Let $X\subset \mathbb P^n$ be a compact manifold over $\mathbb C$, $Y\subset X$ be a submanifold. We know there is a smooth decomposition $$T_X|_Y=T_Y \oplus N_{Y/X}$$
My question is:
is there some known condition to ensure this decomposition to be holomorphic?
In particular I want to know if $Y$ is a hyperplane section, will the decomposition be holomorphic?