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-5
votes
0answers
25 views

Proof in analytic geometry using vector multiplication [migrated]

Let us have a triangle $\Delta ABC$. $H$ is the intersection of heights if and only if $$ \overrightarrow{HA}\cdot \overrightarrow{HB} = \overrightarrow{HB}\cdot \overrightarrow{HC} = ...
6
votes
1answer
333 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
103 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 ...
9
votes
0answers
630 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
153 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
99 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
149 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
479 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
173 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
414 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
132 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
238 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
523 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
511 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
722 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 ...
6
votes
1answer
440 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
335 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
210 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
693 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
882 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
672 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
961 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 ...