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In the First Main Theorem of value distribution for holomorphic maps into a compact Kahler manifold A, the central object is a differential form Xa on A with a singularity at a A. The most satisfactory Xa was given by Wu in [5]. His construction, using the Green function, has the advantage that Xa is automatically invariant under the group of isometries of A; more precisely, if i: A->A is an isometry with i(a) = b, then i*Xb = Xa. This property is difficult to obtain with any other construction. In this note we apply the technical machinery developed by Stoll and the author [1 ], [2 ], [3 ], [4] to Wu's Xa to show that it is suitable for value distribution theory in arbitrary fiber dimension. We also show that the terms of the First Main Theorem are continuous functions. vf will denote the multiplicity of the holomorphic map f; see [2]. Ok will denote any differential form of degree k whose coefficients are locally bounded; the subscript will usually be omitted. Of course it will never mean the same thing twice. On Cn set v1 = (i/2) 2,.=1 dz, Ad p and v-vk/k?I If M is a complex manifold and H an open subset of M, then S= H=-H is a boundary manifold of H, provided that for every point of S there is a Ccoordinate system (xi, * * *, x2n) valid in a neighborhood U of the point, such that in U, S is described by xi = 0 and H by x1 < 0. S will be considered as oriented to the exterior of H.

On Wu’s form of the first main theorem of value distribution