Abstract
We introduce primitive forms with or without higher residue structure and explore their connection with the flat structures with or without a metric [18] and integrable hierarchies of KdV type. Just as the classical case of primitive forms with metric [12], the primitive forms without metrics are constructed as the positive part of the Birkhoff decomposition of formal oscillatory integrals with respect to the descendent variable. The oscilating integrals of a primitive form without metric give rise to a hierarchy of commuting PDE of the KdV type as in the case of primitive forms with metric. This shall be studied in (II).
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