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We study an eigenvalue problem for the Laplacian on a compact K\"{a}hlermanifold. Considering the $k$-th eigenvalue $\lambda_{k}$ as a functional onthe space of K\"{a}hler metrics with fixed volume on a compact complexmanifold, we introduce the notion of $\lambda_{k}$-extremal K\"{a}hler metric.We deduce a condition for a K\"{a}hler metric to be $\lambda_{k}$-extremal. Asexamples, we consider product K\"{a}hler manifolds, compact isotropyirreducible homogeneous K\"{a}hler manifolds and flat complex tori.

Deformation of K\"{a}hler Metrics and an Eigenvalue Problem for the Laplacian on a Compact K\"{a}hler Manifold