Multiplier Ideal Sheaves, Nevanlinna Theory, and Diophantine Approximation

This note 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. This note 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 is equivalent to earlier conjectures in Nevanlinna theory or diophantine approximation, but may be a more convenient formulation for some applications. It also shows how multiplier ideal sheaves may have a role in Nevanlinna theory and diophantine approximation, and therefore may give more information on the structure of the situation. Section 1 briefly describes multiplier ideal sheaves, and gives a variant definition specific to this situation. Section 2 describes proximity functions for sheaves of ideals, using work of Silverman and Yamanoi. Sections 3 and 4 form the heart of the paper, giving the conjectures and showing their equivalence to previous conjectures. Throughout this paper, X is a smooth complete variety over C (in the case of Nevanlinna theory) or over a global field of characteristic zero (in the case of diophantine approximation). Supported by NSF grants DMS-0200892 and DMS-0500512. 1