On the integral cohomology of toric varieties

摘要

It is known that the integral cohomology algebra of any smooth compact toric variety X-Sigma associated to a complete regular fan Sigma is isomorphic to the Stanley-Reisner algebra Z[Sigma] modulo the ideal J(Sigma) generated by linear relations determined by Sigma. The aim of this paper is to show how to determine the integral cohomology algebra of a toric variety (in particular, a projective toric variety) associated to a certain simplicial fan. As a consequence, we confirm our expectation that for a certain simplicial fan the integral cohomology algebra is also given by the same formula as in a complete regular fan.