Topological invariants of a fan associated to a toric variety

Abstract

Associated to a toric variety $X$ of dimension $r$ over a field $k$ is a fan $\Delta$ on $\Bbb R^r$. The fan $\Delta$ is a finite set of cones which are in one-to-one correspondence with the orbits of the torus action on $X$. The fan $\Delta$ inherits the Zariski topology from $X$. In this article some cohomological invariants of $X$ are studied in terms of whether or not they depend only on $\Delta$ and not $k$. Secondly some numerical invariants of $X$ are studied in terms of whether or not they are topological invariants of the fan $\Delta$. That is, whether or not they depend only on the finite topological space defined on $\Delta$. The invariants with which we are mostly concerned are the class group of Weil divisors, the Picard group, the Brauer group and the dimensions of the torsion free part of the \'etale cohomology groups with coefficients in the sheaf of units. The notion of an open neighborhood of a fan is introduced and examples are given for which the above invariants are sufficiently fine to give nontrivial stratifications of an open neighborhood of a fan all of whose maximal cones are nonsimplicial.