p-curvature in non-commutative Hodge theory and the Kontsevich-Soibelman operad

Abstract

Let 𝒞 be a differential ℤ/2-graded category over ℂ. Its periodic cyclic homology HH^per_*(𝒞), when viewed as a vector bundle over the formal punctured disk, is equipped with a canonical connection ∇^𝒞_{∂_t} called the Getzler-Gauss-Manin connection in the t-direction (or the categorical t-connection). Our main result is that when 𝒞 is smooth and proper, this connection has a regular singularity at t=0 and quasi-unipotent monodromy, affirming a conjecture of Katzarkov-Kontsevich-Pantev \cite{KKP}. Our proof follows a reduction mod p argument using a spreading out technique of Toën \cite{To} and a regularity criterion of Katz \cite{Ka1}. The main novelty is the proof of a multiplicative property of the p-curvature of ∇^𝒞_{∂_t} through an interpretation in terms of the two-colored Kontsevich-Soibelman operad. We then explore two applications of the main result. First, we give an explicit description (under additional assumptions) of the non-commutative Hodge filtration on the periodic cyclic homology of a smooth proper d(ℤ/2)g category, following a construction of Shklyarov \cite{Shk}. The second application, which is special to our particular method or proof, is an upper bound on the sizes of Jordan blocks of the monodromy of ∇^𝒞_{∂_t}, which simultaneously generalizes Scherk's local monodromy theorem for isolated hypersurface singularities \cite{Sche} and (partially) a recent result of Pomerleano-Seidel on the quantum connection of a closed monotone symplectic manifold \cite{PS2}. As a specialization, we show that the sizes of these Jordan blocks are bounded above by the diagonal dimension of 𝒞 plus one.

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Cite

BibTeX RIS

BibTeX

@misc{chen2026pcurvature,
  title = {{p-curvature in non-commutative Hodge theory and the Kontsevich-Soibelman operad}},
  author = {Zihong Chen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26765},
  eprint = {2609.26765},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26765},
}

RIS

TY  - UNPB
TI  - p-curvature in non-commutative Hodge theory and the Kontsevich-Soibelman operad
AU  - Zihong Chen
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26765
UR  - https://arxiv.org/abs/2609.26765
AB  - Let 𝒞 be a differential ℤ/2-graded category over ℂ. Its periodic cyclic homology HH^per_*(𝒞), when viewed as a vector bundle over the formal punctured disk, is equipped with a canonical connection ∇^𝒞_∂_t called the Getzler-Gauss-Manin connection in the t-direction (or the categorical t-connection). Our main result is that when 𝒞 is smooth and proper, this connection has a regular singularity at t=0 and quasi-unipotent monodromy, affirming a conjecture of Katzarkov-Kontsevich-Pantev \cite{KKP}. Our proof follows a reduction mod p argument using a spreading out technique of Toën \cite{To} and a regularity criterion of Katz \cite{Ka1}. The main novelty is the proof of a multiplicative property of the p-curvature of ∇^𝒞_∂_t through an interpretation in terms of the two-colored Kontsevich-Soibelman operad. We then explore two applications of the main result. First, we give an explicit description (under additional assumptions) of the non-commutative Hodge filtration on the periodic cyclic homology of a smooth proper d(ℤ/2)g category, following a construction of Shklyarov \cite{Shk}. The second application, which is special to our particular method or proof, is an upper bound on the sizes of Jordan blocks of the monodromy of ∇^𝒞_∂_t, which simultaneously generalizes Scherk's local monodromy theorem for isolated hypersurface singularities \cite{Sche} and (partially) a recent result of Pomerleano-Seidel on the quantum connection of a closed monotone symplectic manifold \cite{PS2}. As a specialization, we show that the sizes of these Jordan blocks are bounded above by the diagonal dimension of 𝒞 plus one.
ER  -

Abstract

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