Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDe Rham comparison and Poincare duality for rigid varieties
200
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Over any smooth algebraic variety over a $p$-adic local field $k$, we construct the de Rham comparison isomorphisms for the etale cohomology with partial compact support of de Rham $mathbb Z_p$-local systems, and show that they are compatible with Poincare duality and with the canonical morphisms among such cohomology. We deduce these results from their analogues for rigid analytic varieties that are Zariski open in some proper smooth rigid analytic varieties over $k$. In particular, we prove finiteness of etale cohomology with partial compact support of any $mathbb Z_p$-local systems, and establish the Poincare duality for such cohomology after inverting $p$.
rate research
Read More
On any smooth algebraic variety over a $p$-adic local field, we construct a tensor functor from the category of de Rham $p$-adic etale local systems to the category of filtered algebraic vector bundles with integrable connections satisfying the Griffiths transversality, which we view as a $p$-adic analogue of Delignes classical Riemann--Hilbert correspondence. A crucial step is to construct canonical extensions of the desired connections to suitable compactifications of the algebraic variety with logarithmic poles along the boundary, in a precise sense characterized by the eigenvalues of residues; hence the title of the paper. As an application, we show that this $p$-adic Riemann--Hilbert functor is compatible with the classical one over all Shimura varieties, for local systems attached to representations of the associated reductive algebraic groups.
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.
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 $p$.
In this short notes, we prove a stronger version of Theorem 0.6 in our previous paper arXiv:1709.01485: Given a smooth log scheme $(mathcal{X} supset mathcal{D})_{W(mathbb{F}_q)}$, each stable twisted $f$-periodic logarithmic Higgs bundle $(E,theta)$ over the closed fiber $(X supset D)_{mathbb{F}_q}$ will correspond to a $mathrm{PGL}_r(mathbb{F}_{p^f})$-crystalline representation of $pi_1((mathcal{X} setminus mathcal{D})_{W(mathbb{F}_q)[frac{1}{p}]})$ such that its restriction to the geometric fundamental group is absolutely irreducible.
We use the Beilinson $t$-structure on filtered complexes and the Hochschild-Kostant-Rosenberg theorem to construct filtrations on the negative cyclic and periodic cyclic homologies of a scheme $X$ with graded pieces given by the Hodge-completion of the derived de Rham cohomology of $X$. Such filtrations have previously been constructed by Loday in characteristic zero and by Bhatt-Morrow-Scholze for $p$-complete negative cyclic and periodic cyclic homology in the quasisyntomic case.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
