We identify a combinatorial quantity (the alternating sum of the
h-vector) defined for any simple polytope as the signature of a
toric variety. This quantity was introduced by R. Charney and M. Davis
in their work, which in particular showed that its nonnegativity is
closely related to a conjecture of H. Hopf on the Euler characteristic
of a nonpositively curved manifold.
¶ We prove positive (or nonnegative) lower bounds for this quantity
under geometric hypotheses on the polytope and, in particular, resolve
a special case of their conjecture. These hypotheses lead to ampleness
(or weaker conditions) for certain line bundles on toric divisors, and
then the lower bounds follow from calculations using the Hirzebruch
signature formula.
¶ Moreover, we show that under these hypotheses on the polytope, the
ith L-class of the corresponding toric variety is (−1)i
times an effective class for any i.