Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T09:58:29.978Z Has data issue: false hasContentIssue false
  • English
  • Français

A defect relation for meromorphic maps on generalized p-parabolic manifolds intersecting hypersurfaces in complex projective algebraic varieties

Part of: Holomorphic mappings and correspondences Complex manifolds Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  21 March 2013

Qi Han
Qi Han*
Affiliation:
Department of Mathematics, The University of Houston, Houston, TX 77204-3008, USA ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We establish a defect relation for algebraically non-degenerate meromorphic maps over generalized p-parabolic manifolds that intersect hypersurfaces in smooth projective algebraic varieties, extending certain results of H. Cartan, L. Ahlfors, W. Stoll, M. Ru, P. M. Wong and Philip P. W. Wong and others.

Keywords

generalized p-parabolic manifoldsprojective algebraic varietiesmeromorphic mapsalgebraic non-degeneratedefect relationChow weight and Hilbert weight

MSC classification

Secondary: 32H30: Value distribution theory in higher dimensions 11K60: Diophantine approximation 32Q15: Kähler manifolds
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 2 , June 2013 , pp. 551 - 574
Copyright
Copyright © Edinburgh Mathematical Society 2013

References

1.Ahlfors, L., The theory of meromorphic curves, Acta Soc. Sci. Fenn. Nova. A 3 (1941), 171–183.Google Scholar
2.Cao, J. G. and Wong, P. M., Finsler geometry of projectivized vector bundles, J. Math. Kyoto Univ. 43 (2003), 369–410.Google Scholar
3.Carlson, J. and Griffiths, Ph., A defect relation for equi-dimensional holomorphic mappings between algebraic varieties, Annals Math. 95 (1972), 557–584.CrossRefGoogle Scholar
4.Cartan, H., Sur les zeros des combinaisons linéaires de p fonctions holomorphes données, Mathematica 7 (1933), 5–31.Google Scholar
5.Eremenko, A. E. and Sodin, M. L., Distribution of values of meromorphic functions and meromorphic curves from the standpoint of potential theory, St Petersburg Math. J. 3 (1992), 109–136.Google Scholar
6.Evertse, J. H. and Ferretti, R. G., Diophantine inequalities on projective varieties, Int. Math. Res. Not. 25 (2002), 1295–1330.CrossRefGoogle Scholar
7.Griffiths, Ph. and King, J., Nevanlinna theory and holomorphic mappings between algebraic varieties, Acta Math. 130 (1973), 145–220.CrossRefGoogle Scholar
8.Gundersen, G. G. and Hayman, W. K., The strength of theory, Bull. Lond. Math. Soc. 36 (2004), 433–454.CrossRefGoogle Scholar
9.Hodge, W. V. D. and Pedoe, D., Methods of algebraic geometry, Volume II (Cambridge University Press, 1952).Google Scholar
10.Liu, Y. and Ru, M., A defect relation for meromorphic maps on parabolic manifolds intersecting hypersurfaces, Illinois J. Math. 49 (2005), 237–257.CrossRefGoogle Scholar
11.Mumford, D., Stability of projective varieties, Enseign. Math. 23 (1977), 39–110.Google Scholar
12.Ru, M., On a general form of the second main theorem, Trans. Am. Math. Soc. 349 (1997), 5093–5105.CrossRefGoogle Scholar
13.Ru, M., A defect relation for holomorphic curves intersecting hypersurfaces, Am. J. Math. 126 (2004), 215–226.CrossRefGoogle Scholar
14.Ru, M., Holomorphic curves into algebraic varieties, Annals Math. 169 (2009), 255–267.CrossRefGoogle Scholar
15.Shiffman, B., On holomorphic curves and meromorphic maps in projective space, Indiana Univ. Math. J. 28 (1979), 627–641.CrossRefGoogle Scholar
16.Stoll, W., Value distribution on parabolic spaces, Lecture Notes in Mathematics, Volume 600 (Springer, 1977).CrossRefGoogle Scholar
17.Stoll, W., The Ahlfors–Weyl theory of meromorphic maps on parabolic manifolds, in Value Distribution Theory, Joensuu, 1981, Lecture Notes in Mathematics, Volume 981, pp. 101–219 (Springer, 1983).Google Scholar
18.Stoll, W., Value distribution theory in several complex variables (Shandong University Press, Jinan, China, 1995).Google Scholar
19.Vojta, P., On Cartan's Am. J. Math. 119 (1997), 1–17.CrossRefGoogle Scholar
20.Wong, P. M. and Stoll, W., Second main theorem of Nevanlinna theory for nonequidi-mensional meromorphic maps, Am. J. Math. 116 (1994), 1031–1071.CrossRefGoogle Scholar
21.Wong, P. M. and Wong, P. P. W., The second main theorem on generalized parabolic manifolds, in Some topics on value distribution and differentiability in complex and p-adic analysis, Mathematical Monographs Series, Volume 11, pp. 3–41 (Science Press, Beijing, 2008).Google Scholar