Holomorfni spreji v kompleksni analizi in geometriji : doktorska disertacija

slovenian

doktorska dela

2013

Univerza v Ljubljani, Fakulteta za matematiko in fiziko
(Obvezni izvod spletne publikacije)

Let ?$\pi \colon Z \to X$? be a holomorphic submersion of a complex manifold ?$Z$? onto a complex manifold ?$X$? and ?$D \Subset X$? a 1-convex domain with strongly pseudoconvex boundary. We prove that under certain conditions there always exists a spray of ?$\pi$?-sections over ?$\bar{D}$? which has prescribed core, fixes the exceptional set ?$E$? of ?$D$?, and is dominating on ?$\bar{D} \setminus E$?. Each section in this spray is of class ?${\mathcal C}^k(\bar{D})$? and holomorphic on ?$D$?. As a consequence we obtain several approximation results for ?$\pi$?-sections. In particular, we prove that ?$\pi$?-sections which are of class ?${\mathcal C}^k(\bar{D})$? and holomorphic on ?$D$? can be approximated in the ?${\mathcal C}^k(\bar{D})$? topology by ?$\pi$?-sections that are holomorphic in open neighborhoods of ?$\bar{D}$?. Under additional assumptions on the submersion we also get approximation by global holomorphic ?$\pi$?-sections and the Oka principle over 1-convex manifolds. We include a result on the existance of proper holomorphic maps from 1-convex domains into ?$q$?-convex manifolds.

12.05.2023