A note on an anabelian open basis for a smooth variety

2020 ◽  
Vol 72 (4) ◽  
pp. 537-550
Author(s):  
Yuichiro Hoshi
Keyword(s):  
Smooth Variety

Continuity of homotopies

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Open Subset ◽  
Small Neighborhood ◽  
Unit Vector ◽  
Zariski Topology ◽  
Simple Point ◽  
Smooth Variety ◽  
Additional Material ◽  
Generic Type ◽  
Zariski Open Subset

This chapter includes some additional material on homotopies. In particular, for a smooth variety V, there exists an “inflation” homotopy, taking a simple point to the generic type of a small neighborhood of that point. This homotopy has an image that is properly a subset of unit vector V, and cannot be understood directly in terms of definable subsets of V. The image of this homotopy retraction has the merit of being contained in unit vector U for any dense Zariski open subset U of V. The chapter also proves the continuity of functions and homotopies using continuity criteria and constructs inflation homotopies before proving GAGA type results for connectedness. Additional results regarding the Zariski topology are given.

scholarly journals Resolutions With Conical Slices and Descent for the Brauer Group Classes of Certain Central Reductions of Differential Operators in Characteristic p

2019 ◽  
Author(s):  
Dmitry Kubrak ◽  
Roman Travkin
Keyword(s):  
Differential Operators ◽  
Closed Field ◽  
Brauer Group ◽  
Affine Variety ◽  
Smooth Variety ◽  
Characteristic P ◽  
Algebraically Closed Field ◽  
Quiver Varieties ◽  
Hypertoric Varieties ◽  
General Reduction

Abstract “Even more so is the word ‘crystalline’, a glacial and impersonal concept of his which disdains viewing existence from a single portion of time and space” Eileen Myles, “The Importance of Being Iceland” For a smooth variety $X$ over an algebraically closed field of characteristic $p$ to a differential 1-form $\alpha $ on the Frobenius twist $X^{\textrm{(1)}}$ one can associate an Azumaya algebra ${{\mathcal{D}}}_{X,\alpha }$, defined as a certain central reduction of the algebra ${{\mathcal{D}}}_X$ of “crystalline differential operators” on $X$. For a resolution of singularities $\pi :X\to Y$ of an affine variety $Y$, we study for which $\alpha $ the class $[{{\mathcal{D}}}_{X,\alpha }]$ in the Brauer group $\textrm{Br}(X^{\textrm{(1)}})$ descends to $Y^{\textrm{(1)}}$. In the case when $X$ is symplectic, this question is related to Fedosov quantizations in characteristic $p$ and the construction of noncommutative resolutions of $Y$. We prove that the classes $[{{\mathcal{D}}}_{X,\alpha }]$ descend étale locally for all $\alpha $ if ${{\mathcal{O}}}_Y\widetilde{\rightarrow }\pi _\ast{{\mathcal{O}}}_X$ and $R^{1}\pi _*\mathcal O_X = R^2\pi _*\mathcal O_X =0$. We also define a certain class of resolutions, which we call resolutions with conical slices, and prove that for a general reduction of a resolution with conical slices in characteristic $0$ to an algebraically closed field of characteristic $p$ classes $[{{\mathcal{D}}}_{X,\alpha }]$ descend to $Y^{\textrm{(1)}}$ globally for all $\alpha $. Finally we give some examples; in particular, we show that Slodowy slices, Nakajima quiver varieties, and hypertoric varieties are resolutions with conical slices.

scholarly journals QUANTIZATION OF A MODULI SPACE OF PARABOLIC HIGGS BUNDLES

2004 ◽  
Vol 15 (09) ◽  
pp. 907-917 ◽  
Author(s):  
INDRANIL BISWAS ◽  
AVIJIT MUKHERJEE
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Symplectic Form ◽  
Dense Subset ◽  
Projective Structure ◽  
Open Dense Subset ◽  
Higgs Bundles ◽  
Smooth Variety ◽  
Structure Formula

Let [Formula: see text] be a moduli space of stable parabolic Higgs bundles of rank two over a Riemann surface X. It is a smooth variety defined over [Formula: see text] equipped with a holomorphic symplectic form. Fix a projective structure [Formula: see text] on X. Using [Formula: see text], we construct a quantization of a certain Zariski open dense subset of the symplectic variety [Formula: see text].

scholarly journals On the center of the ring of differential operators on a smooth variety over ℤ/pnℤ

2012 ◽  
Vol 149 (1) ◽  
pp. 63-80 ◽  
Author(s):  
Allen Stewart ◽  
Vadim Vologodsky
Keyword(s):  
Associative Algebra ◽  
Differential Operators ◽  
Private Communication ◽  
Smooth Variety ◽  
Witt Vectors ◽  
Degeneracy Condition

AbstractWe compute the center of the ring of PD differential operators on a smooth variety over ℤ/pnℤ, confirming a conjecture of Kaledin (private communication). More generally, given an associative algebraA0over ℤpand its flat deformationAnover ℤ/pn+1ℤ, we prove that under a certain non-degeneracy condition, the center ofAnis isomorphic to the ring of length-(n+1) Witt vectors over the center ofA0.

scholarly journals GERSTEN COMPLEX FOR SHEAVES WITH TRANSFERS FOR NOETHERIAN SCHEMES

2017 ◽  
Vol 21 (6) ◽  
pp. 98-101
Author(s):  
A.A. Mingazov
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Smooth Variety ◽  
Fundamental Fact

V. Voevodsky introduced Gersten complex for sheaves with transfers in one of the first papers where the category of motives was constructed. Becides he proved Gersten conjecture which states that the Gersten complex for the local ring of point of smooth variety over field k is resolution of the group of global sections over this ring. Because of this fundamental fact Gersten complex can be used for calculations of cohomologies of sheaves with transfers over smooth k -varieties. In this paper we construct Gersten complex for sheaves with trans- fers, which defined on the category of noetherian k -schemes, where char k = 0. After this we proof the Gersten conjecture in case of local noetherian ring over field k . This generalises Voevodsky’s result.

scholarly journals FUJITA DECOMPOSITION AND MASSEY PRODUCT FOR FIBERED VARIETIES

2021 ◽  
pp. 1-29
Author(s):  
LUCA RIZZI ◽  
FRANCESCO ZUCCONI
Keyword(s):  
General Type ◽  
Structure Theorem ◽  
Smooth Curve ◽  
Local System ◽  
Base Change ◽  
Smooth Variety ◽  
Massey Product

Abstract Let $f\colon X\to B$ be a semistable fibration where X is a smooth variety of dimension $n\geq 2$ and B is a smooth curve. We give the structure theorem for the local system of the relative $1$ -forms and of the relative top forms. This gives a neat interpretation of the second Fujita decomposition of $f_*\omega _{X/B}$ . We apply our interpretation to show the existence, up to base change, of higher irrational pencils and on the finiteness of the associated monodromy representations under natural Castelnuovo-type hypothesis on local subsystems. Finally, we give a criterion to have that X is not of Albanese general type if $B=\mathbb {P}^1$ .

The p-adic Simpson Correspondence (AM-193)

2017 ◽  
Author(s):  
Ahmed Abbes ◽  
Michel Gros ◽  
Takeshi Tsuji
Keyword(s):  
Fundamental Group ◽  
Linear Algebra ◽  
General Framework ◽  
Hodge Theory ◽  
Higgs Bundles ◽  
Smooth Variety ◽  
Trivial Representation ◽  
New Approaches ◽  
New Family ◽  
Systematic Development

The p-adic Simpson correspondence, recently initiated by Gerd Faltings, aims at describing all p-adic representations of the fundamental group of a proper smooth variety over a p-adic field in terms of linear algebra—namely Higgs bundles. This book undertakes a systematic development of the theory following two new approaches. It mainly focuses on generalized representations of the fundamental group that are p-adically close to the trivial representation. The first approach relies on a new family of period rings built from the torsor of deformations of the variety over a universal p-adic thickening defined by J. M. Fontaine. The second approach introduces a crystalline-type topos and replaces the notion of Higgs bundles with that of Higgs isocrystals. The book shows the compatibility of the two constructions and the compatibility of the correspondence with the natural cohomologies. The last part of the book contains results of wider interest in p-adic Hodge theory. The reader will find a concise introduction to Faltings' theory of almost étale extensions and a chapter devoted to the Faltings topos. Though this topos is the general framework for Faltings' approach in p-adic Hodge theory, it remains relatively unexplored.

scholarly journals On the freeness of hypersurface arrangements consisting of hyperplanes and spheres

2018 ◽  
Vol 16 (1) ◽  
pp. 437-446
Author(s):  
Ruimei Gao ◽  
Qun Dai ◽  
Zhe Li
Keyword(s):  
Hyperplane Arrangements ◽  
Local Models ◽  
Smooth Variety

AbstractLet V be a smooth variety. A hypersurface arrangement 𝓜 in V is a union of smooth hypersurfaces, which locally looks like a union of hyperplanes. We say 𝓜 is free if all these local models can be chosen to be free hyperplane arrangements. In this paper, we use Saito’s criterion to study the freeness of hypersurface arrangements consisting of hyperplanes and spheres, and construct the bases for the derivation modules explicitly.

scholarly journals Symmetry and parity in Frobenius action on cohomology

2011 ◽  
Vol 148 (1) ◽  
pp. 295-303 ◽  
Author(s):  
Junecue Suh
Keyword(s):  
Finite Fields ◽  
Characteristic Zero ◽  
Betti Numbers ◽  
Newton Polygons ◽  
Smooth Variety ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Crystalline Cohomology ◽  
Lefschetz Theorem ◽  
Hard Lefschetz Theorem

AbstractWe prove that the Newton polygons of Frobenius on the crystalline cohomology of proper smooth varieties satisfy a symmetry that results, in the case of projective smooth varieties, from Poincaré duality and the hard Lefschetz theorem. As a corollary, we deduce that the Betti numbers in odd degrees of any proper smooth variety over a field are even (a consequence of Hodge symmetry in characteristic zero), answering an old question of Serre. Then we give a generalization and a refinement for arbitrary varieties over finite fields, in response to later questions of Serre and of Katz.

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