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In this paper, we prove some fundamental theorems for holomorphic curves onangular domain intersecting a hypersurface, finite set of fixed hyperplanes ingeneral position and finite set of fixed hypersurfaces in general position oncomplex projective variety with the level of truncation. As applications of thesecond main theorems for an angle, we will discuss the uniqueness problem ofholomorphic curves in an angle instead of the whole complex plane. Detail, weestablish a result for uniqueness problem of holomorphic curve by inverse imageof a hypersurface. In my knowledge, this is the first result for uniquenessproblem of holomorphic curve by inverse image of hypersurface on angulardomain. On complex plane, we obtain a uniqueness result for holomorphic curves,it is improvement of some results before [5, 10] in this trend.

On Nevanlinna - Cartan theory for holomorphic curves with Tsuji characteristics