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HEARING THE WEIGHTS OF WEIGHTED PROJECTIVE MIGUEL ABREU, EMILY B. DRYDEN, PEDRO FREITAS, AND LEONOR GODINHO
 

Summary: HEARING THE WEIGHTS OF WEIGHTED PROJECTIVE
PLANES
MIGUEL ABREU, EMILY B. DRYDEN, PEDRO FREITAS, AND LEONOR GODINHO
Abstract. Which properties of an orbifold can we "hear," i.e., which topo-
logical and geometric properties of an orbifold are determined by its Laplace
spectrum? We consider this question for a class of four-dimensional K¨ahler
orbifolds: weighted projective planes M := CP 2(N1, N2, N3) with three iso-
lated singularities. We show that the spectra of the Laplacian acting on 0-
and 1-forms on M determine the weights N1, N2, and N3. 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 N1, N2, and N3, we can
hear whether M is endowed with an extremal K¨ahler metric.
1. Introduction
An orbifold consists of a Hausdorff topological space together with an atlas of
coordinate charts satisfying certain equivariance conditions (cf. §2.1). We will
be interested in compact K¨ahler orbifolds; in particular, we focus on weighted
projective planes, asking what the Laplace spectrum of such a space can tell us
about its properties.

  

Source: Abreu, Miguel - Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa

 

Collections: Mathematics