On Proper Holomorphic Mappings Of Generalized Balls In Projective Spaces

Generalized balls in projective spaces are one of the classical research topics of proper mappings in complex analysis.In this thesis,we study the rigidity for proper holomorphic mappings among a broader type of generalized balls which are domains on the complex projec-tive spaces with possibly Levi-degenerate boundaries and also give a classification for certain proper monomial rational maps between generalized balls with Levi-nondegenerate boundaries.The full text is mainly divided into three parts.In Chapter 1,we first introduce the historical background of proper holomorphic mappings,and then survey the developments on proper holomorphic maps between bounded symmetric domains and between ordinary generalized balls.Finally,we give our main theorems.In Chapter 2,we mainly investigate the rigidity for local proper holomorphic mappings be-tween generalized balls with Levi-degenerate boundaries.We begin by analyzing the structure of the moduli spaces of compact complex analytic subvarieties contained in these generalized balls.Then,we obtain double fibrations over the Grassmann bundle in projective spaces with the help of the explicit parametrization of these projective linear subspaces.By a maximum principle-type argument and a standard extension lemma,we get the linearity of the maps that concern us.In addition,we provide a complete description of local proper holomorphic map-pings between certain pairs of generalized balls with Levi-degenerate boundaries.The above results can also be regarded as a generalization some earlier results for the ordinary generalized balls with Levi-nondegenerate boundaries.In Chapter 3,we are concerned about the classification problem of proper rational mono-mial mappings between ordinary generalized balls,inspired by the result in Seo[71].After-wards,based on Ng[65] and Seo[71],we introduce the relations between proper holomorphic maps among ordinary generalized balls and proper rational maps among bounded symmetric domains of type I.Accordingly,we give some new examples of proper holomorphic mappings among bounded symmetric domains.