A comparison of logarithmic overconvergent de Rham–Witt and log-crystalline cohomology for projective smooth varieties with normal crossing divisor

2017 ◽  
Vol 137 ◽  
pp. 229-235
Author(s):  
Andreas Langer ◽  
Thomas Zink
Keyword(s):  
Crystalline Cohomology ◽  
Normal Crossing ◽  
De Rham

scholarly journals The Filtered Poincaré Lemma in Higher Level (With Applications to Algebraic Groups)

2008 ◽  
Vol 191 ◽  
pp. 79-110
Author(s):  
Bernard Le Stum ◽  
Adolfo Quirós
Keyword(s):  
Differential Forms ◽  
Algebraic Groups ◽  
Classical Case ◽  
Group Scheme ◽  
Crystalline Cohomology ◽  
Hodge Filtration ◽  
Rham Complex ◽  
De Rham ◽  
Poincaré Lemma ◽  
Invariant Differential

AbstractWe show that the Poincaré lemma we proved elsewhere in the context of crystalline cohomology of higher level behaves well with regard to the Hodge filtration. This allows us to prove the Poincaré lemma for transversal crystals of level m. We interpret the de Rham complex in terms of what we call the Berthelot-Lieberman construction and show how the same construction can be used to study the conormal complex and invariant differential forms of higher level for a group scheme. Bringing together both instances of the construction, we show that crystalline extensions of transversal crystals by algebraic groups can be computed by reduction to the filtered de Rham complexes. Our theory does not ignore torsion and, unlike in the classical case (m = 0), not all invariant forms are closed. Therefore, close invariant differential forms of level m provide new invariants and we exhibit some examples as applications.

scholarly journals Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz theorem

2019 ◽  
Vol 155 (5) ◽  
pp. 1025-1045
Author(s):  
Christopher Lazda ◽  
Ambrus Pál
Keyword(s):  
Perfect Field ◽  
Global Function ◽  
Crystalline Cohomology ◽  
Tate Conjecture ◽  
Global Function Fields ◽  
Counter Example ◽  
Rigid Cohomology ◽  
De Rham ◽  
First Chern Class ◽  
Special Fibre

In this paper we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, showing that for $k$ a perfect field of characteristic $p$ , a rational (logarithmic) line bundle on the special fibre of a semistable scheme over $k\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$ lifts to the total space if and only if its first Chern class does. The proof is elementary, using standard properties of the logarithmic de Rham–Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. Finally, we give a counter-example to show that the variational Tate conjecture for divisors cannot hold with $\mathbb{Q}_{p}$ -coefficients.

scholarly journals Crystalline Cohomology of Towers of Curves

2018 ◽  
Vol 2020 (21) ◽  
pp. 7454-7488
Author(s):  
Jan Vonk
Keyword(s):  
Modular Curves ◽  
De Rham Cohomology ◽  
Crystalline Cohomology ◽  
Etale Cohomology ◽  
Étale Cohomology ◽  
De Rham ◽  
Integral Structures

Abstract We investigate the geometry of finite maps and correspondences between curves, and construct canonical trace and pullback maps between Hyodo–Kato integral structures on de Rham cohomology of curves, which are functorial for finite morphisms of the generic fibres. This leads to a crystalline version of the étale cohomology of towers of modular curves considered by Hida and Ohta, whose ordinary part satisfies $\Lambda $-adic control and Eichler–Shimura theorems.

scholarly journals Smoothing toroidal crossing spaces

2021 ◽  
Vol 9 ◽  
Author(s):  
Simon Felten ◽  
Matej Filip ◽  
Helge Ruddat
Keyword(s):  
Moduli Spaces ◽  
Frobenius Manifold ◽  
Compact Form ◽  
Homotopy Equivalence ◽  
Main Approach ◽  
Fano Manifolds ◽  
Normal Crossing ◽  
De Rham ◽  
Potential Applications ◽  
Infinitesimal Deformations

Abstract We prove the existence of a smoothing for a toroidal crossing space under mild assumptions. By linking log structures with infinitesimal deformations, the result receives a very compact form for normal crossing spaces. The main approach is to study log structures that are incoherent on a subspace of codimension 2 and prove a Hodge–de Rham degeneration theorem for such log spaces that also settles a conjecture by Danilov. We show that the homotopy equivalence between Maurer–Cartan solutions and deformations combined with Batalin–Vilkovisky theory can be used to obtain smoothings. The construction of new Calabi–Yau and Fano manifolds as well as Frobenius manifold structures on moduli spaces provides potential applications.

scholarly journals LOGARITHMIC DE RHAM COMPARISON FOR OPEN RIGID SPACES

2019 ◽  
Vol 7 ◽  
Author(s):  
SHIZHANG LI ◽  
XUANYU PAN
Keyword(s):  
Comparison Theorem ◽  
Local Systems ◽  
Analytic Variety ◽  
Normal Crossing ◽  
Analytic Varieties ◽  
De Rham ◽  
Simple Normal Crossing ◽  
Ramified Coverings

In this note, we prove the logarithmic $p$ -adic comparison theorem for open rigid analytic varieties. We prove that a smooth rigid analytic variety with a strict simple normal crossing divisor is locally $K(\unicode[STIX]{x1D70B},1)$ (in a certain sense) with respect to $\mathbb{F}_{p}$ -local systems and ramified coverings along the divisor. We follow Scholze’s method to produce a pro-version of the Faltings site and use this site to prove a primitive comparison theorem in our setting. After introducing period sheaves in our setting, we prove aforesaid comparison theorem.

scholarly journals LOGARITHMIC DE RHAM–WITT COMPLEXES VIA THE DÉCALAGE OPERATOR

2021 ◽  
pp. 1-64
Author(s):  
Zijian Yao
Keyword(s):  
New Formalism ◽  
Crystalline Cohomology ◽  
De Rham

Abstract We provide a new formalism of de Rham–Witt complexes in the logarithmic setting. This construction generalises a result of Bhatt–Lurie–Mathew and agrees with those of Hyodo–Kato and Matsuue for log-smooth schemes of log-Cartier type. We then use our construction to study the monodromy action and slopes of Frobenius on log crystalline cohomology.

THE DE RHAM COHOMOLOGY OF DIFFERENTIAL SPACES

1989 ◽  
Vol 22 (1) ◽  
pp. 249-272 ◽  
Author(s):  
Wiesław Sasin
Keyword(s):  
De Rham Cohomology ◽  
De Rham ◽  
Differential Spaces

scholarly journals On the Voevodsky motive of the moduli space of Higgs bundles on a curve

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Characteristic Zero ◽  
Rational Point ◽  
Triangulated Category ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

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