All Questions

Filter by
Sorted by
Tagged with
1 vote
1 answer
90 views

Naturality of Lie bracket - alternate proof

Let $M$ and $N$ be smooth manifolds, and let $F: M \to N$ be a smooth map. Let $X$ and $Y$ be vector fields on $M$, and let $\tilde{X}$ and $\tilde{Y}$ be vector fields on $N$. We say that $X$ and $\...
Zhang Yuhan's user avatar
6 votes
1 answer
189 views

Criteria for when Gauss-Manin sheaves are vector bundles

Let $(X,D_X)$ and $(S, D_S)$ be smooth normal crossings pairs over $\mathbb C$; i.e. smooth schemes of finite type over $\mathbb C$ with a normal crossings divisor. If $f:X \to S$ is a proper, flat ...
Aitor Iribar Lopez's user avatar
3 votes
0 answers
156 views

The local global principle for differential equations

Are there any good reference to tackle the problem below? Or, are there any know result? Problem Let $f_1...f_n\in \mathbb{Z}[x_1,..,x_n]$ and $V:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a vector ...
George's user avatar
  • 465
-3 votes
0 answers
96 views

On unitary manifold [closed]

For integer $d$, we can define unitary matrices $\mathbb{U}:=\{U\mid U^*U=UU^*=I_d\}$. This forms a manifold of dimension $d^2$. Suppose we are given a submanifold of $\mathbb{U}$, denoted as $\mathbb{...
gondolf's user avatar
  • 1,487
3 votes
1 answer
118 views

Comparing Kummer maps to étale homotopy at finite level

$\DeclareMathOperator\Mor{Mor}\DeclareMathOperator\Hom{Hom}\newcommand{\et}{\mathrm{et}}\newcommand{\top}{\mathrm{top}}$In Voevodsky's paper "Étale topologies of schemes over fields of finite ...
Thigh High Crocs's user avatar
0 votes
0 answers
73 views

Can a Tangent Space always be expressed with “more structure” than just a vector space (e.g. a choice of basis for Stiefel manifold)

I'm currently trying to read about the Stiefel manifold, or set of all $p$ orthonormal $n$-dimensional vectors embedded in $\mathbb{R}^{n\times p}$. $$\mathcal{V}_p(\mathbb{R}^n) = \{U \in \mathbb{R}^{...
Moo's user avatar
  • 1
0 votes
0 answers
87 views

Expressing the union of principal orbits as a disjoint union of global slices for proper group actions

Setup: I was reading about slices and principal orbit theorems (Theorem 3.4.6) from these notes. Let the Lie group $G$ act on a complete Riemannian manifold $(M,g)$ isometrically on $M$, i.e. $\phi^{*}...
Learning math's user avatar
5 votes
0 answers
46 views

Approximation of triply periodic minimal surfaces with trigonometric level sets

Some triply periodic minimal surfaces are known to be approximated by trigonometric level sets very accurately. To see this, let's sample a gyroid scaled to the bounding box $[0, 1]^3$ exactly through ...
Greg Hurst's user avatar
1 vote
0 answers
78 views

Compactification of a Cartan-Hadamard manifold

Let $X$ be a simply connected manifold with nonpositive sectional curvature. It is standard that $X$ is uniquely geodesic, i.e., for any distinct points $p$ and $q$, there is a unique geodesic ...
ZZZ's user avatar
  • 11
1 vote
0 answers
80 views

Extension of MMP from the central fiber to some neighborhood

I was reading Professor Kollár's paper deformation of varieties of general type (here: https://arxiv.org/abs/2101.10986 ) There is a theorem about the extension of MMP step when the central fiber has ...
yi li's user avatar
  • 173
4 votes
0 answers
89 views

Does the permutohedron satisfy any minimal distortion property for graph metric vs Euclidean distance?

We can look on the permutohedron as a kind of "embedding" of the Cayley graph of $S_n$ to the Euclidean space. (That Cayley graph is constructed by the standard generators, i.e. ...
Alexander Chervov's user avatar
0 votes
0 answers
37 views

Prove the orthogonality of vector spherical harmonics

We define $S_a^{lm} = \Big( - \frac{1}{\sin \theta} Y^{lm}_{,\varphi}, \sin \theta\ Y^{lm}_{,\theta} \Big)$ $Y_a^{lm} = \Big( Y^{lm}_{,\theta}, Y^{lm}_{,\varphi} \Big)$ to be the axial vector ...
AleNekro97's user avatar
5 votes
1 answer
105 views

Finding the point within a convex n-gon that maximizes the least angle subtended there by an edge of the n-gon

For any point P in the interior of a convex polygon, the sum of the angles subtended by the edges of the polygon is obviously 2π. Given a convex polygon, how does one algorithmically find the point (...
Nandakumar R's user avatar
  • 5,473
2 votes
0 answers
61 views

What Cayley graphs arise as nodes+edges from "nice" polytopes and when are these polytopes convex?

The Permutohedron is a remarkable convex polytope in $R^n$, such that its nodes are indexed by permutations and edges correspond to the Cayley graph of $S_n$ with respect to the standard generators, i....
Alexander Chervov's user avatar
3 votes
1 answer
99 views

Concentration of measure on spheres with respect to a unitary of trace approximately zero

Cross-posted from MSE, where it hasn’t received any answer yet: This question arose out of my attempt to understand how a unitary of trace approximately zero acts on the unit sphere of a $n$-...
David Gao's user avatar
  • 1,262
7 votes
0 answers
160 views

Maybe a folklore natural map between reflexive pullbacks

In the introduction of [HK04], it is proposed that for a morphism between varieties $f:X'\to X$, and a coherent sheaf $\mathcal{F}$ on $X$, there is a natural map $\alpha:f^*(\mathcal{F}^{\vee\vee})\...
nariri's user avatar
  • 296
3 votes
0 answers
144 views

A relative Abel-Jacobi map on cycle classes

I have a question about relativizing a classical cohomological construction that I think should be easy for someone well versed in such manipulations. Background: Suppose $X$ is a smooth projective ...
Asvin's user avatar
  • 7,648
0 votes
0 answers
138 views

Why $k((x,t))$ can not be a local field?

If $k$ is a finite field, then $k((x))$ is a local field, and we can define a discrete valuation on $k((x))$ with respect to which it is complete. It is sometimes called a 1-dimensional local field. I ...
MAS's user avatar
  • 872
6 votes
1 answer
150 views

Coarse embeddings and Gromov products in (Gromov) hyperbolic spaces

I am new into geometric group theory and I have recently started reading the book "Sur les Groupes Hyperboliques d’après Mikhael Gromov" by Ghys and de la Harpe. The following inequality ...
Steve's user avatar
  • 61
5 votes
0 answers
197 views

Theories of manifolds w/ extra structure and singularities

Many different objects in mathematics can be described as manifolds with extra structure. Among the most famous examples of these are smooth manifolds, Riemannian manifolds, complex manifolds, and ...
Will Sawin's user avatar
  • 135k
1 vote
1 answer
141 views

Proving that $H^1(X,\mathcal{Hom}(\mathcal{G},\mathcal{E})) \cong \text{Ext}^1(\mathcal{G},\mathcal{E})$ holds for locally free sheaves

The following passage is from a thesis I'm reading: Suppose we have a short exact sequence of vector bundles $$0 \to \mathcal{E} \to \mathcal{F}\to \mathcal{G} \to 0.$$ Since these sheaves are ...
Johannes's user avatar
3 votes
1 answer
182 views
+100

Systems of (hyperbolic) 2nd order PDEs with lower order constraints

Certain surfaces in mechanics are endowed with the fundamental forms \begin{align} \text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\ \text{II} &= \alpha\left(\...
Daniel Castro's user avatar
0 votes
1 answer
99 views

Regular value theorem for Banach manifolds without surjectivity

It is well-known (e.g. Lang, Fundamentals of differential geometry, Prop. 2.3 in Chapter II) that the following extension of the regular value theorem holds for Banach manifolds: Let $\phi : M\...
Martin's user avatar
  • 141
5 votes
1 answer
216 views

Parabolic subgroups of reductive group as stabilizers of flags

$\DeclareMathOperator\GL{GL}$Let $G$ be a linear algebraic group (probably reductive will be needed). Consider a faithful representation $G \to \GL(V)$. Given a parabolic subgroup $P < G$, we can ...
a_g's user avatar
  • 497
0 votes
0 answers
109 views

Projective subvarieties are closed?

I want to show that projective subvarieties of a quasi-projective variety are closed. One possible solution should be the following: Let $W \subseteq \mathbb{P}^n$ be a quasi-projective variety and $V ...
psl2Z's user avatar
  • 101
0 votes
1 answer
118 views

Vector bundles over a homotopy-equivalent fibration

I think this question is related to what is known as "obstruction theory", but I'm not very familiar with this field of mathematics, so I am asking here. Let $\pi:N\rightarrow M$ be a smooth ...
Bence Racskó's user avatar
1 vote
0 answers
138 views

Any "motive"(-like) theory which can catch that cusp $y^2=x^3$ (and similar) are non-trivial?

Consider cusp $y^2=x^3$ which can also be described as $k[z]~without~z$ , taking $x=z^3,y=z^2$. Algebraically its $Spec$ is quite different from $k$. For example: it has plenty non-trivial "line-...
Alexander Chervov's user avatar
3 votes
0 answers
135 views

What d.o. $\sum_i f_i(z)\partial_z^i$ correspond to subalgebras $M$ in polynoms $C[x_i]$ being Langlands dual to motive of $Spec(M) \to X$?

Briefly: The question is about presenting explicit examples of the construction discussed in the recent MO question "Relation between motives and geometric Langlands" and Will Sawin's asnwer ...
Alexander Chervov's user avatar
2 votes
0 answers
81 views

Existence of extensions of a flat projective morphism

Suppose that $S$ is a noetherian integral scheme and $U\subset S$ is an open subscheme. Let $f:X\to U$ be a flat projective morphism. I would like to know whether (or when) $f$ can be extended to a ...
Nachhauseweg's user avatar
0 votes
0 answers
65 views

Convergence of metric implies convergence of eigenvalues?

Consider the metric $g_\varepsilon=d\theta^2 + \varepsilon^2 \sin(\theta)^2 \, d\varphi^2$ on the hemisphere $S^2_+.$ I had two naive questions: Does $g_\varepsilon$ converge to the flat metric on ...
Student's user avatar
  • 653
1 vote
0 answers
105 views

Noetherian local ring with non-lci formal fibers

I am looking for an example of a noetherian local ring $A$ such that its formal fibers are not complete intersection rings in the sense of https://stacks.math.columbia.edu/tag/09PY (i.e., there is a ...
gdb's user avatar
  • 2,851
4 votes
1 answer
175 views

Equivariant projective embeddings with optimal dimension

Let $X$ be a complex projective manifold, and $f\in Aut(X)$ an automorphism, which is linearizable, that is, can be extended to an ambient projective space ${\mathbb P}^m$. I am interested to find ...
Misha Verbitsky's user avatar
4 votes
1 answer
152 views

Why can I take the quotient of a relative elliptic curve by a finite locally free subgroup?

I am currently reading Katz and Mazur’s Arithmetic moduli of elliptic curves and I am puzzled by a statement in the discussion of the $[\Gamma_0(N)]$ moduli problem in Chapter 3. The authors define a $...
Aphelli's user avatar
  • 330
6 votes
0 answers
139 views

$K_0$ of arithmetic surfaces

In his paper "Algebraic K-Theory and classfield theory of arithmetic surfaces", Annals of Mathematics 114 (1981), Spencer Bloch proved the following result: if $A$ is a finitely generated ...
Daniel Schäppi's user avatar
2 votes
0 answers
51 views

Limiting distribution of separated points in a unit square

Let $n$ and $r$ be fixed, and consider the following process, with $S=\emptyset$ to start: For $i\in\{1,\dots,n\}$: Sample a random point $X$ in the unit square. If $X$ is a distance at least $r$ ...
Tom Solberg's user avatar
  • 3,929
1 vote
1 answer
69 views

Simple convergence of convex compact set implies Hausdorff convergence

I am wondering about the following : In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
Anthony's user avatar
  • 31
20 votes
3 answers
561 views

Examples when quantum $q$ equals to arithmetic $q$

First, as a disclaimer, I should say that this post is not about any specific propositions, but is more of some philosophical flavor. In the world of quantum mathematics, the letter $q$ is a standard ...
Estwald's user avatar
  • 1,187
3 votes
1 answer
138 views

When is a del Pezzo surface a conic bundle?

I am considering over a field $k$ which is not algebraically closed, characteristic 0, and perhaps contains all the complex roots of unity that may appear. Feel free to realize it as some function ...
fp1's user avatar
  • 71
2 votes
0 answers
116 views

Classifying stack for finite flat group scheme

Let $G$ be a finite flat non-smooth group scheme over an algebraically closed field $k$, for example, $G$ can be $\operatorname{Spec}(\overline{\mathbb{F}}_p[t]/(t^p))$. Then the classifying stack $\...
mhahthhh's user avatar
  • 381
6 votes
1 answer
198 views

Densities, pseudoforms, absolute differential forms and measures, differential forms, etc

Apologies if this question is too basic, but I figured I first heard of most of these concepts on MO, so perhaps I can ask here. Gelfand’s definition, copied from AlvarezPaiva [My edit, could be ...
D.R.'s user avatar
  • 741
35 votes
1 answer
2k views

Clausen–Scholze's Theorem 9.1 of Analytic.pdf, in view of light condensed sets, AKA is the Liquid Tensor Experiment easier now?

In the recent lecture series run jointly from IHÉS and Bonn, Clausen and Scholze have reworked—again—their foundations of geometry to focus attention on not arbitrary condensed sets and solid modules ...
David Roberts's user avatar
  • 33.8k
3 votes
1 answer
282 views

Does the absolute Frobenius induce the identity on étale topoi?

Let $X$ be a scheme defined over $\mathbb{F}_p$ and denote by $X_{et}$ its étale topos . Associated to $X,$ we can consider the absolute Frobenius map $F_X: X \rightarrow X$ which gives an associated ...
etalemorphisms's user avatar
3 votes
1 answer
186 views

Etale cohomology of relative elliptic curve

Let $E_a: y^2 = x(x-1)(x-a)$ be a smooth proper relative elliptic curve over $\text{Spec}(A)$, with $a\in A$, and assume $\text{Spec}(A)$ is a $\text{Spec}(\mathbb{Q}_p)$-scheme. Let $R^1f_*\mathbb{Q}...
kindasorta's user avatar
  • 1,473
4 votes
2 answers
364 views

“Geometric” vs Homotopical completion

There are two notions of completions of slightly different nature, and I am wondering if there is a precise statement relating them. The first one is the “homotopical” (or maybe it should be called ...
Grisha Taroyan's user avatar
7 votes
1 answer
330 views

Proving the inequality involving Hausdorff distance and Wasserstein infinity distance

Prove the inequality $$d_{H}(\mathrm{spt}(\mu),\mathrm{spt}(\nu))\leq W_{\infty}(\mu,\nu)$$ where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\...
Luna Belle's user avatar
1 vote
0 answers
97 views

Dolbeault class of the curvature of the Chern connection equaling the Atiyah class

The following is a proposition from Complex Geometry by Huybrechts. Proposition $4.3.10$. For the curvature $F_\nabla$ of the Chern connection on an hermitian holomorphic vector bundle $(E,h)$ one ...
Johannes's user avatar
3 votes
0 answers
61 views

Surface terms in the calculus of variations on jet bundles

Let $\pi:N\rightarrow M$ be a fibered manifold with $m=\dim M$ and $m+n=\dim N$. The variational bicomplex on the infinite jet space $J^\infty(\pi)$ is denoted $(\Omega^{k,l}(\pi),\delta,\mathbf d)$ ...
Bence Racskó's user avatar
3 votes
1 answer
241 views

Is there a variety which is not locally set theoretic complete intersection?

A variety $X$ is a locally set theoretic complete intersection in a nonsingular variety $Y \supseteq X$ if each point in $X$ has a neighborhood $U$ in $Y$ such that $X \cap U$ is set theoretically the ...
pinaki's user avatar
  • 5,064
1 vote
1 answer
108 views

Does a Riemannian submersion map horizontal geodesics to geodesics, and a relevant question?

I asked this question on MSE, but I didn't receive a response yet, so I'm asking here. Apologies if the question is not exactly a research level question, but I'm having some trouble in figuring them ...
Learning math's user avatar
6 votes
1 answer
382 views

Reference request: good reduction equivalent to crystalline étale cohomology

Suppose $X$ is an abelian variety over a $p$-adic field $K$, and it's well known that $X$ has good reduction is equivalent to the étale cohomology of $X$ is crystalline, and $X$ has semistable ...
Richard's user avatar
  • 481

1
2 3 4 5
658