Hearing the weights of weighted projective planes

Which properties of an orbifold can we “hear,” i.e., which topological and geometric properties of an orbifold are determined by its Laplace spectrum? We consider this question for a class of four-dimensional Kähler orbifolds: weighted projective planes \(M := {\mathbb{C}}P^2(N_1, N_2, N_3)\) with three isolated singularities. We show that the spectra of the Laplacian acting on 0- and 1-forms on M determine the weights N ₁, N ₂, and N ₃. The proof involves analysis of the heat invariants using several techniques, including localization in equivariant cohomology. We show that we can replace knowledge of the spectrum on 1-forms by knowledge of the Euler characteristic and obtain the same result. Finally, after determining the values of N ₁, N ₂, and N ₃, we can hear whether M is endowed with an extremal Kähler metric.