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 $\mathcal{E}$ 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 $(4r+2)e(S)\ln_2\left(\tfrac{e(S)}{12}\right)$. 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.