In [5] Rhoades found linear relations among the Poincaree series of weight k for [[GAMMA].sub.0](N) given by weakly

holomorphic modular forms of weight 2 - k for [[GAMMA].sub.0] (N).

A central theme is to understand the dynamics of

holomorphic maps on complex domains.

On the order of

holomorphic curves with maximal deficiency sum, II Nobushige TODA Communicated by Masaki KASHIWARA, M.J.A.

Let [OMEGA] be a domain in [C.sup.n] and H([OMEGA]) be the set of

holomorphic mappings from [OMEGA] into [C.sup.n].

Then the difference coordinates [x.sub.j], [[??].sub.j] given by (9)-(10) are

holomorphic functions in the hyperdisc [absolute value of ([xi])] < [square root of (q)], [absolute value of ([eta])] < [square root of (q)] and satisfy (7) if [xi](t), [eta](t) satisfy (8).

where H([OMEGA]) denotes the

holomorphic functions on [beta].

The Privalov class [N.sup.p](U), 1 < p < [infinity], is defined as the set of all

holomorphic functions f on U, satisfying

If f = ([f.sub.1], ..., [f.sub.n]) : D [right arrow] [R.sup.n] is a minimal surface, then there exist

holomorphic functions [g.sub.j] in D such that [f.sub.j] = [Reg.sub.j], 1 [less than or equal to] j [less than or equal to] n.

with f = ([f.sup.1], ..., [f.sup.v]) and F = ([F.sup.1], ..., [F.sup.v]) v-vector functions, [F.sup.i]

holomorphic in a polydisc, say [mathematical expression not reproducible] for some [[rho].sub.1] > [rho] > 0 (here, [D.sub.[rho]]([z.sub.0]) = {z [member of] C : [absolute value of z-[z.sub.0]] < [rho]} denotes an open disc of radius [rho] > 0, centered at [z.sub.0], [[bar.D].sub.[rho]]([z.sub.0]) denotes its closure and [D.sub.[rho]] = [D.sub.[rho]](0)) such that the v x v matrix [A.sub.0] = [F.sub.f](0, 0, 0) is invertible, a condition that makes (1) possess a regular singularity at z = 0.

Now, it is known ([4, 5])--or it could be taken here as a definition--that M is Levi nondegenerate at 0 [member of] M when the local

holomorphic map

It is thanks to the fact that the cohomology class of (2,1)-forms is isomorphic to the cohomology class [H.sup.1/[partial derivative]] (TM), the first Dolbeault cohomology group of M with values in a

holomorphic tangent bundle TM that characterizes infinitesimal complex structure deformations.

Abstract: Suppose that fi is a domain of On , n S 1 , E c f i closed in fi , the Hausdorff measure H2 n -1 (E )= 0 , and f is

holomorphic in fi \ E .