Rendiconti del Seminario Matematico della Università di Padova


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Volume 134, 2015, pp. 47–91
DOI: 10.4171/RSMUP/134-2

Published online: 2015-11-17

On logarithmic nonabelian Hodge theory of higher level in characteristic $p$

Sachio Ohkawa[1]

(1) University of Tokyo, Japan

Given a natural number $m$ and a log smooth integral morphism $X\to S$ of fine log schemes of characteristic $p>0$ with a lifting of its Frobenius pull-back $X'\to S$ modulo $p^{2}$, we use indexed algebras ${\mathcal A}_{X}^{gp}$, ${\mathcal B}_{X/S}^{(m+1)}$ of Lorenzon-Montagnon and the sheaf ${\cal D}_{X/S}^{(m)}$ of log differential operators of level $m$ of Berthelot-Montagnon to construct an equivalence between the category of certain indexed ${\mathcal A}^{gp}_{X}$-modules with ${\mathcal D}_{X/S}^{(m)}$-action and the category of certain indexed ${\mathcal B}_{X/S}^{(m+1)}$-modules with Higgs field. Our result is regarded as a level $m$ version of some results of Ogus-Vologodsky and Schepler.

Keywords: Log geometry, log $\mathcal D$-module, HIggs module, Cartier transform

Ohkawa Sachio: On logarithmic nonabelian Hodge theory of higher level in characteristic $p$. Rend. Sem. Mat. Univ. Padova 134 (2015), 47-91. doi: 10.4171/RSMUP/134-2