After recalling the definitions of the Abel-Radon transformation of currents and of locally residual currents, we show that the Abel-Radon transform R(α) of a locally residual current α remains locally residual. Then a theorem of P. Griffiths, G. Henkin and M. Passare (cf. [7], [9] and [10]) can be formulated as follows : Let U be a domain of the Grassmannian variety G( p, N ) of complex p-planes in P N , U * :=∪ t∈U H t be the corresponding linearly p-concave domain of P N , and α be a locally residual current of bidegree (N , p). Suppose that the meromorphic n-form R(α) extends meromorphically to a greater domainŨ of G( p, N ). If α is of type ω∧[T ], with T an analytic subvariety of pure codimension p in U * , and ω a meromorphic (resp. regular) q-form (q > 0) on T , then α extends in a unique way as a locally residual current to the domainŨ * := ∪ t∈Ũ H t . In particular, if R(α) = 0, then α extends as a ∂-closed residual current on P N . We show in this note that this theorem remains valid for an arbitrary residual current of bidegree (N , p), in the particular case where p = 1.

This paper gives a partial answer to a problem raised by Griffiths in [4], which is a kind of converse of Abel's theorem.Definition 0.1. Abelian forms are the sections of a~'q; abelian one-forms on curves are called abelian differentials. Finite forms are the sections of s Let T be a reduced and irreducible analytic space, I be a reduced analytic space of pure dimension dim T. and ~: I-+T be a proper morphism.Definition 0.2. The proper morphism re is a ramified covering if there is an analytic subset SCT, without interior point, such that re': rr-~(T\S)--+T\S is an analytic covering, and moreover such that rr-l(T\S) is dense in I. The constant cardinal of the fiber rc-l(t), tET\S, is then called the degree of the ramified covering rr.Let rr: I--+T be a ramified covering and let us consider the set Z of parameters t ET over which re1 l(t) is not zero-dimensional (and thus Z C S).Let us observe the following fact. Lemma 0.3. The set E is an analytic subset of codimension at least two in T.Pro@ Let I'cI be the set of points z such that the fiber 7rll(Tr~(z)) at z has dimension > 1. Then we know from Remmert [9] that I' is a closed analytic subset without interior point. Since 7rl is proper, still by Remmert, 7rl(I')=E is an analytic subset of T. Let us assume that E has an irreducible component T ~ of codimension one in T. Since the fibers over T' are disjoint of dimension >1,(1) TMR Research Network, ERBFMRXCT 98063.

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