Holomorfni spreji v kompleksni analizi in geometriji

doktorska disertacija

Kris Stopar (Author), Jasna Prezelj-Perman (Mentor)

Abstract

Naj bo ▫$\pi \colon Z \to X$▫ holomorfna submerzija iz kompleksne mnogoterosti ▫$Z$▫ na kompleksno mnogoterost ▫$X$▫ in ▫$D \Subset X$▫ 1-konveksna domena s strogo psevdokonveksnim robom. V disertaciji dokažemo, da pod določenimi predpostavkami vedno obstaja sprej ▫$\pi$▫-prerezov nad ▫$\bar{D}$▫, ki ima predpisano jedro, fiksira izjemno množico ▫$E$▫ domene ▫$D$▫ in je dominanten na ▫$\bar{D} \setminus E$▫. Vsak prerez v tem spreju je razreda ▫${\mathcal C}^k(\bar{D})$▫ in holomorfen na ▫$D$▫. Kot posledico dobimo več aproksimacijskih rezultatov za ▫$\pi$▫-prereze. Med drugim dokažemo, da lahko ▫$\pi$▫-prereze, ki so razreda ▫${\mathcal C}^k(\bar{D})$▫ in holomorfni na ▫$D$▫ aproksimiramo v ▫${\mathcal C}^k(\bar{D})$▫ topologiji s ▫$\pi$▫-prerezi, ki so holomorfni v odprtih okolicah množice ▫$\bar{D}$▫. Pod dodatnimi predpostavkami na submerzijo dobimo tudi aproksimacijo z globalnimi holomorfnimi ▫$\pi$▫-prerezi in princip Oka nad 1-konveksnimi mnogoterostmi. Vključimo tudi rezultat o obstoju pravih holomorfnih preslikav iz 1-konveksnih domen v ▫$q$▫-konveksne mnogoterosti.

Keywords

1-konveksna domena;1-konveksen Cartanov par;Cartanova lema;sprej;sprej prerezov;aproksimacija;princip Oka;prava holomorfna preslikava;

Data

Language:	Slovenian
Year of publishing:	2013
Typology:	2.08 - Doctoral Dissertation
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[K. Stopar]
UDC:	517.55(043.3)
COBISS:	16765529
Views:	654
Downloads:	187
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary abstract:	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.
Secondary keywords:	1-convex domain;1-convex Cartan pair;Cartan lemma;spray;spray of sections;approximation;Oka principle;proper holomorphic map;
Type (COBISS):	Doctoral dissertation
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 3. stopnja
Pages:	72 str.
ID:	10865388