First eigenvalue estimates on complete Kähler manifolds

Let $ (M,ω_g) $ be a complete Kähler manifold of complex dimension $n$. We prove that if the holomorphic sectional curvature satisfies $\mathrm{HSC} \geq 2 $, then the first eigenvalue $λ_1$ of the Laplacian on $(M,ω_g)$ satisfies $$ λ_1 \geq \frac{320(n-1)+576}{81(n-1)+144}.$$ This result is established through a new Bochner-Kodaira type identity specifically developed for holomorphic sectional curvature.