Di Rocco, Sandra; Sommese, Andrew J. Chern numbers of ample vector bundles on toric surfaces. (English) Zbl 1023.14026 Trans. Am. Math. Soc. 356, No. 2, 587-598 (2004). 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. Cited in 4 Documents 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 Keywords:ample vector bundles; inequalities for Chern numbers; toric surface PDFBibTeX XMLCite \textit{S. Di Rocco} and \textit{A. J. Sommese}, Trans. Am. Math. Soc. 356, No. 2, 587--598 (2004; Zbl 1023.14026) Full Text: DOI arXiv References: [1] Aldo Biancofiore and Elvira Laura Livorni, On the iteration of the adjunction process for surfaces of negative Kodaira dimension, Manuscripta Math. 64 (1989), no. 1, 35 – 54. · Zbl 0687.14030 · doi:10.1007/BF01182084 [2] Mauro C. Beltrametti, Michael Schneider, and Andrew J. Sommese, Applications of the Ein-Lazarsfeld criterion for spannedness of adjoint bundles, Math. Z. 214 (1993), no. 4, 593 – 599. · Zbl 0804.14006 · doi:10.1007/BF02572427 [3] Mauro C. Beltrametti, Michael Schneider, and Andrew J. Sommese, Chern inequalities and spannedness of adjoint bundles, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993) Israel Math. Conf. Proc., vol. 9, Bar-Ilan Univ., Ramat Gan, 1996, pp. 97 – 107. · Zbl 0870.14009 [4] Mauro C. Beltrametti and Andrew J. Sommese, The adjunction theory of complex projective varieties, De Gruyter Expositions in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1995. · Zbl 0845.14003 [5] Takao Fujita, Classification theories of polarized varieties, London Mathematical Society Lecture Note Series, vol. 155, Cambridge University Press, Cambridge, 1990. · Zbl 0743.14004 [6] David I. Lieberman, Holomorphic vector fields and rationality, Group actions and vector fields (Vancouver, B.C., 1981) Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 99 – 117. · doi:10.1007/BFb0101512 [7] Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. · Zbl 0628.52002 [8] Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980. · Zbl 0438.32016 [9] Miles Reid, Bogomolov’s theorem \?\(_{1}\)²\le 4\?\(_{2}\), Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) Kinokuniya Book Store, Tokyo, 1978, pp. 623 – 642. [10] R. Remmert and A. J. H. M. van de Ven, Zur Funktionentheorie homogener komplexer Mannigfaltigkeiten, Topology 2 (1963), 137 – 157 (German). · Zbl 0122.08602 · doi:10.1016/0040-9383(63)90029-6 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.