Stability of tangent bundles of complete intersections and effective restriction

For $n\ge 3$, let $M$ be an $(n+r)$-dimensional irreducible Hermitian symmetric space of compact type and let ${\mathcal{O}}_M(1)$ be the ample generator of $Pic(M)$. Let $Y=H_1\cap \cdots \cap H_r$ be a smooth complete intersection of dimension $n$ where $H_i\in \mathrm{\mid }{\mathcal{O}}_M(d_i)\mathrm{\mid }$ with $d_i\ge 2$. We prove a vanishing theorem for twisted holomorphic forms on $Y$. As an application, we show that the tangent bundle $T_Y$ of $Y$ is stable. Moreover, if $X$ is a smooth hypersurface of degree $d$ in $Y$ such that the restriction $Pic(Y)\to Pic(X)$ is surjective, we establish some effective results for $d$ to guarantee the stability of the restriction $T_Y{\mathrm{\mid }}_X$. In particular, if $Y$ is a general hypersurface in ${\mathbb{P}}^{n+1}$ and $X$ is general smooth divisor in $Y$, we show that $T_Y{\mathrm{\mid }}_X$ is stable except for some well-known examples. We also address the cases where the Picard group increases by restriction.