All Questions
Tagged with hyperbolic-geometry ag.algebraic-geometry
32 questions
3
votes
1
answer
157
views
Geometry and topology of Fuchsian character varieties
Consider the hyperbolic space, $\mathbb H^2$. A Fuchsian group is a discrete subgroup of $\text{PSL}(2,\mathbb R)$. We can generate tessellations, especially $\{p,q\} \;\text{tesellations}$ of $\...
- 357
7
votes
0
answers
124
views
Projections of closed geodesics under the modular function
In the answers to this question it was shown that for closed geodesics on $\mathbb{H}^2/\Gamma(2)$, the projection under the modular function $\lambda$ is an immersed topological component of a real ...
- 68.9k
1
vote
0
answers
143
views
Describing the hyperbolic structure of punctured torus in terms of the period lattice
Let $T$ be a torus, $T^* = T - \{p\}$ be the complement of a point $p$. Let's fix a pair of generators $x,y\in\pi_1(T^*)$. Their images in $\pi_1(T)$ also generate, and will also be denoted by $x,y$.
...
- 7,527
1
vote
1
answer
140
views
Hyperbolicity and inequality for variety of general type
$\DeclareMathOperator\BMY{BMY}$Let $(X,H)$ be a smooth, complex, projective, polarized variety of dimension $n\geq3$, whose canonical bundle $K_X$ is big and nef.
Is it know whether the inequality $\...
- 828
0
votes
0
answers
111
views
The upper bound of hyperbolic cosine function in complex plane
I want to find the upper bound of the following function :
\begin{equation}
M(\lambda)=\max _{E \in \mathbb{C}}\left|L_{+}-L_{-}\right|
\end{equation}
where
\begin{equation}
\begin{aligned}
& 4 \...
1
vote
0
answers
89
views
Almost modularity of Belyi curves and etale fundamental group of non-Belyi curves
Belyi's theorem states that every curve defined over $\mathbb{\bar Q}$ is almost modular (obtained from $\mathbb{H}^2/\Gamma,\ \Gamma$ a finite index subgroup of $PSL(2,\mathbb{Z})$), after ...
- 441
1
vote
0
answers
227
views
Why are compact arithmetic surfaces defined through quaternion algebras (usually) only over $\mathbb{Q}$?
As in Chapter 6.2 of "Introduction to arithmetic groups" (by D.W. Morris), compact arithmetic surfaces could be defined through quaternion algebras $\mathbb{H}^{a,b}_F=\big(\frac{a,b}{F}\big)...
- 128
1
vote
0
answers
140
views
Obstruction in construction of some lattices, related with $K3$ surfaces
I am considering a certain $K3$ surface that is lattice-polarized in two ways.
This leads to the following simple problem in lattice theory:
(Let me borrow notations for lattice from Ch.14 of this ...
- 1,841
1
vote
0
answers
70
views
Classification of principal monodromy elements
Let $(X,0)$ be a germ of normal analytic space with an isolated singularity at $0$, and let $Y:=X\backslash\{0\}$. Suppose $Y$ has a complex-hyperbolic metric which is complete at $0$. Burns-Mazzeo ...
- 1,121
4
votes
1
answer
609
views
About isotopy and homotopy
In the " A Primer on Mapping Class
Groups
Benson Farb and Dan Margalit"
We have :
Proposition 1.10 Let $\alpha$ and $\beta$ be two essential simple closed curves in a surface $S$. Then $\...
- 119
2
votes
1
answer
136
views
Example of maximal multicurve complex
in this paper we have :
" On the Teichmüller tower of mapping class groups By Allen Hatcher at Ithaca, Pierre Lochak at Paris and Leila Schneps."
Definition. The maximal multicurve complex $...
- 119
7
votes
3
answers
655
views
Hyperbolic 3-manifolds inside algebraic varieties
I have a hyperbolic 3-manifold $M$ that I'd like to see sit inside a complex algebraic variety $V$ as beautifully and snugly as possible. I don't want to specify attributes of "beauty" (e.g.,...
- 121
0
votes
0
answers
336
views
Efficiently find at least one lattice point on hyperbola of equation $axy+bx+cy+d=0$
I need an algorithm to efficiently find at least one lattice point on a hyperbola of equation $axy+bx+cy+d=0$.
Lattice point means integer coordinates and equation with integer means diophantine ...
- 101
4
votes
0
answers
129
views
What does hyperbolicity of curves buy us in the arithmetic context?
This is going to be a fairly vague question but hopefully it will have concrete answers:
There is this recurrent phenomenon in the arithmetic of curves (possibly stacky,affine) where there is a "...
- 7,746
3
votes
0
answers
152
views
Riemannian metric over moduli space of Riemann spheres with n punctures
In the paper `Tessellations of moduli spaces and the mosaic operad' by Devadoss (https://arxiv.org/pdf/math/9807010.pdf), on page 5-6, the author identifies hyperbolic planar tree space (or the ...
- 31
4
votes
0
answers
103
views
Quadrics in $\mathbb{H}^3$
Consider a hyperbolic space $\mathbb{H}^3$ in the Beltrami-Klein model:
$$\mathbb{H}^3=\{(x,y,z|x^2+y^2+z^2\leq 1\}\subset \mathbb{R}^3.$$
Let $Q$ be a quadric in $\mathbb{R}^3.$
Question: What is a ...
- 4,316
10
votes
0
answers
379
views
Hyperkähler structure on the moduli space of tetrahedra?
Consider a moduli space of geodesic tetrahedra in the hyperbolic space $\mathbb{H}^3.$ In the Klein's model the hyperbolic space can be presented as the interior of a unit ball
$$
\mathbb{H}^3=\{(x_1,...
- 4,316
8
votes
0
answers
316
views
Lines in upper half-space
A couple of years ago, I taught an undergraduate class introducing various aspects of classical geometry, learning the (beautiful!) subject as I went. I found one thing that really bothered me: the ...
- 6,308
10
votes
1
answer
314
views
Is there a divisor in $\mathbb P^2$ such that all analytic maps into its complement algebraize?
Is there a closed subscheme $D$ in $\mathbb P^2_{\mathbb C}$ pure of codimension one such that, for all algebraic varieties $X$ over $\mathbb C$, any analytic map
$$ \phi: X(\mathbb C) \to \mathbb P^...
- 103
3
votes
1
answer
388
views
Kobayashi distance function on the upper half-space
I asked this question already in mathstackexchange but got no answer, so I ask it again here.
Let $\mathbb{H}_{g}$ be the Siegel upper half space, i.e., the set of complex symmetric $g\times g$ ...
- 637
5
votes
1
answer
307
views
Infinitesimal deformations of fake projective planes (or ball quotients)
This question is related to the answer I gave to this MO question. What I'm asking is probably well-known to the experts in the field, and I apologize in advance if this turns out to be trivial.
By ...
- 66.3k
3
votes
1
answer
328
views
Decomposition of hyperbolic surfaces near cusps into annuli
Let $C=\mathbb{H}/\Gamma$ be a hyperbolic surface and $c$ a cusp of this sruface. In the paper "Billiards and Teichmüller curves on Hilbert modular surfaces" by C. McMullen, it is claimed that near ...
- 2,221
1
vote
0
answers
126
views
Is triple point intersection 'generic' in Teichmuller space?
Let $S$ be a hyperbolic surface of finite type and $\alpha,\beta$ be two closed curves. Consider $X$ to be the set of all those points $\chi$ in the Teichmuller space $\mathcal{T}(S)$ of $S$ such that ...
- 1,713
3
votes
1
answer
253
views
simplex in hyperbolic space and quadrics in projective space
I am trying to understand Goncharov's paper "Volumes of hyperbolic manifolds and mixed Tate motives", http://www.ams.org/journals/jams/1999-12-02/S0894-0347-99-00293-3/S0894-0347-99-00293-3.pdf
In ...
- 31
16
votes
1
answer
1k
views
Borel-Serre compactification of $\mathbb{H}^3 / SL_2(\mathcal{O}_K)$
Let $\mathbb{H}^3$ be the three-dimensional hyperbolic space. Let $K$ be an imaginary quadratic number field and $\mathcal{O}_K$ its ring of integers. Then $SL_2(\mathcal{O}_K)$ acts on $\mathbb{H}^3$ ...
- 163
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 $\pi_1(X) \subset Aut(\tilde X)$.
...
- 103
8
votes
3
answers
942
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 ...
- 81
0
votes
0
answers
70
views
Geometric effects of removing elements of D2n generalizable?
So, if I start with a full Dihedral group D2n to represent a regular, ideal polygon in the hyperbolic plane, then I remove an element (and any subsequently necessary elements so that it is still a ...
6
votes
1
answer
479
views
If rational points are like entire curves, then what do algebraic points correspond to
I read somewhere that if $X$ is a projective variety of general type over a number field $K$, then rational points are an analogue of entire curves $\mathbf{C}\to X^{an}$ (with $X^{an}$ the ...
- 381
24
votes
3
answers
1k
views
Hyperbolic Coxeter polytopes and Del-Pezzo surfaces
Added. In the following link there is a proof of the observation made in this question: http://dl.dropbox.com/u/5546138/DelpezzoCoxeter.pdf
I would like to find a reference for a beautiful ...
- 28.9k
15
votes
6
answers
3k
views
Generalizations of Belyi's theorem
Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent:
1) $X$ is defined over $\overline{\mathbb{Q}};$
2) There exists a meromorphic ...
56
votes
6
answers
6k
views
What do Weierstrass points look like?
As somebody who mostly works with smooth, real manifolds, I've always been a little uncomfortable with Weierstrass points. Smooth manifolds are totally homogeneous, but in the complex category you ...
- 4,014