Projective Crystalline Representations of Étale Fundamental Groups and Twisted Periodic Higgs-de Rham Flow

Abstract

This paper contains three new results. 1.We introduce new notions of projective crystalline representations and twisted periodic Higgs-de Rham flows. These new notions generalize crystalline representations of étale fundamental groups introduced in [7, 10] and periodic Higgs-de Rham flows introduced in [19]. We establish an equivalence between the categories of projective crystalline representations and twisted periodic Higgs-de Rham flows via the category of twisted Fontaine-Faltings module which is also introduced in this paper. 2.We study the base change of these objects over very ramified valuation rings and show that a stable periodic Higgs bundle gives rise to a geometrically absolutely irreducible crystalline representation. 3. We investigate the dynamic of self-maps induced by the Higgs-de Rham flow on the moduli spaces of rank-2 stable Higgs bundles of degree 1 on P 1 with logarithmic structure on marked points D := {x1, ..., xn} for n ≥ 4 and construct infinitely many geometrically absolutely irreducible PGL2(Z ur p )-crystalline representations of π et 1 (P 1We find an explicit formula of the selfmap for the case {0, 1, ∞, λ} and conjecture that a Higgs bundle is periodic if and only if the zero of the Higgs field is the image of a torsion point in the associated elliptic curve C λ defined by y 2 = x(x − 1)(x − λ) with the order coprime to p.2.2. The category of projective representations 2.3. Gluing representations and projective representations 2.4. Comparing representations associated to local Fontaine-Faltings modules underlying isomorphic filtered de Rham sheaves 2.5. The functor D P 3. Twisted periodic Higgs-de Rham flows 3.1. Higgs-de Rham flow over X n ⊂ X n+1 3.2. Twisted periodic Higgs-de Rham flow and equivalent categories 3.3. Twisted Higgs-de Rham self map on moduli schemes of semi-stable Higgs bundle with trivial discriminant 3.4. Sub-representations and sub periodic Higgs-de Rham flows 4. Constructing crystalline representations of étale fundamental groups of p-adic curves via Higgs bundles 4.1. Connected components of the moduli space M 4.2. Self-maps on moduli spaces of Higgs bundles on P 1 with marked points 4.3. An explicit formula of the self-map in the case of four marked points. 4.4. Lifting of twisted periodic logarithmic Higgs-de Rham flow on the projective line with marked points and strong irreducibility 4.5. Examples of dynamics of Higgs-de Rham flow on P 1 with four-marked points 4.6. Question on periodic Higgs bundles and torsion points on the associated elliptic curve. 4.7. Question on ℓ-adic representation and ℓ-to-p companions. 4.8. Projective F -units crystal on smooth projective curves 5. Base change of twisted Fontaine-Faltings modules and twisted Higgs-de Rham flows over very ramified valuation rings 5.1. Notations in the case of Spec k. 5.2. Base change in the small affine case. 5.3.Categories and Functors on proper smooth variety over very ramified valuation ring W π 5.4. degree and slope 6. Appendix: explicit formulas 6.1. Self-map 6.2. Multiple by p map on ellipt...

Nearby cycles and semipositivity in positive characteristic

“…The same error appears independently in the first version of [SYZ,Theorem 3.10], where the authors claimed existence of a certain map on the open subset of the moduli space of semistable sheaves, parameterizing reflexive sheaves. However, in case of [SYZ,Theorem 3.10], it is not so easy to adjust the arguments adding additional assumptions (this would require at least Lemma 1.7 and repeating the proof of [La3,Theorem 11]). So Theorem 2.2 offers in this case the only available proof.…”

“…Theorem 2.2 generalizes [La3,Theorem 11] to the case of logarithmic Higgs sheaves with possibly non-trivial Chern classes. It also generalizes [SYZ,Theorems 3.6 and 3.10], which deal with systems of Hodge sheaves of rank r < p on X defined over k = Fp . In this last case Theorem 2.2 allows to compute higher Chern classes of twisted preperiodic Higgs bundles.…”

“…A special case of Theorem 0.5 (see Theorem 2.2) generalizes [La3,Theorem 11] to the case of logarithmic Higgs sheaves with possibly non-trivial Chern classes and fills in a gap in its proof. The stronger version is indispensable for the proofs of [SYZ,Theorem 3.6,. In this last case Theorem 2.2 allows to compute higher Chern classes of twisted preperiodic Higgs bundles.…”