# Questions tagged [grassmannian]

In mathematics, the Grassmannian $\mathbf{Gr}(r, V)$ is a space which parameterizes all linear subspaces of a vector space $V$ of given dimension $r$.

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### Questions on grassmannians from Harris' and Griffith's principles of algebraic geometry

I have a couple of questions about the explanations on the Grassmannian from Harris' and Griffith's "Principles of algebraic geometry" on page 194/195. We consider for $k \le n$ the $k$-th ...
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### $C^m\to C^{m-1}$ projection map projecting out last factor inducing $G_n(C^{m-1})\to G_n(C^m)$?

Assume $m>n$. Consider projection map $\pi: C^m\to C^{m-1}$ by $(z_1,\dots, z_m)\to (z_1,\dots, z_{m-1})$. Let $G_n(V)$ be the set of codimension $n$ hyperplanes of $V$ and endow the topology as a ...
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### Let $h,h'$ be hermitian metric over vector space $V$, then grassmanian $G_n(V_h)\to G_n(V_{h'})$ is always continuous?

Let $V$ be a finite dimensional vector spaces over complex number and choose hermitian metrics $h,h'$ over $V$. Let $G_n(V)$ be the set of codimension $n$ hyperplanes of $V$. Since $V$ has 2 metrics, ...
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### Grassmannian is homogeneous, isotropic, and symmetric

I'm trying to prove the Grassmann manifold $\mathrm G_k(\mathbb R^n)$ of $k$-dimensional linear subspaces of $\mathbb R^n$ is isotropic and symmetric. By "isotropic" I specifically that for every ...
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### Is the Grassmannian a unique algebraic variety?

I'm fairly new to algebraic geometry, so this is may just be fairly simple question about when two varieties are "the same" variety. In this question, I noted that the Grassmannian can be expressed ...
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### What is the way to write a manifold of manifolds?

I want to write this in mathematical notation: "Let us represent a ball, $B_3$, with a metric $g$ as a point on manifold. Let $M$ be the (infinite dimensional) manifold formed from every ball with all ...
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### What, exactly, is Schubert's symbolic calculus?

Most modern treatments of the Schubert calculus typically write about the cohomology ring of the Grassmannian. They also write, almost as an afterthought, that this is derived from Schubert's "...
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### Stratify Grassmannian based on dimension of intersection

Consider a vector space $V$ of dimension $n$ and an automorphism $A:V\rightarrow V$. Using this we can define, $\Sigma_i = \{W\in Gr_k V| \dim(W\cap AW) = i\}$ for $i=0,\ldots, k$. My question is ...
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### The “true” metric on the Grassmannian: Plücker vs Projective-Frobenius embeddings

There are several references in the literature to some kind of "most natural" metric on the Grassmannian manifold, often called the "geodesic distance" or the "Binet-Cauchy" distance. There are ...
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### Grassmannian as a quotient of orthogonal or general linear group

I'm trying to understand some different ways to construct the Grassmannian of a real vector space, but I'm having trouble getting some of the notation and definitions. One definition that I often see ...
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### Minimal embedding of the Grassmannian into Euclidean (or projective) space

Let $Grass(r,k)$ be the set of all $r$-dimensional subspaces of $\Bbb R^k$. It is well known that $Grass(r,k)$ embeds isometrically as a projective variety into the projectivization of the r'th power ...
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### Ring structure of $\Bbb CP^n$ and Chern class.

In this notes Prop 1.71 in nlab, the author aims to compute $H^*(\Bbb C P^n, \Bbb Z)$. I have two confusions. What makes it justified to use $c_1$ as the generator of $H^2(\Bbb CP^n, \Bbb Z)$? There ...
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### Does the orthogonal complement determine the inner product up to scaling?

Let $V$ be a real $n$-dimensional vector space, and let $g,h$ be two inner products on $V$. Fix some $1\le k\le n-1$, and denote by $\text{Gr}_k(V)$ the Grassmannian of $k$-dimensional subspaces of $V$...
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### Intersection of lines in $\mathbb{P}^3$ can be given as zero locus of homogeneous linear polynomial in $\mathbb{P}^5$

I'm currently stuck at Exercise 8.19 b) in the notes to Algebraic Geometry from Gathmann Let $L \subset \mathbb{P}^3$ be an arbitrary line. Show that the set of lines in $\mathbb{P}^3$ that ...
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### Meaning of stable $CP^2$

I came across the following phrase in arXiv:1903.08904 ....in order to have a stable $CP^2$ , i.e., one in which all the automorphism group is fixed... Can anyone explain to me what one means by ...
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### Does the action of a linear map on $k$-dimensional subspaces determine it up to scaling?

Let $V$ be a real $d$-dimensional vector space, and let $1 \le k \le d-1$ be a fixed integer. Let $A,B \in \text{Hom}(V,V)$, and suppose that $AW=BW$ for every $k$-dimensional subspace $W \le V$. Is ...
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### Which subspaces of exterior power have decomposable bases?

Let $V$ be a real $n$-dimensional vector space, and let $1<k<n,r>1$. I wonder: Is there a way to characterise which $r$-dimensional subspaces of the exterior power $\bigwedge^k V$ have ...
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### Projecting a projective variety away from a linear subspace

I read in Harris' book at page 148, proposition 11.37 and got slightly confused regarding the argument. Harris mentions a projective space $\mathbb{P}^{2n+1}$ and a linear subspace $\mathbb{P}^{n}$ ...