Holomorfni spreji v kompleksni analizi in geometriji : doktorska disertacija

Stopar, Kris (author), Prezelj, Jasna (mentor)

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.

• Language: Slovenian
• Keywords: 1-konveksna domena, 1-konveksen Cartanov par, Cartanova lema, sprej, sprej prerezov, aproksimacija, princip Oka, prava holomorfna preslikava
• Work type: Doctoral dissertation
• Typology: 2.08 - Doctoral Dissertation
• Organization: FMF - Faculty of Mathematics and Physics
• Place of publishing: Ljubljana
• Publisher: [K. Stopar]
• Year: 2013
• Number of pages: 72 str.
• PID: 20.500.12556/RUL-95850
• UDC: 517.55(043.3)
• COBISS.SI-ID: 16765529
• Publication date in RUL: 24.10.2017

STOPAR, Kris, 2013, Holomorfni spreji v kompleksni analizi in geometriji : doktorska disertacija [online]. Doctoral dissertation. Ljubljana : K. Stopar. [Accessed 14 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=95850

Secondary language

• Language: English

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.

• Keywords: 1-convex domain, 1-convex Cartan pair, Cartan lemma, spray, spray of sections, approximation, Oka principle, proper holomorphic map