scholarly journals Étale cohomology of rank one $$\ell $$-adic local systems in positive characteristic

2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Hélène Esnault ◽  
Moritz Kerz
Keyword(s):  
Positive Characteristic ◽  
Vanishing Theorem ◽  
Deformation Spaces ◽  
Local Systems ◽  
Etale Cohomology ◽  
Lefschetz Theorem ◽  
Rank One ◽  
Étale Cohomology ◽  
Generic Vanishing ◽  
Hard Lefschetz Theorem

AbstractWe show that in positive characteristic special loci of deformation spaces of rank one $$\ell $$ ℓ -adic local systems are quasi-linear. From this we deduce the Hard Lefschetz theorem for rank one $$\ell $$ ℓ -adic local systems and a generic vanishing theorem.

scholarly journals On the ramification of étale cohomology groups

2019 ◽  
Vol 2019 (749) ◽  
pp. 295-304 ◽  
Author(s):  
Isabel Leal
Keyword(s):  
Positive Characteristic ◽  
Residue Field ◽  
Cohomology Groups ◽  
Generic Fiber ◽  
Codimension One ◽  
Etale Cohomology ◽  
Rank One ◽  
Étale Cohomology ◽  
Valuation Field ◽  
Dimension One

Abstract Let K be a complete discrete valuation field whose residue field is perfect and of positive characteristic, let X be a connected, proper scheme over \mathcal{O}_{K} , and let U be the complement in X of a divisor with simple normal crossings. Assume that the pair (X,U) is strictly semi-stable over \mathcal{O}_{K} of relative dimension one and K is of equal characteristic. We prove that, for any smooth \ell -adic sheaf \mathcal{G} on U of rank one, at most tamely ramified on the generic fiber, if the ramification of \mathcal{G} is bounded by t+ for the logarithmic upper ramification groups of Abbes–Saito at points of codimension one of X, then the ramification of the étale cohomology groups with compact support of \mathcal{G} is bounded by t+ in the same sense.

scholarly journals Topology of Subvarieties of Complex Semi-abelian Varieties

2020 ◽  
Author(s):  
Yongqiang Liu ◽  
Laurentiu Maxim ◽  
Botong Wang
Keyword(s):  
Euler Characteristic ◽  
Morse Theory ◽  
Characteristic Property ◽  
Abelian Varieties ◽  
Betti Numbers ◽  
Local Systems ◽  
Affine Manifolds ◽  
Cyclic Covers ◽  
Rank One ◽  
Generic Vanishing

Abstract We use the non-proper Morse theory of Palais–Smale to investigate the topology of smooth closed subvarieties of complex semi-abelian varieties and that of their infinite cyclic covers. As main applications, we obtain the finite generation (except in the middle degree) of the corresponding integral Alexander modules as well as the signed Euler characteristic property and generic vanishing for rank-one local systems on such subvarieties. Furthermore, we give a more conceptual (topological) interpretation of the signed Euler characteristic property in terms of vanishing of Novikov homology. As a byproduct, we prove a generic vanishing result for the $L^2$-Betti numbers of very affine manifolds. Our methods also recast June Huh’s extension of Varchenko’s conjecture to very affine manifolds and provide a generalization of this result in the context of smooth closed sub-varieties of semi-abelian varieties.

scholarly journals GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

2013 ◽  
Vol 1 ◽  
Author(s):  
MIHNEA POPA ◽  
CHRISTIAN SCHNELL
Keyword(s):  
Abelian Varieties ◽  
Local Systems ◽  
Coherent Sheaves ◽  
Natural Categories ◽  
Rank One ◽  
Generic Vanishing ◽  
Mixed Hodge Modules ◽  
Irregular Varieties

AbstractWe extend most of the results of generic vanishing theory to bundles of holomorphic forms and rank-one local systems, and more generally to certain coherent sheaves of Hodge-theoretic origin associated with irregular varieties. Our main tools are Saito’s mixed Hodge modules, the Fourier–Mukai transform for $\mathscr{D}$-modules on abelian varieties introduced by Laumon and Rothstein, and Simpson’s harmonic theory for flat bundles. In the process, we also discover two natural categories of perverse coherent sheaves.

scholarly journals The Hard Lefschetz Theorem in Positive Characteristic for the Flag Varieties

2017 ◽  
Vol 2018 (18) ◽  
pp. 5562-5582
Author(s):  
Leonardo Patimo
Keyword(s):  
Positive Characteristic ◽  
Flag Varieties ◽  
Lefschetz Theorem ◽  
Hard Lefschetz Theorem

scholarly journals Vanishing and comparison theorems in rigid analytic geometry

2019 ◽  
Vol 156 (2) ◽  
pp. 299-324
Author(s):  
David Hansen
Keyword(s):  
Structure Theorem ◽  
Comparison Theorems ◽  
Analytic Geometry ◽  
Vanishing Theorem ◽  
Branched Covers ◽  
Etale Cohomology ◽  
Constructible Sheaves ◽  
Analytic Varieties ◽  
Étale Cohomology ◽  
Rigid Analytic Geometry

We prove a rigid analytic analogue of the Artin–Grothendieck vanishing theorem. Precisely, we prove (under mild hypotheses) that the geometric étale cohomology of any Zariski-constructible sheaf on any affinoid rigid space $X$ vanishes in all degrees above the dimension of $X$. Along the way, we show that branched covers of normal rigid spaces can often be extended across closed analytic subsets, in analogy with a classical result for complex analytic spaces. We also prove some new comparison theorems relating the étale cohomology of schemes and rigid analytic varieties, and give some applications of them. In particular, we prove a structure theorem for Zariski-constructible sheaves on characteristic-zero affinoid spaces.

scholarly journals Brauer groups and étale cohomology in derived algebraic geometry

2014 ◽  
Vol 18 (2) ◽  
pp. 1149-1244 ◽  
Author(s):  
Benjamin Antieau ◽  
David Gepner
Keyword(s):  
Algebraic Geometry ◽  
Brauer Groups ◽  
Etale Cohomology ◽  
Derived Algebraic Geometry ◽  
Étale Cohomology

p-adic Etale Cohomology

1983 ◽  
pp. 13-26 ◽  
Author(s):  
S. Bloch
Keyword(s):  
Etale Cohomology ◽  
Étale Cohomology

Etale Cohomology of Rigid Spaces

2004 ◽  
pp. 239-258
Author(s):  
Jean Fresnel ◽  
Marius van der Put
Keyword(s):  
Etale Cohomology ◽  
Étale Cohomology

Ordinary p -adic étale cohomology groups attached to towers of elliptic modular curves. II

2000 ◽  
Vol 318 (3) ◽  
pp. 557-583 ◽  
Author(s):  
Masami Ohta
Keyword(s):  
Modular Curves ◽  
Cohomology Groups ◽  
Etale Cohomology ◽  
Étale Cohomology
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