UNIVERSITY OF CALIFORNIA, IRVINE. On the Asymptotic Expansion of the Bergman Kernel DISSERTATION

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1 UNIVERSITY OF CALIFORNIA, IRVINE On the Asymptotic Expansion of the Bergman Kernel DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in athematics by Shoo Seto Dissertation Committee: Professor Zhiqin Lu, Chair Professor Song-Ying Li Professor Jeffrey Streets 2015

2 c 2015 Shoo Seto

3 DEDICATION To all my family and friends. ii

4 TABLE OF CONTENTS Page ACKNOWLEDGENTS CURRICULU VITAE ABSTRACT OF THE DISSERTATION v vi vii 1 Introduction 1 2 Preliminaries Kähler Geometry Kähler anifolds Calculus on vector bundles Line bundles Curvature First Chern Class Example: Complex Projective Space Kodaira Embedding Polarized Kähler anifolds The Bergman ernel The Bergman Kernel Bergman ernel on manifolds Bergman ernel on the complex projective space Asymptotic expansion Pea Sections Reduction of the Problem Computation up to first order Near-Diagonal Expansion Bargmann-Foc odel Local Construction Local Kernel Estimates of local ernel Local to Global Higher Order Convergence iii

5 5 Off-Diagonal Preliminary Discussion Comparison between Bergman and Green ernel Heat ernel for L section Proof of Theorem Perturbation of the Green Operator Resolving the singularity Estimates Appendix Useful Formulas Bound of normalizing constant for pea section Weitzenböc Formula for L N -valued (0, 1-forms estimate Hodge Theory and Kähler identities Volume Comparison Bibliography 89 iv

6 ACKNOWLEDGENTS First I would lie to than my advisor, Professor Zhiqin Lu, for his guidance, wisdom and patience throughout my graduate years. His constant encouragement and perseverance carried me through and none of this would have been possible without his support. I would also lie to than Professors Jeff Streets and Hamid Hezari for allowing me to join in on their projects. I learnt a great deal from them and am very grateful for their friendliness and willingness to teach me mathematics. I am also very grateful to have been able to tae many courses with Professor Song-Ying Li, who taught my core analysis classes as well as a topics course in several complex variables. I was very lucy to have been enrolled in his courses my first years at UCI. Thans to the UCI athematics Department and the geometry group for their generous support. Finally, thans to all my friends for all of your support throughout the years. I would lie to especially than Hang Xu and Casey Kelleher, for all the math, coffee, fun and adventure that I don t intend on stopping with you two. v

7 CURRICULU VITAE Shoo Seto EDUCATION Doctor of Philosophy in athematics 2015 University of California, Irvine Irvine, California aster of Science in athematics 2011 University of California, Irvine Irvine, California Bachelor of Arts in athematics 2009 University of California, Bereley Bereley, California RESEARCH EXPERIENCE Graduate Research Assistant University of California, Irvine Irvine, California TEACHING EXPERIENCE Teaching Assistant University of California, Irvine Irvine, California vi

8 ABSTRACT OF THE DISSERTATION On the Asymptotic Expansion of the Bergman Kernel By Shoo Seto Doctor of Philosophy in athematics University of California, Irvine, 2015 Professor Zhiqin Lu, Chair Let (L, h (, ω be a polarized Kähler manifold. We define the Bergman ernel for H 0 (, L, holomorphic sections of the high tensor powers of the line bundle L. In this thesis, we will study the asymptotic expansion of the Bergman ernel. We will consider the on-diagonal, near-diagonal and far off-diagonal, using L 2 estimates to show the existence of the asymptotic expansion and computation of the coefficients for the on and near-diagonal case, and a heat ernel approach to show the exponential decay of the off-diagonal of the Bergman ernel for noncompact manifolds assuming only a lower bound on Ricci curvature and C 2 regularity of the metric. vii

9 Chapter 1 Introduction Bacground and motivation Let (L, h (, ω be a positive Hermitian line bundle over a complex manifold. Let L be the th tensor power of L. An active field of research in complex geometry is to analyze the geometry of the manifold when. A classical result in this direction is the celebrated Kodaira embedding theorem which states that such manifolds admitting positive line bundles can be embedded into a projective space of sufficiently high dimension. The main tools in this analysis are the holomorphic sections H 0 (, L, which can be used to construct many objects in geometry, one of them being the Bergman ernel. The Bergman ernel, as defined classically on pseudoconvex domains Ω C n, is the holomorphic integral ernel of the projection operator from square integrable to holomorphic square integrable functions. Its analogue on complex manifolds can be obtained by replacing holomorphic functions and the L 2 inner product of functions with an L 2 inner product induced from the Hermitian metric. In a local neighborhood, we can view holomorphic sections as local holomorphic functions, an observation we use in Chapter 4. 1

10 An explicit formula for the Bergman ernel, except for certain cases (c.f , Lemma 4.1.1, is not possible. However by the wors of Zelditch [33] and independently by Catlin [5], a complete asymptotic expansion of the Bergman ernel on the diagonal was given by using a result by Boutet de onvel-sjöstrand on the asymptotics of the Szegö ernel. The coefficients of the asymptotic expansion carry geometric information as demonstrated by Lu in [19] in which he explicitly computed the first four coefficients. In particular, the second coefficient is half the scalar curvature of the manifold. This fact was used by Donaldson [11] demonstrating the stability of polarized manifolds with constant scalar curvature Kähler metrics. The method employed by Lu in [19] uses certain holomorphic sections called pea sections constructed by Tian in [31]. These are sections which in a small neighborhood can be represented by monomials and by Hörmander s -estimate, extend to a global section with sufficient decay property. Other methods to the analysis of the Bergman ernel include a heat ernel method by Dai, Liu, and a [9], where they obtained the full off-diagonal asymptotic expansion and Agmon-type estimates. Their results hold in a more general setting of the Bergman ernel of the spin c Dirac operator associated to a positive line bundle on a compact symplectic manifold. Another approach, done by Berman, Berndtsson, and Sjöstrand in [3], involves using microlocal analysis techniques inspired by the calculus of pseudodifferential operators and Fourier integral operators with complex phase. In this thesis, we consider the diagonal and the near-diagonal expansion, and the off-diagonal Agmon-type decay estimates. In chapter 2, we begin by reviewing the fundamentals of Kähler geometry and introduce the polarized setting that we will use. We introduce the primary object of this thesis, the Bergman ernel in Chapter 3, and discuss the on-diagonal behavior of the Bergman ernel. We give a computation of the coefficients up to the first order using methods of Tian [31] and Lu [19]. In Chapter 4, we give an elementary proof of the expansion based on a joint wor by the author with Hezari, Kelleher, and Xu [14]. In Chapter 5, we prove an exponential decay 2

11 of the Bergman ernel on the off-diagonal using a perturbation of the operator approach as seen in the wors of [29]. 3

12 Chapter 2 Preliminaries 2.1 Kähler Geometry In this chapter, we review the basics of Kähler geometry and set up notations and conventions which will be used. We focus on the elements of Kähler geometry which will be pertinent to our analysis of the Bergman ernel for positive line bundles. We refer the reader to [30] for further detail Kähler anifolds Let (, J be a complex manifold of complex dimension n, where J End(T is the complex structure. A Hermitian metric on is a Riemannian metric g such that for v, w T, we have g(v, w = g(jv, Jw. Under a holomorphic local coordinate system (z i n i=1, g can be written as g := g( i, j dz i dz j := g ij dz i dz j, 4

13 and (g ij can be viewed as an n n Hermitian matrix. Here and throughout, we will use the Einstein summation convention for repeated indices. Next define the Kähler form (or fundamental 2-form ω by ω(v, w := 1 g(jv, w, v, w T, 2π which in local coordinates is given by ω = 1 2π g ijdz i dz j. We say a metric is Kähler if dω = 0, and call a manifold equipped with such a metric a Kähler manifold. The condition dω = 0 in local coordinates is given by g ij = i g j. An immediate consequence of the symmetries of the metric is the existence of a holomorphic normal coordinate system. Lemma (Holomorphic normal coordinates. At each point x 0 on a Kähler manifold (, g, there exists a holomorphic coordinate chart (U, (z i n i=1 centered at x 0 such that g ij (x 0 = δ ij and g ij (x 0 = 0, for i, j, {1,..., n}. Proof. Let x 0. We first choose coordinates (w i n i=1 centered at x 0 such that g ij (0 = δ ij, which can be found since (g ij (0 is Hermitian symmetric, and expand the metric at 0 to obtain: ω = 1 ( δ ij + l g ij (0w l + l g ij (0w l + O( w 2 dw i dw j. 5

14 We use the following holomorphic change of variables Γ l ij := g lq j g iq, w l := z l 1 2 Γl ij(0z i z j. Then dw l = dz l Γ l ij(0z i dz j, so that ω = ( 1 δ ij + l g ij (0z l + l g ij (0z l Γ j il (0zl Γ i jl (0zl + O( z 2 dz i dz j = 1 ( δ ij + O( z 2 dz i dz j. A crucial point is that for Riemannian metrics normal coordinates can be constructed via the exponential map, however the coordinates may not be holomorphic, hence we require the Kähler condition to construct such holomorphic coordinates. The strategy of the proof of Lemma can be extended further yielding the result below Lemma (K-coordinate system. With the same notation and hypotheses as Lemma 2.1.1, for any choice of p i Z + with p = n i=1 p i, there exists a holomorphic coordinate chart (U, (z i centered at x 0 such that g ij (x 0 = δ ij and p g ij ( z 1 p 1 ( zn pn (x 0 = 0, for i, j {1,..., n}. In other words, we can find a coordinate system where the derivative of the metric vanishes for purely holomorphic and purely antiholomorphic directions. These coordinates play an 6

15 important role in the computation of the coefficients of the Bergman ernel asymptotic expansion. The Kähler condition also implies additional basic results in Kähler geometry, nown as the -lemmas. Lemma (Local -lemma. Let (, ω be a Kähler manifold. For any x 0, there is a neighborhood U and a real function ϕ(z, z, called the local Kähler potential, such that ω = 1 ϕ. It can be derived as a special case of the following global version, Lemma (Global -lemma. Let (, ω be a Kähler manifold and [ω] the cohomology class of (p, q forms. Given ω, ω [ω], there exists ϕ H p 1,q 1 such that ω ω = ϕ. Proof. Since ω, ω [ω] there exists α Λ p+q 1 such that dα = ω ω Decomposing into types α = α p,q 1 + α p 1,q we have α p,q 1 = 0, α p 1,q = 0. By the Hodge decomposition, α p 1,q = h + f 7

16 where h is -harmonic. Furthermore, on Kähler manifolds, h = h = 0 so that h = 0. Applying the same argument, α p,q 1 = h + f, with h = 0, we have ω ω = dα = ( + ( h + f + h + f = (f f Letting ϕ = f f, the result follows. Definition (Holomorphic Vector Bundle. Let be a complex manifold. A holomorphic vector bundle of ran over is a complex manifold E together with a holomorphic projection map π : E satisfying: 1. For any p, the fiber E p := π 1 (p is a complex vector space of complex dimension. 2. There exists an open covering {U i } of and homeomorphic maps ϕ i : π 1 (U i U i C commuting with the projection to U i such that for each p U i, the restriction of ϕ i π 1 (p {p} C is an isomorphism of the complex vector spaces. 3. On the overlap p U i U j, the induced transition maps ϕ ij (x 0 := ϕ i ϕ 1 j (x 0, : C C are holomorphic C-linear maps. The primary object of study on vector bundles are sections, which can be thought of as generalization of functions. 8

17 Definition (Sections. Let E be a complex vector bundle over. A section of E is a map s : E such that (π s(x 0 = x 0 for all x 0. The ey idea is that it is a map taing values in E which maps p to objects in the fiber E p. The space of smooth sections is denoted at Γ(, E. In particular, if E is holomorphic, then we denote H 0 (, E as the space of holomorphic sections. Restricting the vector bundle E to an open set U preserves the vector bundle structure and we can define the space of sections on the restricted bundle, denoted as Γ(U, E := Γ(U, E U. Definition (Local frame. The set of sections {e i }, with each e i Γ(U, E, is called a local frame if for each x 0 the collection {e i (x 0 } forms a basis for the fiber E x0 as a vector space Calculus on vector bundles We now introduce the notion of a Hermitian metric on a vector bundle E. Definition Let E be a holomorphic vector bundle. A Hermitian metric h of E is an assignment of a Hermitian inner product h(, on each E x0, which varies smoothly with respect to x 0. To be explicit, let {e i } be a local frame. Then h ij (x 0 := h(e i, e j x0 where the matrix (h ij (x 0 is a Hermitian n n matrix for each x 0. Connections Here we introduce the notion of a connection, which is a way to differentiate sections by connecting them between different fibers. 9

18 Definition (Connection. Let E be a complex vector bundle. A connection D on E is a C-linear operator D : Γ(, E Γ(, Λ 1 (E satisfying D(fs = df s + fds for f C ( and s Γ(, E. The connection above induces a connection on the bundle Λ p (E given by D : Γ(, Λ p (E Γ(, Λ p+1 (E by D(v s = dv s + ( 1 p v Ds, where v Γ(, Λ p (E. Let (E, h be a holomorphic vector bundle over a Kähler manifold. We will focus mainly on a particular type of connection. Definition Let D be a connection satisfying the following conditions: Dh = 0 DJ = 0 (metric compatibility, (complex structure compatibility. Such a connection is called a Hermitian connection Line bundles The following are two important examples of line bundles. 10

19 Example (Canonical line bundle. Let be a complex manifold of dimension n. The nth exterior power of the holomorphic cotangent bundle forms a line bundle called the canonical bundle of K := Λ n T (1,0 ( = det(t (1,0 (. Let {U α, (z i α } be a holomorphic atlas of. A local holomorphic frame on U α is given by {(dz 1... dz n α } and the transition function is given by (dz 1... dz n α = det ( z i α z j β (dz 1 dz n β on U α U β. If g is a Hermitian metric on, then it induces a metrics a natural Hermitian metric on K is given by h = (det g 1 Its dual, denoted K 1, is called the anti-canonical bundle. Example (Line bundles over CP n. On CP n (defined in 2.1.6, we can construct a line bundle, called the tautological line bundle or O( 1, by assigning to each point the line that the point represents and viewing it as a line subbundle of the trivial bundle CP n C n+1. To be precise, let [p] := [p 0 : : p n ]. Then at [p], attach the line in C n+1 defined by the vector p 0,..., p n. Let U i = {[p] p i 0}. Since p0 p 0,, p n = p i,..., p n p i p i we can use p i as a local trivialization. Then the transition function g ij must satisfy g ij = p i p j. 11

20 Let s H 0 (CP n, O( 1 be a global holomorphic section. Since O( 1 is a subbundle of the trivial bundle, under a global non-vanishing frame, we can view s as a holomorphic map s : CP n C n+1. The components of s are then holomorphic functions on a compact complex manifold, hence they must be constant. According to the transition map, on U α U β = it must satisfy s α = g αβ s β, hence s α p α = s β p β, which for constants is only satisfied when s = 0. The dual of O( 1, sometimes called the hyperplane bundle, is denoted O(1 and its tensor powers O(m := O(1 m. The transition functions for O(m are obtained by inversions from the tautological bundle, i.e. g m ij = ( pj p i m. The global sections of O(m can be thought of as homogeneous polynomials in p 0,..., p n of degree m. By looing at the transition functions, it can also be shown that K CP n = O( n Curvature In this subsection, we define and establish the curvature and sign conventions we will use. Let be the Levi-Civita connection on a Kähler manifold (, ω. Definition (Curvature Tensor. The curvature tensor R is defined as R(v 1, v 2, v 3, v 4 = g( v1 v2 v 3 v2 v1 v 3 [v1,v 2 ]v 3, v 4 12

21 for v i T On Kähler manifolds, the local coordinate formula for the curvature tensor is greatly simplified due to the compatibility with the complex structure. Let v, w T 1,0. Then v w T 1,0, v w = 0 = v w. For local coordinates (z i n i=1 with coordinate holomorphic vector fields { i } n i=1, the curvature tensor is determined completely by terms of the form R ijl := R( i, j,, l = g( j i, l = g ml j Γ m i = i j g l + g pq ( j g pl ( g iq. Definition (Ricci and scalar curvature. The Ricci curvature Ric is the trace of the Riemann curvature tensor and the scalar curvature ρ is the trace of the Ricci curvature, i.e. Ric ij = g l R ijl, ρ = g ij R ij. (Ricci (scalar A useful identity for the Ricci curvature which holds for Kähler manifolds is the following Lemma Ric ij = i j (log det g. (2.1 13

22 Proof. Follows from the two identities, j det g = g l ( j g l det g i g l = g pl g q ( i g pq First Chern Class We will use the identity (2.1 to define an important cohomology class associated to a manifold called the (first Chern class. First define the Ricci form to be Ric(ω = 1 2π 1 Ric ij dz i dz j = log det g 2π Let h be another Kähler metric on. Then det h det g difference of the Ricci forms is given by be a globally defined function and the Ric(h Ric(g = 1 log det h det g Hence [Ric(g] H 2 (, R defines a cohomology class independent of the metric and we define the first Chern class of Definition (First Chern class of. c 1 ( = [Ric(g] ore generally, we can define the first Chern class of a Hermitian line bundle (L, h as the cohomology class of form of the line bundle. Let s be a local non-vanishing holomorphic 14

23 section of L. As in the case for Ricci curvature, the curvature form is locally defined by F (h = 1 log h(s where h(s = s, s h. Given any other Hermitian metric, it can be written as e f h for a globally defined function f and so F (e f h F (h = 1 f thus we can define the first Chern class of the line bundle L to be the cohomology class c 1 (L = 1 [F (h] 2π By the -lemma, we have that every real (1, 1-form in c 1 (L is the curvature of some Hermitian metric on L. With this viewpoint, we see that the first Chern class of is the first Chern class of the anti-canonical bundle of, c 1 ( = c 1 (K Example: Complex Projective Space The model case of a Kähler manifold which we will consider is the complex projective space, CP n. It is constructed from C n+1 {0}/, where the equivalence relation is given by p q if and only if for p = (p 0,, p n, q = (q 0,, q n, there is a nonzero complex number λ C such that p = λq. A local coordinate system is given by the following. Let U i = {[p 0 : : p n ] CP n p i 0} 15

24 for i = 0,, n. Then the map z i given by z i ([p] = ( p0,, p i,, p n p i p i p i where p i p i is removed, gives a local holomorphic coordinate chart. The holomorphic structure is the one induced from C n+1 and the transition maps are easily seen to be holomorphic. Fubini-Study etric A Kähler metric that is commonly equipped to CP n is the Fubini-Study metric. Definition (Fubini-Study etric. Let [p 0 : : p n ] be homogeneous coordinates on CP n. The Fubini-Study metric is defined as the 2-form ω F S := 1 n 2π log( p i 2. i=0 On a neighborhood, say U 0 = {p 0 0}, it can be written in local coordinates z i = p i p 0 ω F S = 1 n 2π log(1 + z i 2. i=1 The metric is a homogeneous metric, that is, for any A U(n + 1 Aut(CP n, the holomorphic automorphism group, acts transitively and leaves the form ω F S invariant. As such, to chec that the form ω F S is indeed a metric, we need to only chec that it is positive definite at one point, say p = [1 : 0 : : 0]. Directly computing, we have ω F S (p = = ( 1 δij (1 + z 2 z j z i dz i dz j 2π (1 + z 2 2 p 1 2π dzi dz i > 0 (2.2 16

25 Curvature of Fubini-Study metric As an example, we will compute the curvature terms of the Fubini-Study metric. Using normal coordinates at the point p = [1 : 0 : : 0], the curvature tensor is given by R ijl = l g ij. Using g ij given in (2.2, computing while dropping the terms which will evaluate to 0, ( δij l 1 + z z j z i p 2 (1 + z 2 2 = ( δij z l (1 + z z j δ il (1 + z 2 2 p = δ ij δ l + δ j δ il Hence the curvature tensor is given by R ijl = g ij g l + g j g il and tracing gives the Ricci curvature Ric ij = (n + 1g ij and the scalar curvature ρ = n(n + 1 The Fubini-Study metric on CP n is in fact a Kähler-Einstein metric with positive first Chern class. 17

26 2.2 Kodaira Embedding Definition (Ample line bundle. A line bundle L over is very ample, if for suitable sections s 0,..., s N of L, the Kodaira map ϕ : CP N p [s 0 (p :... : s N (p] (2.3 defines an embedding of into CP N. A line bundle L is ample if L is very ample for sufficiently large Z +. Theorem (Kodaira Embedding Theorem. Let L be a line bundle over a compact complex manifold. Then L is ample if and only if c 1 (L > 0, i.e. L is positive. Proof for ample implies positive. By the embedding, L can be identified with the restriction of the O(1 bundle of some complex projective space. In particular, L is positive. Let h be the positively curved Hermitian metric of L. Then L 1 is a positively curved Hermitian metric on L. In proving positive implies ample, we first mae the following lemma Lemma Let L be a positive holomorphic line bundle. If for every x, y with x y and every v C n, there exists elements S, T H 0 (, L N such that S(y 0, S(x = 0, T (x = 0, and dt (x = v, then L is ample. Proof. Let s 0,..., s N be a basis for H 0 (, L. Since at each point y, we can find a non-vanishing section s(y 0, we see that the at least one of s i (y 0, hence the Kodaira map ϕ (2.6 defined by the basis is well defined. Suppose ϕ(x = ϕ(y. Then there exists 18

27 nonzero complex number λ C such that (s 0 (y,, s N (y = λ(s 0 (x,..., s N (x but this would contradict the existence of a section such that S(x = 0 and S(y 0, hence ϕ is injective. It remains to prove that dϕ is injective, hence we want to show it has maximal ran. For any x, let T 1,..., T n be sections of L such that T j (x = 0 for 1 j n. Let e N L be a local frame at x. Then each T j has a local representative T j = t j e N L, where t j H 0 (U. Also let T 0 = e N L. By assumption, we can assign for each j, dt j = v j, where {v j } is the standard basis vector on C n. Then define the immersion x [T 0 : T 1 : : T n : 0 : : 0]. Since the T i = a ij S j, we have the ran of ϕ must be n as well. Theorem (L 2 estimate construction. Let p 1,, p s and K 1,, K s Z +. Then for sufficiently large, there is a holomorphic section S H 0 (, L such that at each p i, S has the prescribed derivatives up to order K i. Proof. Let U i be a coordinate neighborhood of p i, with coordinates (z 1,..., z n. By prescribed derivatives, we mean to assign the values for a holomorphic function f i at p i, K i f i z 1 1 z n n (p i Let η i be a smooth cut-off function whose supports are within U i and is 1 in a neighborhood of p i. We then define a global smooth section of L m by w = i η i f i. 19

28 Since we need a global holomorphic section, we consider the equation f = w If the solution f exists, then w f will be holomorphic. To ensure that we have a section with the required conditions for the derivatives, that is, the vanishing order be at least K i +1 at each p i. Consider the Laplace equation ( + u = g = w (2.4 Taing on both sides, we have u = 0. Then 0 = ( u, u = u 2 So that u = 0. Letting u = f, we have f = w. To solve (2.4, we first use the Weitzenboc formula to get a lower bound estimate for the first eigenvalue. Let e η h be a Hermitian metric on L. For any L -valued (0, 1-form, a α dz α, (a α dz α = β β (a α dz α + (Ric αγ a α dz γ + m Ric(h αγ a αγ dz γ + ( α γ ηa α dz γ 20

29 Let ψ be an eigensection corresponding to the lowest eigenvalue, λ 1. Then λ 1 ψ 2 e η dv g = (ψ, ψ e η dv g ψ 2 e η dv g + (C 1 C 2 ψ 2 e η dv g for some C 1, C 2 > 0. Hence λ 1 C 1 C 2. By the first eigenvalue estimate u λ 1 u, hence w 2 L 2,η = u 2 L 2,η λ2 1 u 2 L 2,η (C 1 C 2 u 2 L 2,η Thus we have f 2 e η dv g = u 2 e η dv g = u L 2,η w L 2,η w 2 L 2,η C 1 C 2 u, u e η dv g By shrining the neighborhoods if necessary, we assume that the U i are all disjoint. Then in a small neighborhood of p i, (w = i ( η i f i + η i ( f i = 0. Hence w 2 L 2,η <. Now we choose an appropriate cut-off function η for the vanishing order of f. Consider η to be the function such that at each point near p i, η(x = λ log(r i where r i is the distance d(x, p i. The value of λ will control the vanishing order of the section. 21

30 For the integral on the neighborhood U i, we have f i 2 e η f i = U i Ui 2 r λ 1 0 f i 2 r 2n 1 λ < Since r p < when p < 1. Let f i 2 r 2q. Then we require λ 2q 2n < 1, so choosing λ sufficiently large, we can push the vanishing order of f i to be large. 2.3 Polarized Kähler anifolds Let L be a Hermitian line bundle over with Hermitian metric h. We say that the line bundle L is positive if the curvature Ric(h is positive definite. We define a Kähler form ω g by the curvature form of L, i.e. for a fixed point p with local coordinates (z i n i=1 at p, 1 2 ω g = 2π z i z j log hdz i dz j = 1 2π g ijdz i dz j, where h is the local representation of the Hermitian metric h. We refer to this setting as polarized Kähler manifold with polarization L. For each Z +, the Hermitian metric h induces a Hermitian metric h on L := L } {{ L }. The Hermitian metric h further times induces an L 2 inner product on H 0 (, L, the space of holomorphic global sections of L m as follows: Choose an orthonormal basis {S α} of H 0 (, L. Then define the inner product Sα, Sβ L 2 := (Sα, Sβ ω n h n! (2.5 22

31 We use the following notations L 2 (, L = {f Γ(, L f L 2 := f 2 h dv g < } H 0 (, L = {f L 2 f holomorphic section of L } By the Kodaira embedding theorem, for sufficiently high tensor power, the basis {S α} induces a holomorphic embedding of into CP N, N + 1 = dim H 0 (, L given by the mapping ϕ : CP N x [S 0 (x :... : S N(x] (2.6 Let g F S be the standard Fubini-Study metric on CP N, i.e., for homogeneous coordinate system [Z 0 : : Z N ] of CP N, ω F S = ( N 1 log Z i 2 2π i=0 Definition (Bergman metric. The 1 -multiple of g F S on CP N restricts to a polarized Kähler metric 1 ϕ g F S on, where ϕ is the Kodaira map defined above. The metric is called the Bergman metric with respect to L. The Bergman metric and polarized metrics are related by the following Theorem (Tian. With notation as above, ( 1 ϕ g F S g 1 = O C 2 ( Tian [31] originally proved the C 2 convergence with remainder O 1 and was improved 23

32 to C and O ( 1 by Ruan [26]. Zelditch [33] and Catlin [5] independently generalized the above theorem by giving the asymptotic expansion of the Bergman ernel using the Szegö ernel on the unit circle bundle of L over. 24

33 Chapter 3 The Bergman ernel In this chapter, we will introduce the Bergman ernel and and its expansion on the diagonal. The first section will define the Bergman ernel, the second section will introduce the asymptotic expansion and in the third we will give an analysis of it via Tian s pea sections [31]. 3.1 The Bergman Kernel In this section, we introduce the primary object of our study, the Bergman ernel. We will be concerned with the Bergman ernel on manifolds, which is analogous to the classical Bergman ernel defined on pseudoconvex domains Ω C n. Some literature on the classical theory can be found in [1], [15]. 25

34 3.1.1 Bergman ernel on manifolds Let (, ω be a compact polarized Kähler manifold with a positive line bundle (L, h such that Ric(h = ω. Let h α be the local representation of the Hermitian metric with respect to a local frame e L on some neighborhood U α, that is, h α = h(e L, e L. By the Hodge theorem, this space is finite-dimensional and a normal family argument shows that the space is a closed subspace of L 2 (, L m. We then consider the orthogonal projection P m : L 2 (, L m H 0 (, L m. Definition (Bergman ernel. The holomorphic integral ernel B of the projection is called the Bergman ernel, i.e., for f L 2 (, L, P (f(x = f(y, B (y, x L 2 If we choose an orthonormal basis {S α} of H 0 (, L, then the Bergman ernel is given by B (x, y = α S α(x S α(y Definition Let B (x, x Γ(L L be the Bergman ernel restricted to the diagonal. The Bergman function, sometimes called density function, B(x is given by the point-wise norm, i.e. B(x := B m (x, x h m = B m (x, x e ϕ(x, where B(x, x is the coefficient function of the Bergman ernel with respect to the frame e L e L. We have an extremal characterization of the Bergman ernel which we will be useful. 26

35 Lemma Let B(x be the Bergman function. Then B(x = sup s(x 2 h s L 2 =1 Proof. We have B(x = N S i (x 2 h S(x 2 h i=0 for any S(x 2 L 2 = 1, since we can choose an orthonormal basis {S α } with S 0 = S. For the converse inequality, let x and consider the subset Z x H 0 (, L of sections that vanish at x. Since B(x > 0, we now there exists a nonvanishing section S 0 (x 0. Then for any S H 0 (, L, consider the decomposition S(x = S(x λs 0 (x + λs 0 (x where λ is chosen so that S(x λs 0 (x = 0. We see that Z x has codimension one. Let S 0 be in the orthogonal complement of Z x and S 0 L 2 = 1 and extend to an orthonormal basis on H 0 (, L. Then each section orthogonal to S 0 vanishes at x so B(x = S 0 (x 2 h which gives us the result. We first give a rough upper bound of the Bergman ernel. Lemma (Uniform upper bound on Bergman function. Let B(x be the Bergman function for the Bergman ernel of H 0 (, L. Then there exists C dependent on, and independent of and x such that B(x C n. 27

36 Proof. We use the extremal characterization of the Bergman function (4.9. On the compact Kähler manifold, we fix a finite coordinate cover {U α } and also fix a coordinate (z i n i=1 in U α. For each U α, we have a local Kähler potential ϕ(z. We can assume that U = B(0, 2, and sup z B(0,2 D 2 ϕ(z C, i.e. the second derivatives are uniformly bounded. Since ϕ is plurisubharmonic, we can assume the volume form dv g = ( 1 2π ϕ n (z is equivalent to dv E (z in B(0, C 1 B(z 0, 1 B(z 0, 1 s(z 2 e ϕ(z dv g 1 e sup B(0,2 D2 ϕ C 1 = 1 e ϕ(z 0 C 2 s(z 2 e ϕ(z dv E B(z 0, 1 B(z 0, 1 s(z 2 e ϕ(z 0 ϕ z(z 0 (z z 0 ϕ z(z 0 (z z 0 dv E s(ze ϕz(z 0(z z 0 2 dv E. Since s(ze ϕz(z 0(z z 0 is holomorphic, by the mean-value inequality we have 1 e ϕ(z 0 C 2 B(z 0, 1 1 C 3 n e ϕ(z 0 s(z 0 2. s(ze ϕz(z 0(z z 0 2 dv E So we have e ϕ(z 0 s(z 0 2 C 3 n, where C 3 is uniform for any z 0 B(0, 1 and any s H 0 (, L. Taing the supremum over all such s and a standard finite cover argument yields the desired result. 28

37 3.1.2 Bergman ernel on the complex projective space As a concrete example, we will compute the Bergman ernel on the polarized manifold (CP n, O(1. Consider the open set U 0 = {Z 0 0} of CP n. The homogeneous coordinates [1 : z 1 : : z n ] give local coordinates (z 1,, z n for U 0. Since CP N is a symmetric space, we only need to consider the Bergman ernel at one point, say [1 : 0 :... : 0]. By considering the hyperplane line bundle O(1 with Hermitian metric h = 1 1+ z 2, we consider CP n with the Fubini-Study metric, which on U 0 is represented by ω = 1 2π log(1 + z 2. The tensor powers of the hyperplane line bundle is denoted O( = O(1 and sections of the O( bundle correspond to -th degree homogeneous polynomials. To compute the Bergman ernel, we need to find an orthonormal basis for H 0 (CP n, O(m. Let ρ := 1 + z 2 and dv 0 := n i=1 dzi dz i, where the product is the wedge product. We compute the volume form of the Fubini-Study metric ( n ( 1 dz ωg n i dz i = 2π ρ n ρ ρ ρ 2 = n!dv 0 n! z 2 dv 0 ρ n ρ ( n+1 n 1 n!dv0 = 2π ρ n+1 where we use the fact that ( ρ ρ = 0 for > 1. Next the Hermitian metric h on U 0 can be expressed locally as ( 1 ρ. Let el be a frame on H, write z P = z p 0 0 zn pn e L, P = (p 0, p 1,... p n Z n+1 such that P = p i =, we have 29

38 CPn (z P, z Q z P z Q = ( z i 2 dv +n+1 0 p 1 1 z q 1 1 zn pn z qn n = dv ρ +n+1 0 U 0 z ( n 1 where dv 0 = dz 1 dz 1 dz n dz n. 2π By using polar coordinates, we can see that the set {z P } is orthogonal under the L 2 norm. Hence we compute the norm of each monomial and normalize to form an orthonormal set. Proposition C n z 1 2p1 z n 2pn (1 + z z n 2 dv +n+1 0 = p 0!p 1! p n! (n +! Proof. Changing to polar coordinates, z i = r i e 1θ i, we obtain C n z 1 2p1 z n 2pn (1 + z z n 2 +n+1 dv 0 = r 2p rn 2pn+1 (1 + r rn 2 ( 2π dr (+n+1 0 dθ i n For convenience, we change variables s i = r 2 i to get 1 2 n 0 s p 1 1 s pn n... dv (s 0 (1 + s s n (+n+1 Now consider the integral 0 = ds 1... ds n... 0 (1 + t 1 s t n s n (n+p 0+1 p 0! (n + p 0!t 1 t n 30

39 Then inductively we see that p i ( t i p i ( p 0! = (n + p 0!t 1 t n ( 1 p i p 0!p i! (n + p 0!t 1 t p i+1 i t n On the other hand, we have p i ( t i p i = R n + ds 1... ds n (1 + t 1 s t n s n (n+p 0+1 ( 1 p i (n + p 0 + 1(n + p (n + p 0 + p i s p i i (1 + t 1 s t n s n (n+p 0+p i dv (s +1 ( So ( 1 p 0 (n + p 0 + 1(n + p (n + s p 1 1 s pn n dv (s R+ n (1 + s s n (n++1 ( p1 ( pn ds 1... ds n = t 1 t n (1 + t 1 s t n s n (n+p 0+1 = ( 1( p 0 (p 0!(p 1! (p n! (n + p 0! 0 0 (ti =0 and the result follows. Using the above, we see that the set S P ( + n! = p 0! p n! zp e L, z P = z p 0 0 zn pn, P = (p 0,..., p n, P = p p n = forms an orthonormal basis for H 0 (CP n, O(. The Bergman ernel is then given by B (x, y = P = S P (x S P (y = P = ( + n! x P y P e L e L P! 31

40 Computing the Bergman function, we have B(x = P = ( + n! x P 2 P! Which at the origin is (+n!!. Expanding out the terms in term of the power of, we have ( + n!! ( = n n(n O( 1 2 where we see that the second coefficient is indeed 1/2 of the scalar curvature of CP n. 3.2 Asymptotic expansion In this section, we discuss the asymptotic expansion of the Bergman ernel. We begin with the following theorem Theorem (Zelditch [33], Catlin [5]. There is a complete asymptotic expansion: N α=0 ( Sα(x 2 h = n a 0 (x + a 1(x + a 2(x + 2 for smooth coefficients a j (x with a 0 = 1. ore precisely, for any m N S α(x 2 h a m j (xm n j C R,m n R α=0 j<r C m where C R,m depends on R, m and the manifold. The algorithm to compute the coefficients were given by Lu [19], who also determined the coefficients to be polynomials in the metric and covariant derivatives of the curvature, and computed the first 4 coefficients, and extended his computation with Shiffman to the near- 32

41 diagonal terms. For the near diagonal expansion, it is given in powers of 1. As a corollary of the asymptotic expansion, we give a quic proof of Tian s theorem, Theorem Proof of Theorem Let {S i } be an orthonormal basis of H 0 (, L and let ϕ : CP N be the associated Kodaira map, see (2.6. Consider homogeneous coordinates [Z 0 : : Z N ] and open set U 0 = {Z 0 0}. Then the pullbac of the Fubini Study metric by ϕ is given in local coordinates by ( 1 n ϕ ω F S = ϕ log 1 + z i 2 2π i=1 ( 1 S0 2 = log 2π S 0 + S S S N 2 2 S = 2π log( S 0 2 h + 2π log B 1 = ω + 2π log B where the last line is due to the fact that we are considering a polarized manifold. Using the asymptotic expansion, we have 1 2π log B = 1 2π log ( (1 n + ρ ( 1 + O = O 2 ( 1 Combining with the above, we have ( 1 1 ϕ ω F S ω = O 2 in C Another immediate corollary that can be obtained is an approximation of the dimension of space H 0 (, L for high tensor powers. ore precisely, 33

42 Corollary As, we have dim H 0 (, L = n ω n n! + n 1 ρ ωn 2 n! + O(n 2 Proof. dim H 0 (, L = N i=0 S i 2 h ω n = B n! ( = n 1 + ρ ( 1 ω n 2 + O 2 n! 3.3 Pea Sections We now introduce Tian s pea section method [31] to compute the coefficients of the asymptotic expansion. The pea sections were originally introduced by Tian to prove the following theorem approximating polarized metrics by a sequence of Bergman metrics: Let x 0. Choose local normal coordinates (z i n i=1 centered at x 0 such that the Hermitian matrix (g ij satisfies g ij (x 0 = δ ij p 1+ +p n g ij (x 0 z p 1 1 zn pn = 0 for i, j = 1,..., n and any nonnegative integers p 1,..., p n with p 1 + +p n 0. Next choose a local holomorphic frame e L of L at x 0 such that the local representation function h of the 34

43 Hermitian metric h has the properties h(x 0 = 1, p 1+ +p n h z p (x 1 1 zn pn 0 = 0 for any nonnegative integers (p 1,..., p n with p p n 0. Lemma (Pea Sections. Let c be a sequence such that lim c log( α = 1, α > 1 Suppose Ric(g Kω g, where K > 0 is a constant. For P Z n + and an integer p > P. Then there is a > 0 and a holomorphic section S P, H 0 (, L, satisfying S P, 2 h dv g 1 1 Ce 8 c (3.1 oreover, S P, can be decomposed as S P, = S P, u P, such that ( z 2 λ P z P e S P, (x = λ P η z P e L L = c x { z c 2 0 x \ { z c } and u P, 2 h dv g Ce 1 8 c 35

44 where η is a smoothly cut-off function 1 for 0 t 1 2 η(t = 0 for t 1 satisfying 4 η (t 0 and η (t 8 and λ 2 P = c z z P 2 a dv g Proof. Define the weight function ( ( z 2 z 2 Ψ(z = (n + 2pη log c c Where η : R R is a smooth cut-off function such that 1 for t < 1 2 η(t = 0 for t 1 satisfying 4 η (t 0 and η (t 8 and z 2 = z z n 2. Directly computing we have ( z 1 Ψ = 1(n + 2p {[η 2 2 z 2 z 2 ( z +η 2 z 2 c c ( z +2 Re [η 2 c c ] log c 2 ( z 2 c c z 2 log z 2 ] ( } z 2 + η log z 2 c To obtain a lower bound, we first note that 1 log( z 2 is positive definite, hence can be dropped. According to the support of the cut-off function η and its derivatives, we restrict 36

45 ( our attention to the interval c 2 z 2 c. In that interval, log z 2 c 0, hence ( z η 2 log c ( z 2 1 z 2 0, c thus can be dropped. Continuing the computation, we have ( 8(n + 2p 1 Ψ(z 1 z 2 z 2 + z 2 log z 2 c c 242 (n + 2p 1 z 2 z 2 c 2 24(n + 2p 1dz i dz j = c 48π(n + 2p c ω g. Using the above, we have for any unit vector v T 1,0 and any point p, Ψ + 2π ( Ric(h + Ric(g, v v 1 4 v 2 g For sufficiently large. Define the following 1-form w P = 1 4 (η ( z 2 c z p 1 1 z pn n e L which will serve as the main portion of the pea section. By solving the equation u P = w P, we obtain an L -valued section u P such that u P 2 h e Ψ dv g 4 w P 2 h e Ψ 37

46 Computing out the right hand side to get a more explicit form of the bound, we have 4 w P 2 h e Ψ dv g = c 2 ( z 2 η z P e c L 2 h e Ψ dv g ( z η 2 zi z P dz i e c L 2 e Ψ dv g C cp 1 p c c 2 z 2 c For large, we expand h in K-coordinates z 2p+2 h e Ψ dv g h dv 0. c 2 z 2 c h = e ϕ = e ( z 2 +O( z 4 so that u P 2 h e Ψ dv g C cp 1+n c 2 p+n e. Since c (log α, with α > 1, for sufficiently large, we can obtain the order O ( 1 p for any desired p. Using these pea sections, Tian proved a convergence theorem for Bergman metrics. In computing the coefficient of the Bergman ernel, we require an orthonormal basis. However, if we only want to compute up to a certain order, we only require that the sections be almost orthogonal. In that direction, we have a lemma by Tian [31] and generalized by Ruan [26] Lemma Let x 0 be fixed. Let S P be a pea section and T H 0 (, L. Locally, T = fe L for a holomorphic function f. 38

47 1. If z P is not in the Taylor expansion of f at x 0, then (S P, T h = O ( 1 T L If f contains no terms z Q such that q < p + σ in its Taylor expansion at p, then ( 1 (S P, T h = O T 1+σ/2 L Reduction of the Problem The following reduction used to compute the asymptotic expansion has been given in [19], and is included here for completeness. Let {S 0,..., S d } be a basis for H 0 (, L such that at x 0, S 0 (x 0 0 S i (x 0 = 0 for i = 1,... d (3.2 Let S = (S 0,..., S d and define the Gram matrix F = S S, where the entries are F ij = S i, S j L 2 (3.3 It is positive definite, since xf x = Sx 2, hence admits a decomposition F = GG (3.4 39

48 Let H = G 1. Then the entries of SH, i.e. d { H ij S j }, i = 0,..., d (3.5 j=0 form an orthonormal basis of H 0 (, L. Using this orthonormal basis, the Bergman ernel at x 0 can be reduced to i j H ij S j (x 0 2 h = i H i0 2 S 0 (x 0 2 h (3.6 Furthermore, if I = F 1, then the (0, 0th entry is given by I 00 = i H i0 2. (3.7 Hence to compute the asymptotic expansion, we only need to estimate I 00 and S 0 (x 0 2 h to the desired order. 3.5 Computation up to first order With the above considerations, we will compute the diagonal asymptotic expansion up to first order. It was shown in [19] that I 00 = 1 + O ( 1 3 hence to compute up to the first order, we only need to compute λ 2 0. We will need the following integral identity: Lemma (Lemma 4.1 [19]. Let A be a symmetric function on {1,..., n} p {1,..., n} p. Then for any p > 0, I,J z log ( = A I,I I A I,J z i1 z ip z j1 z jp z 2q e z 2 dv 0 ( p!(n + p + q 1! 1 (p + n 1!m + O, n+p+q m p 40

49 where I = (i 1,..., i p, J = (j 1,..., j p and 1 i 1,..., i p, j 1,..., j p n. Under normal coordinates and frame, S 0 2 h (x 0 = λ 2 = h dv g z 2 c = e ϕ det(gdv 0 z 2 c = e ϕ e log det(g dv 0 z 2 c Taing the Taylor expansions with normal coordinates gives ϕ(z = z 2 Rm ijl z i z z j z l + O( z 5 4 and log det g = Ric ij z i z j + O( z 3. Therefore e ϕ(z = e z 2 e 4 Rm ijl zi z z j z l = e z 2 (1 + 4 Rm ijl zi z z j z l + O( z 5, and e log det g = 1 Ric ij z i z j + O( z 3. 41

50 Applying Lemma and computing up to 1 order, z log (1 Ric ij z i z l + 4 Rm ijl zi z z j z l e z 2 = 1 n (1 ρ 2 + O ( 1 n+2. Inverting the above we have that ( λ 2 0 = n 1 + ρ ( O. 2 42

51 Chapter 4 Near-Diagonal Expansion In this chapter, we provide a proof of the near-diagonal expansion of the Bergman ernel via a perturbation method given in [14]. We use the observation that the Bergman ernel is concentrated in the near-diagonal. The first section will review the calculus of the Bargmann- Foc space. In the second section, we begin by establishing the local setting. There we will construct our candidate local ernel. The third section will show that the difference between the candidate ernel with the asymptotic expansion and the global ernel differ by an decaying term. In the last section, we show the smooth convergence of the asymptotic expansion. 4.1 Bargmann-Foc odel The Bargmann-Foc space is the space of entire functions that satisfy the weighted square integrability condition: f(z 2 e z 2 dv <. C n 43

52 The space F is precisely H 0 (C n, z 2, and is thus a closed linear subspace of the space L 2 (C n, z 2 with inner product given by f, g F := f(zg(ze z 2 dv, C n and thus is a Hilbert space. In fact, it is a reproducing ernel Hilbert space on C n, with reproducing ernel R C n(u, v := e u v. We first show that this ernel has the reproducing property on C and then extend this argument to C n. Lemma On C, the Bargmann-Foc ernel is given by R C (u, v := e uv. Proof. Taing some f H 0 (C, we consider the inner product against R C. We convert the resulting integral to polar coordinates and then apply the Cauchy Integral Formula to obtain f(v, R C F = 1 = 1 π = 1 π dv d v uv v 2 f(ve 2π C 2π 0 = f(u = f(u. 0 0 f(u + re iθ e u(u+re iθ u+re iθ 2 r 2 dθdr 2π re r2 f(u + re iθ e ureiθ dθdr 0 0 2re r2 dr The result follows. 44

53 Corollary On C n, the Bargmann-Foc ernel is given by R C n(u, v := e u v. Proof. Let u, v Z n + with u = (u 1,..., u n and v = (v 1,..., v n. Observe that e u v = n e u iv i v i 2. i=1 To demonstrate the reproducing property, we consider f H 0 (C n and decompose the integrand of the resulting inner product agains R C n. Applying Lemma?? to each dimensional component, we have f(v, R C n F = f(ve u v v 2 dv C n ( n = f(v 1,..., v n e u iv i v i 2 dv C n = f(u. i=i The result follows. The following lemmas demonstrate the Bargmann-Foc projection of monomials of different variables. Lemma Given some multiindex m Z n + the following equality holds. v m e u v v 2 dv = 0. C n 45

54 Proof. By manipulation and an application of Dominated Convergence Theorem, v m e u v v 2 dv = C n = (m u = 0. u (m C n [ [ e u v v 2] dv ] e u v v 2 dv C n Note that the integral is constant with respect to u, hence the derivative vanishes. The result follows. Lemma The following equality holds, for p, q Z + with p q. 0 if p > q, v p v q e u v v 2 dv = C n q! (q p! uq p if p q. Proof. Again by manipulation and an application of Dominated Convergence Theorem, v p v q e u.v v 2 dv = C n = (p u u (p C n [ = (p u [u q ], [ v q e u v v 2] dv ] v q e u v v 2 dv C n therefore 0 if p > q, v p v q e u v v 2 dv = C n q! (q p! uq p if p q. Result follows. 46

55 4.2 Local Construction The idea is to first construct a local ernel, then use Hörmander s L 2 -estimates to compare the local candidate Bergman ernel, which by construction will be in the form of an asymptotic expansion, with the global Bergman ernel. To mae precise what we mean by local construction, we define what we mean by local reproducing property (modulo N+1 2. Definition (Local Reproducing Property. A function Q N (x, y on U x0 U x0 which is holomorphic in x, antiholomorphic in y, is a local reproducing ernel modulo N+1 2 on U x0, if it satisfies the following local reproducing property (modulo N+1 2 f(x = χ (yf(y, Q N (y, x L 2 (U x,ϕ + O ( N+1 n 2 f L 2 (U x,ϕ, f H 0 (U x0, where χ is a cutoff function supported in a scale ball of radius 1 4 ɛ for some ε > 0 The ey here is to show that it is possible to construct such ernel for any N > 0 to be able to say that Bergman ernel has an asymptotic expansion. To construct such a local reproducing ernel, we need to consider Böchner coordinates and frames, Definition (Böchner coordinates. Let x 0 and U x0 be a sufficiently small neighborhood which admits a local holomorphic frame e L of L. Define the local Kähler potential ϕ by h(e L, e L = e ϕ. Böchner coordinates (z i n i=1 centered at x 0 are special coordinates in which ϕ admits the following form ϕ(z = z 2 + R(z, R(z = O( z 4 (4.1 47

56 On a Kähler manifold, we can always find holomorphic Böchner coordinates. To construct the local reproducing ernel modulo N at a point x 0, we begin by choosing Böchner coordinates and a local trivialization of the bundle. Our cutoff function χ is chosen to have shrining support on B( 1 4 ɛ, so that the inner product is localized near the diagonal and also to ensure that the local rescaled Kähler potential admits an asymptotic expansion of the form ( ϕ v := ( 2 v 1 + j=2 a j (v,v j,. This expression is an asymptotic perturbation of 2, hence we propose that the local v Bergman ernel admits an asymptotic expansion of the form K loc ( u v, = n e (1 u v + j=2 c j (u, v j, (4.2 where the c j s depends on a j and satisfy c j (u, v = c j (v, u. The reason we propose such an expansion is that if φ(x = x 2, then a j = 0 for all j 2 and hence c j = 0 for all j 2 yielding K loc ( u, v = n e u v, which is precisely the rescaling of the ernel of Bargmann- Foc model z Local Kernel Existence of coefficients We first establish some notations. For convenience, we write the volume form as ω n n! = ΩdV 48

57 Let a r,s m be the coefficients in the formal power series expansion of the product ( e R v ( Ω v = 2m m=0 r+s=0 a r,s m v r v s m, (4.3 where R(z was defined in (4.1. The following was shown by the author, with Hezari, Kelleher, and Xu in [14] Proposition (Existence of coefficients (Proposition 2.1 [14]. There exist unique coefficients c p,q j F and any N 0, ( u F = C depending only on the Kähler potential ϕ such that for any polynomial C n F ( v e u v v 2 ( N t=0 m+j=t p,q r,s c p,q j a r,s m t u p v q v r v s dv. (4.4 Furthermore, the coefficients c p,q j have the following finiteness and parity property 1. c p,q m = 0 when p + q > 2m, 2. c p,q m = 0 when p + q 2 m. Remar. When N = 0 the equation reduces to the reproducing property of the Bargmann- Foc ernel. Also note that the expression in the parenthesis is precisely the truncation up to N/2 of the product ( j=0 ( c j (u, v e R v ( Ω j v. The ey point is that there exists coefficients that reproduce the polynomials each time we increase the accuracy of the expansion of the volume form. The full proof is provided in [14] and is purely algebraic relying heavily on the identity given in Lemma

58 4.2.2 Estimates of local ernel In this section, we show a series of estimates necessary to show that the coefficients constructed perturbing the Bargmann-Foc ernel satisfies the local reproducing property. Proposition (Local reproducing property. Let f H 0 (B, and c j be the quantities as found in Proposition Then for u B, ( u f = χ ( v f ( + O N+1 n 2 ( v, e u v ( N f L 2 (B,ϕ. j c j (v, ū j L 2 (B(,ϕ (v To show the above, we need to show that we can approximate each term by its series expansion, and the error term depending on the L 2 norm of the function being reproduced. Lemma (Remainder of the exponential term. Let = [ N+1 ] + 1. Then for N 0 2ε and any f H 0 (B, B( χ ( v f ( v = f L 2 (B,ϕO e uv v 2 ( N+1 n 2 N j=0. c j (u,v j ( (e R u ( ( e R u ( 2N+5 Ω v dv Lemma (Remainder of determinant. Let = [ N+1 ] + 1. Then 2ɛ B( ( ( N χ v f v e uv v 2 = f L 2 (B,ϕO ( n N+1 2. j=0 c j (u,v j ( ( v e R 2N+5 ( (Ω Ω 2N+1 v dv (4.5 Lemma (Estimate outside the ball. Let F be a holomorphic polynomial. Then the 50