Algebraic hyperbolicity of subvarieties of homogeneous varieties

We study the algebraic hyperbolicity of certain subvarieties of homogeneous varieties, building on the techniques introduced by Coskun-Riedl, Yeong and Mioranci. This generalizes earlier known results for hypersurfaces to higher codimensions. In particular, we observe that if $X=X_1\cap \cdots \cap X_k$ is a very general complete intersection of degree $d_j$ hypersurfaces $X_j$ in ${\mathbb{P}}^n$ with $k\le n-2$, then $X$ is algebraically hyperbolic if $\sum d_j\ge 2n-k$, and $X$ is not algebraically hyperbolic if $\sum d_j\le 2n-k-2$.