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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.

14J29 Surfaces of general type
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
Full Text: DOI
[1] F. A. Bogomolov, Families of curves on a surface of general type, Dokl. Akad. Nauk SSSR 236 (1977), no. 5, 1041-1044. · Zbl 0415.14013
[2] L. Caporaso, J. Harris and B. Mazur, Uniformity of rational points, J. Amer. Math. Soc. 10 (1997), no. 1, 1-35. 417 · Zbl 0872.14017 · doi:10.1090/S0894-0347-97-00195-1
[3] M. Deschamps, Courbes de genre géométrique borné sur une surface de type général · Zbl 0412.14013 · numdam:SB_1977-1978__20__233_0 · eudml:109921
[4] , in Séminaire Bourbaki, 30e année (1977/78), Exp. 519, pp. 233-247, Lecture Notes in Math., 710, Springer, Berlin, 1979.
[5] M. Green and P. Griffiths, Two applications of algebraic geometry to entire holomorphic mappings, in The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), 41-74, Springer, New York, 1980. · Zbl 0508.32010
[6] J. P. Jouanolou, Hypersurfaces solutions d’une équation de Pfaff analytique, Math. Ann. 232 (1978), no. 3, 239-245. · Zbl 0354.34007 · doi:10.1007/BF01351428 · eudml:163084
[7] S. Lang, Hyperbolic and Diophantine analysis, Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 159-205. · Zbl 0602.14019 · doi:10.1090/S0273-0979-1986-15426-1
[8] S. S.-Y. Lu and Y. Miyaoka, Bounding curves in algebraic surfaces by genus and Chern numbers, Math. Res. Lett. 2 (1995), no. 6, 663-676. · Zbl 0870.14020 · doi:10.4310/MRL.1995.v2.n6.a1
[9] M. McQuillan, Diophantine approximations and foliations, Inst. Hautes Études Sci. Publ. Math. No. 87 (1998), 121-174. · Zbl 1006.32020 · doi:10.1007/BF02698862 · numdam:PMIHES_1998__87__121_0 · eudml:104127
[10] , Non-commutative Mori theory, IHES, preprint, IHES/M/01/42.
[11] Y. Miyaoka, On the Chern numbers of surfaces of general type, Invent. Math. 42 (1977), 225-237. · Zbl 0374.14007 · doi:10.1007/BF01389789 · eudml:142501
[12] , The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268 (1984), no. 2, 159-171. · Zbl 0521.14013 · doi:10.1007/BF01456083 · eudml:182912
[13] N. Nakayama, Zariski-decomposition and abundance, Math. Soc. Japan, Tokyo, 2004. · Zbl 1061.14018
[14] F. Sakai, Semistable curves on algebraic surfaces and logarithmic pluricanonical maps, Math. Ann. 254 (1980), no. 2, 89-120. · Zbl 0431.14011 · doi:10.1007/BF01467073 · eudml:163485
[15] S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Amp‘ ere equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339-411. · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
[16] O. Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2) 76 (1962), 560-615. · Zbl 0124.37001 · doi:10.2307/1970376
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