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Chern numbers of ample vector bundles on toric surfaces. (English) Zbl 1023.14026

Summary: 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 \((4r+2)e(S)\ln_2(\frac{e(S)}{12})\). 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.

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
57R20 Characteristic classes and numbers in differential topology
14J25 Special surfaces
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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References:

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