The sharp first-eigenvalue conjecture for compact Kähler manifolds with HSC at least 2

Let $(M,\omega_g)$ be an $n$-dimensional compact Kähler manifold with holomorphic sectional curvature $\operatorname{HSC}\geq 2$, and let $\lambda_1$ denote the first eigenvalue of the Laplacian.

First-eigenvalue conjecture. One has

$$\lambda_1\geq 4.$$

Moreover, $\lambda_1=4$ if and only if $(M,\omega_g)$ is isometric to $(\mathbb{C}\mathbb{P}^1,\omega_{\mathrm{FS}})$.

The product examples in the source show that the optimal lower bound is independent of the complex dimension and lies between $320/81$ and $4$. The conjecture proposes that the upper endpoint is the sharp universal lower bound, with equality characterized by the Fubini–Study projective line. The source does not provide evidence that it has been resolved.