Weyl's Law for Singular Algebraic Varieties

[摘要] 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.

[发布机构] University of Michigan

[效力级别] geometric analysis

[关键词] geometry;geometric analysis;Mathematics;Science;Mathematics