Differential forms and currents on non-reduced complex spaces with applications to divergent integrals and the dbar-equation

Sammanfattning: This thesis consists of three papers in which we study differential forms and currents on complex spaces. An important tool for us is the theory of residue currents. In Paper I we study divergent integrals over singular differential forms on a complex manifold. The differential form should have a pole along a complex hypersurface. To such a differential form we associate a residue form and a current with properties similar to residue currents. We connect the residue form and the current in a formula which can be thought of as a residue formula in this setting. In Paper II we solve the ¯ ∂-equation for (p, q)-forms on nonreduced complex spaces. It is not obvious what smooth differential forms and currents should be on a non-reduced space. We define these objects using residue calculus and show that we can (locally) solve the ¯ ∂-equation. In Paper III the setting is similar to that of Paper I but we now allow the differential form to be singular on a complex submanifold of higher codimension. iii