A Topos-Theoretic Approach to the Logarithmic Cartier Transform

@misc{fersi2026topostheoretic,
  title = {{A Topos-Theoretic Approach to the Logarithmic Cartier Transform}},
  author = {Sami Fersi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27549},
  eprint = {2609.27549},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27549},
}

TY  - UNPB
TI  - A Topos-Theoretic Approach to the Logarithmic Cartier Transform
AU  - Sami Fersi
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.27549
UR  - https://arxiv.org/abs/2609.27549
AB  - This article is the second of three articles whose goal is to generalize the Cartier transform of Ogus and Vologodsky to the logarithmic setting. We generalize a topos-theoretic version of this transform, due to Oyama. Let k be a perfect field of positive characteristic p and equip S=Speck with the trivial log structure. For a log smooth morphism of logarithmic schemes X → S, we construct crystalline-like ringed topoi ℰ' and underlineℰ and subcategories of crystals of quasi-coherent modules 𝒞' and underline𝒞, equivalent respectively, under some lifting assumption, to modules with Higgs fields and integrable connections, both satisfying certain nilpotence conditions, and a morphism of topoi underlineℰ → ℰ'. We then prove that the pullback functor of this morphism of topoi preserves quasi-coherent crystals and hence induces a functor 𝒞' → underline𝒞, generalizing the Cartier transform. We finally use a log flat descent theorem for morphisms, that we proved in the first article, to prove that this functor is fully faithful.
ER  -