On Remmert's conjecture
This article, based on the ``mémoire de DEA'' of the author, describes a number of results around a question raised by Remmert. Fix a number \(n\). The question is whether the dimension of the automorphism group of a homogeneous compact complex manifold of dimension \(n\) can be bounded by \(n(n+2)\) (which is the dimension of the automorphism group of the projective space \({\mathbb P}_n\)). By a result of \textit{A. Borel} and \textit{R. Remmert} [Math. Ann. 145, 429--439 (1962; Zbl 0111.18001)] this is true for Kähler manifolds. In 1983 \textit{D. N. Akhiezer} proved that there is some (fairly large) bound \(d(n)\) [Soobshch. Akad. Nauk Gruz. SSR 110, 469--472 (1983; Zbl 0579.32049). Later, in 1998 \textit{D. M. Snow} and \textit{J. Winkelmann} constructed a series of homogeneous compact complex manifolds which show that this bound \(d(n)\) necessarily grows faster than any polynomial [Invent. Math. 134, No. 1, 139--144 (1998; Zbl 0901.32022)]. Thus the question of Remmert has a negative answer. The article of Deroin summarizes these and some related results.
scientific article; zbMATH DE number 2063639