scholarly journals On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties

1998 ◽  
Vol 88 (1) ◽  
pp. 5-95 ◽  
Author(s):  
Michael Kapovich ◽  
John J. Millson
Keyword(s):  
Fundamental Groups ◽  
Artin Groups ◽  
Algebraic Varieties ◽  
Smooth Complex ◽  
Representation Varieties

scholarly journals GENERALIZED ZARISKI–VAN KAMPEN THEOREM AND ITS APPLICATION TO GRASSMANNIAN DUAL VARIETIES

2010 ◽  
Vol 21 (05) ◽  
pp. 591-637 ◽  
Author(s):  
ICHIRO SHIMADA
Keyword(s):  
Hyperplane Section ◽  
Fundamental Groups ◽  
Algebraic Varieties ◽  
Smooth Complex ◽  
Section Theorem ◽  
Dual Varieties

We formulate and prove a generalization of Zariski–van Kampen theorem on the topological fundamental groups of smooth complex algebraic varieties. As an application, we prove a hyperplane section theorem of Lefschetz–Zariski–van Kampen type for the fundamental groups of the complements to the Grassmannian dual varieties.

scholarly journals On the fundamental groups of normal varieties

2016 ◽  
Vol 18 (04) ◽  
pp. 1550065 ◽  
Author(s):  
Donu Arapura ◽  
Alexandru Dimca ◽  
Richard Hain
Keyword(s):  
Fundamental Groups ◽  
Algebraic Varieties ◽  
Local Systems ◽  
Normal Variety ◽  
Normal Complex ◽  
Normal Varieties ◽  
Rank One

We show that the fundamental groups of normal complex algebraic varieties share many properties of the fundamental groups of smooth varieties. The jump loci of rank one local systems on a normal variety are related to the jump loci of a resolution and of a smoothing of this variety.

scholarly journals Rank 3 rigid representations of projective fundamental groups

2018 ◽  
Vol 154 (7) ◽  
pp. 1534-1570 ◽  
Author(s):  
Adrian Langer ◽  
Carlos Simpson
Keyword(s):  
Irreducible Representation ◽  
Projective Variety ◽  
Fundamental Groups ◽  
Projective Varieties ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Geometric Origin ◽  
Rank 2 ◽  
Complex Projective ◽  
Smooth Projective Varieties

Let$X$be a smooth complex projective variety with basepoint$x$. We prove that every rigid integral irreducible representation$\unicode[STIX]{x1D70B}_{1}(X\!,x)\rightarrow \operatorname{SL}(3,\mathbb{C})$is of geometric origin, i.e., it comes from some family of smooth projective varieties. This partially generalizes an earlier result by Corlette and the second author in the rank 2 case and answers one of their questions.

scholarly journals FOLDINGS, GRAPHS OF GROUPS AND THE MEMBERSHIP PROBLEM

2005 ◽  
Vol 15 (01) ◽  
pp. 95-128 ◽  
Author(s):  
ILYA KAPOVICH ◽  
RICHARD WEIDMANN ◽  
ALEXEI MYASNIKOV
Keyword(s):  
Solvable Groups ◽  
Fundamental Groups ◽  
Membership Problem ◽  
Artin Groups ◽  
Graphs Of Groups ◽  
Uniform Subgroup ◽  
Right Angled Artin Groups ◽  
Subgroup Membership Problem

We introduce a combinatorial version of Stallings–Bestvina–Feighn–Dunwoody folding sequences. We then show how they are useful in analyzing the solvability of the uniform subgroup membership problem for fundamental groups of graphs of groups. Applications include coherent right-angled Artin groups and coherent solvable groups.

scholarly journals Fundamental groups of open algebraic varieties

Topology ◽  
1995 ◽  
Vol 34 (3) ◽  
pp. 509-531 ◽  
Author(s):  
Ichiro Shimada
Keyword(s):  
Fundamental Groups ◽  
Algebraic Varieties

scholarly journals H1–semistability for projective groups

2016 ◽  
Vol 162 (1) ◽  
pp. 89-100 ◽  
Author(s):  
INDRANIL BISWAS ◽  
MAHAN MJ
Keyword(s):  
Fundamental Groups ◽  
Projective Varieties ◽  
Smooth Complex ◽  
Universal Covers ◽  
Complex Projective

AbstractWe initiate the study of the asymptotic topology of groups that can be realised as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers (these are called here as holomorphically convex groups). We prove the H1-semistability conjecture of Geoghegan for holomorphically convex groups. In view of a theorem of Eyssidieux, Katzarkov, Pantev and Ramachandran [EKPR], this implies that linear projective groups satisfy the H1-semistability conjecture.

scholarly journals RANK TWO TOPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI OF QUASI-PROJECTIVE MANIFOLDS

2018 ◽  
Vol 19 (2) ◽  
pp. 451-485 ◽  
Author(s):  
Stefan Papadima ◽  
Alexander I. Suciu
Keyword(s):  
Algebraic Group ◽  
Borel Subgroup ◽  
Linear Algebraic Group ◽  
The Other ◽  
Fundamental Groups ◽  
Flat Connections ◽  
Other Hand ◽  
Projective Manifolds ◽  
Representation Varieties

We study the germs at the origin of $G$-representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. Using the Morgan–Dupont model associated to a convenient compactification of such a manifold, we relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on those models. When the linear algebraic group $G$ is either $\text{SL}_{2}(\mathbb{C})$ or its standard Borel subgroup and the depth of the jump locus is 1, this dictionary works perfectly, allowing us to describe in this way explicit irreducible decompositions for the germs of these embedded jump loci. On the other hand, if either $G=\text{SL}_{n}(\mathbb{C})$ for some $n\geqslant 3$, or the depth is greater than 1, then certain natural inclusions of germs are strict.

A GENERALIZATION OF LEFSCHETZ-ZARISKI THEOREM ON FUNDAMENTAL GROUPS OF ALGEBRAIC VARIETIES

1995 ◽  
Vol 06 (06) ◽  
pp. 921-932 ◽  
Author(s):  
ICHIRO SHIMADA
Keyword(s):  
Fundamental Groups ◽  
Natural Homomorphism ◽  
Algebraic Varieties

Let X and Y be the complements of divisors on non-singular irreducible closed subvarieties [Formula: see text] and [Formula: see text] in ℙn, respectively. Suppose that dim X+ dim Y≥n+2. Then, for a general g∈PGL (n+1), the natural homomorphism π1(g(X)∩Y)→π1(Y) induces a surjection from Ker (π1(g(X)∩Y)→π1(g(X))) onto π1(Y), and there is a surjection to its kernel from the cokernel of π2(X)→π2(ℙn). In particular, if E⊂ℙn is a hypersurface and 2·dim [Formula: see text], then [Formula: see text] is isomorphic to π1(ℙn\E) for a general g∈PGL (n+1).

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