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Équations différentielles motiviques et au–delà

Generalized cross-ratios, functions on Hilbert schemes, and Abel's problem

L'Institut Poincaré

Suppose Y=Y_1-Y_2 and Z=Z_1-Z_2 are algebraic cycles with Y of codim. r and Z of dim. r-1 on a smooth, projective variety P of dimension n over the complex numbers. We assume Y and Z are homologous to 0 and the supports of Y and Z are disjoint. The method of Hain yields a biextension which is a mixed Hodge structure with weight graded pieces Q(1), H^{2r-1}(P,\Q(r)), and Q(0). We study the degenerate case when 
H^{2r-1}(P,\Q(r)) = (0). The resulting extension of Q(0) by Q(1) (Kummer extension) yields an invariant in C^\times. In the simplest case P=P^1 and Y,Z given by differences of points, this invariant is the cross-ratio. In general it is difficult to calculate. Working with families of cycles yields solutions to a classical problem of Abel and also to rational functions on Hilbert schemes