TITLE: how to compute the Euler characters of a Grassmannian?
QUESTION [8 upvotes]: Let $G(n,m)$ be the Grassmannian of all n-dim subspaces of an m-dim vector space over $\mathbb{C}$. How to compute the Euler characters of $G(n,m)$? For example, $G(1, 2)$ is $\mathbb{C}P^1$ which is $S^2$. So the Euler characters of $G(1,2)$ is $2$. But how to compute $G(n,m)$ in general? Thank you.

REPLY [5 votes]: This is a computation based on a straightforward application of the the Atiyah-Bott localization theorem.
Using the version given for example in the paper by Aleksey Zinger:
Given a torus acting smoothly on a on a manifold with isolated fixed points, then the Euler characteristic is equal to thethe number of the fixed points of its action.
The Grassmannian $Gr(n,m)$ can be identified with the space of n-dimensional orthogonal projectors in $\mathbb{R}^m$. The projectors can be represented by Hermitian $m \times m $ projection matrices of rank $n$. The unitary roup $U(m)$ acts by conjugation on the set of projectors. The fixed points of the subgroup $\mathbb{T}^m$ of diagonal unitary matrices are the diagonal $m \times m $ projection matrices of rank $n$. Since these matrices have $n$ units and $m-n$ zeros along the diagonal. Their number is equal to the number of distinct ways we can arrange $n$ units and $m-n$ zeros along the diagonal which is equal to: $\binom{m}{n}$.