Holomorphic synthesis of monogenic functions of several quaternionic variables

The system of differential equations for polymonogenic functions of several quaternionic variables is an analogue of the ∂̄-equation in complex analysis. We give a representation of polymonogenic functions by means of integration of a family of σ-holomorphic functions as σ runs over the variety ∑ of all complex structures ℍ ≅ ℂ² which are consistent with the metric and an orientation in ℍ. The variety ∑ is isomorphic to the manifold of all proper right ideals in the complexified quaternionic algebra and has a natural complex analytic structure. We construct a ∂̄-complex on ∑ that provides a resolvent for the sheaf of polymonogenic functions.