All Questions
Tagged with fundamental-group ag.algebraic-geometry
115 questions
6
votes
1
answer
354
views
Fundamental groups of complements of divisors in $\mathbb P^2$
Let $D$ be a divisor in $\mathbb P^2_{\mathbb C}$ and let $X= \mathbb P^2_{\mathbb C} - D$.
Under what condition on $D$ is the fundamental group of $X$ infinite?
- 71
6
votes
1
answer
471
views
Étale fundamental group of multiplicative group over an algebraically/separably closed field
This is a repost of my question here.
Do we know the structure of the étale fundamental group $\pi^\text{et}_1(\mathbb{G}_{m,K^\text{sep}})$ of the multiplicative group, for a given field $K$? For ...
- 311
6
votes
1
answer
292
views
Finite étale covers of concentrated schemes and extension of base field
Let $k'/k$ be an extension of algebraically closed fields of characteristic $0$, and $X$ a concentrated (i.e. quasi-compact and quasi-separated) scheme over $k$.
Question: is the pullback functor ...
- 1,000
6
votes
1
answer
289
views
Relationship between the holonomy pseudogroup and holonomy homomorphism (foliation)
This question is surely a duplication of https://math.stackexchange.com/questions/4343635/relationship-between-the-holonomy-pseudogroup-and-holonomy-homomorphism-foliati , however, I got no replies. ...
- 295
6
votes
1
answer
393
views
finite quotients of fundamental groups in positive characteristic
For affine smooth curves over $k=\bar{k}$ of char. $p,$ Abhyankar's conjecture (proved by Raynaud and Harbater) tells us exactly which finite groups can be realized as quotients of their fundamental ...
- 4,265
6
votes
0
answers
377
views
Fundamental group of a product in characteristic 0
It is proven in SGA1 that if $k$ is an algebraically closed field, if $X$ is a proper $k$-scheme and if $Y$ is a locally noetherian $k$-scheme (say, $X$ and $Y$ are non-empty and connected) then $\...
- 1,584
6
votes
0
answers
428
views
generating the etale fundamental group by sections?
Let $X$ be a proper smooth scheme over a field $k$ of characteristic zero (well you can naturally weaken the assumption to normal integral scheme over some "nice" base like $\mathbb{Z}$, $\mathbb{F}_q$...
- 1,305
5
votes
1
answer
2k
views
What is the arithmetic fundamental group?
Let $X$ be an irreducible variety defined over $\mathbf{F}_p$. What's the "arithmetic fundamental group" of $X$? How does this relate to the algebraic fundamental group of a scheme? What's a good ...
user97187
5
votes
2
answers
457
views
Finite etale covers of products of curves
Probably this question can be phrased in a much greater generality, but I will just state it in the generality I require. I work over $\mathbb{C}$.
Let $C_1, C_2 \subset \mathbb{P}^1$ be non-empty ...
- 21.4k
5
votes
1
answer
905
views
Abelianized fundamental group of a curve over a finite field
Let $X$ be a smooth, projective, and geometrically connected curve over a finite field $\mathbb{F}_q$ and fix a geometric point $\overline{x} : \text{Spec } \overline{\mathbb{F}_q} \to X$. Then there ...
- 3,605
5
votes
2
answers
399
views
Conjugation of homogeneous spaces
Let $X$ be a smooth irreducible algebraic variety
over the field of complex numbers ${\mathbb{C}}$.
Let $x\in X({\mathbb{C}})$.
Let $\tau$ be an automorphism of ${\mathbb{C}}$ (not necessarily ...
- 14.2k
5
votes
1
answer
584
views
Reconstruction of hyperbolic curves using the fundamental group
In the paper Curves and their Fundamental Groups written by Gerd Faltings, Mochizuki's proof of Grothendick's conjecture on anabelian curves is explained.
In the proof, he shows that for two ...
- 527
5
votes
0
answers
392
views
Complex conjugation inducing a trivial map on the fundamental group
Let $V$ be a smooth projective complex variety defined over the reals such that $G=\pi_1(V)$ is a non-abelian finite simple group. Assume that $V$ has a real point. Can the map $G\to G$ induced by ...
user145520
5
votes
0
answers
189
views
Nonabelian Hodge structure for noncompact curves and Hodge structure on the fundamental group
Nonabelian Hodge theory, introduced by C. Simpson and others, may be interpreted as a description of the (real) Hodge structure on the fundamental group (say, of a compact curve) in terms of some ...
- 743
5
votes
0
answers
199
views
Algebraic fundamental group without regularity at infinity
Suppose $X$ is a smooth (connected) variety over $\mathbb{C}$. Let $\mathscr{C}$ be the category of finite rank vector bundles on $X$ equipped with an integrable connection, and let $\mathscr{C}'$ be ...
- 9,061
5
votes
0
answers
287
views
Nori fundamental group and etale fundamental group in positive characteristic
Let $X$ be a smooth projective surface over an algebraically closed field of char $p > 0$. Suppose that $\pi_{1}^{et}(X) = \{1\}$. Can Nori fundamental group scheme of $X$ be non-trivial?
- 163
4
votes
1
answer
297
views
Fundamental group of the smooth locus of a normal algebraic surface is a quotient of that of a Zariski open subset
Let $X$ be a normal algebraic surface (over $\mathbb{C}$) and $Y$ its smooth locus, i.e., the complement of the singularities of $X$. Suppose $Z\subset Y$ is a Zariski open subset of $X$. Then is it ...
- 335
4
votes
1
answer
264
views
Isomorphism étale fundamental group
Given a birational proper morphism $f\colon X \rightarrow Y$
( Assume $X$ and $Y$ irreducible )
of complex algebraic varieties.
It is always true that $f^* \colon \pi^{et}_1 (X)\rightarrow \pi^{et}_1(...
- 1,595
4
votes
2
answers
729
views
Reference request: the comparison theorem for the étale fundamental group
I am looking for exact references for the comparison theorem for the étale fundamental group.
I mean the following result:
Theorem (Grothendieck). For a pointed algebraic variety $(X,x)$ over $\...
- 14.2k
4
votes
2
answers
1k
views
About "de-Rham" and "l-adic" local systems - comparison
Hello,
Suppose that $k$ is an algebraically closed field of char. 0.
Let $X$ be a smooth connected variety over $k$.
Then I have the category $A$ of Regular Singular smooth $D$-modules on $X$ (i.e. ...
- 5,562
4
votes
1
answer
250
views
Galois action on the pro-algebraic completion of the singular fundamental group
Let $X$ be a smooth variety over a field $K \subset \mathbb{C}$. The singular fundamental group $\pi_1(X^{\text{an}}, x)$ generally does not carry an action of the absolute Galois group $\operatorname{...
- 199
4
votes
1
answer
604
views
A question about the Tannakian etale fundamental group of a curve
Let $X$ be a smooth connected quasi-projective curve over $\mathbf{Q}$. Let $U$ be the pro-unipotent etale fundamental group of $X$ over $\mathbf{Q}_p$.
$U^1 = U$ and let $U^n =[U,U^{n-1}]$.
Let $n\...
- 1,213
4
votes
1
answer
1k
views
Fundamental Group and Etale Cohomology
I encountered the following statement without a reference many times. For a smooth variety $X$ over a perfect field $k$.
$Hom(H^1_{et}(X, \mathbb{Z}/n), \mathbb{Z}/n) \cong \pi^{ab}_1(X)/n$
Is there ...
- 235
4
votes
2
answers
349
views
Number of connected components of the Hurwitz space $H_n^o$ and subgroups of the fundamental group
A cover (of $\mathbf{P}^1_{\overline{\mathbf{Q}}}$) is a finite morphism $X\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$, where $X$ is a smooth projective connected curve over $\overline{\mathbf{Q}}$. ...
- 163
4
votes
1
answer
481
views
local fundamental group of elliptic singularities
Is the local fundamental group of an elliptic singularity virtually solvable ? Here (the terminology is sometimes divergent) an elliptic singularity is a (germ of) normal surface $(X,x)$ such that $X$ ...
- 190
4
votes
1
answer
1k
views
Computing fundamental groups of the complement of plane curves
This paper of Zariski contains this statement: If $C$ is a curve in $\mathbb{CP}^2$, and $L$ is a generic line, then the injection $L\setminus C \hookrightarrow \mathbb{CP}^2\setminus C$ induces an ...
- 5,359
4
votes
1
answer
210
views
Conjugacy-invariance of sections of etale homotopy exact sequence
My questions arise on page xiv of Stix's Rational Points and Arithmetic of Fundamental Groups. Here is an excerpt:
Given a geometrically connected variety $X/k$, a fixed separable closure $\bar{k}/k$,...
- 1,338
4
votes
1
answer
369
views
Structure of fundamental groups arising from smooth projective morphisms
Let $f:X\to B$ be a smooth projective morphism of complex algebraic varieties.
If $f$ is of relative dimension zero, i.e., $f$ is a finite etale cover, then the image of the topological fundamental ...
- 41
4
votes
0
answers
100
views
Fundamental groups of Hirzebruch's line arrangement varities
Let $\Lambda$ be a line arrangement in $\mathbb{P}^2$ and $n > 0$ an integer. Then Hirzebruch defined a smooth projective surface $H(\Lambda, n)$ as the minimal desingularization of a covering $Y \...
- 3,730
4
votes
0
answers
195
views
Geometric fundamental group and algebraically closed residue field
my questions relates to the following talk of Tsuji:
https://www.youtube.com/watch?v=2brDj26phP0
At around 10:30 of the video, Tsuji is interrupted by a man stating that his construction does not ...
- 455
4
votes
0
answers
517
views
Exact sequence of the fundamental group of the general fiber
Let $f\colon X\rightarrow Y$ be a morphism of complex algebraic varieties.
Let $y\in Y$ be a general point, then we have a sequence of homomorphisms
of fundamental groups induced by the inclusion of ...
- 1,595
4
votes
0
answers
152
views
local systems with cyclic monodromy
In their book "Lectures on vanishing theorems", Esnault and Viehweg used finite cyclic covering of varieties constructed as follows:
Let $X$ be a smooth projective variety over some field $k$ of ...
- 41
3
votes
1
answer
321
views
A complex variety with a finite non-abelian simple fundamental group
Does there exist a complex smooth proper variety whose fundamental group is finite non-abelian simple?
user145520
3
votes
1
answer
294
views
Is the fundamental group of an open arithmetic Riemann surface contained in $\Gamma(2)$
Let $X$ be a non-compact Riemann surface with universal covering $\mathbb H$ and suppose that the fundamental group of $X$ is an arithmetic subgroup of $\mathrm{Aut}(\mathbb H) = \mathrm{PSL}_2(\...
- 31
3
votes
2
answers
2k
views
homotopy exact sequence for the étale fundamental group
I have been trying to understand the homotopy exact sequence for the étale fundamental group which says
$$ 1 \rightarrow \pi_1 (\bar{X},\bar{x_0})\rightarrow \pi_1 (X,x_0)\rightarrow Gal(k)\...
- 319
3
votes
1
answer
1k
views
The (topological) fundamental group of (quasi)-projective algebraic varieties
I would like to know:
What does the fundamental group of a quasi-projective algebraic variety look like?
I remember that I have seen somewhere that for a connected, finite-type CW-complex $X$, ...
- 169
3
votes
1
answer
600
views
Grothendieck's section conjecture and base change: restricting sections
Let $X$ be a smooth projective geometrically connected curve over $\mathbf{Q}$ of genus at least two. Fix an algebraic closure $\overline{\mathbf{Q}}$ of $\mathbf{Q}$ and let $G_{\mathbf{Q}}$ be the ...
- 125
3
votes
1
answer
209
views
Is $\pi_1(\widetilde{X/G})$ always finite if $\pi_1(X)$ is finite?
Let $X$ be a smooth complex manifold with finite fundamental group. Suppose that a finite group $G$ acts on $X$ and let $\widetilde{X/G}$ be a resolution of singularities. Is $\pi_1(\widetilde{X/G})$ ...
- 1,663
3
votes
0
answers
164
views
Pro-algebraic fundamental groups
Let $X$ be a smooth projective variety over an algebraically closed field $K$ of characteristic zero and fix a point $x\in X(K)$.
We can associate to $X$ two Tannakian categories: the category of ...
- 3,446
3
votes
0
answers
228
views
Is there a reasonable notion of universal cover for schemes over arbitrary fields?
Let $X$ be smooth projective variety, such as an elliptic curve. We know that $X$ does not have a universal cover in the category of schemes.
However, if $k = \mathbf{C}$, then $\tilde{X}$ exists as ...
- 4,164
3
votes
0
answers
246
views
First thoughts about fundamental group of a topological (Lie) groupoid
I am reading the paper Chern-Weil map for principal bundles over groupoids.
In page number $13$, authors say
let us recall the definition of fundamental group of a topological groupoid.
But, they ...
- 6,023
3
votes
0
answers
135
views
How does the fundamental group of $\mathbb G_{m,S}$ depend on the base scheme
Let $S$ be an integral noetherian regular scheme, and let $X =\mathbb{G}_{m,S}$.
How to compute $\pi_1^{et}(X)$?
Note. I am only interested in the part not coming trivially from the finite etale ...
- 39
3
votes
0
answers
618
views
Monodromy representations are "quasi-unipotent"
Let $S$ be a smooth complex algebraic variety, let $b$ be a closed point of $X$, and let $f:X\to S$ be a polarized family of smooth projective varieties over $S$. This induces a monodromy ...
- 113
2
votes
3
answers
651
views
question about the induced homomorphism of etale fundamental groups
Background/Setup
For any connected scheme $S$, let $\text{FEt}_S$ denote the category of finite etale $S$-schemes. Let $f : X\rightarrow Y$ be a morphism of connected schemes, then for any finite ...
- 10.7k
2
votes
2
answers
503
views
Lie Algebras and Simple Connectivity for general algebraic groups
In the representation theory of Lie groups (say, over $\mathbb{R}$ or $\mathbb{C}$), one can show that a Lie algebra homomorphism between the Lie algebras of two algebraic groups $G$ and $H$ always ...
- 15.4k
2
votes
1
answer
375
views
Does the fundamental group of the normalization of a scheme inject into the fundamental group of the scheme
Let $X$ be an integral noetherian finite type scheme over an algebraically closed field $k$. Let $X'\to X$ be its normalization. Is the induced homomorphism of etale fundamental groups $\pi_1(X')\to\...
- 29
2
votes
1
answer
270
views
Motivation of Zariski–Van Kampen theorem
The Zariski–Van Kampen theorem gives the presentation of the fundamental group of the complement of the plane curve of degree $d$. But what's the motivation of this theorem? More generally, why are ...
- 197
2
votes
1
answer
184
views
The fundamental group of quotient space of 3-folds
Let $S$ be a K3 surface with an involution $\iota_S$, $E$ an elliptic curve with an involution $\iota_E$. Assume the fixed locus of $S$ under $\iota_S$ contains $N>0$ disjoint curves. Note the ...
- 199
2
votes
1
answer
368
views
fundamental group and torus action
Let $T$ be the complex torus acting on a complex connected algebraic variety $X$
and let $p \colon X\rightarrow Y$ be a good quotient for this action.
For any $y\in Y$ we have a sequence $p^{-1}(y) \...
- 1,595
2
votes
1
answer
200
views
Extending étale covers from the regular locus to a resolution of singularities
Let $X$ be a normal proper variety with rational singularities (or terminal if that is necessary) and $X_{\text{reg}} \to X$ the regular locus. Let $\pi : \tilde{X} \to X$ be a resolution of ...
- 3,730