Chern numbers of ample vector bundles on toric surfaces

Abstract

This article shows a number of strong inequalities that hold for the Chern numbers c 1 2 c_1^2 , c 2 c_2 of any ample vector bundle E \mathcal {E} of rank r r on a smooth toric projective surface, S S , whose topological Euler characteristic is e ( S ) e(S) . One general lower bound for c 1 2 c_1^2 proven in this article has leading term ( 4 r + 2 ) e ( S ) ln 2 ⁡ ( e ( S ) 12 ) (4r+2)e(S)\ln _2\left (\tfrac {e(S)}{12}\right ) . Using Bogomolov instability, strong lower bounds for c 2 c_2 are also given. Using the new inequalities, the exceptions to the lower bounds c 1 2 > 4 e ( S ) c_1^2> 4e(S) and c 2 > e ( S ) c_2>e(S) are classified.