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Let X be a smooth complex projective variety with basepoint x. We prove that every rigid integral irreducible representation $pi_1(X,x)to SL (3,{mathbb C})$ is of geometric origin, i.e., it comes from some family of smooth projective varieties. This partially generalizes an earlier result by K. Corlette and the second author in the rank 2 case and answers one of their questions.
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This paper contains three new results. {bf 1}.We introduce new notions of projective crystalline representations and twisted periodic Higgs-de Rham flows. These new notions generalize crystalline representations of etale 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. {bf 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. {bf 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 $D:={x_1,,...,x_n}$ for $ngeq 4$ and construct infinitely many geometrically absolutely irreducible $mathrm{PGL_2}(mathbb Z_p^{mathrm{ur}})$-crystalline representations of $pi_1^text{et}(mathbb{P}^1_{{mathbb{Q}}_p^text{ur}}setminus D)$. 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 $p$.
We prove a generalization of a conjecture of C. Marion on generation properties of finite groups of Lie type, by considering geometric properties of an appropriate representation variety and associated tangent spaces.
We propose a new method to construct rigid $G$-automorphic representations and rigid $widehat{G}$-local systems for reductive groups $G$. The construction involves the notion of euphotic representations, and the proof for rigidity involves the geometry of certain Hessenberg varieties.
We construct examples of smooth proper rigid-analytic varieties admitting formal model with projective special fiber and violating Hodge symmetry for cohomology in degrees $geq 3$. This answers negatively a question raised by Hansen and Li.
Algebraic hyperbolicity serves as a bridge between differential geometry and algebraic geometry. Generally, it is difficult to show that a given projective variety is algebraically hyperbolic. However, it was established recently that a very general surface of degree at least five in projective space is algebraically hyperbolic. We are interested in generalizing the study of surfaces in projective space to surfaces in smooth projective toric threefolds with Picard rank 2 or 3. Following Kleinschmidt and Batyrev, we explore the combinatorial description of smooth projective toric threefolds with Picard rank 2 and 3. We then use Haase and Iltens method of finding algebraically hyperbolic surfaces in toric threefolds. As a result, we determine many algebraically hyperbolic surfaces in each of these varieties.
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