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 by Faltings [Algebraic Analysis, Geometry, and Number Theory (1989)] and Fontaine and Laffaille [Ann. Sci. Éc. Norm. Supér. (4) 15 (1983)] and periodic Higgs–de Rham flows introduced by Lan, Sheng and Zuo [J. Eur. Math. Soc. (JEMS) 21 (2019)]. 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 \mathbb{P}^1 with logarithmic structure on marked points D:=\{x_1, \dots,x_n\} for n\geq 4 and construct infinitely many geometrically absolutely irreducible \operatorname{PGL}_2(\mathbb{Z}_{p}^{ur} )-crystalline representations of {\pi_1^{\textup{\'{e}t}}(\mathbb{P}^{1}_{\mathbb{Q}_p^{\mathrm{ur}}}\setminus D)} . We find an explicit formula of the self-map for the case \{0, 1, \infty, \lambda\} 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 \mathcal{C}_\lambda defined by y^2=x(x-1)(x-\lambda) with the order coprime to p .

<abstract><p>Let $ f:\, X\to Y $ be a semistable non-isotrivial family of $ n $-folds over a smooth projective curve with discriminant locus $ S \subseteq Y $ and with general fiber $ F $ of general type. We show the strict Arakelov inequality</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ {\deg f_*\omega_{X/Y}^\nu \over {{{\rm{rank\,}}}} f_*\omega_{X/Y}^\nu} &lt; {n\nu\over 2}\cdot\deg\Omega^1_Y(\log S), $\end{document} </tex-math></disp-formula></p> <p>for all $ \nu\in \mathbb N $ such that the $ \nu $-th pluricanonical linear system $ |\omega^\nu_F| $ is birational. This answers a question asked by Möller, Viehweg and the third named author ^{[<xref ref-type="bibr" rid="b1">1</xref>]}.</p></abstract>

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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...

As a corollary of nonabelian Hodge theory, Simpson proved a strong Lefschetz theorem for complex polarized variations of Hodge structure. We show an arithmetic analog. Our primary technique is p-adic nonabelian Hodge theory.