Normality And Value Distribution Of Holomorphic And Meromorphic Mappings Into The Complex Projective Space

This dissertation is devoted to studying holomorphic or meromorphic map-pings of several complex variables into PN(C), the N-dimensional complex pro-jective space. We get a number of new results which greatly improve some earlier related theorems.In the third chapter, we focus on normal families of holomorphic mappings concerning shared hypersurfaces. This chapter contains two distinct results:(i) a family of holomorphic mappings which share 2t+1 hyperplanes located in t-subgeneral position is normal, and (ii) two families of holomorphic mappings into an arbitrary closed set of PN(C) share some hypersurfaces have the same normal-ity. An example is included to complement our theory.In the next chapter, the definition of the derivative of meromorphic functions is extended to holomorphic mappings from planar domains into PN(C). We then use it to study the normality criteria for families of holomorphic mappings. The results derived generalize and improve Schwick’s theorem for normal families of meromorphic functions.In 1974, H. Fujimoto introduced the notion of a meromorphically normal family of meromorphic mappings into PN(C) which improves the notion of quasi-normal family, and gave some sufficient conditions for a family of meromorphic mappings to be meromorphically normal. In Chapter 5, we investigate normal-ity for meromorphic mappings of several complex variables into PN(C), Firstly, by some normality criteria obtained in Chapter 3, we weaken the requirements of Fujimoto’s theorem. And then an example is shown to say that a meromor-phically normal family (?) fail to be holomorphic normal even in case all map-pings in (?) are holomorphic against intuition. So we introduce a new concept called M-normal (in contrast to meromorphically normal). For a family of holo-morphic curves, M-normal is nothing but holomorphic normal. Moreover, some M-normality criteria are obtained also.In the final chapter, our main consideration is value distribution of holomor-phic curves. We illustrate that the best posibble truncated level in Cartan’s second main theorem for holomorphic curves into PN(C) is N with an example relat-ed to Femart-type equation. A second main theorem for holomorphic mappings intersecting a fixed hypersurface is also given.