Cohomology of Higgs isocrystals

2017 ◽  
Author(s):  
Takeshi Tsuji
Keyword(s):  
Fundamental Group ◽  
Higgs Bundles ◽  
Faithful Functor

This chapter describes the cohomology of Higgs isocrystals, which are introduced to replace the notion of Higgs bundles. The link between these two notions uses Higgs envelopes and calls to mind the link between classical crystals and modules with integrable connections. After discussing Higgs isocrystals and Higgs crystals, cohomology of Higgs isocrystals, and representations of the fundamental group, the chapter presents the main result: the construction of a fully faithful functor from the category of Higgs (iso)crystals satisfying an overconvergence condition to that of small generalized representations. It also proves the compatibility of this functor with the natural cohomologies and concludes by comparing the cohomology of Higgs isocrystals with Faltings cohomology.

scholarly journals Parabolic Higgs bundles and representations of the fundamental group of a punctured surface into a real group

2020 ◽  
Vol 372 ◽  
pp. 107305
Author(s):  
Olivier Biquard ◽  
Oscar García-Prada ◽  
Ignasi Mundet i Riera
Keyword(s):  
Fundamental Group ◽  
Higgs Bundles ◽  
Real Group

scholarly journals Higgs bundles and fundamental group schemes

2019 ◽  
Vol 19 (3) ◽  
pp. 381-388
Author(s):  
Indranil Biswas ◽  
Ugo Bruzzo ◽  
Sudarshan Gurjar
Keyword(s):  
Fundamental Group ◽  
Vector Bundles ◽  
Projective Variety ◽  
Group Scheme ◽  
Smooth Projective Variety ◽  
Higgs Bundles ◽  
Chern Classes ◽  
Group Schemes ◽  
Tannakian Category

Abstract Relying on a notion of “numerical effectiveness” for Higgs bundles, we show that the category of “numerically flat” Higgs vector bundles on a smooth projective variety X is a Tannakian category. We introduce the associated group scheme, that we call the “Higgs fundamental group scheme of X,” and show that its properties are related to a conjecture about the vanishing of the Chern classes of numerically flat Higgs vector bundles.

Facsimile : A p-adic Simpson correspondence

2017 ◽  
Author(s):  
Gerd Faltings
Keyword(s):  
Fundamental Group ◽  
Exponential Function ◽  
Full Subcategory ◽  
Multiplicative Group ◽  
Hodge Theory ◽  
Higgs Bundles ◽  
Definition Of

This chapter presents the facsimile of Gerd Faltings' article entitled “A p-adic Simpson Correspondence,” reprinted from Advances in Mathematics 198(2), 2005. In this article, an equivalence between the category of Higgs bundles and that of “generalized representations” of the étale fundamental group is constructed for curves over a p-adic field. The definition of “generalized representations” uses p-adic Hodge theory and almost étale coverings, and it includes usual representations which form a full subcategory. The equivalence depends on the choice of an exponential function for the multiplicative group. The method used in the proofs is the theory of almost étale extensions. A nonabelian Hodge–Tate theory is also developed.

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.

Various Generalizations and Deformations of PSL(2,ℝ) Surface Group Representations and their Higgs Bundles

2018 ◽  
pp. 471-502
Author(s):  
Brian Collier
Keyword(s):  
Fundamental Group ◽  
Vector Bundles ◽  
Surface Group ◽  
Closed Surface ◽  
Connected Components ◽  
Character Variety ◽  
Higgs Bundles ◽  
Group Representations ◽  
Symmetric Powers ◽  
Higher Rank

The goal of this chapter is to examine the various ways in which Fuchsian representations of the fundamental group of a closed surface of genus g into PSL(2, R) and their associated Higgs bundles generalize to the higher-rank groups PSL(n, R), PSp(2n, R), SO0(2, n), SO0(n,n+1) and PU(n, n). For the SO0(n,n+1)-character variety, it parameterises n(2g−2) new connected components as the total spaces of vector bundles over appropriate symmetric powers of the surface, and shows how these components deform in the character variety. This generalizes results of Hitchin for PSL(2, R).

scholarly journals Picard Group and Fundamental Group of the Moduli of Higgs Bundles on Curves

2018 ◽  
Vol 5 (1) ◽  
pp. 146-149
Author(s):  
Sujoy Chakraborty ◽  
Arjun Paul
Keyword(s):  
Fundamental Group ◽  
Algebraic Group ◽  
Moduli Space ◽  
Topological Type ◽  
Picard Group ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve

Abstract Let X be an irreducible smooth projective curve of genus g ≥ 2 over ℂ. Let MG, Higgsδbe a connected reductive affine algebraic group over ℂ. Let Higgs be the moduli space of semistable principal G-Higgs bundles on X of topological type δ∈π1(G). In this article,we compute the fundamental group and Picard group of

PROJECTIVE STRUCTURES AND HIGGS BUNDLES ON SURFACES

2011 ◽  
Vol 08 (02) ◽  
pp. 367-379
Author(s):  
INDRANIL BISWAS ◽  
JACQUES HURTUBISE
Keyword(s):  
Fundamental Group ◽  
Conformal Structure ◽  
Higgs Bundles ◽  
Projective Structures ◽  
Geometric Structures

There are two families of geometric structures associated to a surface, with both structures related to representations of the fundamental group of the surface into SL(2, ℂ). These are projective structures on the surface, and Higgs bundles for a given conformal structure of the surface. This note discusses the links between the two.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

Representations of the fundamental group and the torsor of deformations. Global aspects

2017 ◽  
Author(s):  
Ahmed Abbes ◽  
Michel Gros
Keyword(s):  
Fundamental Group ◽  
Koszul Complex ◽  
Finiteness Conditions ◽  
Inverse Systems ◽  
Logarithmic Geometry

This chapter continues the construction and study of the p-adic Simpson correspondence and presents the global aspects of the theory of representations of the fundamental group and the torsor of deformations. After fixing the notation and general conventions, the chapter develops preliminaries and then introduces the results and complements on the notion of locally irreducible schemes. It also fixes the logarithmic geometry setting of the constructions and considers a number of results on the Koszul complex. Finally, it develops the formalism of additive categories up to isogeny and describes the inverse systems of a Faltings ringed topos, with a particular focus on the notion of adic modules and the finiteness conditions adapted to this setting. The chapter rounds up the discussion with sections on Higgs–Tate algebras and Dolbeault modules.

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