An inverse Cartier transform via exponential in positive characteristic

TL;DR

This paper introduces a new method to associate de Rham bundles to nilpotent Higgs bundles in positive characteristic using exponential functions, connecting to the inverse Cartier transform and leveraging Frobenius liftings.

Contribution

It proposes a novel construction of de Rham bundles via exponential functions for nilpotent Higgs bundles, relating to existing inverse Cartier transforms in positive characteristic.

Findings

Establishes the association's equivalence with Sheng-Xin-Zuo's construction in the geometric case.

Uses cocycle property of Frobenius liftings over W_2(k).

Supports the degeneration of the Hodge to de Rham spectral sequence.

Abstract

Let $k$ be a perfect field of odd characteristic $p$ and $X_0$ a smooth connected algebraic variety over $k$ which is assumed to be $W_2(k)$-liftable. In this short note we associate a de Rham bundle to a nilpotent Higgs bundle over $X_0$ of exponent $n\leq p-1$ 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 $W_2(k)$, which plays the key role in the proof of $E_1$-degeration of the Hodge to de Rham spectral sequence of $X_0$ due to P. Deligne and L. Illusie.