Weyl's Law for Singular Algebraic Varieties

Abstract

It is a classical result that the spectrum of the Laplacian on a compact Riemannian manifold forms a sequence going to positive infinity and satisfies an asymptotic growth rate known as Weyl's law determined by the volume and dimension of the manifold. Weyl's law motivated Kac's famous question, "Can one hear the shape of a drum?" which asks what geometric properties of a space can be determined by the spectrum of its Laplacian? I will show Weyl's law also holds for the non-singular locus of embedded, irreducible, singular projective algebraic varieties with the metric inherited from the Fubini-Study metric of complex projective space. This non-singular locus is a non-complete manifold with finite volume that comes from a very natural class of spaces which are extensively studied and used in many different disciplines of mathematics. Since the volume of a projective variety in the Fubini-Study metric is equal to its degree times the volume of the complex projective space of the same dimension, the result of this thesis shows the algebraic degree of a projective variety can be "heard" from its spectrum. The proof follows the heat kernel method of Minakshisundaram and Pleijel using heat kernel estimates of Li and Tian. Additionally, the eigenfunctions of the Laplacian on a singular variety will also be shown to satisfy a bound analogous to the known bound for the eigenfunctions of the Laplacian on a compact manifold.