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Weyl's Law for Singular Algebraic Varieties

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Weyl's Law for Singular Algebraic Varieties Weyl's Law for Singular Algebraic Varieties by John Kilgore A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2019 Doctoral Committee: Professor Lizhen Ji, Chair Professor Daniel Burns Professor Richard Canary Associate Professor Indika Rajapakse Professor Alejandro Uribe John Kilgore [email protected] ORCID iD: 0000-0003-0103-693X c John Kilgore 2019 ACKNOWLEDGEMENTS This research was partially supported by NSF EMSW21-RTG 1045119. I would also like to thank my advisor Lizhen Ji for his guidance throughout this project. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS :::::::::::::::::::::::::::::::::: ii LIST OF APPENDICES ::::::::::::::::::::::::::::::::::: iv ABSTRACT ::::::::::::::::::::::::::::::::::::::::::: v CHAPTER I. Introduction ....................................... 1 II. The Laplacian on singular varieties ......................... 4 III. The heat kernel approach to Weyl's Law ..................... 19 IV. Minakshisundaram-Pleijel asymptotic expansion . 25 R V. The asymptotics of N H(x; x; t)dx .......................... 39 VI. Eigenfunction bounds ................................. 41 VII. An application ...................................... 49 APPENDICES :::::::::::::::::::::::::::::::::::::::::: 51 BIBLIOGRAPHY :::::::::::::::::::::::::::::::::::::::: 72 iii LIST OF APPENDICES APPENDIX A. Tauberian theorem . 52 B. Sobolev spaces and inequalities . 66 C. Volume estimate . 68 D. The Euclidean Heat Kernel . 70 iv ABSTRACT It is a classical result that the spectrum of the Laplacian on a compact Rieman- nian 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 pro- jective 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. v CHAPTER I Introduction Weyl's law dates back to Weyl's 1911 paper \Ueber die asymptotische Verteilung der Eigenwerte" (About the Asymptotic Distribution of Eigenvalues), in which Weyl proved in [Wey] Theorem X i Area(Ω) lim = i!1 λi 4π 2 for any bounded domain Ω ⊂ R with smooth boundary, where λi is the i-th eigen- value of the Laplacian with Dirichlet boundary condition. Weyl also stated that his method can be generalized to higher dimensions. Minakshisundaram and Pleijel proved Weyl's law for any compact Riemannian manifold M n (see pages 244 and 255- 256 of [Min].) If M n has a boundary, then Weyl's law holds for both the Dirichlet and Neumann boundary conditions. In these cases, i ! V (M) lim = n i!1 n=2 (2π)n λi n where !n is the volume of the unit ball in R . This can be reformulated as X !nV (M) 1 ∼ c · λn=2 as λ ! 1; c = : M M (2π)n λi≤λ Here, cM is a constant which depends only on n and V (M), and f(λ) ∼ g(λ) as λ ! a f(λ) is the equivalence relation meaning limλ!a g(λ) = 1. Observe that if Spec(∆) := fλig 1 is known, then Weyl's law determines first the dimension of M from the exponenent, and then V (M) from cM . This suggests one can `hear' the volume of M. This is one of the first answers to the following question: What properties of a domain are determined by the spectrum of its Laplacian? Mark Kac [Kac] famously phrased this as \Can One Hear the Shape of a Drum?" The field of inverse problems in spectral geometry attempts to answer this question. In general, it is known that the spectrum cannot completely determine the Riemannian manifold. A counterexample was given by Milnor [Mil] in 1964 (this paper is one page.) Singular, projective algebraic varieties give examples of non-compact manifolds by considering the non-singular locus. In fact, with the Fubini-Study metric inheirited from an embedding in CPd, the non-singular locus is not even a complete Riemannian manifold. Still, this metric and the corresponding Laplacian share many properties with those on compact manifolds. The volume is finite, and the spectrum of the unique self-adjoint extension of Laplacian in L2(M) is discrete (see Theorems 4.1 and 5.3 of [LT].) The main result of this thesis is that Weyl's law holds for this non-singular locus with the Fubini-Study metric. Weyl's law is already known in the special case when the singular variety is a curve (complex dimension 1) due to Br¨uningand Lesch's paper [BL2]. Theorem I.1. (Weyl's Law) d n Let V ⊂ CP be an irreducible, projective variety of complex dimension 2 , and define N := V n Sing(V ) with the Fubini-Study metric from CPd. Define the counting X function N(λ) = 1. Then λi≤λ n ! V (N)λ 2 N(λ) ∼ n as λ ! 1; (2π)n 2 hence n n (2π) i (λi) 2 ∼ as i ! 1: !nV (N) As will be shown, the irreducible condition can be relaxed, so long as all of the irreducible components are the same dimension. The proof comes from comparing local asymptotic properties of the heat kernel to a global description. Additionally, the eigenfunctions of the Laplacian on a singular, projective alge- braic variety are bounded and satisfy the same bound in terms of the eigenvalues which the eigenfunctions of the Laplacian of a compact Riemannian manifold are known to satisfy (see equation (10.7) of [Li]). 1 2 Theorem I.2. Let fφigi=0 be an orthonormal basis of L (N) consisiting of eigen- fuctions of ∆ with ∆φi = λiφi and λi+1 ≥ λi. There exists a constant C depending only on N such that for all i > 0, n 4 jjφijj1 ≤ C · λi : 3 CHAPTER II The Laplacian on singular varieties d n Let V ⊂ CP be an 2 −(complex) dimensional, irreducible, singular subvariety. Let N denote the non-singular locus of V , so N is an n−(real) dimensional, smooth, oriented, connected submanifold of CPd. The Fubini-Study metric on CPd restricts to a Riemannian metric on N, making it a K¨ahlermanifold. This restricted metric is also referred to as the Bergmann metric. With this metric, it is easy to see the volume of N is finite, as will be shown using a key volume estimate in Appendix C. Let ∆ = (d + dt)2 = ddt + dtd = dtd = −div ◦ r be the Laplacian on functions. Here, dt := −∗d∗ is the formal adjoint of d, following the notation of Grieser and Lesch [GL]. In local coordinates (see page 5 of [Ch]), ∆ is given by 1 X p ∆u = −p @ (gjk g)@ u g j k j;k jk p where gjk is the metric in local coordinates, g is its inverse, and g is the square root of the determinant of this matrix. This definition makes ∆ a positive operator since t t t 1 h∆u; ui2 = h(d + d )u; (d + d )ui2 = jj(d + d )ujj2 ≥ 0; u 2 C0 (N): 4 Further, ∆ is an elliptic operator with principal symbol 1 σ 2 C1(N; S2(TN) ⊗ Hom( ; )); σ (ξ )(a) := − ∆(φ2 · u)(x) = −||ξ jj2a: ∆ R R ∆ x 2 x ∗ 1 where ξx 2 Tx N; a 2 R; and u 2 C (N) such that u(x) = a: Here, the norm on the cotangent bundle is the one induced from the Fubini-Study metric on the tangent bundle (see page 126 of [Voi] and page 183 of [BB].) Because the manifold N is not compact, a priori there are multiple closed ex- 1 1 2 tensions of ∆ : C0 (N) ! C0 (N) in L (N). This is discussed extensively in [GL]. Following [LT], we define the domain of ∆ to be the following: 2 2 2 ∗ 2 2 D(∆) := fu 2 C (N) \ L (N): d(u) 2 L (N; T N) and ∆(u) 2 L (N)g ⊂ L (N; R): This can be stated in terms of domains of d and dt by defining the domains D(d) = fu 2 C1(N) \ L2(N): d(u) 2 L2(N; T ∗N)g; D(dt) = fu 2 C1(N; T ∗N) \ L2(N; T ∗N): dt(u) 2 L2(N)g; so D(∆) = fu 2 C2(N) \ L2(N): u 2 D(d) and d(u) 2 D(dt)g: This definition makes ∆ an essentially self-adjoint operator as proven by Li and Tian in Theorem 4.1 of [LT]. In the process of Li and Tian's proof, they prove an important integration by parts result known as the L2 Stokes Theorem (L2ST ) which is a condition which is the topic of investigation of the paper [GL]. This theorem states the following. 5 Theorem II.1. (L2ST ) (Li-Tian) 1;2 1;2 W0 (N) = W (N) and t hdu; wi2 = hu; d wi2 for all u 2 W 1;2(N); w 2 W 1;2(N; T ∗N).
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