Narrow Results

In 1983, Nochka proved a conjecture of Cartan on defects of holomorphic curves in ℂPⁿ relative to a possibly degenerate set of hyperplanes. In this paper, we generalize Nochka's theorem to the case of curves in a complex projective variety intersecting hypersurfaces in subgeneral position. Further work will be needed to determine the optimal notion of subgeneral position under which this result can hold, and to lower the effective truncation level which we achieved.

The purpose of this paper is twofold. The first is to show a second main theorem with truncated counting function for holomorphic curves from punctured disks into semi-Abelian varieties. The second is to give an alternative proof of a Big Picard's theorem for algebraically non-degenerate holomorphic curves by Nevanlinna theory.

This paper derives several new results, as well as gives a partial survey of the recent development, on the value sharing for algebraic and holomorphic curves into ${\mathbb{P}}^n(ℂ)$.

In this paper, under the refinement of the subgeneral position, we give an improvement for the Second Main Theorem with truncated counting functions of algebraically non-degenerate holomorphic curves into algebraic varieties $V$ intersecting divisors in subgeneral position with some index.

In this paper, we establish a new second main theorem for meromorphic mappings of ${ℂ}^m$ into ${\mathbb{P}}^n(ℂ)$ and moving hypersurfaces with truncated counting functions in the case, where the meromorphic mappings may be algebraically degenerate. A version of the second main theorem with weighted counting functions is also given. Our results improve the recent results on this topic. As an application, an algebraic dependence theorem for meromorphic mappings sharing moving hypersurfaces is given.

In this paper, we first establish a degeneracy second main theorem for algebraic curves from compact complex Riemann surfaces into projective varieties intersecting hypersurfaces in subgeneral position. We then use it to study the ramified values for the Gauss map of the complete (regular) minimal surfaces in ${ℝ}^m$ with finite total curvature in the case where the Gauss map may be algebraic degenerate. Our results generalize and improve the previous results in the field.

In this paper, we study a second main theorem for holomorphic curves from finite ramified coverings of the complex line to complex projective varieties intersecting hypersurfaces in subgeneral position.

In this paper, we incorporate the beta constants into the defect relation and deduce a non-integrated defect relation for meromorphic maps from a Kähler manifold whose universal covering is a ball into a projective variety intersecting general divisors properly.

In this paper, we establish a Schmidt’s subspace theorem for moving hypersurfaces in weakly subgeneral position. Our result generalizes the previous results on Schmidt’s theorem for the case of moving hypersurfaces.

In this paper, we show some Second Main Theorems for zero-order meromorphic mappings intersecting slowly moving targets in ${\mathbb{P}}^m(ℂ)$ by considering their $p$-Casorati determinant. Our results are $p$-difference analogues of Cartan’s Second Main Theorem for moving targets. As an application, we give an unicity theorem for meromorphic mappings of ${ℂ}^m$ into ${\mathbb{P}}^m(ℂ)$ under the growth condition “order $=0$”.

In this paper, we show some $q$-difference analogues of the second main theorems for algebraically nondegenerate meromorphic mappings over the field ${\varphi }_q^0$ of zero-order meromorphic functions in ${ℂ}^m$ satisfying $f(qz)=f(z)$ intersecting hypersurfaces, located in subgeneral position in ${\mathbb{P}}^n(ℂ)$, where $q=(q_1,\dots ,q_m)\in {ℂ}^m$ and $q_j$ may be different. As an application, we give some unicity theorems for meromorphic mappings of ${ℂ}^m$ into ${\mathbb{P}}^n(ℂ)$ under the growth condition “order $=0$”, which are analogous to Picard’s theorems.