Holomorfni spreji v kompleksni analizi in geometriji - Stopar, Kris (visokošolska dela, 2013)
Let ?$π \colon Z → X$? 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 ?$π$?-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 ?$π$?-sections. In particular, we prove that ?$π$?-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 ?$π$?-sections that are holomorphic in open neighborhoods of ?$\bar{D}$?. Under additional assumptions on the submersion we also get approximation by global holomorphic ?$π$?-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.
Keywords: 1-convex Cartan pair; 1-convex domain; 1-konveksen Cartanov par; 1-konveksna domena; approximation; aproksimacija; Cartan lemma; Cartanova lema; Disertacije; Holomorfni spreji (matematika); Oka principle; prava holomorfna preslikava; princip Oka; proper holomorphic map; spray; spray of sections; sprej; sprej prerezov