In this paper,we prove a Second Main Theorem for holomorphic mappings in a disk whose image intersects some families of nonlinear hypersurfaces(totally geodesic hypersurfaces with respect to a meromorphic connection) ...In this paper,we prove a Second Main Theorem for holomorphic mappings in a disk whose image intersects some families of nonlinear hypersurfaces(totally geodesic hypersurfaces with respect to a meromorphic connection) in the complex projective space P^(k).This is a generalization of Cartan’s Second Main Theorem.As a consequence,we establish a uniqueness theorem for holomorphic mappings which intersect O(k^(3)) many totally geodesic hypersurfaces.展开更多
The main result of this article is an extension of the Second Main Theorem,of Halburd and Korhonen,for meromorphic functions of finite order.Their result replaces the counting function of the ramification divisor Nram...The main result of this article is an extension of the Second Main Theorem,of Halburd and Korhonen,for meromorphic functions of finite order.Their result replaces the counting function of the ramification divisor Nramf(r)in the classical Second Main Theorem by the counting function of a finite difference divisor Npair(r).In this article,the Second Main Theorem of Halburd and Korhonen is extended to the case of holomorphic maps into Pn of finite order.展开更多
In this paper, we establish a Second Main Theorem for an algebraically degenerate holomorphic curve f : C → Pn(C) intersecting hypersurfaces in general position. The related Diophantine problems are also considered.
In this paper,by using Seshadri constants for subschemes,the author establishes a second main theorem of Nevanlinna theory for holomorphic curves intersecting closed subschemes in(weak)subgeneral position.As an applic...In this paper,by using Seshadri constants for subschemes,the author establishes a second main theorem of Nevanlinna theory for holomorphic curves intersecting closed subschemes in(weak)subgeneral position.As an application of his second main theorem,he obtain a Brody hyperbolicity result for the complement of nef effective divisors.He also give the corresponding Schmidt’s subspace theorem and arithmetic hyperbolicity result in Diophantine approximation.展开更多
In this paper, a weak Cartan-type second theorem for holomorphic curve f ： C→P^n（C） intersecting hypersurfaces Dj, 1≤j≤q, in P^n（C） in general position with degree dj is given as follows： For every ε〉0, the...In this paper, a weak Cartan-type second theorem for holomorphic curve f ： C→P^n（C） intersecting hypersurfaces Dj, 1≤j≤q, in P^n（C） in general position with degree dj is given as follows： For every ε〉0, there exists a positive integer M such that ｜｜（q - （n ＋ 1） ε）Tf（r）≤∑j^q=1 1/dj Nf^M（r,Dj）＋o（Tf（r））, where ＂｜｜＂ means the estimate holds for all large r outside a set of finite Lebesgue measure.展开更多
In 2002, in the paper entitled ＂A subspace theorem approach to integral points on curves＂, Corvaja and Zannier started the program of studying integral points on algebraic varieties by using Schmidt＇s subspace theo...In 2002, in the paper entitled ＂A subspace theorem approach to integral points on curves＂, Corvaja and Zannier started the program of studying integral points on algebraic varieties by using Schmidt＇s subspace theorem in Diophantine approximation. Since then, the program has led a great progress in the study of Diophantine approximation. It is known that the counterpart of Schmidt＇s subspace in Nevanlinna theory is H. Cartan＇s Second Main Theorem. In recent years, the method of Corvaja and Zannier has been adapted by a number of authors and a big progress has been made in extending the Second Main Theorem to holomorphic mappings from C into arbitrary projective variety intersecting general divisors by using H. Cartan＇s original theorem. We call such method ＂a Cartan＇s Second Main Theorem approach＂. In this survey paper, we give a systematic study of such approach, as well as survey some recent important results in this direction including the recent work of the author with Paul Voja.展开更多
We give an improvement for the second main theorem of algebraically non-degenerate holomorphic curves into a complex projective variety V intersecting hypersurfaces in subgeneral position, obtained by Chen et al.(2012...We give an improvement for the second main theorem of algebraically non-degenerate holomorphic curves into a complex projective variety V intersecting hypersurfaces in subgeneral position, obtained by Chen et al.(2012). An explicit estimate for the truncation level is also obtained in the projective normal case.展开更多
Let f:C→P^(n)be a holomorphic curve of order zero.The authors establish a Jackson difference analogue of Cartan’s second main theorem for the Jackson q-Casorati determinant and introduce a truncated second main theo...Let f:C→P^(n)be a holomorphic curve of order zero.The authors establish a Jackson difference analogue of Cartan’s second main theorem for the Jackson q-Casorati determinant and introduce a truncated second main theorem of Jackson difference operator for holomorphic curves.In addition,a Jackson difference Mason’s theorem is proved by using a Jackson difference radical of a polynomial.Furthermore,they extend the Mason’s theorem for m+1 polynomials.Some examples are constructed to show that their results are accurate.展开更多
Riemann Hypothesis was posed by Riemann in early 50’s of the 19th century in his thesis titled “The Number of Primes less than a Given Number”. It is one of the unsolved “Supper” problems of mathematics. The Riem...Riemann Hypothesis was posed by Riemann in early 50’s of the 19th century in his thesis titled “The Number of Primes less than a Given Number”. It is one of the unsolved “Supper” problems of mathematics. The Riemann Hypothesis is closely related to the well-known Prime Number Theorem. The Riemann Hypothesis states that all the nontrivial zeros of the zeta-function lie on the “critical line” . In this paper, we use Nevanlinna’s Second Main Theorem in the value distribution theory, refute the Riemann Hypothesis. In reference [7], we have already given a proof of refute the Riemann Hypothesis. In this paper, we gave out the second proof, please read the reference.展开更多
The author proves that there are at most two meromorphic mappings of C^m into P^n(C)(n ≥ 2) sharing 2 n+ 2 hyperplanes in general position regardless of multiplicity,where all zeros with multiplicities more than cert...The author proves that there are at most two meromorphic mappings of C^m into P^n(C)(n ≥ 2) sharing 2 n+ 2 hyperplanes in general position regardless of multiplicity,where all zeros with multiplicities more than certain values do not need to be counted. He also shows that if three meromorphic mappings f^1, f^2, f^3 of Cminto P^n(C)(n ≥ 5) share2 n+1 hyperplanes in general position with truncated multiplicity, then the map f^1×f^2×f^3 is linearly degenerate.展开更多
基金partially supported by a graduate studentship of HKU,the RGC grant(1731115)the National Natural Science Foundation of China(11701382)partially supported by the RGC grant(1731115 and 17307420)。
文摘In this paper,we prove a Second Main Theorem for holomorphic mappings in a disk whose image intersects some families of nonlinear hypersurfaces(totally geodesic hypersurfaces with respect to a meromorphic connection) in the complex projective space P^(k).This is a generalization of Cartan’s Second Main Theorem.As a consequence,we establish a uniqueness theorem for holomorphic mappings which intersect O(k^(3)) many totally geodesic hypersurfaces.
基金supported by Natural Science Foundation of USA(Grant No.DMS0713348)Research Grant Council of Hong Kong(Grant No.HKU7053/06P)
文摘The main result of this article is an extension of the Second Main Theorem,of Halburd and Korhonen,for meromorphic functions of finite order.Their result replaces the counting function of the ramification divisor Nramf(r)in the classical Second Main Theorem by the counting function of a finite difference divisor Npair(r).In this article,the Second Main Theorem of Halburd and Korhonen is extended to the case of holomorphic maps into Pn of finite order.
基金supported by National Natural Science Foundation of China (Grant Nos. 11171255, 10901120)Doctoral Program Foundation of the Ministry of Education of China (Grant No.20090072110053)US National Security Agency (Grant Nos. H98230-09-1-0004, H98230-11-1-0201)
文摘In this paper, we establish a Second Main Theorem for an algebraically degenerate holomorphic curve f : C → Pn(C) intersecting hypersurfaces in general position. The related Diophantine problems are also considered.
基金supported by the National Natural Science Foundation of China(No.11801366)。
文摘In this paper,by using Seshadri constants for subschemes,the author establishes a second main theorem of Nevanlinna theory for holomorphic curves intersecting closed subschemes in(weak)subgeneral position.As an application of his second main theorem,he obtain a Brody hyperbolicity result for the complement of nef effective divisors.He also give the corresponding Schmidt’s subspace theorem and arithmetic hyperbolicity result in Diophantine approximation.
基金the National Natural Science Foundation of China (No.10571135)Doctoral Program Foundation of the Ministry of Education of China (No.20050240711)Foundation of Committee of Science and Technology of Shanghai(03JC14027)
文摘In this paper, a weak Cartan-type second theorem for holomorphic curve f ： C→P^n（C） intersecting hypersurfaces Dj, 1≤j≤q, in P^n（C） in general position with degree dj is given as follows： For every ε〉0, there exists a positive integer M such that ｜｜（q - （n ＋ 1） ε）Tf（r）≤∑j^q=1 1/dj Nf^M（r,Dj）＋o（Tf（r））, where ＂｜｜＂ means the estimate holds for all large r outside a set of finite Lebesgue measure.
基金supported in part by the Simons Foundation Mathematics and Physical Sciences-Collaboration Grants for Mathematicians
文摘In 2002, in the paper entitled ＂A subspace theorem approach to integral points on curves＂, Corvaja and Zannier started the program of studying integral points on algebraic varieties by using Schmidt＇s subspace theorem in Diophantine approximation. Since then, the program has led a great progress in the study of Diophantine approximation. It is known that the counterpart of Schmidt＇s subspace in Nevanlinna theory is H. Cartan＇s Second Main Theorem. In recent years, the method of Corvaja and Zannier has been adapted by a number of authors and a big progress has been made in extending the Second Main Theorem to holomorphic mappings from C into arbitrary projective variety intersecting general divisors by using H. Cartan＇s original theorem. We call such method ＂a Cartan＇s Second Main Theorem approach＂. In this survey paper, we give a systematic study of such approach, as well as survey some recent important results in this direction including the recent work of the author with Paul Voja.
基金supported by National Natural Science Foundation of China(Grant No.11371139)National Security Agency of the USA(Grant No.H98230-11-1-0201)
文摘We give an improvement for the second main theorem of algebraically non-degenerate holomorphic curves into a complex projective variety V intersecting hypersurfaces in subgeneral position, obtained by Chen et al.(2012). An explicit estimate for the truncation level is also obtained in the projective normal case.
基金supported by the National Natural Science Foundation of China(Nos.12071047,11871260)the Fundamental Research Funds for the Central Universities(No.500421126)
文摘Let f:C→P^(n)be a holomorphic curve of order zero.The authors establish a Jackson difference analogue of Cartan’s second main theorem for the Jackson q-Casorati determinant and introduce a truncated second main theorem of Jackson difference operator for holomorphic curves.In addition,a Jackson difference Mason’s theorem is proved by using a Jackson difference radical of a polynomial.Furthermore,they extend the Mason’s theorem for m+1 polynomials.Some examples are constructed to show that their results are accurate.
文摘Riemann Hypothesis was posed by Riemann in early 50’s of the 19th century in his thesis titled “The Number of Primes less than a Given Number”. It is one of the unsolved “Supper” problems of mathematics. The Riemann Hypothesis is closely related to the well-known Prime Number Theorem. The Riemann Hypothesis states that all the nontrivial zeros of the zeta-function lie on the “critical line” . In this paper, we use Nevanlinna’s Second Main Theorem in the value distribution theory, refute the Riemann Hypothesis. In reference [7], we have already given a proof of refute the Riemann Hypothesis. In this paper, we gave out the second proof, please read the reference.
基金supported by the Vietnam National Foundation for Science and Technology Development(No.101.04-2018.01)
文摘The author proves that there are at most two meromorphic mappings of C^m into P^n(C)(n ≥ 2) sharing 2 n+ 2 hyperplanes in general position regardless of multiplicity,where all zeros with multiplicities more than certain values do not need to be counted. He also shows that if three meromorphic mappings f^1, f^2, f^3 of Cminto P^n(C)(n ≥ 5) share2 n+1 hyperplanes in general position with truncated multiplicity, then the map f^1×f^2×f^3 is linearly degenerate.
