Nonequidimensional value distribution theory and meromorphic connections

Abstract

This is a sequel to the paper [Si2-1 in whichwe introduced the use ofmeromorphic connections to handle nonequidimensional value distribution theory. The meromorphic connection is chosen so that the divisor has zero second fundamental form with respect to the meromorphic connection. When we have more than one divisor, they must have zero second fundamental form with respect to the same meromorphic connection. Such a condition is so stringent that it is difficult to find examples to which the method can be applied and which cannot be handled by other means. One example is a collection of Fermat curves in P2 of the form aowg + atw + a2wg 0 for the same homogeneous coordinate system [Wo, wt, w2]. Shift’man I-Sh2] told me that for such a collection of Fermat curves defect relations can be obtained by using a technique of H. Cartan [C2] (see 9), whereas the general case of allowing Fermat curves with respect to different homogeneous coordinates cannot be handled by any of the known methods. The only known result on the defect of general divisors is obtained by using the Veronese map and is given in [Shl] in the following form. Let f: C Pn be a holomorphic curve such that f(C) is not contained in any algebraic hypersurface in Pn. Let {Sj} be a collection of algebraic hypersurfaces of degree d in P. The sum of the defects of f for the