Abstract
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.
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Communicated by: V. Guillemin (Cambridge).
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Abreu, M., Dryden, E.B., Freitas, P. et al. Hearing the weights of weighted projective planes. Ann Glob Anal Geom 33, 373–395 (2008). https://doi.org/10.1007/s10455-007-9092-6
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DOI: https://doi.org/10.1007/s10455-007-9092-6