First eigenvalue estimates on complete K\"ahler manifolds

Let $(M,\omega_g)$ be a complete K\"ahler manifold of complex dimension $n$. We prove that if the holomorphic sectional curvature satisfies $\mathrm{HSC} \geq 2$, then the first eigenvalue $\lambda_1$ of the Laplacian on $(M,\omega_g)$ satisfies $$\lambda_1 \geq \frac{320(n-1)+576}{81(n-1)+144}.$$ This result is established through a new Bochner-Kodaira type identity specifically developed for holomorphic sectional curvature.