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| Format: | Preprint |
| Publié: |
2026
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| Accès en ligne: | https://arxiv.org/abs/2604.20054 |
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| _version_ | 1866910156835520512 |
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| author | Barz, Michael |
| author_facet | Barz, Michael |
| contents | Let $k$ be a field of characteristic $p,$ and $f : X \to S$ a smooth proper morphism of smooth $k$-schemes. Katz's formula gives a relationship between the Kodaira--Spencer map of $f,$ and an invariant called the $p$-curvature of the Gauss--Manin connection associated to $f.$ Recently, Lam--Litt proved a variant of Katz's formula in non-abelian Hodge theory, and suggested that it should be possible to give a more conceptual proof of their formula using the stacky approach to $p$-adic Hodge theory.
In this article, we realize their suggestion, explaining how the rather concrete phenomena observed by Katz and Lam--Litt can be explained in a conceptual way using sheared de Rham stacks, as developed by Bhatt--Kanaev--Vologodsky--Zhang and Drinfeld (though we prove a slightly different statement than Lam--Litt do). We do not assume the reader has any background in the theory of de Rham stacks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20054 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-Abelian p-Curvature and a Non-Abelian Katz's Formula Barz, Michael Algebraic Geometry Let $k$ be a field of characteristic $p,$ and $f : X \to S$ a smooth proper morphism of smooth $k$-schemes. Katz's formula gives a relationship between the Kodaira--Spencer map of $f,$ and an invariant called the $p$-curvature of the Gauss--Manin connection associated to $f.$ Recently, Lam--Litt proved a variant of Katz's formula in non-abelian Hodge theory, and suggested that it should be possible to give a more conceptual proof of their formula using the stacky approach to $p$-adic Hodge theory. In this article, we realize their suggestion, explaining how the rather concrete phenomena observed by Katz and Lam--Litt can be explained in a conceptual way using sheared de Rham stacks, as developed by Bhatt--Kanaev--Vologodsky--Zhang and Drinfeld (though we prove a slightly different statement than Lam--Litt do). We do not assume the reader has any background in the theory of de Rham stacks. |
| title | Non-Abelian p-Curvature and a Non-Abelian Katz's Formula |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2604.20054 |