Nonabelian Hodge theory in positive characterstic via exponential twisting

Abstract

Let $k$ be a perfect field of odd 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 $\le p-1$ over $X/k$ and the category of nilpotent flat sheaves of exponent $\le p-1$ over $X/k$, and 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 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 $p$-adic Simpson correspondence in nonabelian Hodge theory and Langer's algebraic proof of Bogomolov inequality for semistable Higgs bundles and Miyaoka-Yau inequality.

13 pages. Subsume "An inverse Cartier transform via exponential in positive characteristic" arxiv 1205.6599

Cited by in corpus (4)

• Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups
• Intersection de Rham complexes in positive characteristic
• Uniformization of $p$-adic curves via Higgs-de Rham flows
• The $p$-curvature conjecture for the non-abelian Gauss-Manin connection