The second main theorem of holomorphic curves into projective spaces

Hu, Pei-Chu; Yang, Chung-Chun

doi:10.1090/S0002-9947-2011-05394-8

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6850(online) ISSN 0002-9947(print)

The second main theorem of holomorphic curves into projective spaces

Authors: Pei-Chu Hu and Chung-Chun Yang
Journal: Trans. Amer. Math. Soc. 363 (2011), 6465-6479
MSC (2010): Primary 32H02, 32H25
Posted: July 26, 2011
MathSciNet review: 2833564
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: By utilizing Jacobian sections introduced by Stoll, we prove a second main theorem of holomorphic curves into projective spaces for hypersurfaces under certain conditions on the jets of the curves. The precise conditions are listed in three equations in the introduction. This solves a weaker form of Griffiths' conjecture and only for a special class of holomorphic curves.

1. Aldo Biancofiore, A hypersurface defect relation for a class of meromorphic maps, Trans. Amer. Math. Soc. 270 (1982), no. 1, 47–60. MR 642329 (83h:32005), http://dx.doi.org/10.1090/S0002-9947-1982-0642329-1
2. A. Bloch, Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires, Ann. Sci. École Norm. Sup. (3) 43 (1926), 309–362 (French). MR 1509274
3. James Carlson and Phillip Griffiths, A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. of Math. (2) 95 (1972), 557–584. MR 0311935 (47 #497)
4. Henri Cartan, Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires et leurs applications, Ann. Sci. École Norm. Sup. (3) 45 (1928), 255–346 (French). MR 1509288
5. Karen Chandler and Pit-Mann Wong, Finsler geometry of holomorphic jet bundles, A sampler of Riemann-Finsler geometry, Math. Sci. Res. Inst. Publ., vol. 50, Cambridge Univ. Press, Cambridge, 2004, pp. 107–196. MR 2132659 (2006i:32022)
6. Jacques Dufresnoy, Théorie nouvelle des familles complexes normales. Applications à l’étude des fonctions algébroïdes, Ann. Sci. École Norm. Sup. (3) 61 (1944), 1–44 (French). MR 0014469 (7,289f)
7. Hirotaka Fujimoto, Extensions of the big Picard’s theorem, Tôhoku Math. J. (2) 24 (1972), 415–422. MR 0333253 (48 #11578)
8. Hirotaka Fujimoto, Families of holomorphic maps into the projective space omitting some hyperplanes, J. Math. Soc. Japan 25 (1973), 235–249. MR 0328131 (48 #6473)
9. Mark L. Green, Holomorphic maps into complex projective space omitting hyperplanes, Trans. Amer. Math. Soc. 169 (1972), 89–103. MR 0308433 (46 #7547), http://dx.doi.org/10.1090/S0002-9947-1972-0308433-6
10. Mark Lee Green, Some Picard theorems for holomorphic maps to algebraic varieties, Amer. J. Math. 97 (1975), 43–75. MR 0367302 (51 #3544)
11. Mark L. Green, Some examples and counter-examples in value distribution theory for several variables, Compositio Math. 30 (1975), no. 3, 317–322. MR 0374498 (51 #10698)
12. Mark Green and Phillip Griffiths, Two applications of algebraic geometry to entire holomorphic mappings, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer, New York, 1980, pp. 41–74. MR 609557 (82h:32026)
13. Phillip A. Griffiths, Holomorphic mappings: Survey of some results and discussion of open problems, Bull. Amer. Math. Soc. 78 (1972), 374–382. MR 0294718 (45 #3786), http://dx.doi.org/10.1090/S0002-9904-1972-12905-7
14. Phillip Griffiths and James King, Nevanlinna theory and holomorphic mappings between algebraic varieties, Acta Math. 130 (1973), 145–220. MR 0427690 (55 #721)
15. Pei Chu Hu, Holomorphic mapping into algebraic varieties of general type. I, II, Nagoya Math. J. 120 (1990), 155–170, 171–180. MR 1086576 (92b:32033)
16. Pei Chu Hu, Holomorphic mappings between spaces of different dimensions. I, Math. Z. 214 (1993), no. 4, 567–577. MR 1248113 (94k:32043), http://dx.doi.org/10.1007/BF02572425
Pei Chu Hu, Holomorphic mappings between spaces of different dimensions. II, Math. Z. 215 (1994), no. 2, 187–193. MR 1259455 (95a:32046), http://dx.doi.org/10.1007/BF02571708
17. Pei-Chu Hu, Ping Li, and Chung-Chun Yang, Unicity of meromorphic mappings, Advances in Complex Analysis and its Applications, vol. 1, Kluwer Academic Publishers, Dordrecht, 2003. MR 1988768 (2004m:32031)
18. Pei-Chu Hu and Chung-Chun Yang, Meromorphic functions over non-Archimedean fields, Mathematics and its Applications, vol. 522, Kluwer Academic Publishers, Dordrecht, 2000. MR 1794326 (2002a:11085)
19. Pei-Chu Hu and Chung-Chun Yang, Value distribution theory related to number theory, Birkhäuser Verlag, Basel, 2006. MR 2245631 (2008c:32021)
20. Shoshichi Kobayashi, Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 318, Springer-Verlag, Berlin, 1998. MR 1635983 (99m:32026)
21. Serge Lang, Introduction to complex hyperbolic spaces, Springer-Verlag, New York, 1987. MR 886677 (88f:32065)
22. Alan Michael Nadel, Hyperbolic surfaces in 𝑃³, Duke Math. J. 58 (1989), no. 3, 749–771. MR 1016444 (91a:32036), http://dx.doi.org/10.1215/S0012-7094-89-05835-3
23. Bernard Shiffman, Holomorphic curves in algebraic manifolds, Bull. Amer. Math. Soc. 83 (1977), no. 4, 553–568. MR 0440075 (55 #12956), http://dx.doi.org/10.1090/S0002-9904-1977-14323-1
24. Yum Tong Siu, Defect relations for holomorphic maps between spaces of different dimensions, Duke Math. J. 55 (1987), no. 1, 213–251. MR 883671 (89a:32030), http://dx.doi.org/10.1215/S0012-7094-87-05512-8
25. Yum Tong Siu, Nonequidimensional value distribution theory and meromorphic connections, Duke Math. J. 61 (1990), no. 2, 341–367. MR 1074301 (92f:32044), http://dx.doi.org/10.1215/S0012-7094-90-06115-0
26. Yum Tong Siu, Nonequidimensional value distribution theory, Complex analysis, Joensuu 1987, Lecture Notes in Math., vol. 1351, Springer, Berlin, 1988, pp. 285–311. MR 982094 (90j:32028), http://dx.doi.org/10.1007/BFb0081264
27. Yum-Tong Siu and Sai-Kee Yeung, Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees, Amer. J. Math. 119 (1997), no. 5, 1139–1172. MR 1473072 (98h:32044)
28. Yum-Tong Siu and Sai-Kee Yeung, Addendum to: “Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees” [Amer. J. Math 119 (1997), no. 5, 1139–1172; MR1473072 (98h:32044)], Amer. J. Math. 125 (2003), no. 3, 441–448. MR 1980629 (2004e:32028)
29. Wilhelm Stoll, Value distribution on parabolic spaces, Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 2, Williams Coll., Williamstown, Mass., 1975), Amer. Math. Soc., Providence, R.I., 1977, pp. 259–263. MR 0450626 (56 #8919)
30. Wilhelm Stoll, The Ahlfors-Weyl theory of meromorphic maps on parabolic manifolds, Value distribution theory (Joensuu, 1981) Lecture Notes in Math., vol. 981, Springer, Berlin, 1983, pp. 101–219. MR 699135 (85c:32045), http://dx.doi.org/10.1007/BFb0066385
31. Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451 (91k:11049)
32. Paul Vojta, On the 𝐴𝐵𝐶 conjecture and Diophantine approximation by rational points, Amer. J. Math. 122 (2000), no. 4, 843–872. MR 1771576 (2001i:11094)
33. Paul Vojta, On the 𝐴𝐵𝐶 conjecture and Diophantine approximation by rational points, Amer. J. Math. 122 (2000), no. 4, 843–872. MR 1771576 (2001i:11094)
Paul Vojta, Correction to: “On the 𝐴𝐵𝐶 conjecture and Diophantine approximation by rational points” [Amer. J. Math. 122 (2000), no. 4, 843–872; MR1771576 (2001i:11094)], Amer. J. Math. 123 (2001), no. 2, 383–384. MR 1828227 (2002d:11095)
34. Julie T.-Y. Wang, The truncated second main theorem of function fields, J. Number Theory 58 (1996), no. 1, 139–157. MR 1387732 (97c:11074), http://dx.doi.org/10.1006/jnth.1996.0071
35. Wong, P. M. and Wong, Philip P. W., The second main theorem on generalized parabolic manifolds, preprint.
36. Wong, P. M., Nevanlinna theory for holomorphic curves in projective varieties, preprint.
37. Wong, P. M. and Stoll, W., On holomorphic jet bundles, math. AG/0003226, preprint.
38. Hung-hsi Wu, The equidistribution theory of holomorphic curves, Annals of Mathematics Studies, No. 64, Princeton University Press, Princeton, N.J., 1970. MR 0273070 (42 #7951)
39. Zhuan Ye, On Nevanlinna’s second main theorem in projective space, Invent. Math. 122 (1995), no. 3, 475–507. MR 1359601 (96j:32030), http://dx.doi.org/10.1007/BF01231453

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|