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The orbibundle Miyaoka-Yau-Sakai inequality and an effective Bogomolov-McQuillan theorem. (English) Zbl 1162.14026
Let \((g,X)\) be a pair consisting of a nonnegative integer \(g\) and a minimal complex projective surface \(X\) of general type with \(K_X^2>c_2(X)\). F. A. Bogomolov [Dokl. Akad. Nauk SSSR 236, 1041–1044 (1977; Zbl 0415.14013)] proved that the irreducible curves of genus \(g\) on \(X\) form a bounded family; in particular, \(X\) contains only finitely many rational/elliptic curves. Moreover, the ideas in his proof lead to M. McQuillan’s partial solution [Publ. Math., Inst. Hautes Études Sci. 87, 121–174 (1998; Zbl 1006.32020)] of the Green–Griffiths conjecture [M. Green, P. Griffiths, Two applications of algebraic geometry to entire holomorphic mappings, Differential geometry, Proc. int. Chern Symp., Berkeley 1979, 41–74 (1980; Zbl 0508.32010)] concerning the algebraicity and finitedness of entire holomorphic curves on a surface of general type. However, it seems that the result of Bogomolov–McQuillan is not effective. To clear up the problem, in the paper under review the author proves that the canonical degree of an irreducible curve of genus \(g\) on \(X\) is bounded from above by a function of \(g\), \(K_X^2\) and \(c_2(X)\). As remarked by the author, this result has been obtained independently also by McQuillan.

MSC:
14J29 Surfaces of general type
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
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