A structure theorem for holomorphic curves in \(Gr(3,{\mathbb{C}}^ 6)\) (Q1118836)

A holomorphic curve f in \(Gr(n,{\mathbb{C}}^{2n})\) is called generic, if the curvature of the canonical connection of \(f^*(S(n,{\mathbb{C}}^{2n}))\) has distinct eigenvalues, where \(S(n,{\mathbb{C}}^{2n})\) is the universal subbundle over \(Gr(n,{\mathbb{C}}^{2n})\). A holomorphic curve in \(Gr(n,{\mathbb{C}}^{2n})\) is completely split, if it is the orthogonal direct sum of n holomorphic curves in the projective plane. These two types of curves are both relatively simple. In this paper, we prove that a 1-nondegenerate holomorphic curve in \(Gr(3,{\mathbb{C}}^ 6)\) is either generic or completely split.