Seminar: Thomas Kurbach (G.15.25)
The classical Relative Riemann-Hilbert Theorem on submersions states that flat relative connections are entirely determined by their kernel. In this talk, some progress is presented, where the submersion is replaced by a locally trivial morphism with singular fibers. The methods developed in the singular setting relying on considerations of reducedness, torsion-freeness, maximality and/or homogenity of the fibers. The existence of weak solutions is established in the general setting and further it is discussed under which conditions the weak solutions are actually strong solutions. Moreover, it is discussed that the case of general fibers can be reduced to the case of curves. As an application it is presented, that real analytic generalized d-bar-operators can be viewed as relative connections on the complexification and hence real analytic Newlander-Nirenberg Theorems on singular spaces can be proven by solving the associated Relative Riemann-Hilbert Theorems.
