ⊗-generating canonical bundles. Finally, we study subvarieties obstructing ⊗-generation of canonical bundles and implications to autoequivalences of derived categories.
ER  -