Why does the Euler characteristic of a toric variety equal the number of vertices in the defining polytope? - MathOverflow

Why does the Euler characteristic of a toric variety equal the number of vertices in the defining polytope?

2010-05-10T22:36:56Z

In this link, Corollary 3.2.2, page 59 the author claims that: The Euler characteristic of the toric variety $X_K$ associated to a convex polytope $K$ is the number of vertices of $K$.

I want to see how it works. Could someone please illustrate this for me by using this method to compute the Euler characteristic of $\mathbb{P}^{2}$ and $\mathbb{P}^{1}\times \mathbb{P}^{1}$? Thanks so much.

Answer by VA for Why does the Euler characteristic of a toric variety equal the number of vertices in the defining polytope?

2010-05-10T23:04:12Z

Merely observe that a toric variety is the union of torus orbits $(\mathbb C^*)^r$ for various dimensions $r$, and that the Euler characteristic of $(\mathbb C^*)^r$ is zero if $r>0$ and $1$ if $r=0$.

Vertices of a polytope correspond to 0-dimensional orbits, $r$-dimensional faces -- to $r$-dimensional orbits.

$\mathbb P^2$ corresponds to a triangle, $\mathbb P^1\times\mathbb P^1$ to a square. It is not very hard to count their vertices.