Second Main Theorems for Holomorphic Curves in the Projective Space with Slowly Moving Hypersurfaces

Let \(f : {\mathbb C} \rightarrow {{\mathbb P}^{N}}({\mathbb C})\) $$f:\mathbb{C}\to {\mathbb{P}}^N(\mathbb{C})$$ be a nonconstant holomorphic curve and \({\cal K}_{f}\) $${\mathcal{K}}_f$$ be the subfield of meromorphic function field on ℂ consisting of all meromorphic functions of slow growth with respect to f. Let D₁,…,D_q be slowly moving hypersurfaces defined by homogeneous polynomials in \({\cal K}_{f}[x_{0},\ldots,x_{N}]\) $${\mathcal{K}}_f[x_0,\dots ,x_N]$$ . In this paper, the second main theorems for nonconstant holomorphic curve f and slowly moving hypersurfaces D₁,…,D_q with respect to f are given. The motivation comes from the replacing hypersurfaces technique posed by Si Duc Quang and Nochka weights method.