Abstract
This paper states a conjecture for Nevanlinna theory or diophantine approximation, with a sheaf of ideals in place of the normal crossings divisor. This is done by using a correction term involving a multiplier ideal sheaf. This new conjecture trivially implies earlier conjectures in Nevanlinna theory or diophantine approximation, and in fact is equivalent to these conjectures. Although it does not provide anything new, it may be a more convenient formulation for some applications.
Mathematics Subject Classification (2010): 11J97 (primary); 14G25, 32H30, 14F18 (secondary)
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References
Fulton, W.: Introduction to toric varieties, Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton, NJ (1993). The William H. Roever Lectures in Geometry.
Lang, S.: Fundamentals of Diophantine Geometry. Springer-Verlag, New York, 1983.
Lazarsfeld, R.: Positivity in Algebraic Geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Vol. 49. Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals
Noguchi, J.: Nevanlinna Theory in Several Variables and Diophantine Approximation [Japanese]. Kyoritsu Publ., Tokyo, 2003.
Silverman, J.H.: Arithmetic distance functions and height functions in Diophantine geometry. Math. Ann. 279(2), 193–216 (1987). DOI10.1007/BF01461718. URL http://dx.doi.org/10.1007/BF01461718.
Vojta, P.: A more general abc conjecture. Internat. Math. Res. Notices 1998(21), 1103–1116 (1998)
Vojta, P.: Diophantine approximation and Nevanlinna theory. In: Arithmetic Geometry (Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, September 10–15, 2007), Lecture Notes in Math., vol. 2009, pp. 111–230. Springer, 2010.
Yamanoi, K.: Algebro-geometric version of Nevanlinna’s lemma on logarithmic derivative and applications. Nagoya Math. J. 173, 23–63, 2004.
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Supported by NSF grants DMS-0200892 and DMS-0500512.
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Dedicated to the memory of Serge Lang. No one could hope for a better mentor.
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Vojta, P. (2012). Multiplier ideal sheaves, Nevanlinna theory, and Diophantine approximation. In: Goldfeld, D., Jorgenson, J., Jones, P., Ramakrishnan, D., Ribet, K., Tate, J. (eds) Number Theory, Analysis and Geometry. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1260-1_28
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DOI: https://doi.org/10.1007/978-1-4614-1260-1_28
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