Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups · arXivDesk
1311.6424Nov 21, 201360 pages (Updated Version). Subsume the manuscript with title "Semistable Higgs bundles and representations of algebraic fundamental groups: Positive characteristic case", arXiv:1210.8280. Accepted for publication in "Journal of European Mathematical Society"
Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups
Let k be the algebraic closure of a finite field of odd characteristic p and X a smooth projective scheme over the Witt ring W(k) which is geometrically connected in characteristic zero. We introduce the notion of Higgs-de Rham flow and prove that the category of periodic Higgs-de Rham flows over X/W(k)
Nearby in the stack
is equivalent to the category of Fontaine modules, hence further equivalent to the category of crystalline representations of the étale fundamental group
π1(XK)
of the generic fiber of
X
, after Fontaine-Laffaille and Faltings. Moreover, we prove that every semistable Higgs bundle over the special fiber
Xk
of
X
of rank
≤p
initiates a semistable Higgs-de Rham flow and thus those of rank
≤p−1
with trivial Chern classes induce
k
-representations of
π1(XK)
. A fundamental construction in this paper is the inverse Cartier transform over a truncated Witt ring. In characteristic
p
, it was constructed by Ogus-Vologodsky in the nonabelian Hodge theory in positive characteristic; in the affine local case, our construction is related to the local Ogus-Vologodsky correspondence of Shiho.