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3
votes
0answers
92 views

Galois categories and the connected components functor

In stacks 0BMQ, a Galois category is defined to be a functor $F:\mathsf C\longrightarrow \mathsf{FinSet}$ such that $\mathsf C$ is finitely bicomplete, every object ...
5
votes
2answers
209 views

Constructively, are all fibrations cloven?

A "cloven fibration" is a fibration for which we have an explicit choice of cartesian liftings; this is often phrased as, "We can pick a lifting without using the axiom of choice". Firstly, I'm a bit ...
6
votes
0answers
187 views

Homotopical interpretation of flatness?

I have read a discussion (in a less common language) which discussed a homotopical interpretation of flatness, which went something like: A map of commutative algebras is flat if pushing it out ...
3
votes
0answers
99 views

Some questions on Kontsevich's moduli space

Motivation: Work of Eisenbud, Harris, and Mumford shows that $\mathcal M_g$ is of general type when $g≥24$. Moreover, by Logan's function $f(g)$ , $\overline {\mathcal M_{g,n}}$ is of general type for ...
2
votes
0answers
138 views

local description of $\mathbb{P}^2$-fibrations over $\mathbb{P}^1$

Let $X$ be a rational threefold (over the field of complex numbers) with terminal singularities. It is well-known that $X$ has only finitely many singular points $x_1,x_2, \ldots,x_n$. To be more ...
0
votes
0answers
97 views

Generalization of the fiber changing trick for principal bundles?

We know that a principal bundle can induce a fiber bundle as follows: if $F$ is a space which admits a $G$-action then a principal $G$-bundle $p: E \to B$ induces a fiber bundle $p: E \times_G F \to B$...
4
votes
1answer
167 views

On push-forward of the constant sheaf for fibrations

Let $f\colon E\to B$ be a fiber bundle with a connected fiber $F$, $f$ is proper. Let $\underline{\mathbb{C}}_E$ be the constant sheaf on $E$. Let $f_*(\underline{\mathbb{C}}_E)$ denote its direct ...
4
votes
1answer
287 views

Shafarevich conjecture for abelian varieties

In the paper "Arakelov's theorem for abelian varieties" Faltings proves the Shafarevich conjecture for abelian varieties. The statement is the following: Let B be smooth projective a curve, S a ...
1
vote
0answers
90 views

Orders of zeros of section of sheaf

We have a semistable family (fibers has normal crossings and they are reduced, multp 1) $f: X \rightarrow Y,$ of complex curves over a smooth curve $Y.$ The family is smooth over the set $U=Y\...
-1
votes
1answer
115 views

Construction of fibration over Riemannian Manifold

Let $\pi: E \rightarrow B$ be a fibration over a Riemannian manifold $B$, with $\pi^{-1}[b]$ homeomorphic to $\mathbb{R}$. More precisely: I want each fiber $\pi^{-1}[b]=Im(f_b)$ for some $C^{\...
8
votes
2answers
276 views

Circle Action on Quaternionic Projective Space

Quoting from Wikipedia article on quaternionic projective space: Therefore the quotient manifold $$ \mathbb{HP}^{2}/\mathrm{U}(1) $$ may be taken, writing $U(1)$ for the circle group. It has ...
1
vote
2answers
144 views

Four Sphere Fibrations

Does there exist a manifold $M$ and a compact Lie group $H$ such that we have a fibration $H \to S^4 \to M$, where $S^4$ is the four sphere?
2
votes
0answers
99 views

Elliptic fibration arising from a higher genus linear system

Let $H$ be a very ample linear system on a smooth compact complex surface $X$ whose Kodaira dimension is $\geq 0$. A general element of $H$ is smooth and has genus $\geq 2$. Let $L\subset H$ be a ...
1
vote
1answer
157 views

Elliptic fibrations with few singular fibers

It is known that non-isotrivial fibrations of genus $g>0$ curves over the projective line have a bunch of singular fibers. There are at least three of them. It is not difficult to prove that an ...
7
votes
1answer
257 views

Pullback-stable model of fibrewise suspension of fibrations (in simplicial sets, or similar setting)

Given a fibration $p : Y \to X$ in simplicial sets (or any other model category), there are various ways to construct its fibrewise suspension, i.e. its suspension as an object of the slice $\...
3
votes
2answers
399 views

Cup product of cohomology in a Serre spectral sequence

How to use Serre spectral sequence to compute cup product structures? Let $F\to E\to B$ be a fibration. Suppose all the differentials of the corresponding Serre spectral sequence of cohomology are ...
3
votes
3answers
330 views

Does there exist a holomorphic fibration of genus two over $\mathbb{P}^{1}$ with $7$ nodal singularities?

This is a problem about the holomorphic fibration on a complex manifold. Does there exist a holomorphic fibration of genus two over $\mathbb{CP}^{1}$ with 7 nodal singularities? If you are aware of ...
4
votes
1answer
194 views

Homotopy fibers and stratified fibrations

Suppose I have a map $f:X \to Y$ of topological spaces and a nice stratification of $X$ ( say such that the inclusion of each stratum is a Hurewicz cofibration) such that the restriction of $f$ to ...
6
votes
2answers
355 views

cup product and Steenrod operations in Serre spectral sequence

Let $F\to E\to B$ be a fibration with $B$ simply-connected. Suppose all differentials in the cohomology Serre spectral sequence (corresponding to the above fibration) are zero maps. Then as a graded ...
5
votes
0answers
170 views

When is a circle fibration a circle bundle?

Let $\pi : E \to B$ be a Serre fibration over a CW complex, with circle fibers. In the orientable case, it is easy to see that $\pi$ is fiber homotopy equivalent to a principal $SO(2)$--bundle. ...
0
votes
0answers
113 views

Characterization of Singular locus

Let $A$ be a complete regular local ring over a field $k$ and $B$ be a complete normal local ring over a field $k$. We assume that the Krull-dimension of $A$ is greater than 1. We consider the ...
2
votes
0answers
58 views

cohomology ring of cross-section space of one-point compactification of tangent bundle

Let $M$ be an $m$-manifold whose cohomology is known. Let $TM$ be the tangent bundle of $M$ and $\xi$ be the fibre-wise one-point compactification of $TM$. Then $\xi$ is a $m$-sphere bundle over $M$. ...
4
votes
2answers
347 views

Infinitesimal deformations of a fibration

Let $f:X\rightarrow Y$ be a morphism of normal projective varieties over an algebraically closed field with connected fibers. Assume that both $Y$ and the general fiber of $f$ admit a non-trivial ...
3
votes
1answer
128 views

group completion theorem by using homology fibrations

In the paper Homology fibrations and group completion theorem, McDuff-Segal (www.maths.ed.ac.uk/~aar/papers/mcdsegal.pdf), page 281: Let $M$ be a topological monoid such that $\pi_0M$ is generated by ...
3
votes
1answer
233 views

Automorphisms of surfaces

Let $X$ be a projective surface with a morphism $f:X\rightarrow\mathbb{P}^1$. Assume that $f^{-1}(t)\cong\mathbb{P}^1$ for any $t\neq 0$ but $f^{-1}(0)$ is the union of two $\mathbb{P}^1$'s ...
9
votes
2answers
557 views

Pull-back of a fibration along a homotopy equivalence and homotopy classes of sections

I previously asked this on Math.SE but didn't receive a satisfactory answer. Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: B'\...
2
votes
0answers
152 views

A homomorphism in the long exact sequence of a fibration for a homogeneous space of a Lie group

Let $G$ be a connected Lie group, and let $H\subset G$ be a (closed) Lie subgroup, not necessarily connected. Set $X=G/H$. The fibration $j\colon G\to X$ with fiber $H$ induces an exact sequence $$ \...
9
votes
2answers
328 views

Fibrations of projective varieties

Let $f:X\rightarrow Y$ be a flat morphism of normal projective varieties with fibers of positive dimension (in particular all the fibers are connected and of the same dimension). Let $g:X\rightarrow ...
3
votes
1answer
268 views

Reference for Arakelov's theorem: $K^2_f=0$ iff $f$ is locally trivial

Let $f:X\longrightarrow B$ be a family of curves, with $f$ relatively minimal, over a fixed curve $B$ ($B$ is projective, irreducible and smooth). The fibration $f$ is said locally trivial if all ...
2
votes
1answer
150 views

Fibration $p : \tilde Y \to Y$ with discrete fiber induces bijection $p_*:[X, \tilde Y]_* \to [X, Y]_*$

If $X$ is simply connected, locally path connected space and $p : \tilde Y \to Y$ is a covering map then it is easy to show that it induces bijection $p_*:[X, \tilde Y]_* \to [X, Y]_*$. Let's weak ...
2
votes
1answer
186 views

fiber sequence of principal bundles

Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle $$ G\to E\to B,$$ then $B=E/G$, the orbit space under action of $G$. Let $BG$ be the classifying space of $G$. ...
3
votes
1answer
168 views

What's the geometric statement of this fibrewise integration on a symplectic manifold with Lagrangian fibration?

I understand this statement from the physics side. Consider an $n-$dimensional manifold $\cal M$ ("configuration space") and its cotangent bundle ${\cal P} = T^*\cal M$ ("phase space"), a symplectic ...
2
votes
1answer
148 views

Open Books $( \Sigma, \Phi) $ living in Lefschetz Fibrations over the disk $D^2$

I have a question about open books and Lefschetz fibrations over the 2-disk $D^2$. Please let me set it up first, before going on. Setup: Say we have a Lefschetz fibration $f: W^4 \rightarrow D^2 $ ,...
6
votes
2answers
574 views

Isotrivial fibrations over $\mathbb P^1$

First of all I want to say that algebraic geometry is not "my field of research" so I apologize if the notation is not standard. $S$ is a smooth complex projective surface with a fibration $f$ over ...
4
votes
1answer
380 views

Classifying space for fibrations with Eilenberg-MacLane space as fibers

The following result seems to be frequently quoted: Consider the fibration $K(\pi,n)=\Omega K(\pi,n+1)\to PK(\pi,n+1)\to K(\pi,n+1)$. Let $B$ be any topological space (which is not too pathologic). ...
2
votes
2answers
356 views

A fibration of classifying spaces

Let $G$ be a Lie group, $N$ a closed connected normal subgroup. Let $BG$, $BN$, $B(G/N)$ be the classifying spaces of $G,N$ and $G/N$. Is there a fibration $BN\to BG\to B(G/N)$ ? It seems that such a ...
3
votes
0answers
128 views

Recognizing Simplicial (Quasi)Fibrations

Let's say we are given two finite simplicial complexes, which I will suggestively call $E$ and $B$. We'd like an algorithm for the following decision problem: Does there exist a simplicial map $p:...
2
votes
1answer
102 views

Going Back-and-Forth Between Different Expressions/“Representations” for Open Books.

I am trying to have a better understanding of how one goes , "travels" between the different formats/layouts of open books for a fixed given 3-manifold M; between the abstract type and the "actual" ...
4
votes
0answers
296 views

Global geometry of discriminant locus

Let $X$ be a smooth projective threefold, and $\pi : X \to S$ an elliptic fibration over a surface (i.e. flat, with general fiber an elliptic curve). I'm interested in constructing such fibrations ...
3
votes
1answer
143 views

Equivalence of the total spaces of two Serre fibrations with equivalent fibers

Let $B$ be a connected pointed CW complex, let $E$ and $E'$ be two CW complexes and let $f\colon E\to B$ and $f'\colon E'\to B$ be two Serre fibrations. Let $g\colon E\to E'$ be a continuous map such ...
2
votes
1answer
148 views

modify a fibration with a fiber of higher multiple

Suppose we have an elliptic fibration $f:X\to \mathbb{P}^1$, with a singular fiber $F$, can we construct an elliptic fibration over $\mathbb{P}^1$ with fiber $nF$?
6
votes
1answer
280 views

Where is simpleness used in the proof of existence of Postnikov towers of principal fibrations?

I've read one proof, rather long, in Allen Hatcher's book. There the key is Lemma 4.70, which uses the relative Hurewicz Theorem. But there is another, shorter proof in J.P.May's book "A concise ...
8
votes
4answers
665 views

What does it mean to speak of a homotopy fibration sequence?

I'm reading a paper in which the following is done. We have a certain particular map of spaces $f:X\to Y$ and then it is said something along the lines of "let $Z_f$ denote the space whose defining ...
8
votes
2answers
499 views

Uniformization of Kodaira fibered surfaces

Consider a Kodaira fibration. i.e. a smooth non-isotrivial fibration $X\rightarrow C$ with $X$ a smooth complex surface and $C$ a smooth complex curve, such that both the genus of $C$ and genus of the ...
5
votes
1answer
261 views

$\Pi$, $\Sigma$, and identity types without $\eta$ in comprehension categories

In comprehension categories, dependent sums are defined as a choice of left adjoints for all reindexing functors along display maps, satisfying a Beck-Chevalley condition. Dependent products are ...
34
votes
1answer
786 views

Is an affine fibration over an affine space necessarily trivial?

Let $X$ be an algebraic variety over an alg. closed field with zero char. and let $f:X\to \mathbb{A}^n$ be a smooth surjective morphism, such that all fibers (at closed points) are isomorphic to $\...
6
votes
2answers
331 views

Why is the path fibration a strong Hurewicz fibration?

In May and Sigurdsson "Parametrized homotopy theory" there is a general treatment of Hurewicz style model structures in Chapter 4, see definitions 4.2.1 and 4.2.2. I am trying to adapt these to a more ...
0
votes
0answers
117 views

on the fibers over closed points

Let $X$ and $S$ $k$-schemes of finite type . ($k$ a field) and $U$ an open subset of $X$ Let $f:X\rightarrow S$ a $k$-morphism of finite type. We assume that for any closed point $s\in S(\bar{k})$, $...
9
votes
1answer
474 views

What is the difference between internal presheaves and presheaves on a total space?

Suppose that $\mathbb{C}$ is a category with finite limits and that $\mathcal{D}$ is a category internal to $\mathbb{C}$. We can also represent $\mathcal{D}$ as a fibration $\mathbb{D}\to\mathbb{C}$. ...
23
votes
3answers
884 views

Is the counit of geometric realization a Serre fibration?

Recall that a Serre fibration between topological spaces is a map which has the homotopy lifting property (HLP) for all CW complexes (equivalently for all disks $D^k$). The Serre fibrations are the ...