2
votes
0answers
98 views

when is an irreducible SL_2(C) representation of a cusped hyperbolic 3-manifold scheme reduced or smooth

Let $M$ be an orientable cusped hyperbolic 3-manifold. Let $\rho \in hom(\pi_1(M),SL_2(\mathbb{C}))$ be an irreducible representation. Is $\rho$ scheme reduced ? What can prevent $\rho$ from being ...
0
votes
0answers
67 views

moduli space of equivariant holomorphic embeddings into the quintic

I'd like to understand better the following problem, whether it is mathematically well-posed, trivial, etc. Fix a non-negative integer $g$ and consider the space ...
4
votes
1answer
172 views

Relation between Milnor ring and middle dimensional homology of hypersurface

I have suspected that the following is well-known: If $P$ is a homogeneous polynomial of degree $d$ in $n$ variables (for example, Fermat quintic $x_1^5 + \cdots + x_5^5$). The Milnor ring is ...
2
votes
2answers
277 views

Origin of the name “Torelli group”

The genus $g$ Torelli group $I_g$ is the kernel of the action of the mapping class group of a genus $g$ surface on the first homology group of the surface. The first paper I am aware of that uses the ...
9
votes
6answers
1k views

Is there a good notion of morphism between orbifolds?

Following Thurston, an orbifold is a topological space which looks locally like a finite quotient of $\mathbb R^n$ by a finite group of $O(n)$: this is expressed using charts as for differentiable ...
-2
votes
2answers
221 views

topology on varieties

Let $X, Y$ be varieties over $\mathbb{C}$, and the topology I am talking about is the Eucliden topology. I am not sure if the following two results are true, and where can I find the references: (1) ...
2
votes
1answer
320 views

looking for an identity for higher jet bundle $J^kM$?

We know this fact that the first jet Bundle $J^1M$ is diffeomorphic with $T^*M×\mathbb{R}$.i.e, ($J^1M=T^*M×\mathbb{R}$) Is there something like this identity for higher jet bundle $J^kM$? I editted ...
9
votes
1answer
497 views

moduli spaces are kahler?

I often heard from experts that "moduli spaces are Kahler". This sounds as a meta-theorem asserting that every time one defines reasonable moduli spaces, then there is a standard strategy to see ...
0
votes
0answers
166 views

Fractional degree of a map?

Is there some natural notion of a fractional degree of a map? The degree of a map is a generalization of the winding number, and fractional winding numbers appear in the (mathematical physics) ...
3
votes
1answer
266 views

On complex surfaces with Kodaira dimension 1

Let $S$ be a complex surface of Kodaira dimension $1$ and $\pi_{1}(S) \neq 1 $. What is known on possible diffeomorphism types of such $complex$ surfaces with a given fundamental group? Is it true ...
2
votes
1answer
216 views

Covering seifert manifolds

Let $M$ be a 3-manifold with boundary. If $M$ has an orientable finite cover that is a Seifert fiber space, then is $M$ also a Seifert fiber space?
6
votes
3answers
557 views

Complex structure of the Teichmüller space in terms of Fenchel-Nielsen coordinates

The Teichmüller space $T_g$ of genus $g$ Riemann surfaces can be parameterized in terms of Fenchel-Nielsen coordinates, taking values in $\mathbb{R}^{3g-3}\times \mathbb{R}_+^{3g-3}$. The ...
7
votes
1answer
253 views

Generators for exact representations of 3-manifold groups

Does anyone have a list of matrix generators for a bunch of hyperbolic 3-manifold groups? I am testing an algorithm and am looking for a collection of test cases. I am looking for exact values of ...
4
votes
0answers
202 views

Relationship between virtual cohomological dimension and tautological rings for moduli spaces of curves

Here's the short version of the question. For $M_{g,n}$, $M_{g,n}^{rt}$, $M_{g,n}^{ct}$ and $\overline M_{g,n}$ it seems that the virtual cohomological dimension is given by the complex dimension plus ...
1
vote
1answer
391 views

Is surface $x^2+z^2=2\cdot y^2$ something of a Möbius strip?

This question is naive. My association with Möbius strip comes from not being able to smoothly extract positive solutions of the diophantine equation $$x^2+z^2=2\cdot y^2$$ I got a parametrization ...
0
votes
0answers
151 views

Casson's invariant and intersection homology

EDIT: Immediately after I wrote this question, I remembered the elegant paper "An intersection homology invariant for knots in a rational homology 3-sphere" by Frohman and Nicas, which I believe does ...
1
vote
1answer
90 views

On the simply connectedness of Symmetric products and Hilbert schemes of points

My first question is whether $m$-th symmetric product of $\mathbb{C}^{n}$ is simply connected, where $n\geq 3$. The second question is whether $Hilb^{m}(\mathbb{C}^{n})$ is simply connected, where ...
10
votes
2answers
1k views

What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...
9
votes
2answers
308 views

Algebraic Stratifications of $G$-varieties

My question is simple: Given an algebraic group $G$ acting on a variety $X$ algebraically. If the orbits are of finite number then they form what is called an algebraic stratification of $X$. Now my ...
1
vote
0answers
65 views

Complements to reducible plane projective curves

Hello, Suppose that $C_1,C_2\subset\mathbb P^2$ are projective curves (over $\mathbb C$); $C_1$ and $C_2$ may be reducible but they must not have a common component. Let $L\subset \mathbb P^2$ be a ...
11
votes
1answer
487 views

When are Brieskorn Manifolds Homeomorphic?

Let $a_0, \dots, a_n, b_0, \dots, b_n \in \mathbb{N}$ and consider two polynomials $f = \sum_{i = 0}^{n} z_{i}^{a_i}$, $g = \sum_{i = 0}^{n} z_{i}^{b_i}$. Given two Brieskorn manifolds $\Sigma(a_0, ...
3
votes
1answer
186 views

Vanishing of !-restriction of constructible sheaves

If $\mathcal F$ is a constructible sheaf (say of $\mathbb C$-modules) on a (real) manifold concentrated in degree $0$ and $i\colon Z \hookrightarrow X$ is a submanifold, can I say anything about ...
16
votes
2answers
371 views

A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface?

In a 1986 paper, Harer and Zagier proved the recursion: $$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$ where e(g,n) is the number of ways of grouping sides $S_1...S_{2n}$ of a 2n-gon ...
1
vote
1answer
166 views

Relation between the Character variety of a knot $K\subset M$ and that of $M$

Suppose there is a knot $K\subset M$,$M$ is a closed 3-manifold. What's the relation between $\chi(\pi_{1}(M-K))$ and $\chi(\pi_{1}(M))$? ($\chi(G)$ means the $\text{SL}_{2}(C)$ representation of $G$ ...
15
votes
1answer
451 views

Can the SL_2 character variety of a three-manifold be nonreduced?

Let $M^3$ be a three-manifold and consider the representation variety and the character variety of $M$: $$Y=\operatorname{Hom}(\pi_1(M^3),\operatorname{SL}(2,\mathbb C))$$ ...
4
votes
1answer
206 views

Smooth projective toric varieties which are quotients of product of spheres and torii by a free torus action?

My question is just as in the box. Is every smooth projective toric variety diffeomorphic to a quotient of $\prod_i S^{n_i} \times T^k$ (I know torus is a one-sphere but I just wanted to make clear I ...
5
votes
2answers
266 views

Given the vertices of a convex polytope, How can we construct its Half-Space representation

HI, I have a question regarding convex polytopes. Let us say I have the vertices of a polytope which I name as $ V = \{v_1,\cdots,v_k\}$. Each of the $v_k$ are n-dimensional vectors, i.e. ...
14
votes
0answers
597 views

Horrible sets and blowups in Hubbard's Teichmuller theory

Edit: I can rephrase this question this way: When blowing up every point in the $x$-axis in $\mathbb{C}^2$ by means of an inverse limit of finite blowups, how can anything be 'left over'? The horrible ...
0
votes
1answer
298 views

K3 surfaces as double covers of P2

Consider a K3 surface given as the double branched cover over P^2, branched along a smooth sextic. What is the ramification divisor in the K3 surface?
0
votes
1answer
356 views

A Question about SO(n)

My question is: How to find out all the finite subgroup of SO(n)? Or just for the simple case SO(4) SO(5)? With more discribe: If $S^n\backslash \Gamma$ is a manifold, I just want to know that ...
10
votes
4answers
2k views

Geometric invariant theory for geometers

I am trying to learn "Geometric invariant theory" like it was introduced by Mumford. But I do not have a strong background in algebraic geometry since I work in geometric topology and geometry. So ...
4
votes
1answer
238 views

braids and dynamics of roots of a polynomial

The 2-variable polynomial equation $f(z,t) = 0$ with $z = \mathbb{C}, \\,t \in \mathbb{S^1}$ has $n = \mathrm{deg}_z f$ solutions each fixed $t$. I wanted to follow the roots as they travel with time ...
7
votes
1answer
403 views

Why is Heegaard Floer Homology defined in terms of Sym$^g\Sigma_g$ instead of Pic$^g\Sigma_g$?

Recall the definition of Heegaard Floer homology: $\Sigma_g$ is a closed surface, and $\{\alpha_1,\ldots,\alpha_g\}$ and $\{\beta_1,\ldots,\beta_g\}$ are sets of attaching circles. Then Heegaard ...
5
votes
1answer
183 views

Finite index subgroups of the mapping class group with geometric meaning

I have got a question that is perhaps not precise in a mathematical sense. Is there a classification of all coverings of the moduli space of Riemann surfaces which are moduli spaces themselves, that ...
18
votes
2answers
1k views

Is there anything special about the Riemann surface $y^2 = x(x^{10}+11x^5-1)$?

I stumbled upon the fact that the Bolza surface can be obtained as the locus of the equation, $y^2 = x^5-x$ Its automorphism group has the highest order for genus $2$, namely $48$. I recognized ...
2
votes
2answers
611 views

One tetrahedron inside another tetrahedron

Consider two tetrahedrons one placed inside the other. Prove that the sum of all 6 sides of inner tetrahedron is at most the sum of the 6 sides of exterior tetrahedron.
15
votes
2answers
646 views

When does the blow-up of $CP^2$ at N points embed in $CP^4$?

Write $X_N$ for this blow up. Place the N points in 'general position' as needed. Then $X_6$ embeds in $CP^2$ as a smooth cubic surface. (See, eg, Griffiths and Harris.) But there is no other ...
4
votes
2answers
311 views

Twisted cohomology of the mapping class group

Let $M_{g,1}$ be the mapping class group of surfaces of genus g $\geq 1$ with one boundary component. By $S_g$ we denote a closed surface of genus $g$. In the paper "Families of jacobian manifolds ...
8
votes
3answers
824 views

Two rectangular parallelepiped

Prove that if we have two rectangular parallelepiped (cuboids) such that one of them is placed inside the other then the sum of the three lengths of the inner parallelepiped is at most the sum of the ...
8
votes
2answers
368 views

Cohomology of representation varieties

Perhaps this question is too general then I am sorry about this. My question is the following. Let $\pi$ be the fundamental group of a compact surface of genus $g$ (with if necessary $n$ punctures) ...
7
votes
1answer
561 views

Manifolds and Polynomials

Given a compact smooth manifold $M \subset R^k$ there is a Polynom $f\in R[x_1,..x_n]$ such that the zero set of $f$ is diffeomorphic to $M$. Can the coefficients of $f$ be pertubated slightly to a ...
13
votes
3answers
1k views

When are (finite) simplicial complexes (smooth) manifolds?

Hi, is there an algorithm that determines if a given simplicial complex is a.) a manifold b.) a smooth manifold c.) homotopy equivalent to a manifold d.) a real algebraic variety ?
14
votes
0answers
414 views

Do mapping classes have gonality?

(This question was discussed by people at the PCMI workshop on moduli spaces, without any clear resolution, so I thought I'd throw it open to MO.) The hyperelliptic mapping class group is (by ...
12
votes
1answer
548 views

Are surface bundles over a surface with non-zero signature necessarily complex (or algebraic)?

By "surface bundle over a surface" I mean a compact, oriented 4-manifold $X$ which is the total space of an oriented fiber bundle $X\to B $ over an oriented 2-manifold $B$. Assume that the signature ...
7
votes
1answer
764 views

Is $Sym^g$ of a Riemann Surface of genus $g$ Calabi-Yau?

The $g$-fold symmetric product of a Riemann surface of genus $g$ naturally has both a symplectic structure as well as a complex structure. Is it in fact Calabi-Yau? If so, is anything known about a ...
3
votes
0answers
292 views

What is the behaviour of a smooth 3-manifold acting by a circle?

As Mumford pointed out in his paper 'Topology of Normal Singularities and a Criterion for Simplicity'(1961), every point $p$ on a normal complex surface $V$ has an associated 3-manifold $M$ which is ...
8
votes
3answers
642 views

orbifold covering

Given two compact surfaces $S_1$ and $S_2$ of genus at least $2,$ it is easy to tell when $S_1$ covers $S_2:$ whenever $\chi(S_2)$ divides $\chi(S_1).$ Now, suppose I have two orbifolds of negative ...
4
votes
1answer
446 views

Orbifold Beauville-Siu

The Beauville-Siu theorem states that for a compact Kahler manifold the following two statements are equivalent: $M$ admits a surjective holomorphic map with connected fibers to a compact Riemann ...
1
vote
0answers
138 views

Generalized vector bundles with singularities on Riemann surfaces

Let $X$ be a Riemann surface of genus $g \geq 2$ or in other words a complex curve. Let $P_1, \ldots, P_m$ be points in $X$ and $E \rightarrow X$ surjective map such that is is a complex ...
5
votes
0answers
493 views

Computing the Chern class for a flat line bundle using the holonomy group?

Let $X$ be a Riemann surface of genus $g \geq 2$. I would like to consider flat line bundles on $X$. Flat line bundles can be identified with representations $\pi_1(X) \rightarrow U(1)$. From ...