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In this paper we give an explicit formula for the solution of the non-homogeneous complex Cauchy problem with Cauchy data given on a bounded smooth strictly convex domain in a non-characteristic hyperplane. These formulas are obtained using the explicit version of the fundamental principle given in terms of residue currents; moreover, we characterize the domain of definition of the solution and we generalize these techniques to the non-homogeneous Goursat problem.

We give first of all a new criterion for Bergman completeness in terms of the pluricomplex Green function. Among several applications, we prove in particular that every Stein subvariety in a complex manifold admits a Bergman complete Stein neighborhood basis, which improves a theorem of Siu. Secondly, we give for hyperbolic Riemann surfaces a sufficient condition for when the Bergman and Poincaré metrics are quasi-isometric. A consequence is an equivalent characterization of uniformly perfect planar domains in terms of growth rates of the Bergman kernel and metric. Finally, we provide a noncompact Bergman complete pseudoconvex manifold without nonconstant negative plurisubharmonic functions.

Using the Nakayama functor, we construct an equivalence from a Serre quotient category of a category of finitely generated modules to a category of finite-dimensional modules. We then apply this result to the categories FI_G and VI_q, and answer positively an open question of Nagpal on representation stability theory.