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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.

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I added the 'toric-variety' label. – VA. May 11 '10 at 2:31
I've changed the title and some wording. I hope this is acceptable. – S. Carnahan May 12 '10 at 5:19
up vote 15 down vote accepted

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.

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More generally, if a circle $S^1$ acts on a manifold or the 1-dimensional torus $C^*$ acts on a complex algebraic variety then the Euler characteristic of the "whole" = the Euler characteristic of the fixed point set. This is a very powerful tool in topology. – Victor Protsak May 11 '10 at 3:47

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