Title
Existenz semiuniverseller Deformationen in der komplexen Analysis (Aspects of Mathematics, D 5) (German Edition),Used
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Jede komplexe Mannigfaltigkeit ist auf natrliche Weise eine differenzierbare !>1annigfaltigkeit. Sei umgekehrt M eine differenzierbare Mannigfaltigkeit. Es erhebt sich die Frage, ob auf M eine komplexe Struktur existiert. Falls dies der Fall ist, besteht dasnchste Problern darin, eine bersicht ber "alle" komplexen Strukturen auf M zu gewinnen. Sei L(M): =Menge der quivalenzklassen von komplexen Strukturen auf M Menge der zu M diffeornorphen, komplexen Mannigfalt keiten/biholornorphe quivalenz. Das Modulproblern, das seinen Ursprung in der Arbeit [67] von B. Riernann hat, besteht darin, auf L(M) eine "natrliche" komplexe Struktur einzufhren. Beispiel 1. Im Falle, da M = ist, besteht L(M) aus zwei 1 Punkten, falls M = F ist, besteht L(M) nur aus einem Punkt (Riernannscher Abbildungssatz) . Beispiel 2. Sei w E mit Im w > 0 und Gw: = {rnw+nlrn, nE }. Dann ist Tw: = /Gw ein Torus. Zwei Tori Tw' und T sind w genau dann biholornorph zueinander, wenn ganze Zahlen a, b, c, d mit ad bc = 1 existieren, so da + b w' = aw cw + d ist. Jeder Torus hat also einen Reprsentanten T mit w wEr: = {aE i Im Ct > 0, /Real:5. 2' Iai .:: . 1} . VIII Identifiziert man entsprechende Punkte in r, so kann man zeiaen. da fr jeden Torus T gilt r(T) ""a: Man vergleiche dazu [39], Example 2.14. Beispiel 3. Satz (Riemann, Teichmller, Rauch, Ahlfors, Bers).
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