# An inverse Cartier transform via exponential in positive characteristic

Guitang Lan, Mao Sheng, Kang Zuo

Number | 12 |
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Authors |
Dr. Mao Sheng
Guitang Lan Kang Zuo |

Year | 2012 |

Let k be a perfect field of odd characteristic p and X0 a smooth connected algebraic variety over k which is assumed to be W2(k)-liftable. In this short note we associate a de Rham bundle to a nilpotent Higgs bundle over X0 of exponent n≤p−1 via the exponential function. Presumably, the association is equivalent to the inverse Cartier transform of A. Ogus and V. Vologodsky for these Higgs bundles. However this point has not been verified in the note. Instead, we show the equivalence of the association with that of Sheng-Xin-Zuo in the geometric case. The construction relies on the cocycle property of the difference of different Frobenius liftings over W2(k), which plays the key role in the proof of E1-degeration of the Hodge to de Rham spectral sequence of X0 due to P. Deligne and L. Illusie.