First eigenvalue estimates on complete Kähler manifolds

Let $(M,{\omega }_g)$ be a complete Kähler manifold of complex dimension $n$. We prove that if the holomorphic sectional curvature satisfies $\mathrm{H}\mathrm{S}\mathrm{C}\ge 2$, then the first eigenvalue ${\lambda }_1$ of the Laplacian on $(M,{\omega }_g)$ satisfies $${\lambda }_1\ge \frac{320(n-1)+576}{81(n-1)+144}\mathrm{.}$$ This result is established through a new Bochner-Kodaira type identity specifically developed for holomorphic sectional curvature.