All Questions
Tagged with fundamental-group ag.algebraic-geometry
115 questions
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
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
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
1
vote
0
answers
85
views
Companions for positive characteristic arithmetic representations viewed as representations of the topological fundamental group?
Suppose $X / K$ is a variety over a finitely generated field over $\mathbb{Q}$. Fix an embedding $K \subset \mathbb{C}$ and let $\pi := \pi_1(X(\mathbb{C}), x)$ be the topological fundamental group. ...
- 3,730
11
votes
1
answer
415
views
Why can we take the colimit over the category of elements?
I'm trying to understand J. P. Murre's Tata notes on Grothendieck's theory of the fundamental group. For a Galois category $\mathcal C$ (which I'm taking to be locally small) with fundamental functor $...
1
vote
0
answers
98
views
Does there exist a simply connected surface with CM whose cotangent bundle is ample?
Does there exist a smooth projective complex surface $X$ such that,
(1) $\pi_1(X) = 0$
(2) $\Omega_X^1$ is ample
(3) the Mumford-Tate group of $H^2(X)$ is a torus
There exist examples with any two of ...
- 3,730
8
votes
1
answer
255
views
Can "fake rational surfaces" be simply-connected?
I say that a smooth projective complex algebraic surface $X$ is a "fake rational surface" if its Hodge diamond looks like:
and $X$ is of general type.
It is well-known that fake projective ...
- 3,730
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
2
votes
0
answers
111
views
Is the connecting map $\pi_2(B) \to \pi_1(F)$ ever nonzero in smooth proper families?
Suppose that $X, B$ are smooth irreducible varieties over $\mathbb{C}$ and $f : X \to B$ is a smooth proper morphism. Then we can consider the homotopy exact sequence:
$$ \pi_2(B) \to \pi_1(F) \to \...
- 3,730
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
1
vote
0
answers
81
views
Behaviour of cycles modulo algebraic equivalence on an etale covering
I found a neat result in Beauville's paper "VARIÉTÉS DE PRYM ET JACOBIENNES INTERMÉDIAIRES" : if $U \subset \mathbb{P}^n$ is an open and $V \to U$ is a conic bundle whose fibres are all ...
- 679
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
0
answers
93
views
Unramified section associated to a rational point
This is a question for those familiar with the section conjecture, so I'll do away with the definition of a ramification map in this case. Here is the definition of a ramification map from an etale ...
- 895
7
votes
0
answers
330
views
Künneth formula for $\pi_1$-proper morphisms
Context:
Let $X$ and $Y$ be connected qcqs schemes over an algebraically closed field $k$. Denote by $\pi_1(X)$, $\pi_1(Y)$ their étale fundamental groups (base points omitted). Grothendieck proved ...
- 71
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
6
votes
3
answers
1k
views
Motivation of the fundamental theorem of covering spaces
The fundamental theorem of covering spaces states that for a nice topological space $X$, there is an equivalence of categories between covering spaces over $X$ and left $\pi_1(X)$-sets. "...
- 459
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
47
votes
3
answers
5k
views
"Cute" applications of the étale fundamental group
When I was an undergrad student, the first application that was given to me of the construction of the fundamental group was the non-retraction lemma : there is no continuous map from the disk to the ...
- 7,320
2
votes
0
answers
179
views
Isocrystals on simply connected varieties
Esnault and Shiho - Convergent isocrystals on simply connected varieties proves that there are no non-trivial convergent isocrystals on simply connected varieties. There is another similar result in ...
- 5,901
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
2
votes
0
answers
176
views
Outer Galois representations and Tate modules of Jacobian varieties
Let $X$ be a proper smooth curve over a field $k$. Then we have an exact sequence of profinite groups
\begin{equation*}
1 \to \pi_1(X_{\overline k}) \to \pi_1(X) \to G_k \to 1,
\end{equation*}
...
- 988
11
votes
3
answers
1k
views
Are "large enough" finite etale covers arithmetic?
Let $X$ be a variety over a number field $K$. Then it is known that for any topological covering $X' \to X(\mathbb{C})$, the topological space $X'$ can be given the structure of a $\overline{K}$-...
- 437
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
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
1
vote
0
answers
68
views
Glueing local systems over union of compact Riemann surfaces
Let $X,Y$ be two connected, non-singular compact Riemann surfaces such that $X$ intersects $Y$ transversely at two distinct points. Let $L$ be a $\mathbb{C}$-local system on $X$. Let $L'$ be the ...
- 1,593
1
vote
0
answers
175
views
Canonical étale path between a point and its ''nearby'' point
Consider the punctored line $X=\Bbb{A}^1_k\setminus \{s_1,\ldots,s_n\}$ over some field $k$. A(n étale) path in $X$ between two geometric points $x$ and $y$ is, by definition, an isomorphism between ...
- 311
8
votes
1
answer
339
views
The direct product of the geometric fundamental group and the absolute Galois group
Given a geometrically connected variety $X$ over $\mathbb{Q}$ we have a short exact sequence
$$
1\to \pi_1(X_{\overline{\mathbb{Q}}})\to \pi_1(X)\to Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to 1.
$$
A ...
user145520
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
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
0
votes
0
answers
392
views
Galois cover corresponding to finite quotient of the étale fundamental group
Let $X$ be a connected scheme,$\pi_1(X,\bar{x})$ its étale fundamental group for some geometric point $\bar{x} : Spec(K) \rightarrow X$
and $E = \pi_1(X,\bar{x})/N$ a finite quotient of $\pi_1(X,\bar{...
- 181
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
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
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
8
votes
0
answers
294
views
Relationships among constructions of fundamental group for schemes
There seem to be several constructions of fundamental group for schemes and stacks: by Grothendieck, Deligne, Nori, Noohi, Esnault-Hai, Vakil-Wickelgren, perhaps others as well. I am trying to ...
- 447
8
votes
2
answers
721
views
Galois categories for topological spaces?
Can the theory of Galois categories (as developed in SGA1) be modified to produce the usual fundamental group of a topological space (maybe assumed to be path connected and locally path connected)?
...
- 3,284
19
votes
2
answers
1k
views
Can one compute the fundamental group of a complex variety? Other topological invariants? [duplicate]
Given a system of polynomial equations with rational coefficients, is there an algorithm to compute the geometric fundamental group of the variety defined by these equations? I'm interested in both ...
- 15.4k
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
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
9
votes
1
answer
1k
views
Galois theory, topos vs fundamental groups
Classical Galois theory states that the etale topos X of a field k is equivalent to the classifying topos of the absolute Galois group of k.
(Marc Hoyois, Higher Galois theory, $\S$3, arXiv:1506....
- 91
9
votes
1
answer
501
views
Mapping class group and representation of fundamental group of Riemann surfaces
Let $S$ be a Riemann surface with genus $g>0$. Let $M$ be the mapping class group of $S$. $Hom(\pi_1(S),Gl(n, \mathbb{C}))$ is the representation space of fundamental group of $S$
Question: Is ...
- 1,081
1
vote
1
answer
277
views
What does the group of automorphisms corresponding to $\mathfrak{g}$
I am reading a book titled "Lectures on An Introduction to Grothendieck's Theory of the Fundamental Group" by J.P. Murre. I am in the chapter 4 titled "Fundamental groups". Here he fixes a base ...
- 215
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
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
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
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 ...
- 113
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
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 ...
- 83
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 $\...
- 14.2k
16
votes
0
answers
591
views
Are there any examples of hyperbolic curves over finite fields such that the action of frobenius on its prime-to-$p$ fundamental group is known?
Let $X$ smooth curve over a finite field $\mathbb{F}_q$ of type $(g,n)$ - that is, $X$ is an open subscheme of its genus $g$ compactification obtained by removing $n$ points.
Any such curve ...
- 10.7k