The analytic-geometry tag has no wiki summary.
11
votes
3answers
424 views
Why does inconstructibility of $\sqrt[3]{2}$ imply impossibility of cube doubling? [closed]
In this question "constructing" and "doubling" is meant in the compass-and-straightedge sense.
On my desk I have five Basic Algebra texts treating constructability in the plane $\mathbb{C}$ or ...
7
votes
1answer
350 views
Groups and pregeometries
Definition.
For an infinite structure $\mathcal{A}$ and $cl : P(dom(\mathcal{A})) \longrightarrow P(dom(\mathcal{A}))$ , we say
that $(\mathcal{A}, cl)$ is a structure carrying an $\omega$-homogeneous ...
3
votes
0answers
110 views
Universal covering space of a Zariski open subset of projective space
Let $U$ be a Zariski open subset of $\mathbb P^n_{\mathbb C}$. Assume $U$ is the complement of some divisors.
Have the possible universal covering spaces of $U$ been classified?
Do we know when the ...
10
votes
0answers
727 views
Has there been any progress on the standard conjectures on algebraic cycles?
What's the current state of these conjectures?
Who is working on them?
In "Standard conjectures on algebraic cycles" Grothendieck says:
"They would form the basis of the so-called "theory of ...
4
votes
1answer
165 views
Simple maps: Flat versus locally trivial
In deformation of complex analytic spaces, one usually considers an analytic proper simple surjective map $\varpi: \mathscr{M} \twoheadrightarrow \mathscr{P}$ as an analytic family. However the term ...
0
votes
1answer
100 views
Can an analytic set admit such a foliation?
I confess to be not an expert of analytic geometry, but I have come across the following problem, for which I need an help from experts in this specific field.
I was wondering myself if it is ...
2
votes
1answer
155 views
Tangent cone and embedded components
Is it possible for a reduced, equidimensional germ of complex analytic singularity to have a tangent cone with embedded components but without multiple irreducible components?
If it is, how can you ...
2
votes
2answers
528 views
supporting facts to fujita conjecture
I came across the Fujita conjecture which is perhaps very widely known. I want to know what are the supporting facts to the truth of the conjecture.
http://en.wikipedia.org/wiki/Fujita_conjecture
1
vote
0answers
180 views
Asymptotics vs Puiseux series
Define asymptotic as a class of sequences {$ x_i$},$_{i\in\mathbb N}$ modulo equivalence {$x_i$}={$y_i$} if $\lim_{i\to\infty} (x_i/y_i)=c\in\mathbb R,c\ne 0$.
More, we define $X= \{x_i\} \lt Y= \{ ...
17
votes
0answers
433 views
On Determinants of Laplacians on Riemann Surfaces
History of the Formula: In their famous paper "On Determinants of Laplacians on Riemann Surfaces" (1986), D'Hoker and Phong computed the determinant of the Laplacian $\Delta_n^+$ on the space $T^n$ of ...
2
votes
3answers
179 views
How to tell if a second-order curve goes below the $x$ axis?
Suppose we have a second-order curve in general form:
(1) $a_{11}x^{2}+2a_{12}xy+a_{22}y^{2}+2a_{13}x+2a_{23}y+a_{33}=0$.
I'd like to know if there is a simple condition that ensures that the curve ...
0
votes
0answers
136 views
Asymptotes of hyperbolic sections of a given cone
A book I'm reading (Companion to Concrete Math Vol. I by Melzak) mentions, "...any ellipse occurs as a plane section of any given cone. This is not the case with hyperbolas: for a fixed cone only ...
2
votes
0answers
242 views
Topology of theta nulls
Siegel upper half-space, $\mathfrak{h}_g$, consists of symmetric $g\times g$ complex matrices with positive-definite imaginary part. From an element $Z\in \mathfrak{h}_g$ we can construct a theta ...
10
votes
3answers
530 views
Geometric realization of Hochschild complex
Let $A$ be a commutative $\mathbb{C}$-algebra, and consider $C_{\bullet}(A,A)$ the simplicial Hochschild homology module of $A$ with respect to itself (i.e. $C_{n}(A,A)=A^{\otimes (n+1)}$). This is a ...
9
votes
0answers
519 views
Cohesive ∞-toposes for analytic geometry
There is a class of big ∞-toposes that come with a good supply of intrinsic notions of differential geometry and differential cohomology: called cohesive ∞-toposes (after Lawvere's cohesive toposes).
...
4
votes
3answers
764 views
Lattice points close to a line
Take a sheet of grid paper and draw a straight line in any direction from the origin. What is the closest non-zero grid point $\boldsymbol{p}\in\mathbb{Z}^2$ within a distance $\epsilon>0$ of the ...
7
votes
1answer
450 views
Pathologies of analytic (non-algebraic) varieties.
Note: By an "analytic non-algebraic" surface below I mean a two dimensional compact analytic variety $X$ (over $\mathbb{C}$) which is not an algebraic variety.
A property of Nagata's example (see ...
2
votes
2answers
339 views
If $K$ and $L$ are compact convex sets with smooth boundary, does their union have piecewise-smooth boundary?
Clarification: by "piecewise", I mean a finite number of pieces.
I'm sure this must be true, but my search for a citation was in vain (although I did learn the new term "polyconvex").
Thanks!
2
votes
1answer
213 views
Is a compact subset of a Stein space admitting a fundamental system of Stein neighbourhoods necessarily holomorphically convex?
Let X be a Stein manifold and let K be a compact subset of X. Suppose that K possesses in X a fundamental system of neighbourhoods which are Stein spaces. Then, it is a result by Rossi that such a ...
2
votes
1answer
266 views
Is the closure of an open holomorphically convex subset of a Stein space holomorphically convex?
Let X be a Stein manifold and U an open, connected, relatively compact, holomorphically convex subset of X. Is the closure of U in X holomorphically convex?
Also, if X is a Stein space with a finite ...
4
votes
0answers
704 views
Generalized GAGA
So, I have heard GAGA works for Rigid Analytic spaces. I know next to nothing about this, but it made me curious as to whether there are any other contexts in which GAGA "works". Of course, this is a ...
13
votes
1answer
903 views
Are flat morphisms of analytic spaces open?
Let $f:X\to Y$ be a morphism of complex analytic spaces. Assume $f$ is flat (or, more generally, that there is a coherent sheaf on $X$ with support $X$ which is $f$-flat). Is $f$ an open map?
The ...
6
votes
2answers
686 views
Embeddings and triangulations of real analytic varieties
This is a follow up question to my answer here How do you define the Euler Characteristic of a scheme?
A real analytic space is a ringed space locally isomorphic to $(X,O/I)$ where $X$ is the zero ...
7
votes
1answer
975 views
Did Apollonius invent co-ordinate geometry?
When I read descriptions of Apollonius' treatise on conics, some of them say that he invented co-ordinate geometry, some say that he kind of did and others are silent on the matter. Or is it the case ...
