Nonabelian Hodge theory in positive characteristic via exponential twisting

Let $k$ be a perfect field of positive characteristic and $X$ a smooth algebraic variety over $k$ which is $W_2$-liftable. We show that the exponent twisiting of the classical Cartier descent gives an equivalence of categories between the category of nilpotent Higgs sheaves of exponent $\leq p$ over $X/k$ and the category of nilpotent flat sheaves of exponent $\leq p$ over $X/k$, by showing that it is equivalent up to sign to the inverse Cartier and Cartier transforms for these nilpotent objects constructed in the nonabelian Hodge theory in positive characteristic by Ogus–Vologodsky. In view of the crucial role that Deligne–Illusie’s lemma has ever played in their algebraic proof of $E_1$-degeneration of the Hodge to de Rham spectral sequence and Kodaira vanishing theorem in abelian Hodge theory, it may not be overly surprising that again this lemma plays a significant role via the concept of Higgs–de Rham flow in establishing a $p$-adic Simpson correspondence in the nonabelian Hodge theory and Langer’s algebraic proof of Bogomolov inequality for semistable Higgs bundles and Miyaoka–Yau inequality.

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Accepted 8 July 2014

Published 20 May 2015