Hearing the weights of weighted projective planes

Abstract

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 := CP 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.

Geodesics on weighted projective spaces

“…In [1], Abreu, Dryden, Freitas, and Godinho raise the interesting question of whether the spectrum of the Laplace-Beltrami operator on X determines the weights N 1 , . .…”

“…Applying the heat invariants and using equivariant cohomology, they showed that this is indeed the case for d = 3, if one knows the spectrum of the Laplacian on 0-and 1-forms. Theorem 1.1 [1] Let M := CP 2 (N 1 , N 2 , N 3 ) be a four-dimensional weighted projective space with isolated singularities, equipped with a Kähler metric. Then the spectra of the Laplacian acting on 0-and 1-forms on M determine the weights N 1 , N 2 , N 3 .…”

Functional equations for orbifold wreath products

“…The proof is identical to [28, Proposition 6-1] and hence omitted. Recall that C G(Sn) (τ ) ∼ = Aut Pτ Γ, G for a Γ-G-principal bundle associated by Theorem 2.4(i) to the conjugacy class (τ ) of τ ; see Equation (2).…”

“…The most well-known examples are the weighted complex projective spaces, see e.g. [23,2,3,20,22]. Characteristic numbers of wreath symmetric products for non-global quotient orbifolds have been studied e.g.…”

“…In [12], this relationship was used to extend Tamanoi's generating functions for the extension by free abelian groups Γ = Z ℓ of the Euler and Euler-Satake characteristics of wreath symmetric product orbifolds to the case of closed effective orbifolds by using their presentation as the quotient of a closed manifold by the locally free action of a compact, connected Lie group [25,Theorem 2.19]. See also [31] for the generating function for the stringy orbifold Euler characteristic, corresponding to Γ = Z 2 .…”