Chern numbers of ample vector bundles on toric surfaces

This article shows a number of strong inequalities that hold for the Chern numbers c(1)(2), c(2) of any ample vector bundle epsilon of rank r on a smooth toric projective surface, S, whose topological Euler characteristic is e(S). One general lower bound for c(1)(2) proven in this article has leading term (4r + 2)e(S) ln(2) (e(S)/12). Using Bogomolov instability, strong lower bounds for c(2) are also given. Using the new inequalities, the exceptions to the lower bounds c(1)(2) > 4e(S) and c(2) > e(S) are classified.