Skip to main content

# 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 $r$.

420 questions
Filter by
Sorted by
Tagged with
4 votes
1 answer
77 views
+50

• 157
3 votes
2 answers
84 views

### Why the tautological bundle of the Grassmannian has only zero as global sections?

I'm in a course on Complex Geometry, and I have been studying the Grassmannian $G_r(\mathbb{C}^N)$ as a complex manifold and the construction of its tautological bundle. As in the case of the ...
• 323
1 vote
0 answers
34 views

### The universal bundle $\gamma_k\rightarrow BO_k$ is a real vector bundle

For all $n$, let $\gamma_k^n$ bet the tautological bundle over $Gr_k(\mathbb R^n)$, i.e. $$\gamma_k^n=\{(V,v):V\in Gr_k(\mathbb R^n), v\in V\}$$ This is also naturally identified with the associated ...
• 3,441
2 votes
1 answer
33 views

### What is the action of $O(k)$ on $V_k(\mathbb R^n)$ making it a principal bundle?

Let $V_k(\mathbb R^n)$ be the Stiefel manifold of ordered $k$-tuples of vectors in $\mathbb R^n$. I have seen in many places that $V_k(\mathbb R^n)$ is an $O(k)$ principal bundle over the Grassmanian ...
• 3,441
0 votes
1 answer
70 views

### Blowup of diagonal in $\mathbb{P}^r \times \mathbb{P}^r$

Let $X= \mathbb{P}^r \times \mathbb{P}^r$. Suppose we blowup $X$ along $\Delta$ the diagonal to get $\tilde X$. I want to show that this is isomorphic to the fibre product which I describe below - Let ...
• 2,438
5 votes
0 answers
59 views

### Can we construct the exterior algebra just from simple multivectors?

$\newcommand\K{\mathbb K} \newcommand\Ext{\mathop{\textstyle\bigwedge}} \newcommand\Lip{\mathrm{Lip}} \newcommand\ev{\mathrm{ev}} \newcommand\Gr{\mathrm{Gr}}$Let $V$ be a finite-dimensional $\K$-...
• 7,539
0 votes
0 answers
18 views

### Induced morphism of smooth variety to the Grassmannian on global sections?

In Huybrechts & Lehn, The Geometry of Moduli Spaces of Sheaves, page 143, it reads Let $X$ be a smooth variety. Suppose $E$ is a locally free sheaf of rank $r$ which is generated by its space of ...
• 347
0 votes
1 answer
48 views

### $H^i\left(BO_m ; \mathbb{Z}_2\right) \cong H^i\left(B O_n ; \mathbb{Z}_2\right)$ for $i \leq m$ and $i \leq n.$

The notes I am reading say that groups in the title are isomorphic. Could someone explain to me why it is the case? Here by $BO_n$ I mean the infinite Grassmann manifold of $n$-dimensional subspaces. ...
• 830
3 votes
1 answer
42 views

### Dimers from Postnikov diagrams

I am reading several papers on Postnikov diagrams, dimer models, and quivers. From the Postnikov diagram, we can draw a dimer model and an ice quiver. My question is as follows. Does every dimer model(...
• 103
2 votes
1 answer
64 views

• 143
3 votes
2 answers
89 views

• 312
1 vote
1 answer
73 views

• 3,270
2 votes
0 answers
122 views

0 votes
1 answer
34 views

### An invariant that determines membership in an element of $Gr(2,\mathbb{C}^3)$

Consider two basis vectors for a two-dimensional subspace of $\mathbb{C}^3,$ $v_1=(z_1,z_2,z_3),v_2=(z'_1,z'_2,z'_3)$. I am looking for a geometric or algebraic invariant that determines if another ...