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It is well known that a (smooth complete) fan $\Delta$ corresponds to a (smooth proper) toric variety $X= X_\Delta$.

My question is whether there is a relationship between the number of maximal cones in $\Delta$ and a geometric invariant of $X$. (as the number of 1-dimensional cones is the number of torus invariant prime divisor)

Here, we call a cone in $\Delta$ maximal if it is not contained in any other cone in $\Delta$.

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  • $\begingroup$ The reason why I asked this question is that $\# \Delta_{\textrm{max}}$ appeared in the Tamagawa number (in the sense of Peyre) of a universal torsor over the toric variety. $\endgroup$ Mar 23 '17 at 0:50
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Yes, the number of maximal cones is the same as the topological Euler characteristic.

This is the result of the natural generalization of the thing you already pointed out: the number of codimension $r$ irreducible $T$-invariant subvarieties is the number of $r$-dimensional cones in $\Delta$.

So more precisely, maximal cones give the $T$-fixed points. The other $T$-orbits all have topological Euler characteristic zero, and so they contribute zero to the overall Euler characteristic (since for $Y\subset X$ an open subspace of a space $X$, $\chi(X)=\chi(U)+\chi(X\setminus U)$).

Fulton's book explains this all very well.

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