One of the main question in the cohomological study of a space $X$ is whether $X$
has a torsion free and vanishing odd degree cohomology. In particular, if a toric variety associated
to a lattice polytope $P$ has such a property, then its equivariant cohomology is isomorphic to
the piecewise algebra over $P$. In this talk, we study a sufficient condition for a toric variety to
have a vanishing odd degree cohomology, then we apply this to a certain singular toric variety,
which gives us some information about Schubert calculus of a full flag variety.
