scholarly journals The local fundamental group of a Kawamata log terminal singularity is finite

2021 ◽  
Author(s):  
Lukas Braun
Keyword(s):  
Fundamental Group ◽  
Terminal Singularity ◽  
Smooth Locus ◽  
Log Terminal Singularity

AbstractWe prove a conjecture of Kollár stating that the local fundamental group of a klt singularity x is finite. In fact, we prove a stronger statement, namely that the fundamental group of the smooth locus of a neighbourhood of x is finite. We call this the regional fundamental group. As the proof goes via a local-to-global induction, we simultaneously confirm finiteness of the orbifold fundamental group of the smooth locus of a weakly Fano pair.

scholarly journals Finiteness of algebraic fundamental groups

2014 ◽  
Vol 150 (3) ◽  
pp. 409-414 ◽  
Author(s):  
Chenyang Xu
Keyword(s):  
Fundamental Group ◽  
Fano Variety ◽  
Fundamental Groups ◽  
Terminal Singularity ◽  
Smooth Locus ◽  
Log Terminal Singularity

AbstractWe show that the algebraic local fundamental group of any Kawamata log terminal singularity as well as the algebraic fundamental group of the smooth locus of any log Fano variety are finite.

scholarly journals Finiteness of fundamental groups

2017 ◽  
Vol 153 (2) ◽  
pp. 257-273
Author(s):  
Zhiyu Tian ◽  
Chenyang Xu
Keyword(s):  
Fundamental Group ◽  
Three Dimensional ◽  
Fano Variety ◽  
Fundamental Groups ◽  
Canonical Singularities ◽  
Smooth Locus

We show that the finiteness of the fundamental groups of the smooth locus of lower dimensional log Fano pairs would imply the finiteness of the local fundamental group of Kawamata log terminal (klt) singularities. As an application, we verify that the local fundamental group of a three-dimensional klt singularity and the fundamental group of the smooth locus of a three-dimensional Fano variety with canonical singularities are always finite.

Existence of Ball Quotients Covering Line Arrangements

2017 ◽  
Author(s):  
Paula Tretkoff
Keyword(s):  
Blow Up ◽  
Terminal Singularity ◽  
Line Arrangements ◽  
Covering Group ◽  
Log Canonical ◽  
Finite Covers ◽  
Ball Quotients ◽  
Weight Data ◽  
Log Terminal Singularity ◽  
Log Canonical Singularity

This chapter justifies the assumption that ball quotients covering line arrangements exist. It begins with the general case on the existence of finite covers by ball quotients of weighted configurations, focusing on log-canonical divisors and Euler numbers reflecting the weight data on divisors on the blow-up X of P2 at the singular points of a line arrangement. It then uses the Kähler-Einstein property to prove an inequality between Chern forms that, when integrated, gives the appropriate Miyaoka-Yau inequality. It also discusses orbifolds and b-spaces, weighted line arrangements, the problem of the existence of ball quotient finite coverings, log-terminal singularity and log-canonical singularity, and the proof of the main existence theorem for line arrangements. Finally, it considers the isotropy subgroups of the covering group.

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.

scholarly journals TOPOLOGICAL 4-MANIFOLDS WITH 4-DIMENSIONAL FUNDAMENTAL GROUP

2021 ◽  
pp. 1-8
Author(s):  
DANIEL KASPROWSKI ◽  
MARKUS LAND
Keyword(s):  
Fundamental Group ◽  
Homotopy Equivalent ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Canonical Map ◽  
Duality Space ◽  
Simply Connected Manifolds ◽  
Simply Connected ◽  
Poincaré Duality Space

Abstract Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

scholarly journals Matroid connectivity and singularities of configuration hypersurfaces

2021 ◽  
Vol 111 (1) ◽  
Author(s):  
Graham Denham ◽  
Mathias Schulze ◽  
Uli Walther
Keyword(s):  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Smooth Locus

AbstractConsider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. The corresponding configuration hypersurface and its non-smooth locus support the respective first and second degeneracy scheme of the bilinear form. We show that these schemes are reduced and describe the effect of matroid connectivity: for (2-)connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen–Macaulay of codimension 3. If the matroid is 3-connected, then also the second degeneracy scheme is integral. In the process, we describe the behavior of configuration polynomials, forms and schemes with respect to various matroid constructions.

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