# Tagged Questions

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### extension of Riemannian metric on real affine variety

Given a Riemannian metric $g$ on the real part $X_R$ of a real affine variety $X$,
is there a "natural" way to extend $g$ to be a Riemannian metric on $X$?

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36 views

### how to construct 3D curve in highway geometric design

give you some control points ,also give you the initial point and final point ,their curvature ,torsion and coordinate are kown.How to construct a three-dimensional space curve under the constaint of ...

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votes

**1**answer

78 views

### Question about conjugate points

If there exist two geodesics from $p$ to $q$ that are not only different from each other but also infinitesimally close to each other, then it implies that $q$ is conjugate to $p$.
Can anyone give an ...

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vote

**1**answer

104 views

### If there exists a nontrivial vector field $V$ such that $\nabla_{X}V=0$ for any vector field $X$, the manifold must be flat?

If there exists a nontrivial vector field $V\not=0$ in Riemannian manifold $M$ and an open set $U\subset M$ such that $\nabla_{X}V=0$ in $U$ for any vector field $X$ in $M$, then dose $U$ have to be ...

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86 views

### Find the Range of Function

What is the range of the map $\mathbb{C}^m\to\mathbb{C}^m$,
$$(z_1,\ldots,z_m)\mapsto (b_1,\ldots,b_m),$$
where $b_k=\prod_{j\neq k}(z-z_j)$ for $1\leq k\leq m$ ?

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23 views

### Factor of 2 In the Definition of Metric Contact Structure

In Blair's book and many many literatures, I see definition of a contact metric manifold which involves a relation
\begin{equation}
d\kappa \left( {X,Y} \right) = g\left( {X,\Phi Y} \right)
...

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44 views

### Integral straight-line embeddings of planar graphs

Wikipedia says (in the article on Fáry's theorem),
"Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The ...

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**1**answer

196 views

### Why is the kth cohomology group of the DM-compactification of the moduli space of curves pure of weight k?

I'm trying to understand the paper
Arbarello, Enrico, Cornalba, Maurizio,
Calculating cohomology groups of moduli spaces of curves via algebraic geometry.
Inst. Hautes Études Sci. Publ. Math. No. 88 ...

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53 views

### Profinite Local Ring inside Polynomial Ring

This is a "technical" question that I came across in my research.
Let $A = \textbf{Z}_{p}[\![t_1, \cdots, t_a ]\!]<z_1, \cdots, z_b>$ be the $(p, t_1, \cdots, t_a)$-adic completion of the ...

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259 views

### Is it meaningful to work on convergencies, integration, etc. on the Zariski topology? [on hold]

Since I have studied analysis as well as algebra recently, I am familiar to work on integrablities, and such concepts when I look at topologies. Currently, I am studying algebraic geometry, and I want ...

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23 views

### Depth of multigraded modules

Can any one please give me some references on depth of multigraded module over a standard multigraded ring.

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**1**answer

112 views

### Does normalization of projective varieties preserve very ampleness

Let $f:\tilde{X} \to X$ be a normalization of projective variety. Let $L$ be a very ample line bundle on $X$. Is $f^*L$ a very ample line bundle on $\tilde{X}$? If not true in general, is there any ...

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88 views

### Parabolic bundles on elliptic curves

as a warm up for his thesis I would like a student of mine to read something on parabolic bundles. He is reading the famous Atiyah paper on vector bundles on elliptic curves, so I think it would be ...

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33 views

### Quick question about conjugate equivalence bimodules and inner products

let $A$ and $B$ be $W^{*}$-algebras, let $X$ be an $A-B$-equivalence bimodule (according to the definition given in "Morita equivalence for $C^*$-algebras and $W^*$-algebras" by Rieffel, ...

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63 views

### Diameter of $n$-unit-vector closed scribble

Suppose one creates a random, closed, likely self-crossing polygon
from $n$ unit-length vectors arranged head-to-tail,
randomly oriented except for the requirement
that their sum is zero (so the ...

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**2**answers

140 views

### When distance nonincreasing map is an isometry

Let $f: M \to M$ be a distance nonincreasing map between a closed Riemannian manifold $M$
and $f$ is homotopic to the idendity map. Is it then $f$ an isometry?

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154 views

### Quotienting $SU(3)$ by $U(1)$?

As is well-known, if we quotient $SU(2)$ by the action of $U_1$, embedded in the diagonal as $(e^{i \theta}, e^{-i \theta})$, we get the $2$-sphere. As is also well-known, if we quotient $SU(3)$ on ...

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**1**answer

148 views

### Generalization of notion of convexity

I am searching for the correct term for the following, if it exists.
A set $X\subset \mathbb{R}^2$ is called $r$-convex if for any two points $x_1, x_2\in X$ such that there exists an arc of radius ...

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**1**answer

57 views

### Length inequalities in trees and CAT(0) spaces

I have a family of possibly related questions. Let me start with an elementary one:
Question 1. Fix an integer $n$. For which collections of real numbers $a_{ij}$, $i, j = 1, \dots, n$, is it true ...

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83 views

### Jacobian change of basis matrix for different dimensions

I am considering a real Lie group $G$ acting transitively on an open set $U$ in a real Euclidean space of lower dimension. Given a smooth, compactly supported function $f: U \rightarrow \mathbb{R}$ ...

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158 views

### Lifting Abelian Varieties to p-adic fields

Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic ...

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129 views

### Intersection theory on M_{g,n}

Is there a paper\book that lists the top intersections of Hodge classes and tautological classes on $\overline{\mathcal{M}}_{g,n}$ for small $g$ and $k$, e.g. $g=2,3$ and $k=0,1,2$ ?

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83 views

### Veronese surface [on hold]

I have a question(Hartshorne ,page 13,exercise 13):
If Y be the image of the 2-uple embedding(described in exercise 12) of P^2 in P^5.
and if Z(a subset of Y) is a closed curve(variety of dim 1) show ...

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**1**answer

111 views

### Bott formula for projective bundles

For a projective space one has Bott formula to compute $h^q(ℙ^n,Ω^p(k))$, where $Ω^p(k)$ is the k-twisted sheaf of sections in the p-th power of the cotangent bundle of $ℙ^n$. I am wondering if there ...

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39 views

### Euler number of the complex of basic forms

Let $G$ be a compact Lie group and $\pi:P \to M$ a principal $G$-bundle. I would like to understand the geometry of $M$ through $P$ with the $G$-action.
I am trying to understand the Hopf bundle ...

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**1**answer

104 views

### Kaehler form on weighted projective space

The Kaehler potential for the standard Fubini-Study Kaehler form in projective space $\mathbb{C} P^n$ is given by:
$$\log(\sum_{i=0}^n |z_i|^2)).$$
What is the analogous formula for a Kaehler ...

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42 views

### DGBV algebra of symplectic manifold

Let $(M,\omega)$ be a simply connected closed symplectic manifold. Then we have the symplectic codifferential operator $d^{\star}$. Furthermore, $(\Omega^{*}(M),d,d^{\star})$ is a differential ...

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662 views

### Can we construct cohomolgy theory on noetherian separated schemes without Axiom of Choice?

The usual cohomology theory on schemes uses injective or flasque resolutions of quasi-coherent sheaves. Hence it uses Axiom of Choice.
However, if the base scheme is a noetherian separated scheme, the ...

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395 views

### Is there a sideways-walking rolling convex body?

Let $K$ be a solid, homogenous convex body in $\mathbb{R}^3$.
Place $K$ on an inclined plane, and let it roll down the plane,
under some reasonable assumptions of friction between $K$ and
the plane, ...

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39 views

### additive support functions of a convex set

Let $K \subset \mathbb{R}^d$ be a compact, convex set. It could be uniquely determined by its support function (for $u$ on the unit-sphere $S^{d-1}$), given by
$$h_K(u) = \sup \{ \sum_{i=1}^d x_i ...

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314 views

### What are the invariants of $U\otimes V\otimes W$ under action of $GL(U)\times GL(V) \times GL(W)$

The tensor product of some (finite dimensional real) vector spaces is acted on by the direct product of their general linear groups. I would like to know if there are explicit invariants in the case ...

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**1**answer

96 views

### Zero currents localized along a submanifold

Let $\mathcal{D}(\mathbb{R})$ be the continuous dual of $C^\infty_c(\mathbb{R})$, the space of compactly-supported smooth functions. There is a nice characterization of distributions ...

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90 views

### Simple example of isolated critical point with non-semisimple monodromy

Consider a polynomial map $f :\mathbb{C}^{n+1} \rightarrow \mathbb{C}$ with $f(0)=0$ (no constant term) and with isolated critical point at $0 \in \mathbb{C}^{n+1}$. We can choose a disc $D$ of some ...

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73 views

### Comparison of K-groups of (affine) singular schemes with K'=G-groups

It is well known that Quillen K-theory coincides with $K'=G$-theory for regular schemes, and can be distinct from it for singular ones. Are there any methods for studying this distinctions? In ...

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38 views

### Information geometry divergence [on hold]

on http://en.wikipedia.org/wiki/Information_geometry
How to derive this equation. I tried but always got 0 for each item.
$$
D[\partial_i\partial_j||\cdot]= ...

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**4**answers

278 views

### Is any quadric birational to a product of Brauer-Severi varieties?

Let $k$ be a field with algebraic closure $\bar k$. Assume that $k$ is perfect and not of characteristic $2$ for simplicity. Let
$$X: \quad Q(x)=0, \quad \subset \mathbb{P}^n_k,$$
be a non-singular ...

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6 views

### Contraction between basis vectors and basis one-forms [migrated]

Discretion: The title may be misleading, because I am not certain whether the one-forms are actually basis one-forms.
I always thought by definition, $dx^i (e_j) =\delta^i_j $.
But, I am confused ...

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**1**answer

77 views

### Weyl algebras $A_n(k)$ as tensor product of the first Weyl algebra

In afew threads I've read that the Weyl algebra $A_{n+1}(k)$ is isomorphic to the $k$-tensor product of $A_n(k)$ with $A_1(k)$, why is this true?

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**2**answers

273 views

### The moduli space of special Lagrangian submanifolds

Given a special Lagrangian fibration $f:M \rightarrow B$ of a Calabi-Yau manifold $M$, one can associate to it two affine structures (symplectic and complex) on the base space $B$. A theorem of ...

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**1**answer

149 views

### Blow-up of $\mathbb{P}^4$ along a quadric surface

Let $Q\subset\mathbb{P}^3\subset\mathbb{P}^4$ be a smooth quadric surface, and let $X = Bl_Q\mathbb{P}^4$ the blow-up of $\mathbb{P}^4$ along $Q$. Let $H$ be the pull-back of the hyperplane section of ...

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49 views

### Quasi-finite morphisms of stacks

Let $f:X\to Y$ be a morphism of ``nice" stacks over $\mathbf C$ such that the induced morphism on coarse moduli spaces is quasi-finite. Is $f$ quasi-finite?
By a "nice" stack I mean a smooth finite ...

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**1**answer

189 views

### Can eta invariant be written in terms of topological data?

The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In ...

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46 views

### plotting parametrized algebraic curves near singularities

I have a parametrized algebraic curve:
x(t)=A(t)/D(t);
y(t)=B(t)/D(t);
with A(t) and B(t) being polynomials in t. The curve is solution of a linear system in two unknowns x and y with Cramer's ...

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64 views

### Local system over $\mathcal A_{g,[n]}$ with unipotent monodromy

Let $\mathcal A_{g,[n]}$ denote the moduli space of principal polarized abelian varieties with level-[n] structure and
$\bar {\mathcal A}_{g,[n]}\supset \mathcal A_{g,[n]}$ a smooth Toroida ...

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82 views

### Degree and quasi projective family

Let $V$ be a quasi-projective variety in $\mathbb{P}^{n}\times\mathbb{P}^m$. If $p\in \mathbb{P}^m$, we define the degree of $V_p$ as the degree of its closure in $\mathbb{P}^n$.
Question : $\exists ...

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56 views

### Can an acyclic continuum be metrically homogenous? (I'd say: no way! :-)

I asked recently on MO about algebraic structures admitted by topologically homogenous continua like the Hilbert cube $\ I^{\mathbb N}\ $ or the Knaster pseudo-arc. There is a relation between the ...

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**2**answers

187 views

### How to define the square root of $1-\Delta $?

If $M$ is a Riemannian manifold with $\Delta $ its Laplacian, how can we define $(1-\Delta)^{1/2}$?
The book I am reading says that $(1-\Delta)^{1/2}$ is an invertible first-order pseudo-differential ...

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**1**answer

77 views

### Normal coordinates near the boundary

Let $M$ be an Riemannian manifold with boundary $\partial M$ and $e_n$ be a unit normal vector on $\partial M$. With respect to $e_n$, around a point $p$ on boundary, we have the usual normal ...

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149 views

### Limit of distance between two random points in a unit-radius $n$-sphere

This is a companion contrast to the earlier analogous question for unit $n$-cubes,
where the answer (provided by several respondents) is $\infty$ .
What is the limit, as $n \to \infty$, of the ...

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votes

**1**answer

293 views

### What is the “complex third derivative”?

Background
I am including this information about real higher order derivatives because it does not seem to be common knowledge. I also include a review of the complex Hessian.
If $f:\mathbb{R}^n ...