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 $\Delta$.



Relation between the number of maximal cones in a fan and the geometry of corresponding toric variety
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.


