Some New Results About Value Distribution And Normal Family Theory For Holomorphic Curves

The theory of value distribution and normal family for holomorphic mappings into complex projective space are the main subjects of our study. In the process of exploring, we obtain some new results, and these results greatly improved the former theorems in the corresponding research field.In chapter 2, we give an improvement for the second main theorem of alge-braically non-degenerate holomorphic curves into a complex projective variety V intersecting hypersurfaces in subgeneral position, obtained by Chen-Ru-Yan, and give an explicit estimate for the truncation level in the projective normal case. The following two results are obtained:(1). Let X be a smooth complex projective variety of dimension k. Let D1, …, Dq be effective divisors on X, located in m-subgeneral position on X with m+k-2> 0. Assume that Dj～dj A for some ample divisor A for j=1,…, q, where dj are positive integers. Letf:Câ†'X be an algebraically non-degenerate holomorphic curve. Then, for every e> 0, there exists a positive integer M such that(2). Let X be a smooth complex projective variety of dimension k. Let Dl,…,Dq be effective divisors on X, located in m-subgeneral position on X. Assume that Dj-djA for some ample divisor A, for j=1,…, q, where dj are positive integers. We further assume that X is projective normal with respect to NoA, where No> 0 is an integer with No4 being very ample. Letf:C â†'X be an algebraically non-degenerate holomorphic curve. Then, for every 0< e< 1, we have with M≥△ek ([5me-12k+1 (k+â–³)]+l)k. Here â–³ is the degree of the variety φdNoA(X), d= lcm(di,…,dq), and φdNoA:Xâ†'PM is the canonical morphism associated to the divisor dNoA.In chapter 3, we study a Montel-type criterion for normal families of holo-morphic mappings into complex projective space for moving hyperplanes in gen-eral position. This result improves a previous result with respect to the moving hyperplanes in pointwise general position. We have the following result:Let F be a family of holomorphic mappings of a domain D(?)C into PN(C), and let Ai(z)= (aio(z),…, aiN(z)) (i= 1,…,2N+1) be the moving hyper-planes in PN(C) located in general position on D. If each f in F omits Ai(z) (i=1,…,2N+1), then F is normal on D.In chapter 4, by constructing counterexamples, we prove that the truncation level in Cartan’s truncated second main theorem is at least n. Secondly, we establish a second main theorem for algebraically non-degenerate holomorphic mappings into Pn(C) intersecting a special kind of hypersufaces.Let k be a positive integer. If for any linearly non-degenerate holomorphic curves f:Câ†' Pn(C) and an arbitrary set of hyperplanes Hl,…, Hq (q≥n+2) in Pn(C) in general position, we have then k≥n.