All Questions
Tagged with ag.algebraic-geometry fundamental-group
115 questions
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
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\...
- 4,111
10
votes
0
answers
239
views
On tangent space to the fundamental group scheme
Let $X$ be a smooth, projective complex curve of genus at least $2$. If I understand correctly, after choosing a base point, one can associate to $X$, a fundamental group scheme $\pi$. I am trying to ...
- 2,126
8
votes
1
answer
308
views
Do complex varieties have a dense open subset with residually finite fundamental group?
Let $S$ be a smooth connected variety over the complex numbers. The fundamental group might not be residually finite (i.e., the homomorphism $\pi_1(S(\mathbb C)) \to \pi_1^{\mathrm{et}}(S)$ might not ...
- 35.2k
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
23
votes
5
answers
7k
views
Grothendieck's Galois Theory today
I have recently become aware of, and started to study in my free time (abundant in these summer months) Grothendieck's Galois Theory (GGT), as formulated in SGA 1 and later by Grothendieck's ...
- 23.4k
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 ...
CommunityBot
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32
votes
3
answers
4k
views
Fundamental groups of topoi
Just yesterday I heard of the notion of a fundamental group of a topos, so I looked it up on the nLab, where the following nice definition is given:
If $T$ is a Grothendieck topos arising as category ...
- 43
8
votes
1
answer
813
views
Inverse galois problem and étale homotopy
Is there any relation between étale homotopy theory (Grothendieck-Galois theory) and the inverse Galois problem?...I mean...in classical homotopy theory, every finite group $G$ realizes as a "Galois ...
- 149k
8
votes
1
answer
403
views
Is $H_{et}^1(X,F) = H^1(\pi_1^{et}(X), F(\bar{k}))$ true?
Let $X$ be a smooth geometrically connected scheme over a field $k$ of characteristic 0 (but not necessarily algebraically closed, I can take it to be a number field). Let $F$ be a finite algebraic ...
- 639
11
votes
1
answer
386
views
Is there a presentation to the kernel of the prime-to-$p$ fundamental short exact sequence of curves over finite fields?
Let $X$ be $\mathbb{P}^1_{\mathbb{F}_q}\smallsetminus \{a_1,...,a_r\}$, where $a_1,...,a_r$ are some $\mathbb{F}_q$-rational points. Let $\bar X :=X_{\bar{\mathbb{F}}_q}$. There is a short exact ...
- 149k
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?
- 666
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 $\bar{k}/k$,...
- 661
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 ...
- 10.7k
10
votes
0
answers
430
views
Discretifications of the fundamental group functor
Grothendieck calls a "discretification" of a profinite group $\widehat G$, a
discrete group $G$ whose profinite completion is isomorphic to $\widehat G$.
Does Grothendieck also define a notion of ...
- 17.6k
8
votes
0
answers
373
views
$p$-adic representations of the fundamental group of a smooth proper curve over a finite field
This question is very general. Let $C$ be a smooth and proper curve over a finite field ${\bf F}_p$. Are there any general results or conjectures on continuous non abelian representations
$$
\pi_1(C)\...
- 3,076
9
votes
1
answer
2k
views
Under what conditions is the induced map of etale fundamental groups surjective?
Let $f:X \to Y$ be a morphism of schemes. I am interested in sufficient conditions on $f$ which would ensure that the induced map $\pi_1^{et}(X) \to \pi_1^{et}(Y)$ of etale fundamental groups is ...
CommunityBot
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5
votes
0
answers
189
views
Nonabelian Hodge structure for noncompact curves and Hodge structure on the fundamental group
Nonabelian Hodge theory, introduced by C. Simpson and others, may be interpreted as a description of the (real) Hodge structure on the fundamental group (say, of a compact curve) in terms of some ...
- 743
7
votes
1
answer
325
views
Etale fundamental of a parahoric group scheme
Let $p:X\rightarrow Y$ be a double cover of curves, denote by $$SU_n:=(p_*SL_n(\mathcal O_X))^{\tilde{\sigma}}$$
i.e. the $\tilde{\sigma}-$invariant part, the action of $\tilde{\sigma}$ is given by $$\...
- 149k
11
votes
2
answers
2k
views
locally constant constructible sheaves and finite etale coverings
Maybe it is well known to experts or maybe it is just a stupid idea, but I will ask any way.
We know that if $X$ is a topological space, then there is an equivalence of categories between the ...
- 10.7k
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 ...
CommunityBot
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7
votes
1
answer
362
views
Does there exist an algebraic space with large fundamental group but no finite etale covers by schemes
Does there exits a smooth proper algebraic space $X$ over $\mathbb C$ with "large" fundamental group such that no finite etale cover of $X$ is a scheme?
By "large" fundamental group I mean that $X$ ...
- 4,111
14
votes
2
answers
951
views
Relationship between étale and topological $K(\pi,1)$s
I was trying to find a proof, or a counterexample to the claim that if $X/\mathbb{C}$ is connected smooth projective, then $X$ is a $K(\pi^{\mathrm{\acute{e}t}},1)$ if and only if $X^\mathrm{an}$ is a ...
- 35.2k
8
votes
0
answers
220
views
Does the fundamental group identify group structure on subvarieties of products of curves?
Let $C_1,\dots, C_n$ be smooth curves over $\overline{\mathbb F}_p$, not necessarily proper. Let $X$ be a subvariety of $C_1 \times \dots \times C_n$. I'm interested in the natural map:
$$ \pi_1^{ab}(...
- 149k
20
votes
0
answers
2k
views
Etale fundamental group of a curve in characteristic $p$
Let $C$ be a connected, smooth, proper curve of genus $g$ over an algebraically closed field $k$ of characteristic $p>0$. Let $\pi_1(C)$ be the etale fundamental group of $C$ - I only care about ...
- 2,834
3
votes
1
answer
294
views
Is the fundamental group of an open arithmetic Riemann surface contained in $\Gamma(2)$
Let $X$ be a non-compact Riemann surface with universal covering $\mathbb H$ and suppose that the fundamental group of $X$ is an arithmetic subgroup of $\mathrm{Aut}(\mathbb H) = \mathrm{PSL}_2(\...
- 1,990
1
vote
0
answers
176
views
which sections of elliptic curves are conjugate?
Suppose you have a relative elliptic curves $f : E\rightarrow S$ (say $S$ is connected). Then suppose you have two sections $g,g' : S\rightarrow E$, corresponding to two sections $g_*,g'_*$ to the map ...
- 10.7k
3
votes
2
answers
2k
views
homotopy exact sequence for the étale fundamental group
I have been trying to understand the homotopy exact sequence for the étale fundamental group which says
$$ 1 \rightarrow \pi_1 (\bar{X},\bar{x_0})\rightarrow \pi_1 (X,x_0)\rightarrow Gal(k)\...
- 5,504
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
2
votes
0
answers
881
views
Question about the specialization map for Etale Fundamental Groups
Let $A$ be a complete, discrete valuation ring, and let $s$ (resp $\eta$) be the special (resp. generic) point of $S=Spec(A)$. Let $\phi:X \rightarrow S$ be a proper morphism and fix geometric base ...
- 133
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
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(...
- 38k
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
1
vote
0
answers
136
views
algebraic varieties whose fundamental group is subgroup separable wrt subvariety subgroups
Call an algebraic variety $\pi_1$-subgroup separable iff,
for every $Y\subseteq X$ a closed subvariety and $\hat Y\xrightarrow{i} Y$ a normalisation of $Y$,
and subgroup $\Gamma=Im(\pi_1(\hat Y,y)\...
- 1,299
10
votes
1
answer
761
views
fundamental groups of smooth projective variety.
Is there a discrete group G which is the fundamental group of a compact Kahler
manifold but which is not the fundamental group of any smooth projective complex algebraic variety?
It is known that ...
CommunityBot
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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
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) \...
- 4,920
20
votes
2
answers
2k
views
Unipotency in realisations of the motivic fundamental group
Deligne, in his 1987 paper on the fundamental group of $\mathbb{P}^1 \setminus \{0,1,\infty\}$ (in "Galois Groups over $\mathbb{Q}$"), defines a system of realisations for a motivic fundamental group. ...
CommunityBot
- 1
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
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
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 $X$ ...
- 128
9
votes
1
answer
627
views
Do all varieties have only finitely many etale covers of fixed degree
I've been wondering about the following "finiteness statement" concerning etale covers for a while.
Let $K$ be a field of characteristic zero, not necessarily algebraically closed. A variety over $K$ ...
- 9,497
8
votes
1
answer
573
views
Do elements of the fundamental group give rise to isometries
Let $X$ be a complex algebraic variety, and let $\tilde X\to X$ be its universal cover. Suppose that there exists a Kahler-Einstein metric on $\tilde X$. Note that $\pi_1(X) \subset Aut(\tilde X)$.
...
- 2,727
8
votes
3
answers
943
views
Smooth projective varieties with infinite abelian fundamental group and finite $\pi_2$
Let $X$ be a smooth projective complex algebraic variety of general type. Suppose that the (topological) fundamental group of $X$ is an infinite abelian group and that $\pi_2(X^{an})$ is finite.
What ...
- 19.4k
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 ...
- 23k
7
votes
1
answer
1k
views
Algebraic numbers and the complex projective line minus three points
Deligne's monograph “Le Groupe Fondamental de la Droite Projective Moins Trois Points” begins by remarking that when X is the projective line over the complex numbers, minus three points: "every ...
- 11.2k
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 $\pi_1(\widetilde{X/G})$ ...
CommunityBot
- 1
25
votes
2
answers
2k
views
Profinite groups as étale fundamental groups
Does every profinite group arise as the étale fundamental group of a connected scheme?
Equivalently, does every Galois category arise as the category of finite étale covers of a connected scheme?
...
- 61
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. ...
CommunityBot
- 1
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 ...
CommunityBot
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