We consider the vanishing problem for higher cohomology groups on certain infinite-dimensional complex spaces: good branched coverings of suitable projective spaces and subvarieties with a finite free resolution in a projective space P ( V ) (e.g. complete intersections or cones over finitedimensional projective spaces). In the former case we obtain the vanishing result for H 1 . In the latter case the corresponding results are only conditional for sheaf cohomology because we do not have the corresponding vanishing theorem for P ( V ).
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