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Article: The concepts of general position and a second main theorem for non-linear divisors

TitleThe concepts of general position and a second main theorem for non-linear divisors
Authors
KeywordsGeneral Positions
Non-Linear Divisors
Second Main Theorem
Issue Date2011
PublisherTaylor & Francis Ltd. The Journal's web site is located at http://www.tandf.co.uk/journals/titles/02781077.asp
Citation
Complex Variables And Elliptic Equations, 2011, v. 56 n. 1-4, p. 375-398 How to Cite?
AbstractThe recent works of Evertse-Ferretti (Evertse and Ferretti, A generalization of the subspace theorem with polynomials of high degree, Dev. Math. (2008), pp.175-198) and Corvaja-Zannier (Corvaja and Zannier, On a general Theu's equation, Ann. Math. 160 (2004), pp. 705-726; Corvaja and Zannier, On integral points on surfaces, Amer. J. Math. 126 (2004), pp. 1033-1055) in diophantine approximations (Ru, A defect relation for holomorphic curves intersecting hypersurfaces, Amer. J. Math. 126 (2004), pp. 215-266; An, A defect relation for non-Archimedean analytic curves in arbitrary projective varieties, Proc. Amer. Math. Soc. 135 (2007), pp. 1255-1261) in complex and p-adic Nevanlinna theory extend the classical subspace theorem and the classical second main theorem to the case of non-linear divisors in general position. These had been long standing problems in diophantine approximation and Nevanlinna theory. However, their results when specialized to the case of hyperplanes are weaker than the classical results. In this article, we refine the concept of general position to the concepts of p-jet general position. These concepts of general position involve jets of order p and coincide with the usual concept of general position for hyperplanes, but are different for hypersurfaces of higher degrees. With the assumption that the hypersurfaces are in n-jet general position, a second main theorem, with ramification term, for non-linear divisors and d-non-degenerate map f:CgP n is obtained. The result when specialized to hyperplanes is precisely the classical result of Ahlfors (The theory of meromorphic curves, Acta Soc. Sci. Fenn. 3 (1941), pp. 1-31) (see also Cartan, Sur les zeros des combinations lineares de p fonctions holomorphes donees, Mathematica 7 (1933), pp. 5-31; Stoll, About the value distribution of holomorphic maps into the projective space, Acta Math. 123 (1969), pp. 83-114; Cowen and Griffiths, Holomorphic curves and metrics of nonnegative curvature, J. Anal. Math. 29 (1976), pp. 93-153; for small functions, see Yamanoi, The second main theorem for small functions and related problems, Acta Math. 192 (2004), pp. 225-294). In fact the proof is a modification of Ahlfors proof. There are a number of variations of Ahlfors proof, we choose the 'approximate negatively curved' approach used in Cowen and Griffiths (Holomorphic curves and metrics of non-negative curvature, J. Anal. Math. 29 (1976), pp. 93-153), Wong (Defect relation for meromorphic maps on parabolic manifolds and Kobayashi metrics on P n omitting hyperplanes, Ph.D. thesis, University of Notre Dame (1976)), Wong (Holomorphic curves in spaces of constant curvature, in Complex Geometry (Proceedings of the Osaka Conference, 1990), Lecture Notes in Pure and Applied Mathematics, Vol. 143, Marcel Dekker, New York, Basel, Hong Kong, 1993, pp. 201-223), Wong (On the second main theorem of Nevanlinna theory, Amer. J. of Math. 111 (1989), pp. 549-583) and Cowen (The Kobayashi metric on P n \ (2 n+1) hyperplanes, in Value Distribution Theory, Part A, R.O. Kujala and A.L. Vitter III, eds., Marcel Dekker, New York, 1974, pp. 205-223). © 2011 Taylor & Francis.
Persistent Identifierhttp://hdl.handle.net/10722/156264
ISSN
2015 Impact Factor: 0.466
2015 SCImago Journal Rankings: 0.599
ISI Accession Number ID
Funding AgencyGrant Number
NSFDMS 0713348
RGC/HKU 7053/06P
Funding Information:

This research was supported in part by NSF grant # DMS 0713348 and by grant # RGC/HKU 7053/06P.

References

 

DC FieldValueLanguage
dc.contributor.authorLaw, HFen_US
dc.contributor.authorWong, PMen_US
dc.contributor.authorWong, PPWen_US
dc.date.accessioned2012-08-08T08:41:05Z-
dc.date.available2012-08-08T08:41:05Z-
dc.date.issued2011en_US
dc.identifier.citationComplex Variables And Elliptic Equations, 2011, v. 56 n. 1-4, p. 375-398en_US
dc.identifier.issn1747-6933en_US
dc.identifier.urihttp://hdl.handle.net/10722/156264-
dc.description.abstractThe recent works of Evertse-Ferretti (Evertse and Ferretti, A generalization of the subspace theorem with polynomials of high degree, Dev. Math. (2008), pp.175-198) and Corvaja-Zannier (Corvaja and Zannier, On a general Theu's equation, Ann. Math. 160 (2004), pp. 705-726; Corvaja and Zannier, On integral points on surfaces, Amer. J. Math. 126 (2004), pp. 1033-1055) in diophantine approximations (Ru, A defect relation for holomorphic curves intersecting hypersurfaces, Amer. J. Math. 126 (2004), pp. 215-266; An, A defect relation for non-Archimedean analytic curves in arbitrary projective varieties, Proc. Amer. Math. Soc. 135 (2007), pp. 1255-1261) in complex and p-adic Nevanlinna theory extend the classical subspace theorem and the classical second main theorem to the case of non-linear divisors in general position. These had been long standing problems in diophantine approximation and Nevanlinna theory. However, their results when specialized to the case of hyperplanes are weaker than the classical results. In this article, we refine the concept of general position to the concepts of p-jet general position. These concepts of general position involve jets of order p and coincide with the usual concept of general position for hyperplanes, but are different for hypersurfaces of higher degrees. With the assumption that the hypersurfaces are in n-jet general position, a second main theorem, with ramification term, for non-linear divisors and d-non-degenerate map f:CgP n is obtained. The result when specialized to hyperplanes is precisely the classical result of Ahlfors (The theory of meromorphic curves, Acta Soc. Sci. Fenn. 3 (1941), pp. 1-31) (see also Cartan, Sur les zeros des combinations lineares de p fonctions holomorphes donees, Mathematica 7 (1933), pp. 5-31; Stoll, About the value distribution of holomorphic maps into the projective space, Acta Math. 123 (1969), pp. 83-114; Cowen and Griffiths, Holomorphic curves and metrics of nonnegative curvature, J. Anal. Math. 29 (1976), pp. 93-153; for small functions, see Yamanoi, The second main theorem for small functions and related problems, Acta Math. 192 (2004), pp. 225-294). In fact the proof is a modification of Ahlfors proof. There are a number of variations of Ahlfors proof, we choose the 'approximate negatively curved' approach used in Cowen and Griffiths (Holomorphic curves and metrics of non-negative curvature, J. Anal. Math. 29 (1976), pp. 93-153), Wong (Defect relation for meromorphic maps on parabolic manifolds and Kobayashi metrics on P n omitting hyperplanes, Ph.D. thesis, University of Notre Dame (1976)), Wong (Holomorphic curves in spaces of constant curvature, in Complex Geometry (Proceedings of the Osaka Conference, 1990), Lecture Notes in Pure and Applied Mathematics, Vol. 143, Marcel Dekker, New York, Basel, Hong Kong, 1993, pp. 201-223), Wong (On the second main theorem of Nevanlinna theory, Amer. J. of Math. 111 (1989), pp. 549-583) and Cowen (The Kobayashi metric on P n \ (2 n+1) hyperplanes, in Value Distribution Theory, Part A, R.O. Kujala and A.L. Vitter III, eds., Marcel Dekker, New York, 1974, pp. 205-223). © 2011 Taylor & Francis.en_US
dc.languageengen_US
dc.publisherTaylor & Francis Ltd. The Journal's web site is located at http://www.tandf.co.uk/journals/titles/02781077.aspen_US
dc.relation.ispartofComplex Variables and Elliptic Equationsen_US
dc.subjectGeneral Positionsen_US
dc.subjectNon-Linear Divisorsen_US
dc.subjectSecond Main Theoremen_US
dc.titleThe concepts of general position and a second main theorem for non-linear divisorsen_US
dc.typeArticleen_US
dc.identifier.emailWong, PPW:ppwwong@maths.hku.hken_US
dc.identifier.authorityWong, PPW=rp00810en_US
dc.description.naturelink_to_subscribed_fulltexten_US
dc.identifier.doi10.1080/17476930903394804en_US
dc.identifier.scopuseid_2-s2.0-79951806918en_US
dc.identifier.hkuros201378-
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-79951806918&selection=ref&src=s&origin=recordpageen_US
dc.identifier.volume56en_US
dc.identifier.issue1-4en_US
dc.identifier.spage375en_US
dc.identifier.epage398en_US
dc.identifier.eissn1747-6941-
dc.identifier.isiWOS:000287493500027-
dc.publisher.placeUnited Kingdomen_US
dc.identifier.scopusauthoridLaw, HF=35264431800en_US
dc.identifier.scopusauthoridWong, PM=7403978366en_US
dc.identifier.scopusauthoridWong, PPW=12752716000en_US

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