ASYMPTOTICS OF ALMOST HOLOMORPHIC SECTIONS OF AMPLE LINE BUNDLES ON SYMPLECTIC MANIFOLDS

BERNARD SHIFFMAN AND STEVE ZELDITCH

Abstract. In their work on symplectic manifolds, Donaldson and Auroux use analogues of holomorphic sections of an ample line bundle L over a symplectic manifold M to create symplectically embedded zero sections and almost holomorphic maps to various spaces. Their analogues were termed ‘asymptotically holomorphic’ sequences {s_{N}} of sections of L^{N}. We study another analogue H_{J}^{0}(M, L^{N}) of holomorphic sections, which we call ‘almostholomorphic’ sections, following a method introduced earlier by Boutet de Monvel - Guillemin [BoGu] in a general setting of symplectic cones. By definition, sections in H_{J}^{0}(M, L^{N}) lie in the range of a Szegö projector Π_{N}. Starting almost from scratch, and only using almostcomplex geometry, we construct a simple parametrix for Π_{N} of precisely the same type as the Boutet de Monvel-Sjöstrand parametrix in the holomorphic case [BoSj]. We then show that Π_{N}(x, y) has precisely the same scaling asymptotics as does the holomorphic Szegö kernel as analyzed in [BSZ1]. The scaling asymptotics imply more or less immediately a number of analogues of well-known results in the holomorphic case, e.g. a Kodaira embedding theorem and a Tian almost-isometry theorem. We also explain how to modify Donaldson’s constructions to prove existence of quantitatively transverse sections in H_{J}^{0}(M, L^{N}).

Contents

Introduction 2 1. Circle bundles and almost CR geometry 4 1.1. Almost complex symplectic manifolds 4 1.2. The circle bundle and Heisenberg coordinates 5 2. Almost holomorphic sections and Szegö kernels 8 2.1. Description of the algebra I^{∗}(X × X, C) 10 2.2. The positive Lagrangean ideal 11 2.3. The complex canonical relation 14 2.4. The Szegö projector 19 2.5. Almost holomorphic sections 21 3. Scaling asymptotics for Szegö kernels 22 4. Kodaira embedding and Tian almost isometry theorem 28 5. Transversal sections 31 5.1. Quantitative transversality 31 5.2. The Adjunction Formula 35 References 36

Date: March 27, 2001. Research partially supported by NSF grants DMS-9800479 (first author) and DMS-9703775, DMS-0071358

(second author).

Introduction

This paper is concerned with asymptotically holomorphic sections of ample line bundles over almost-complex symplectic manifolds (M, J, ω). Such line bundles and sections are symplectic analogues of the usual objects in complex algebraic geometry. Interest in their properties has grown in recent years because of their use by Donaldson [Don1, Don2], Auroux [Aur1, Aur2] and others [AuKa, Sik] in proving symplectic analogues of standard results in complex geometry. These results involve properties of asymptotically holomorphic sections of high powers of the bundle, particularly those involving their zero sets and the maps they define to projective space.

We take up the study of asymptotically holomorphic sections from the viewpoint of the microlocal analysis of the ¯∂ operator on a symplectic almost-complex manifold, and define a class of ‘almost holomorphic sections’ by a method due to Boutet de Monvel and Guillemin [Bout, BoGu]. Sections s of powers of a complex line bundle L^{N} → M over M are identified with equivariant functions ŝ on the associated S^{1}-bundle X, and the ¯∂ operator on sections is identified with the ¯∂_{b} operator on X. In the non-integrable almost-complex symplectic case there are in general no solutions of ¯∂_{b}s = 0. To define an ‘almost holomorphic section’ s, Boutet de Monvel and Guillemin define a certain (pseudodifferential) D̄-complex over X [BoGu] [BoSj]. The space H_{J}^{0}(M, L^{N}) of almost holomorphic sections is then the space of sections corresponding to solutions of D̄_{0}s = 0. The operator D̄_{0} is not uniquely or even canonically defined, and it is difficult to explicitly write down these almost holomorphic sections. However, our main results show that on small length scales (the same ones used in [Don1]), these almost holomorphic sections have the same asymptotic behavior as holomorphic sections in the N → ∞ limit. Furthermore, as we shall show in a related paper [ShZe2], typical sections have globally similar properties to the asymptotically holomorphic sections of Donaldson and Auroux.

The fundamental ingredient of these statements is the scaling asymptotics of the almostholomorphic Szegö kernels Π_{N}(z, w), i.e. the orthogonal projections onto H_{J}^{0}(M, L^{N}). As our results will show, the ‘peak sections’ z ↦ Π_{N}(z, w) about points w ∈ M are (up to normalization) intrinsically defined almost holomorphic sections with the main properties of Donaldson’s exponentially localized sections (cf. [Don1], Proposition 11). Analytically, it is more convenient to work with the lift of Π_{N} as an equivariant kernel Π_{N}(x, y) on X ×X than as a section of L^{N} ⊗ L^{∗N}, so henceforth we will use the term Szegö kernel for the equivariant kernel.

The scaling asymptotics of Π_{N}(x, y) must be given in special ‘Heisenberg’ coordinates on X, defined as follows: Choose ‘normal’ local coordinates {z_{j}} centered at a point P_{0} ∈ M as in Section 1.1 and choose a ‘preferred’ local frame for L, which together with the coordinates on M give us ‘Heisenberg coordinates’ x = (z, θ), y = (w, φ) on X (see §1.2). We obtain the following near-diagonal asymptotic formula for the Szegö kernel in terms of these Heisenberg coordinates:

Theorem 1. We have:

N^{−m}Π_{N}(P_{0} + \frac{u}{\sqrt{N}} , \frac{θ}{N}; P_{0} + \frac{v}{\sqrt{N}} , \frac{φ}{N})

= \frac{1}{π^{m}} e^{i(θ−φ)+u·v̄− \frac{1}{2} (|u|^{2}+|v|^{2})} [1 + ∑_{r=1}^{K} N^{− \frac{r}{2}} b_{r}(P_{0}, u, v) + N^{− \frac{K+1}{2}} R_{K}(P_{0}, u, v, N)] ,
where ∥R_{K}(P_{0}, u, v, N)∥_{C^{j}({|u|+|v|≤ρ})} ≤ C_{K,j,ρ} for j = 1, 2, 3, ….

A more precise statement will be given in Theorem 3.1. The proof of Theorem 1 is based on the construction of explicit parametrices for Π and its Fourier coefficients Π_{N}. These parametrices closely resemble those of Boutet de Monvel - Sjöstrand [BoSj] in the holomorphic case. The construction is new but closely follows the work of Menikoff and Sjöstrand [MenSj, Sjö] and of Boutet de Monvel and Guillemin [Bout, BoGu]. For the sake of concreteness, we will give a fairly detailed exposition of the construction of the zeroth term of the D̄_{j} complex and of the Szegö kernel.

As more or less immediate corollaries of the scaling asymptotics, we prove the following symplectic analogues of the holomorphic Kodaira embedding theorem and Tian almostisometry theorem [Tian]. These concern the analogue of the usual Kodaira map Φ_{N} : M → ℂℙ^{d_{N} −1} determined by the space H_{J}^{0}(M, L^{N}). Theorem 2. Let L → (M, ω) be the pre-quantum line bundle over a 2m-dimensional symplectic manifold, and let {Φ_{N}} be its Kodaira maps. Then:

(a) There exists a complete asymptotic expansion:

Π_{N}(z, 0; z, 0) = a_{0}N^{m} + a_{1}(z)N^{m−1} + a_{2}(z)N^{m−2} + …

for certain smooth coefficients a_{j}(z) with a_{0} = π^{−m}. Hence, the maps Φ_{N} are well-defined for N ≫ 0.

(b) Let ω_{F S} denote the Fubini-Study form on ℂℙ^{d_{N} −1}. Then

∥\frac{1}{N} Φ_{N}^{∗}(ω_{F S}) − ω∥_{C^{k}} = O(\frac{1}{N})

for any k. Theorem 3. The Kodaira map Φ_{N} : M → ℂℙ^{d_{N} −1} is an embedding, for N sufficiently large.

We should draw attention to related work of Borthwick-Uribe [BoUr1, BoUr2], which defines ‘almost holomorphic’ in a somewhat different way, and proves analogues of the Tian almost-isometry and Kodaira embedding theorems in their setting. The Borthwick-Uribe proof of the almost-complex Tian theorem was in turn motivated by a similar proof in the holomorphic case in [Zel]. The approach we take here, based on the off-diagonal scaling asymptotics of the Szegö kernel, is new and we believe it will have further geometric applications.

Finally, we use the scaling asymptotics of the Szegö kernel Π_{N}(x, y) to show that Donaldson’s symplectic embedding theorem and the genus calculation of embedded symplectic curves can be adapted to the setting of almost holomorphic sections. We will see in §5 that the proof in [Don1] of the existence of quantitatively transversal asymptotically holomorphic sections can be adapted to prove the existence of quantitatively transverse sections in H_{J}^{0}(M, L^{N}).

Further applications of the scaling asymptotics in Theorems 1 and 3.1 will be given in a forthcoming paper [ShZe2], in which we discuss typical properties of almost holomorphic sections (in the probabilistic sense of [BSZ1, BSZ2]). In particular, we show that almost every sequence of L^{2}-normalized sections {s_{N}} ∈ SH_{J}^{0}(M, L^{N}) satisfies the estimates:

∥s_{N}∥_{∞} = O(\sqrt{log N}),  ∥∇^{k}s_{N}∥_{∞} = O(N^{k/2}\sqrt{log N}),
∥¯∂s_{N}∥_{∞} = O(\sqrt{log N}), ∥∇^{k} ¯∂s_{N}∥_{∞} = O(N^{k/2}\sqrt{log N}), (k ≥ 1) .

Consequently, the sequence of almost-holomorphic sections \{\frac{1}{\sqrt{log N}} s_{N}\} is almost surely asymptotically holomorphic in the sense of Donaldson and Auroux [Don1, Aur1].

We also use Theorem 3.1 to show, in joint work with P. Bleher [BSZ2], that our prior result [BSZ1, Theorem 3.6] on the universality of scaling limits of correlations of zeros of sections of positive line bundles over complex manifolds extends to almost holomorphic sections on symplectic manifolds. Thus, on small balls (of radius 1/\sqrt{N}), almost holomorphic sections have the same asymptotic probabilities as holomorphic sections of taking on specified values and derivatives.

For the readers’ convenience, we conclude this introduction with a brief outline of the paper. We begin in §1 by describing some terminology from symplectic geometry and introducing our Heisenberg coordinates. In §2, we use the method of Boutet de Monvel, Guillemin, and Sjöstrand [Bout, BoGu, BoSj] to construct Szegö projectors Π_{N}, which we show are complex Fourier integral operators of the same type as in the holomorphic case, and then to provide a Boutet de Monvel-Guillemin complex of pseudodifferential operators, which replaces the ¯∂_{b} complex in the symplectic setting. The zeroth term of this complex is used to define sequences of almost holomorphic sections analogous to the integrable complex case (§2.5). In §3, we obtain the scaling asymptotics of Π_{N}(P_{0} + \frac{u}{\sqrt{N}} , \frac{θ}{N}; P_{0} + \frac{v}{\sqrt{N}} , \frac{φ}{N}), which we apply in §4 to prove Theorems 2 and 3 on the symplectic Kodaira map. Finally, in §5, we show how concentrated almost holomorphic sections can be used in place of asymptotically holomorphic sections to construct quantitatively transversal sections by Donaldson’s method.

We would like to thank Michael Christ for informative comments concerning the offdiagonal decay of the Szegö kernel in the holomorphic case.

1. Circle bundles and almost CR geometry

We denote by (M, ω) a compact symplectic manifold such that [\frac{1}{π}ω] is an integral cohomology class. As is well known (cf. [Woo, Prop. 8.3.1]; see also [GuSt]), there exists a hermitian line bundle (L, h) → M and a metric connection ∇ on L whose curvature Θ_{L} satisfies \frac{i}{2}Θ_{L} = ω. We denote by L^{N} the N^{th} tensor power of L. The ‘quantization’ of (M, ω) at Planck constant 1/N should be a Hilbert space of polarized sections of L^{N} ([GuSt, p. 266]). In the complex case, polarized sections are simply holomorphic sections. The notion of polarized sections is problematic in the non-complex symplectic setting, since the Lagrangean subbundle T^{1,0}M defining the complex polarization is not integrable and there usually are no ‘holomorphic’ sections. A subtle but compelling replacement for the notion of polarized section has been proposed by Boutet de Monvel and Guillemin [Bout, BoGu], and it is this notion which we adopt in this paper. For the asymptotic analysis, it is best to view sections of L^{N} as functions on the unit circle bundle X ⊂ L^{∗}; we shall describe the ‘almost CR geometry’ of X in §1.2 below.

1.1. Almost complex symplectic manifolds. We begin by reviewing some terminology from almost complex symplectic geometry. An almost complex symplectic manifold is a symplectic manifold (M, ω) together with an almost complex structure J satisfying the compatibility condition ω(Jv, Jw) = ω(v, w) and the positivity condition. ω(v, Jv) > 0. We give M the Riemannian metric g(v, w) = ω(v, Jw). We denote by T^{1,0}M, resp. T^{0,1}M, the holomorphic, resp. anti-holomorphic, sub-bundle of the complex tangent bundle T M; i.e., J = i on T^{1,0}M and J = −i on T^{0,1}M. We give M local coordinates (x_{1}, y_{1}, …, x_{m}, y_{m}), and we write z_{j} = x_{j} + iy_{j}. As in the integrable (i.e., holomorphic) case, we let \{\frac{∂}{∂z_{j}} , \frac{∂}{∂z̄_{j}} \} denote the dual frame to {dz_{j}, dz̄_{j}}. Although in our case, the coordinates z_{j} are not holomorphic and consequently \frac{∂}{∂z_{j}} is generally not in T^{1,0}M, we nonetheless have

\frac{∂}{∂z_{j}} = \frac{1}{2} \frac{∂}{∂x_{j}} − \frac{i}{2} \frac{∂}{∂y_{j}} , \frac{∂}{∂z̄_{j}} = \frac{1}{2} \frac{∂}{∂x_{j}} + \frac{i}{2} \frac{∂}{∂y_{j}} .

At any point P_{0} ∈ M, we can choose a local frame {Z̄_{1}^{M}, …, Z̄_{m}^{M}} for T^{0,1}M near P_{0} and coordinates about P_{0} so that

Z̄_{j}^{M} = \frac{∂}{∂z̄_{j}} + ∑_{k=1}^{m} B_{jk}(z)\frac{∂}{∂z_{k}} , B_{jk}(P_{0}) = 0 ,  (1)

and hence ∂/∂z_{j}|_{P_{0}} ∈ T^{1,0}(M). This is one of the properties of our ‘preferred coordinates’ defined below.

Definition: Let P_{0} ∈ M. A coordinate system (z_{1}, …, z_{m}) on a neighborhood U of P_{0} is preferred at P_{0} if

∑_{j=1}^{m} dz_{j} ⊗ dz̄_{j} = (g − iω)|_{P_{0}} .

In fact, the coordinates (z_{1}, …, z_{m}) are preferred at P_{0} if an only if any two of the following conditions (and hence all three) are satisfied:

i) ∂/∂z_{j}|_{P_{0}} ∈ T^{1,0}(M), for 1 ≤ j ≤ m,

ii) ω(P_{0}) = ω_{0}, iii) g(P_{0}) = g_{0}, where ω_{0} is the standard symplectic form and g_{0} is the Euclidean metric:

ω_{0} = \frac{i}{2} ∑_{j=1}^{m} dz_{j} ∧ dz̄_{j} = ∑_{j=1}^{m}(dx_{j} ⊗ dy_{j} − dy_{j} ⊗ dx_{j}) , g_{0} = ∑_{j=1}^{m}(dx_{j} ⊗ dx_{j} + dy_{j} ⊗ dy_{j}) .

(To verify this statement, note that condition (i) is equivalent to J(dx_{j}) = −dy_{j} at P_{0}, and use g(v, w) = ω(v, Jw).) Note that by the Darboux theorem, we can choose the coordinates so that condition (ii) is satisfied on a neighborhood of P_{0}, but this is not necessary for our scaling results.

1.2. The circle bundle and Heisenberg coordinates. We now let (M, ω, J) be a compact, almost complex symplectic manifold such that [\frac{1}{π}ω] is an integral cohomology class, and we choose a hermitian line bundle (L, h) → M and a metric connection ∇ on L with \frac{i}{2}Θ_{L} = ω. In order to simultaneously analyze sections of all positive powers L^{N} of the line bundle L, we work on the associated principal S^{1} bundle X → M, which is defined as follows: let π : L^{∗} → M denote the dual line bundle to L with dual metric h^{∗}, and put X = {v ∈ L^{∗} : ∥v∥_{h^{∗}} = 1}. We let α be the the connection 1-form on X given by ∇; we then have dα = π^{∗}ω, and thus α is a contact form on X, i.e., α ∧ (dα)^{m} is a volume form on X.

We let r_{θ}x = e^{iθ}x (x ∈ X) denote the S^{1} action on X and denote its infinitesimal generator by \frac{∂}{∂θ}. A section s of L determines an equivariant function ŝ on L^{∗} by the rule ŝ(λ) = (λ, s(z)) (λ ∈ L_{z}^{∗}, z ∈ M). It is clear that if τ ∈ ℂ then ŝ(z, τλ) = τŝ. We henceforth restrict ŝ to X and then the equivariance property takes the form ŝ(r_{θ}x) = e^{iθ}ŝ(x). Similarly, a section s_{N} of L^{N} determines an equivariant function ŝ_{N} on X: put

ŝ_{N}(λ) = (λ^{⊗N}, s_{N}(z)) , λ ∈ X_{z} ,  (2)

where λ^{⊗N} = λ ⊗ ⋯ ⊗ λ; then ŝ_{N}(r_{θ}x) = e^{iNθ}ŝ_{N}(x). We denote by L_{N}^{2}(X) the space of such equivariant functions transforming by the N^{th} character.

In the complex case, X is a CR manifold. In the general almost-complex symplectic case it is an almost CR manifold. The almost CR structure is defined as follows: The kernel of α defines a horizontal hyperplane bundle H ⊂ T X. Using the projection π : X → M, we may lift the splitting T M = T^{1,0}M ⊕ T^{0,1}M to a splitting H = H^{1,0} ⊕ H^{0,1}. The almost CR structure on X is defined to be the splitting T X = H^{1,0} ⊕ H^{0,1} ⊕ ℂ\frac{∂}{∂θ}. We also consider a local orthonormal frame Z_{1}, …, Z_{n} of H^{1,0} , resp. Z̄_{1}, …, Z̄_{m} of H^{0,1}, and dual orthonormal coframes ϑ_{1}, …, ϑ_{m}, resp. ϑ̄_{1}, …, ϑ̄_{m}. On the manifold X we have d = ∂_{b} + ¯∂_{b} + \frac{∂}{∂θ} ⊗ α, where ∂_{b} = ∑_{j=1}^{m} ϑ_{j} ⊗ Z_{j} and ¯∂_{b} = ∑_{j=1}^{m} ϑ̄_{j} ⊗ Z̄_{j}. We define the almost-CR ¯∂_{b} operator by ¯∂_{b} = df|_{H^{1,0}}. Note that for an L^{2} section s^{N} of L^{N}, we have

(∇_{L^{N}} s^{N}) = d^{h}ŝ^{N} ,  (3)

where d^{h} = ∂_{b} + ¯∂_{b} is the horizontal derivative on X.

Our near-diagonal asymptotics of the Szegö kernel (§3) are given in terms of the Heisenberg dilations, using local ‘Heisenberg coordinates’ at a point x_{0} ∈ X. To describe these coordinates, we first need the concept of a ‘preferred frame’:

Definition: A preferred frame for L → M at a point P_{0} ∈ M is a local frame e_{L} in a neighborhood of P_{0} such that

i) ∥e_{L}∥_{P_{0}} = 1; ii) ∇e_{L}|_{P_{0}} = 0; iii) ∇^{2}e_{L}|_{P_{0}} = −(g + iω) ⊗ e_{L}|_{P_{0}} ∈ T_{M}^{∗} ⊗ T_{M}^{∗} ⊗ L.

(A preferred frame can be constructed by multiplying an arbitrary frame by a function with specified 2-jet at P_{0}; any two such frames agree to third order at P_{0}.) Once we have property (ii), property (iii) is independent of the choice of connection on T_{M}^{∗} used to define ∇ : C^{∞}(M, L⊗T_{M}^{∗}) → C^{∞}(M, L⊗T_{M}^{∗} ⊗T_{M}^{∗}). In fact, property (iii) is a necessary condition for obtaining universal scaling asymptotics, because of the ‘parabolic’ scaling in the Heisenberg group. Note that if e_{L} is a preferred frame at P_{0} and if (z_{1}, …, z_{m}) are preferred coordinates at P_{0}, then we compute the Hessian of ∥e_{L}∥:

(∇^{2}∥e_{L}∥_{h})_{P_{0}} = ℜ (∇^{2}e_{L}, e_{L})_{P_{0}} = −g(P_{0}) ;

thus if the preferred coordinates are ‘centered’ at P_{0} (i.e., P_{0} = 0), we have

∥e_{L}∥_{h} = 1 − \frac{1}{2}|z|^{2} + O(|z|^{3}) .  (4)

Remark: Recall ([BSZ1, §1.3.2]) that the Bargmann-Fock representation of the Heisenberg group acts on the space of holomorphic functions on (M, ω) = (ℂ^{m}, ω_{0}) that are square integrable with respect to the weight h = e^{−|z^{2}|}. We let L = ℂ^{m} × ℂ be the trivial bundle.

Then the trivializing section e_{L}(z) := (z, 1) is a preferred frame at P_{0} = 0 with respect to the Hermitian connection ∇ given by

∇e_{L} = ∂ log h ⊗ e_{L} = − ∑_{j=1}^{m} z̄_{j}dz_{j} ⊗ e_{L} .

Indeed, the above yields ∇^{2}e_{L}|_{0} = − ∑ dz̄_{j} ⊗ dz_{j} ⊗ e_{L}(0) = −(g_{0} + iω_{0}) ⊗ e_{L}(0).

The preferred frame and preferred coordinates together give us ‘Heisenberg coordinates’:

Definition: A Heisenberg coordinate chart at a point x_{0} in the principal bundle X is a coordinate chart ρ : U ≈ V with 0 ∈ U ⊂ ℂ^{m} × ℝ and ρ(0) = x_{0} ∈ V ⊂ X of the form

ρ(z_{1}, …, z_{m}, θ) = e^{iθ}a(z)^{− \frac{1}{2}} e_{L}^{∗}(z) ,  (5)

where e_{L} is a preferred local frame for L → M at P_{0} = π(x_{0}), and (z_{1}, …, z_{m}) are preferred coordinates centered at P_{0}. (Note that P_{0} has coordinates (0, …, 0) and e_{L}^{∗}(P_{0}) = x_{0}.)

We now give some computations using local coordinates (z_{1}, …, z_{m}, θ) of the form (5) for a local frame e_{L}. (For the moment, we do not assume they are Heisenberg coordinates.) We write

a(z) = ∥e_{L}^{∗}(z)∥_{h^{∗}}^{2} = ∥e_{L}(z)∥_{h}^{−2} ,
α = dθ + β ,  β = ∑_{j=1}^{m}(A_{j}dz_{j} + Ā_{j}dz̄_{j}) ,
∇e_{L} = φ ⊗ e_{L} ,  hence ∇e_{L}^{⊗N} = Nφ ⊗ e_{L}^{⊗N} .

We let \frac{∂^{h}}{∂z_{j}} ∈ H^{1,0}X denote the horizontal lift of \frac{∂}{∂z_{j}}. The condition (\frac{∂^{h}}{∂z_{j}} , α) = 0 yields

\frac{∂^{h}}{∂z_{j}} = \frac{∂}{∂z_{j}} − A_{j} \frac{∂}{∂θ} , \frac{∂^{h}}{∂z̄_{j}} = \frac{∂}{∂z̄_{j}} − Ā_{j} \frac{∂}{∂θ} .  (6)

Suppose s_{N} = fe_{L}^{⊗N} is a local section of L^{N}. Then by (2) and (5),

ŝ_{N}(z, θ) = f(z)a(z)^{− \frac{1}{2}} e^{iNθ} .  (7)

Differentiating (7) and using (3), we conclude that

φ = −\frac{1}{2}d log a − iβ
= − ∑_{j=1}^{m} (\frac{1}{2} \frac{∂ log a}{∂z_{j}} + iA_{j}) dz_{j} − ∑_{j=1}^{m} (\frac{1}{2} \frac{∂ log a}{∂z̄_{j}} + iĀ_{j}) dz̄_{j} .  (8)

Now suppose that (z_{1}, …, z_{m}, θ) are Heisenberg coordinates at P_{0}; i.e., e_{L} is a preferred frame at P_{0} and (z_{1}, …, z_{m}) are preferred coordinates centered at P_{0} (with P_{0} = 0). By property (ii) of preferred frames, we have φ(0) = 0, and hence by (8)

da|_{0} = d log a|_{0} = 0,  (9)
A_{j}(0) = 0 , (1 ≤ j ≤ m) .  (10)

By differentiating (8) and applying the properties of preferred coordinates and frames, we further obtain

∑_{j=1}^{m} dz̄_{j} ⊗ dz_{j} = −∇φ = ∑_{j=1}^{m} d (\frac{1}{2} \frac{∂ log a}{∂z_{j}} + iA_{j}) ⊗ dz_{j} + ∑_{j=1}^{m} d (\frac{1}{2} \frac{∂ log a}{∂z̄_{j}} + iĀ_{j}) ⊗ dz̄_{j} at 0.

Thus the following four equations are satisfied at P_{0} = 0:

\frac{1}{2} \frac{∂^{2} log a}{∂z_{j}∂z_{k}} + i\frac{∂A_{j}}{∂z_{k}} = 0 , \frac{1}{2} \frac{∂^{2} log a}{∂z_{j}∂z̄_{k}} + i\frac{∂A_{j}}{∂z̄_{k}} = δ_{k}^{j} ,
\frac{1}{2} \frac{∂^{2} log a}{∂z̄_{j}∂z_{k}} + i\frac{∂Ā_{j}}{∂z_{k}} = 0 , \frac{1}{2} \frac{∂^{2} log a}{∂z̄_{j}∂z̄_{k}} + i\frac{∂Ā_{j}}{∂z̄_{k}} = 0 ,

(11)

at P_{0}. Solving (11) and recalling that a(0) = 1, da|_{0} = 0, we obtain

\frac{∂^{2}a}{∂z_{j}∂z_{k}} (0) = 0 ,  \frac{∂^{2}a}{∂z_{j}∂z̄_{k}} (0) = δ_{k}^{j} ,  (12)

\frac{∂A_{j}}{∂z_{k}} (0) = 0 ,  \frac{∂A_{j}}{∂z̄_{k}} = −\frac{i}{2}δ_{k}^{j} .  (13)

Hence A_{j} = −\frac{i}{2} z̄_{j} + O(|z|^{2}) and

\frac{∂^{h}}{∂z_{j}} = \frac{∂}{∂z_{j}} + [\frac{i}{2}z̄_{j} + O(|z|^{2})] \frac{∂}{∂θ} , \frac{∂^{h}}{∂z̄_{j}} = \frac{∂}{∂z̄_{j}} − [\frac{i}{2}z_{j} + O(|z|^{2})] \frac{∂}{∂θ} .  (14)

2. Almost holomorphic sections and Szegö kernels

In this section, we define the space H_{J}^{0}(M, L^{N}) of almost holomorphic sections of L^{N}, or equivalently the space H_{N}^{2}(X) of almost CR functions in H^{2}(X) which transform by e^{iNθ} under the S^{1} action. The Szegö kernels Π_{N} are the orthogonal projections onto the spaces H_{N}^{2}(X).

The order of events in the definition of H_{J}^{0}(M, L^{N}) is almost opposite to that in the holomorphic case. There, one defines H^{2}(X) as the kernel of the ¯∂_{b} operator. As mentioned above, the kernel is generically empty in the non-integrable almost complex case. It is correct but somewhat misleading to say that one defines almost holomorphic by deforming ¯∂_{b} in the non-integrable case to a pseudodifferential operator D̄_{0} (which is the zeroth step of a D̄- complex), setting H^{2}(X) equal to its kernel, and defining Π as the associated orthogonal projection with respect to a metric on L. The misleading aspect is that actually one first defines Π as a complex Fourier integral projection operator associated to a certain canonical relation C which naturally generalizes the one in the integrable case. Having defined Π, one then defines D̄ as a deformation of ¯∂_{b} that annihilates Π.

Existence of a projection Π with a prescribed principal symbol in the algebra I^{∗}(X ×X, C) of Fourier integral kernels with wave front along C follows from symbolic and functional analytic arguments, but uniqueness does not. Indeed, Π is clearly not unique, since it could be replaced by e^{−iA}Πe^{iA} for any pseudodifferential operator A of order −1 on X. We use the term ‘Szegö kernels’ rather than ‘the Szegö kernel’ to remind the reader that Π is by no means uniquely defined as in the holomorphic case. The space H_{J}^{0}(M, L^{N}) therefore also involves non-unique choices.

The non-geometric construction of the spaces of almost holomorphic sections may seem less odd if one compares it to the construction of asymptotically holomorphic sections in [Don1]. There, only ¯∂_{b} is defined and asymptotically holomorphic sections are sequences of sections {s_{N}} which are annihilated by ¯∂_{b} modulo small errors. For a fixed power N, no space of sections is thereby singled out. On the other hand, our construction gives linear spaces H_{J}^{0}(M, L^{N}) of almost holomorphic sections, but these spaces depend on the choice of the projection operator Π.

Let us briefly review the holomorphic case from our perspective. It was proved in [BoSj] that the Szegö kernel of a strictly pseudoconvex domain (including disk bundles of positively curved line bundles) is a complex Fourier integral operator of the form

Π(x, y) ∼ ∫_{ℝ^{+}} e^{itψ(x,y)}s(x, y, t)dt,

with s ∼ ∑_{n=0}^{∞} t^{m−n}s_{n}(x, y) and with tψ(x, y) a phase of positive type. (Complex Fourier integral operators are discussed below.) Note that the phase is linear in the one-dimensional phase variable t. This is a ‘hard’ result because it shows that an object of complex geometry, the Szegö kernel, has a special singularity structure. It requires estimates of J. J. Kohn—in particular, the closed range property of □_{b} (cf. [BoSj]).

In [BSZ1], we related ψ to the hermitian metric on L, which defines the circle bundle X. Let us recall the result. Fix a local holomorphic section e_{L} of L over U ⊂ M and define a ∈ C^{∞}(U) by a = |e_{L}|_{h}^{−2}. Since L^{∗}|_{U} ≈ U × ℂ we can define local coordinates on L^{∗} by (z, λ) ≈ λe_{L}(z). Then a defining function of X ⊂ L^{∗} is given by ρ(z, λ) = 1 − |λ|^{2}a(z). Define the function a(z, w) as the almost analytic extension of a(z), i.e. the solution of ¯∂_{z}a = 0 = ∂_{w}a, a(z, z) = a(z) and put ψ(x, y) = i(1 − λμ̄a(z, w)). Then tψ is a phase for Π. Because Π is a projection, the complex canonical relation C ⊂ T^{∗}(X̃ × X̃) parametrized by tψ must be idempotent (i.e., C^{2} = C = C^{∗}). As will be explained below, it is the flow-out of the diagonal in X × X under the joint Hamiltonian flow of the components of ¯∂_{b}, which commute by integrability of the complex structure. Such a flow-out is an equivalence relation (x ∼ y if they belong to the same orbit of the commuting flows), and in this way one also sees that C is idempotent.

Our purpose now is to construct a complex Fourier integral projection kernel Π in the almost complex (non-integrable) case with the same properties as in the complex case. As discussed above, this is not the same kind of ‘hard’ result as in the complex case because there is no a priori definition of Π dictated by almost complex geometry, so we simply construct one with analogous microlocal properties. Consequently, we do not need any difficult results of analysis, and the construction involves only symbol algebra and geometry. Indeed, the main step is to define the canonical relation C. Since ¯∂_{b} is no longer integrable, we cannot use its components to define the flow-out. Instead, we begin in §2.2 by defining a new positive Lagrangean ideal I corresponding to the ideal generated by the elements of H^{0,1}X in the complex case. We then show in §2.3 that I defines a unique canonical relation C (which lives in the almost analytic extension of T^{∗}X × T^{∗}X) and we describe how C can be parametrized by a phase defined on X × X × ℝ^{+}, as in the holomorphic case. In §2.4, we describe the algebra I^{∗}(X ×X, C) of complex Fourier integral operators with wave front sets along C, and construct a Szegö projector Π ∈ I^{∗}(X × X, C). Finally we introduce in §2.5 the Boutet de Monvel - Guillemin D̄ complex of first order pseudodifferential operators with ker D̄_{0} = H^{2}(X). the operator D̄_{0} is the replacement for ¯∂_{b} in the symplectic case, and the N^{th} Fourier component of H^{2}(X) is the space H_{J}^{0}(M, L^{N}) of almost holomorphic sections.

2.1. Description of the algebra I^{∗}(X × X, C). In order to construct an almost holomorphic Szegö kernel, we must first construct the algebra I^{∗}(X × X, C) of complex Fourier integral operators in which it lies. The definition of the algebra I^{∗}(X × X, C) of complex Fourier integral operators is essentially given in [Hör] (see Definition 25.5.1), although there it is denoted I^{∗}(X × X, I) where I is a positive conic Lagrangean ideal. (We use the older notation of [MelSj] and the language of almost analytic extensions for ease of comparison with our basic references [MelSj, BoSj, MenSj].) It would take us too far afield to review the definitions and properties of complex Fourier integral operators, so we only briefly recall some basic ideas and refer the reader to [Hör] or [MelSj] for background.

The most intuitive definition of a Fourier integral kernel A(x, y) ∈ I^{n}(X × X, Λ) is that it is a distribution (generalized function) defined by a complex oscillatory integral

A(x, y) = ∫_{ℝ^{N}} e^{iφ(x,y,θ)}s(x, y, θ)dθ

where φ is a phase function of positive type parametrizing Λ, and where s(x, y, θ) is a symbol of order n + (dim X − N)/2 (cf., [Hör], Proposition 25.1.5). More precisely, we assume φ(x, y, θ) is a regular phase function in the sense of ([MelSj, Def. 3.5]), i.e. that it has no critical points, is homogeneous of degree one in θ, that the differentials d\frac{∂φ}{θ_{j}} are linearly independent over ℂ on the set

C_{φℝ} = {(x, y, θ) : d_{θ}φ = 0}

and such that ℑφ ≥ 0. We then let φ̃(x̃, ỹ, θ̃) be an almost analytic extension of φ to X̃ × X̃ × ℂ^{N}. Here, Ỹ denotes the almost analytic extension of a C^{∞} manifold Y. When Y is real analytic, Ỹ is the usual complexification of Y, i.e. a complex manifold in which Y sits as a totally real submanifold. The reader may find it simpler to make the extra assumption that X is real analytic. For background on almost analytic extensions, we refer to [MelSj, MenSj].

We put

Cφ̃ = {(x̃, θ̃) : dθ̃φ̃ = 0}

and define the Lagrange immersion

ιφ̃ : (x̃, θ̃) ∈ Cφ̃ → (x̃, dx̃φ̃(x̃, θ̃)).

We say that the phase φ parametrizes Λ if Λ̃ is the image of this map.

By symbols, we mean the following: the space of classical symbols of order k, denoted S^{k}(X × X × ℝ^{N}), consists of elements of the form s ∼ ∑_{j=0}^{∞} s_{j}(x, y, θ), where s_{j} is a smooth function (defined near x = y) of (x, y) and is homogeneous of order k − j in θ ∈ ℝ^{N}.

Finally, we note that if Λ is an equivalence relation, then I^{∗}(X × X, Λ) is an algebra [BoSj, Hör].

2.2. The positive Lagrangean ideal. We begin with some microlocal analysis of ¯∂_{b} in order to introduce the characteristic variety Σ of ¯∂_{b}. In general, we denote by σ_{A} the principal symbol of a pseudodifferential operator A. To describe the principal symbol of ¯∂_{b}, we introduce convenient local coordinates and frames. Recalling that HX = H^{1,0}X ⊕H^{0,1}X, we again consider local orthonormal frames Z_{1}, …, Z_{n} of H^{1,0}X, resp. Z̄_{1}, …, Z̄_{m} of H^{0,1}X, and dual orthonormal coframes ϑ_{1}, …, ϑ_{m}, resp. ϑ̄_{1}, …, ϑ̄_{m}. Then we have ¯∂_{b} = ∑_{j=1}^{m} ϑ̄_{j} ⊗ Z̄_{j}. Let us define complex-valued functions on T^{∗}X by:

p_{j}(x, ξ) = ⟨Z_{j}(x), ξ), p̄_{j}(x, ξ) = ⟨Z̄_{j}(x), ξ⟩.

Then

σ ¯∂_{b}(x, ξ) = ∑_{j=1}^{m} p_{j}(x, ξ)ϵ(ϑ̄_{j})

where ϵ denotes exterior multiplication. We note that {p̄_{j}, p̄_{k}} = ⟨[Z̄_{j}, Z̄_{k}], ξ⟩.

To state results, it is convenient to introduce the operator □_{b} := ¯∂_{b}^{∗} ¯∂_{b} = ∑_{j=1}^{m} Z̄_{j}^{∗}Z̄_{j} where Z̄_{j}^{∗} is the adjoint of the vector field regarded as a linear differential operator. To conform to the notation of [BoGu] we also put q = σ(□_{b}) = ∑_{j=1}^{m} |p̄_{j}|^{2}. The characteristic variety Σ = {q = 0} of ¯∂_{b} is the same as that of □_{b}, namely the vertical sub-bundle of T^{∗}X → M. It is the conic submanifold of T^{∗}X parametrized by the graph of the contact form, Σ = {(x, rα_{x}) : r > 0} ∼ X × ℝ^{+}. It follows that Σ is a symplectic submanifold. It is the dual (real) line bundle to the vertical subbundle V ⊂ T X, since α(X) = G(X, \frac{∂}{∂θ}).

We are now ready to introduce the positive Lagrangean ideal I whose generators will define a canonical relation C underlying the Szegö kernels and the principal symbol of the operator D̄_{0}, which will replace ¯∂_{b}. For background on positive Lagrangean ideals, see [Hör].

Proposition 2.1. There exists a unique positive Lagrangean ideal I with respect to Σ containing q. That is, there exists a unique ideal I ⊂ I_{Σ} (where I_{Σ} is the ideal of functions vanishing on Σ) satisfying:

• I is closed under Poisson bracket;

• Σ is the set of common zeros of f ∈ I;

• There exist local generators ζ_{1}, …, ζ_{m} such that the matrix (\frac{1}{i}\{ζ_{j}, ζ̄_{k}\}) is positive definite on Σ and that q = ∑_{j,k} λ_{jk̄}ζ_{j}ζ̄_{k}, where {λ_{jk̄}} is a hermitian positive definite matrix of functions.

Proof. In the holomorphic case, I is generated by the linear functions ζ_{j}(x, ξ) = ⟨ξ, Z̄_{j}⟩. In the general almost complex (or rather almost CR) setting, these functions do not Poisson commute and have to be modified. Since the deviation of an almost complex structure from being integrable (i.e. a true complex structure) is measured by the Nijenhuis bracket, it is not surprising that the generators ζ_{j} can be constructed from the linear functions ⟨ξ, Z̄_{j}⟩ and from the Nijenhuis tensor. We now explain how to do this, basically following the method of [BoGu].

As a first approximation to the ζ_{j} we begin with the linear functions ζ_{j}^{(1)} = p̄_{j} on T^{∗}X. As mentioned above, the ζ_{j}^{(1)} do not generate a Lagrangean ideal in the non-integrable almost complex case, indeed

{ζ_{j}^{(1)}, ζ_{k}^{(1)}} = ⟨ξ, [Z̄_{j}(x), Z̄_{k}(x)]⟩ .  (15)

However we do have that

{ζ_{j}^{(1)}, ζ_{k}^{(1)}} = {⟨ξ, Z̄_{j}(x)⟩, ⟨ξ, Z̄_{k}(x)⟩} = 0 on Σ.

Indeed, for (x, ξ) ∈ Σ, we have ξ = rα_{x} for some r > 0 so that

\begin{matrix}\{⟨ξ, Z̄_{j}(x)⟩, ⟨ξ, Z̄_{k}(x)⟩\} = rα_{x}([Z̄_{j}(x), Z̄_{k}(x)]) \\ = rdα_{x}(Z̄_{j}(x), Z̄_{k}(x)) = rπ^{∗}ω(Z̄_{j}(x), Z̄_{k}(x)) = 0 \end{matrix}  (16)

since {Z̄_{j}} forms a Lagrangean subspace for the horizontal symplectic form π^{∗}ω. Here, π : X → M is the natural projection. Moreover if we choose the local horizontal vector fields Z_{j} to be orthonormal relative to π^{∗}ω, then we also have:

\begin{matrix}\{ζ_{j}^{(1)}, ζ̄_{k}^{(1)}\}(x, ξ) = ⟨ξ, [Z̄_{j}(x), Z_{k}(x)]⟩ = rπ^{∗}ω(Z̄_{j}(x), Z_{k}(x)) \\ = irδ_{j}^{k} = iδ_{j}^{k}p_{θ}(x, ξ) ,  (x, ξ) ∈ Σ . \end{matrix}  (17)

Here, p_{θ}(x, ξ) = ⟨ξ, \frac{∂}{∂θ}⟩.

Finally, we have

q = ∑_{j=1}^{m} |⟨ξ, Z_{j}⟩|^{2} = ∑_{j=1}^{m} |ζ_{j}^{(1)}|^{2} .

Hence the second and third conditions on the ζ_{j} are satisfied by the functions ζ_{j}^{(1)}. Furthermore, equation (15) tells us that the first condition is satisfied to zero-th order for the ideal I_{1} = (ζ_{1}^{(1)}, …, ζ_{m}^{(1)}). In fact, let us precisely describe the error. We consider the orthonormal (relative to ω) vector fields Z_{j}^{M} = π_{∗}Z_{j} of type (1,0) on M. Recall that the Nijenhuis tensor is given by

N(V, W) = \frac{1}{2}([JV, JW] − [V, W] − J[V, JW] − J[JV, W]) .

Hence,

N(Z_{j}^{M}, Z_{k}^{M}) = (−1 − iJ)[Z_{j}^{M}, Z_{k}^{M}] = −2[Z_{j}^{M}, Z_{k}^{M}]_{(0,1)} =^{def} ∑_{p=1}^{m} N_{jk}^{p}Z̄_{p}^{M} .  (18)

We note that by definition,

N_{jk}^{p} = N_{kj}^{p} .  (19)

Furthermore, by the Jacobi identity

{{ζ_{j}, ζ_{k}}, ζ_{p}} + {{ζ_{p}, ζ_{j}}, ζ_{k}} + {{ζ_{k}, ζ_{p}}, ζ_{j}} = 0

applied to (x, α_{x}) ∈ Σ, we have

N_{jk}^{p} + N_{pj}^{k} + N_{kp}^{j} = 0 .  (20)

By (16) and (18), we have

{ζ_{j}^{(1)}, ζ_{k}^{(1)}} = ∑_{p=1}^{m} f_{p}^{1}ζ_{p}^{(1)} + ∑_{p=1}^{m} N̄_{jk}^{p}ζ̄_{p}^{(1)} .  (21)

We now argue, following [BoGu], that these functions can be successively modified to satisfy the same conditions to infinite order on Σ. The next step is to modify the functions

ASYMPTOTICS OF ALMOST HOLOMORPHIC SECTIONS ON SYMPLECTIC MANIFOLDS 13

ζ_{j}^{(1)} by quadratic terms so that they satisfy the conditions {ζ_{j}, ζ_{k}} ∈ I to first order and the condition q = ∑_{j} |ζ_{j}|^{2} to order 3 on Σ. So we try to construct new functions

ζ_{p}^{(2)} = ζ_{p}^{(1)} + R_{p} , R_{p} = ∑_{j,k} ν_{p}^{jk}ζ̄_{j}^{(1)}ζ̄_{k}^{(1)}

so that

{ζ_{j}^{(2)}, ζ_{k}^{(2)}} = ∑_{p} f_{p}^{2}ζ_{p}^{(2)} + ∑_{α_{1},α_{2}} μ_{jk}^{α_{1}α_{2}} ζ̄_{α_{1}}^{(2)}ζ̄_{α_{2}}^{(2)} ;  (22)
q = ∑_{p} v_{p}^{2}ζ_{p}^{(2)} + ∑_{α} φ_{p}^{α}ζ̄_{α_{1}}^{(2)}ζ̄_{α_{2}}^{(2)}ζ̄_{α_{3}}^{(2)}ζ̄_{α_{4}}^{(2)} , (α = (α_{1}, …, α_{4})) .  (23)

Let us now solve (22)–(23) for the ν_{p}^{jk}. First of all, we choose ν_{p}^{jk} = ν_{p}^{kj}. We have

{ζ_{j}^{(2)}, ζ_{k}^{(2)}} = ∑_{p=1}^{m} f_{p}^{1}ζ_{p}^{(1)} + ∑_{p=1}^{m} N̄_{jk}^{p}ζ̄^{(1)} + {ζ_{j}^{(1)}, R_{k}} − {ζ_{k}^{(1)}, R_{j}} mod I_{Σ}^{2} .

By (17), we have

{ζ_{j}^{(1)}, ζ̄_{k}^{(1)}} = iδ_{j}^{k}p_{θ} mod I_{Σ} ,  (24)

and thus

{ζ_{j}^{(1)}, R_{k}} = ∑_{p=1}^{m} 2iν_{k}^{pj}p_{θ}ζ̄_{p}^{(1)} mod I_{Σ}^{2} .

Therefore,

{ζ_{j}^{(2)}, ζ_{k}^{(2)}} = ∑_{p=1}^{m} f_{p}^{1}ζ_{p}^{(2)} + ∑_{p=1}^{m} (N̄_{jk}^{p} + 2i(ν_{k}^{pj} − ν_{j}^{pk})p_{θ}) ζ̄_{p}^{(1)} mod I_{Σ}^{2} .  (25)

Hence

N̄_{jk}^{p} = 2i(ν_{j}^{pk} − ν_{k}^{pj})p_{θ} on Σ ,

or equivalently,

ν_{j}^{pk} − ν_{k}^{pj} = \frac{i}{2p_{θ}} N̄_{jk}^{p} mod I_{Σ} .  (26)

On the other hand,

q = ∑_{p} |ζ_{p}^{(2)} − R_{p}|^{2} = ∑_{p} v_{p}^{2}ζ_{p}^{(2)} − R_{p}ζ̄_{p}^{(2)}
= ∑_{p} v_{p}^{2}ζ_{p}^{(2)} − ∑_{j,k,p} ν_{p}^{jk}ζ̄_{j}^{(2)}ζ̄_{k}^{(2)}ζ̄_{p}^{(2)} + ∑_{α} φ_{p}^{α}ζ̄_{α_{1}}^{(2)}ζ̄_{α_{2}}^{(2)}ζ̄_{α_{3}}^{(2)}ζ̄_{α_{4}}^{(2)} .

Hence (23) is equivalent to

ν_{p}^{jk} + ν_{k}^{pj} + ν_{j}^{kp} = 0 .  (27)

Using (19)–(20), we can solve the equations (24) and (26) to obtain

ν_{p}^{jk} = \frac{i}{6p_{θ}} (N̄_{pj}^{k} + N̄_{pk}^{j}) .  (28)

Indeed, the solution (28) is unique (modulo I_{Σ}) and hence the R_{p} are unique modulo I_{Σ}^{3}. In summary,

ζ_{p}^{(2)} = ζ_{p}^{(1)} + \frac{i}{3p_{θ}} ∑_{j,k} N̄_{pj}^{k}ζ̄_{j}^{(1)}ζ̄_{k}^{(1)} .  (29)

The passage from the n^{th} to the (n + 1)^{st} step is similar, and we refer to [BoGu, pp. 147– 149].

Remark: Define p_{θ}(x, ξ) = ⟨ξ, \frac{∂}{∂θ}⟩. Since the joint zero set of \{ζ_{1}, …, ζ_{m}\} equals Σ and since p_{θ} ≠ 0 on Σ − 0 it follows that {ζ_{1}, …, ζ_{m}, p_{θ}} is an elliptic system of symbols.

2.3. The complex canonical relation. We now construct the underlying complex canonical relation C and we show that C can be parametrized by a phase of the form tψ(x, y) as in the complex case (Theorem 2.3).

As mentioned above, C does not live in T^{∗}X × T^{∗}X but rather in its almost analytic extension T^{∗}X̃ × T^{∗}X̃. Since π : X → M is an S^{1} bundle over M, its complexification π̃ : X̃ → M̃ defines a ℂ^{∗} bundle over M̃. The connection form α has an (almost) analytic continuation to a connection α̃ to this bundle and we may split T X̃ = H̃⊕Ṽ, where Ṽ → T M̃ is the vertical subbundle of the fibration X̃ → M̃ and where H̃ → T M̃ is the kernel of α̃.

The (almost) complexification of T^{∗}X is of course T^{∗}(X̃). We denote the canonical symplectic form on T^{∗}X by σ and that on T^{∗}(X̃ by σ̃; the notation is consistent because it is the complexification of σ. The symplectic cone Σ complexifies to Σ̃ and it remains symplectic with respect to σ̃. It is given by {(x̃, λ̃α̃x̃) : λ̃ ∈ ℂ^{∗}}. We have a natural identification L^{∗} ⟺ Σ given by rx → (x, rα_{x}). We further note that the ℂ^{∗} bundle L^{∗} → M is the fiberwise complexification of the S^{1} bundle X → M, hence L^{∗} → M is the restriction of π̃ to π̃^{−1}(M). We will therefore view L^{∗} as a submanifold of X̃.

We let ζ̃_{j} be almost analytic extensions to X̃ of the functions ζ_{j}. We then define

J_{+} = {(x̃, ξ̃) ∈ T^{∗}X̃ : ζ̃_{j} = 0 ∀j} ,  (30)

which is an involutive manifold of T^{∗}X̃ with the properties:

(i) (J_{+})_{ℝ} = Σ
(ii) q|_{J_{+}} ∼ 0
(iii) \frac{1}{i}σ(u, ū) > 0, ∀u ∈ T (J_{+})^{⊥}
(iv) T_{ρ}(J_{+}) = T_{ρ}Σ̃ ⊕ W_{ρ}^{+}.

(31)

Here, W_{ρ}^{+} is the sum of the eigenspaces of F_{ρ}, the normal Hessian of q, corresponding to the eigenvalues {iλ_{j}}. The null foliation of J_{+} is given by the joint Hamilton flow of the ζ̃_{j}’s.

The following proposition, proved in [MenSj] and in ([BoGu]), Appendix, Lemma 4.5), adapts to our context and defines the complex canonical relation C:

Proposition 2.2. There exists a unique strictly positive almost analytic canonical relation C satisfying

diag(Σ) ⊂ C ⊂ J_{+} × \overline{J_{+}}.

Indeed,

C = \{(x̃, ξ̃, ỹ, η̃) ∈ J_{+} × \overline{J_{+}} : (x̃, ξ̃) ∼ (ỹ, η̃)\},  (32)

where ∼ is the equivalence relation of ‘belonging to the same leaf of the null foliation of J_{+}. Thus, C is the flow-out of its real points, diag(Σ), under the joint Hamilton flow of the ζ̃_{j}’s. It is clear from the description that C ∘ C = C^{∗} = C, i.e. that C is an idempotent canonical relation.

We now prove that C can be parametrized by a phase tψ(x, y) defined on X×X×ℝ^{+}. This is helpful in analyzing the scaling limit of Π_{N}(x, y). In the following we use local coordinates (z, λ) on L^{∗} coming from a choice of local coordinates z on M and a local frame e_{L}(z) of L, and a corresponding local trivialization (z̃, λ) of X̃ → M̃. As before, we let a = ∥e_{L}^{∗}∥^{2}.

Theorem 2.3. There exists a unique regular phase function tψ(x, y) ∈ C^{∞}(X × X × ℝ^{+}) of positive type such that:

• id_{x}ψ|_{x=y} = −id_{y}ψ|_{x=y} = α;

• The almost analytic extension ψ̃ ∈ C^{∞}(X̃ × X̃) of ψ has the form ψ̃(x̃, ỹ) = i(1 − λμ̄ã(z̃, w̃)) with ã(z, z) = a(z) and ã(z̃, w̃) = \overline{ã(w̃, z̃)};

• tψ̃ parametrizes C.

Proof. We need to construct a function a(z, w) so that tψ̃ as above parametrizes the canonical relation C, i.e. that C is the image of the Lagrange immersion

\begin{matrix}ιψ̃ : C_{tψ̃} = ℝ^{+} × \{ψ̃ = 0\} → T^{∗}(X̃ × X̃) \\ (t, x̃, ỹ) ↦ (x̃, tdx̃ψ̃; ỹ, −tdỹψ̃) \end{matrix}  (33)

Since C is the unique canonical relation satisfying diag(Σ) ⊂ C ⊂ J_{+} × \overline{J_{+}}, the conditions that tψ̃ parametrize C are the following:

i) {(x, y) ∈ X × X : ψ(x, y) = 0} = diag(X); ii) d_{x}ψ|_{x=y} = −d_{y}ψ|_{x=y} = rα for x, y ∈ X and for some function r(x) > 0; iii) ζ̃_{j}(x̃, dx̃ψ̃) = 0 = ζ̃_{j}(ỹ, dỹψ̃) on {ψ̃ = 0}. Such a ψ̃ is not unique, so we require that r ≡ 1 in condition (ii), i.e.,

d_{x}ψ|_{x=y} = −d_{y}ψ|_{x=y} = α .

Suppose we have ψ̃(x̃, ỹ) = i(1 − λμ̄ã(z̃, w̃)). We observe that

ψ̃ = 0 ⟺ ã(z̃, w̃) = (λμ̄)^{−1},

and hence

\begin{matrix}idx̃ψ̃ = μ̄ã(z̃, w̃)dλ + λμ̄dz̃π^{∗}ã(z̃, w̃) \\ = λ^{−1}dλ + ã^{−1}dz̃ã(z̃, w̃) ⟺ ψ̃ = 0. \end{matrix}  (34)

The conditions on a are therefore:

a(z, w)λμ̄ = 1 ⟺ (z, λ) = (w, μ) ∈ X;

(a^{−1}d_{z}a + λ^{−1}dλ)|_{diag(X)} = −(a^{−1}d_{w}a + λ^{−1}dλ)|_{diag(X)} = α

ζ̃_{j} (z̃, λ, λ^{−1}dλ + ã^{−1}dz̃ã(z̃, w̃)) = 0 = ζ̃_{j} (w̃, μ̃, μ^{−1}dμ + ã^{−1}dw̃ã(z̃, w̃)) , ∀(z, w, λ, μ)

A solution a(z, w) satisfying the first condition must satisfy a(z, z)|λ|^{2} = 1 on X, so that a(z, z)|λ|^{2} is the local hermitian metric on L^{∗} with unit bundle X, i.e. a(z, z) = a(z).

We now prove that these conditions have a unique solution near the diagonal. We do this by reducing the canonical relation C by the natural S^{1} symmetry. The reduced relation C_{r} has a unique generating function log a; the three conditions above on a will follow automatically from this fact.

The S^{1} action of X lifts to T^{∗}X as the Hamiltonian flow of the function p_{θ}(x, ξ) := ⟨ξ, \frac{∂}{∂θ}⟩. The ζ_{j} are invariant under this S^{1} action, hence

{p_{θ}, ζ_{j}} = 0 ∀j.  (35)

Now consider the level set {p_{θ} = 1} ⊂ T^{∗}X. Dual to the splitting T X = H ⊕ V we get a splitting T^{∗}X = H^{∗} ⊕ V^{∗}, where

V^{∗}(X) = ℝα = H^{o}, H^{∗}(X) = V^{o}

where E^{o} denotes the annihilator of a subspace E, i.e. the linear functionals which vanish on E. Thus, p_{θ} = 0 on the horizontal space H^{∗}(X) and p_{θ}(α) = 1. Since p_{θ} is linear on the fibers of T^{∗}X, the set {p_{θ} = 1} has the form {α + h : h ∈ H^{∗}(X)}. We also note that p_{θ}(dθ) = 1 in the local coordinates (z, θ) on X defined by λ = e^{iθ}. Hence {p_{θ} = 1} may also be identified with {dθ + h : h ∈ H^{∗}(X)}.

Since {p_{θ} = 1} is a hypersurface, its null-foliation is given by the orbits of the Hamiltonian flow of p_{θ}, i.e. by the S^{1} action. We use the term ‘reducing by the S^{1}-action’ to mean setting p_{θ} = 1 and then dividing by this action. The reduction of T^{∗}X is thus defined by (T^{∗}X)_{r} = p_{θ}^{−1}(1)/S^{1}. Since p_{θ}^{−1}(1) is an affine bundle over X with fiber isomorphic to H^{∗}(X) ≈ T^{∗}M, it is clear that (T^{∗}X)_{r} ≈ T^{∗}M as vector bundles over M. We can obtain a symplectic equivalence using the local coordinates (z, θ) on X. Let (p_{z}, p_{θ}) be the corresponding symplectically dual coordinates, so that the natural symplectic form σ_{T^{∗}X} on T^{∗}X is given by σ_{T^{∗}X} = dz ∧ dp_{z} + dθ ∧ dp_{θ}. The notation p_{θ} is consistent with the above. Moreover, the natural symplectic form on T^{∗}M is given locally by σ_{T^{∗}M} = dz ∧ dp_{z}. Now define the projection

χ : p_{θ}^{−1}(1) → T^{∗}M, χ(z, p_{z}, 1, p_{θ}) = (z, p_{z}).

This map commutes with the S^{1} action and hence descends to the quotient to define a local map over U, still denoted χ, from (T^{∗}X)_{r} → T^{∗}M. Clearly χ is symplectic.

We now reduce the canonical relation C. Thus we consider the ℂ^{∗} × ℂ^{∗} action on T^{∗}X̃ × T^{∗}X̃ − 0 generated by p_{θ}(x, ξ), p_{θ}(y, η). The reduction of C is given by

C_{r} = C ∩ (p_{θ} × p_{θ})^{−1}(1, 1)/ℂ^{∗} × ℂ^{∗}.

We then use χ × χ to identify C_{r} with a (non-homogeneous) positive canonical relation in T^{∗}(M̃ × M̃). Thus in coordinates,

C_{r} = {(z̃, p̃_{z}, w̃, p̃_{w}) ∈ T^{∗}(M̃ × M̃) : ∃λ, μ, (z̃, λ, p̃_{z}, 1; w̃, μ, p̃_{w}, 1) ∈ C}.  (36)

Since reduction preserves real points, it is clear that

(C_{r})_{ℝ} = C_{ℝ} ∩ (p_{θ} × p_{θ})^{−1}(1, 1)/ℂ^{∗} × ℂ^{∗}
= {(z, p_{z}, z, p_{z}) ∈ diag(T^{∗}(M × M)) : ∃θ such that α_{z,e^{iθ}} = dθ + p_{z}}.

Let us denote by ζ̃_{jr} the reductions of the functions ζ̃_{j} by the S^{1} symmetry. Then ζ̃_{jr} = 0 on either pair of cotangent vectors in C_{r}. Moreover, by the uniqueness statement on C it follows that C_{r} is the unique canonical relation in T^{∗}(M̃ × M̃) with the given set of real points and in the zero set of the ζ̃_{jr}’s.

We now observe that C_{r} has, at least near the diagonal, a unique global generating function. This holds because the natural projection

C_{r} ⊂ T^{∗}(M̃ × M̃) → M̃ × M̃  (37)

is a local diffeomorphism near the diagonal. Indeed, its derivative gives a natural isomorphism

T_{ρ,ρ}C_{r} ≈ H^{∗} ⊕ H^{∗} ≈ T (M̃ × M̃) .  (38)

Therefore, there exists a global generating function log ã ∈ C^{∞}(M̃ × M̃) i.e.

C_{r} = {(z̃, dz̃ log ã, w̃, dw̃ log ã), z̃, w̃ ∈ M̃}.  (39)

Since C^{∗} = C it follows that C_{r}^{∗} = C^{r} and hence that a(w, z) = \overline{a(z, w)}.

Working backwards, we find that the function ψ̃(x̃, ỹ) = i(1 − λμ̄ã(z̃, w̃)) satisfies the equations ζ̃_{j}(x̃, dx̃ψ̃) = ζ̃_{j}(ỹ, dỹψ̃) = 0 on ψ̃ = 0. Therefore the Lagrange immersion

\begin{matrix}iψ̃ : C_{tψ̃} = ℝ^{+} × \{ψ̃ = 0\} → T^{∗}(X̃ × X̃) \\ (t, x̃, ỹ) → (x̃, tdx̃ψ̃; ỹ, −tdỹψ̃) \end{matrix}  (40)

takes its image inside J_{+} × \overline{J_{+}} and reduces to C_{r} under the S^{1}-symmetry. To conclude the proof it is only necessary to show that the real points of the image of iψ̃ equal diag(Σ). We know however that these real points reduce to (C_{r})_{ℝ} and hence that z = w at real points. But we have

1 = λμ̄a(z, w) = e^{i(θ−φ)} \frac{a(z, w)}{\sqrt{a(z)}\sqrt{a(w)}}, on \{ψ̃ = 0\}

hence when z = w we have e^{i(θ−φ)} = 1 and hence x = y. Since dx̃ψ̃(x, y)|_{x=y} = α_{x}, it follows that the real points indeed equal diag(Σ). Therefore tψ̃ parametrizes C.

To show that the phase is of positive type, we need to describe the asymptotics of a(z, w) near the diagonal. Note that in the almost-complex case, we cannot describe a(z, w) as the almost analytic extension of a(z, z). (Of course, ã(z̃, w̃) is the almost analytic extension of a(z, w), by definition.) For our near-diagonal asymptotics in the nonintegrable case, we instead use the following second order expansion of a at points on the diagonal:

Lemma 2.4. Suppose that (z_{1}, …, z_{m}) are preferred coordinates and e_{L} is a preferred frame at a point P_{0} ∈ M. Then the Taylor expansion of a(z, w) at z = w = 0 is

a(z, w) = 1 + z · w̄ + · · · .

Proof. To begin, we recall that a(0, 0) = a(0) = ∥e_{L}^{∗}(P_{0})∥^{2} = 1. To compute the first and second order terms, we return to the equation

ζ_{j}(z, λ, \frac{dλ}{λ} + d_{z} log a(z, w)) = 0, ∀(z, λ; w) ∈ X × M.  (41)

Let us write ζ_{j} = ζ_{j}^{(1)} + R_{j}^{(2)}, where R_{j}^{(2)} vanishes to second order on Σ and we recall that ζ_{j}^{(1)}(ξ) = (Z̄_{j}, ξ). Let us also Taylor expand log a:

log a = L(z, w) + Q(z, w) + ⋯,

where L is linear and Q is quadratic. Since e_{L} is a preferred frame at P_{0}, it follows from (4) that a(z, z) = 1 + |z|^{2} + ⋯ and hence

L(z, z) = 0 ,  Q(z, z) = |z|^{2} .  (42)

Since d_{z} log a|_{z=w} + \frac{dλ}{λ} = α ∈ Σ, it follows from (41) that

ζ_{j}^{(1)}(z, λ, \frac{dλ}{λ} + d_{z} log a) = −R_{j}^{(2)}(z, λ, \frac{dλ}{λ} + d_{z} log a) = O(|z − w|^{2}).  (43)

Since a(z, w) = \overline{a(w, z)}, we can write

L(z, w) = ∑_{j=1}^{m}(b_{j}z_{j} + c_{j}z̄_{j} + c̄_{j}w_{j} + b̄_{j}w̄_{j}) .

Since the z_{j} are preferred coordinates and e_{L} is a preferred frame at P_{0}, we can choose the Z̄_{j} so that Z̄_{j}(0) = \frac{∂}{∂z̄_{j}} and hence by (43),

0 = ζ_{j}^{(1)} (z, λ, \frac{dλ}{λ} + d_{z} log a)|_{z=w=0,λ=1} = (\frac{∂}{∂z̄_{j}} , d_{z} log a)|_{(0,0)} = c_{j} ∀j .

Since L(z, z) = 0, we have b_{j} + c̄_{j} = 0, and hence L = 0.

To investigate the quadratic term Q in (42), we write

(\frac{dλ}{λ} + d_{z} log a)|_{(z,w)} = α_{z} + ∑_{j=1}^{m} [z_{j}U′_{j} + z̄_{j}U′′_{j} + w_{j}V′_{j} + w̄_{j}V′′_{j}] + O(|z|^{2} + |w|^{2}) , (44)

where

\begin{aligned}U′_{j} & = & ∑_{k=1}^{m} (\frac{∂^{2}Q}{∂z_{j}∂z_{k}} dz_{k} + \frac{∂^{2}Q}{∂z_{j}∂z̄_{k}} dz̄_{k}) , & U′′_{j} & = & ∑_{k=1}^{m} (\frac{∂^{2}Q}{∂z̄_{j}∂z_{k}} dz_{k} + \frac{∂^{2}Q}{∂z̄_{j}∂z̄_{k}} dz̄_{k}) , \\ V′_{j} & = & ∑_{k=1}^{m} (\frac{∂^{2}Q}{∂w_{j}∂z_{k}} dz_{k} + \frac{∂^{2}Q}{∂w_{j}∂z̄_{k}} dz̄_{k}) , & V′′_{j} & = & ∑_{k=1}^{m} (\frac{∂^{2}Q}{∂w̄_{j}∂z_{k}} dz_{k} + \frac{∂^{2}Q}{∂w̄_{j}∂z̄_{k}} dz̄_{k}) . \end{aligned}

Applying ζ_{k}^{(1)} to (44) and using (43) and the fact that ζ_{k}^{(1)}(α_{z}) = 0, we have

∑_{j=1}^{m} [z_{j}(Z̄_{k}|_{z}, U′_{j}) + z̄_{j}(Z̄_{k}|_{z}, U′′_{j}) + w_{j}(Z̄_{k}|_{z}, V′_{j}) + w̄_{j}(Z̄_{k}|_{z}, V′′_{j})] = O(|z|^{2} + |w|^{2}) . (45)

By (1) and (6),

Z̄_{k}|_{z} = \frac{∂}{∂z̄_{k}} + ∑_{l=1}^{m} B_{kl}(z)\frac{∂}{∂z_{l}} + C_{k}(z)\frac{∂}{∂θ} , B_{kl}(0) = 0 .

Hence by (45),

\frac{∂^{2}Q}{∂z_{j}∂z̄_{k}} = (\frac{∂}{∂z̄_{k}} , U′_{j}) = (Z̄_{k}|_{0}, U′_{j}) = 0 .

Similarly, \frac{∂^{2}Q}{∂z̄_{j}∂z̄_{k}} = \frac{∂^{2}Q}{∂w_{j}∂z̄_{k}} = \frac{∂^{2}Q}{∂w̄_{j}∂z̄_{k}} = 0. Thus Q(z, w) has no terms containing z̄_{k}. Since

Q(z, w) = \overline{Q(w, z)}, the quadratic function Q also has no terms containing w_{k}, so we can write

Q(z, w) = B(z, z) + H(z, w̄) + \overline{B(w, w)} ,

where B, resp. H, is a bilinear, resp. hermitian, form on ℂ^{m}. Since Q(z, z) = |z|^{2} (recall (42)), we conclude that B(z, z) = 0 and hence Q(z, w) = H(z, w̄) = z · w̄.

We now complete the proof that the phase is of positive type; i.e., ℑψ ≥ 0 on some neighborhood of the diagonal in X × X. Let x ∈ X be arbitrary and choose Heisenberg coordinates (z, θ) at P_{0} = π(x) (so that x has coordinates (0, 0)). Recalling that λ = a(z)^{− \frac{1}{2}} e^{iθ} on X, we have by Lemma 2.4,

\frac{1}{i}ψ(0, 0; z, θ) = 1 − \frac{a(0, z)}{\sqrt{a(z)}}e^{−iθ} = (1 − e^{−iθ}) + e^{−iθ} [\frac{1}{2}|z|^{2} + O(|z|^{3})] .

Thus,

ℜ [\frac{1}{i}ψ(0, 0; z, θ)] ≥ 0 for |θ| < \frac{π}{2} , |z| < ε ,

where ε is independent of the point P_{0} ∈ M.

2.4. The Szegö projector. Having defined C, we denote by I^{∗}(X × X, C) the space of complex Fourier integral operators with wave fronts along C, as described in §2.1. We define a Szegö projector Π associated to Σ and C to be a self-adjoint projection Π ∈ I^{0}(X × X, C) with principal symbol σ_{Π} equal to the canonical 1/2-density of C. Our purpose now is to describe the method of [BoSj, §4] [BoGu] (Appendix) for producing a canonical Szegö projector modulo finite rank operators. We refer there for further discussion of the method.

Since by Theorem 2.3, C is parametrized by a function of the form tψ(x, y), the space I^{n}(X × X, C) consists of Fourier integral operators of the form

∫_{0}^{∞} e^{itψ(x,y)}s(x, y, t)dt, s ∼ ∑_{k=0}^{∞} t^{m+n−k}s_{k}(x, y) ∈ S^{m+n}(X × X × ℝ^{+}) ,

where s_{k} is a smooth function of (x, y) defined near x = y. Because C is an equivalence relation, I^{∗}(X × X, C) is a ∗-algebra. Hence there is an induced algebra product (i.e., a ∗-product) defined on the amplitudes by

I(q_{1}, ψ) ∘ I(q_{2}, ψ) ∼ I(q_{1} ∗ q_{2}, ψ),  (46)

where ∼ means roughly that the difference is a smoothing operator. To be more precise, one uses the method of stationary phase as in [BoSj, §4, (4.14)] to determine the symbol expansion of the ∗-product ∗ : I^{n_{1}} × I^{n_{2}} → I^{n_{1}+n_{2}} as a formal infinite series. We do not wish to consider its convergence, hence we will use ∼ in the weaker sense that the two sides agree up to any desired order I^{−K} with a well-defined remainder of lower order. The leading order term in the ∗-product is simply the product of the principal symbols.

The first step in constructing a Szegö projector Π is to construct an approximate projector S, satisfying

S^{2} ∼ S ∼ S^{∗}, S ∈ I^{0}(X × X, C), s_{0}(x, x) = π^{−m} det L_{X}||dρ|| (mod ψ). (47)

The very first term of s ∗ s ∼ s gives the equation ([BoSj], (4.10))

s_{0} = π^{−m}(h_{φ})^{− \frac{1}{2}} mod ψ,  (48)

where h_{φ} is the (non-vanishing) Hessian determinant in (σ, w) of the function φ(x, y, w, σ) = ψ(x, w) + σψ(w, y). On the diagonal x = y one has (h_{φ})^{\frac{1}{2}} = (det L_{X})||dρ||.

We now observe that the amplitude of S is not unique, since we may add to it any amplitude of the form (D_{t}−ψ)a where D_{t} = \frac{∂}{i∂t}. This follows from the fact that (D_{t}−ψ)e^{itψ} = 0 and from the fact that we can integrate by parts in an oscillatory integral. We note that I(q+(D_{t}−ψ)a, ψ) = I(q, ψ) so the product descends to the quotient by the ideal (D_{t}−ψ)S^{M}. We abbreviate the statement I(q+(D_{t}−ψ)a, ψ) = I(q, ψ) by q ≡ q+(D_{t}−ψ)a. Since we may absorb the terms ψ(s_{0} − π^{−m}(h_{φ})^{− \frac{1}{2}} ) into t^{m−1}s_{1}, the difference being of the form (D_{t} − ψ)a, we may take s_{0} to be defined by (48).

We now apply an argument given in [BoSj] to determine the rest of the amplitude s from s_{0} and by the equations s^{∗} = s ∼ s ∗ s. Indeed, since S^{2} ∼ S ∼ S^{∗}, there exists r ∈ S^{−1}(X × X × ℝ^{+}) with

s_{0} ∗ s_{0} = s_{0} + r.  (49)

Clearly, r ∗ s_{0} = s_{0} ∗ r. We now construct E(r) so that

s = s_{0} + E(r) ⟹ s ∗ s ∼ s, s^{∗} = s.  (50)

Omitting the ∗ since the product is commutative, we get:

\begin{matrix}(2s_{0} − 1)E(r) + E(r)^{2} = −r ⟹ 2\sqrt{r + \frac{1}{4}}E(r) + E(r)^{2} = −r \\ ⟹ E(r) ∼ \frac{1}{2} − \sqrt{r + \frac{1}{4}}. \end{matrix}  (51)

The right hand side is a formal power series in ∗ products of r. Given a prescribed symbol order in advance, we may truncate the ∗ product to obtain a convergent expansion for E(r) plus a remainder which is lower than the prescribed order. The resulting amplitude s will satisfy s^{∗} = s ∼ s ∗ s to the prescribed order and hence be an amplitude for an approximate Szegö projector S. Thus, for any K we can construct an approximate Szegö projector S_{K} such that S_{K}^{2} = S_{K} + E_{K} = S_{K}^{∗} + E_{K}^{∗} where E_{K}(x, y) lies in C^{K}(X × X). As discussed in [BoSj], one can in fact produce an approximate projector modulo smoothing operators.

We then modify S by a smoothing operator E to obtain a true projection (see [BoGu], Appendix A.4): Since S^{2} ∼ S and S = S^{∗}, it follows that the spectrum of S is concentrated near {0, 1}. Let U_{0}, U_{1} be disjoint open sets containing the points of the spectrum near 0, 1, respectively, and let F be the analytic (locally constant) function on U_{0} ∪ U_{1} given by F (U_{0}) = {0}, F (U_{1}) = {1}. Hence F (S) := Π is a true projection. More concretely, since S is self-adjoint it has an eigenfunction expansion S(x, y) = ∑_{n=1}^{∞} λ_{n}f_{n}(x)f_{n}(y). We may collect (modulo a finite dimensional ambiguity) the eigenvalues which cluster at {0} and those which cluster at {1} to obtain S(x, y) = ∑_{n:λ_{n}∈U_{0}} λ_{n}f_{n}(x)f_{n}(y) + ∑_{n:λ_{n}∈U_{1}} λ_{n}f_{n}(x)f_{n}(y). Then F (S)(x, y) = ∑_{n:λ_{n}∈U_{1}} f_{n}(x)f_{n}(y).

To summarize the above discussion, a Szegö projector Π can be written in the form:

\begin{matrix}Π(x, y) = S(x, y) + E(x, y) , \\ S(x, y) = ∫_{0}^{∞} e^{itψ(x,y)}s(x, y, t)dt ,  E(x, y) ∈ C^{∞}(X × X) , \end{matrix}  (52)

where ψ is given by Theorem 2.3 and s ∼ ∑_{k=0}^{∞} t^{m−k}s_{k}(x, y) ∈ S^{m}(X×X×ℝ^{+}) is constructed as above. Although Π is not unique, the above construction defines a canonical choice of Π modulo smoothing operators. In the complex case, the construction produces the usual Szegö projector Π onto the kernel of ¯∂_{b}, and (52) is the Boutet de Monvel-Sjöstrand oscillatory integral formula for it ([BoSj, Th. 1.5 and §2.c]).

2.5. Almost holomorphic sections. In the complex case, a holomorphic section s of L^{N} lifts to a ŝ ∈ L_{N}^{2}(X) which satisfying ¯∂_{b}ŝ = 0. The operator ¯∂_{b} extends to a complex satisfying ¯∂_{b}^{2} = 0, which is a necessary and sufficient condition for having a maximal family of CR holomorphic coordinates. In the non-integrable case ¯∂_{b}^{2} ≠ 0, and there may be no local solutions of ¯∂_{b}f = 0. We now quote a result of Boutet-de-Monvel and Guillemin which replaces ¯∂_{b} with a pseudodifferential operator D̄_{0} so that D̄_{0}Π = 0. Indeed, Boutet de Monvel [Bout] and Boutet de Monvel - Guillemin [BoGu] defined a complex D̄_{j}, which is a good replacement for ¯∂_{b} in the non-integrable case. Their main result is:

Theorem 2.5. (see [BoGu], Lemma 14.11 and Theorem A 5.9) There exists an S^{1}-invariant complex of first order pseudodifferential operators D̄_{j} over X

0 → C^{∞}(Λ_{b}^{0,0}) →^{D̄_{0}} C^{∞}(Λ_{b}^{0,1}) →^{D̄_{1}} ⋯ ⟶^{D̄_{m−1}} C^{∞}(Λ_{b}^{0,m}) → 0 ,

where Λ_{b}^{0,j} = Λ^{j}(H^{0,1}X)^{∗}, such that:

i) σ(D̄_{j}) = σ(¯∂_{b}) to second order along Σ := {(x, rα_{x}) : x ∈ X, r > 0} ⊂ T^{∗}X; ii) The Szegö kernel Π is the orthogonal projector onto the kernel of D̄_{0}; iii) (D̄_{0}, \frac{∂}{∂θ}) is jointly elliptic.

Let us briefly summarize the construction of D̄_{0} (following [BoGu, Appendix]). We begin with any S^{1}-equivariant symmetric first order pseudodifferential operator D̄′_{0} with principal symbol equal to ∑_{j=1}^{m} ζ_{j}ϑ̄_{j}. Then D̄′_{0}Π is of order ≤ 0 so one may find a zeroth order pseudodifferential operator Q_{0} such that D̄′_{0}Π ∼ Q_{0}Π (modulo smoothing operators). Then put: D̄_{0} = (D̄′_{0} − Q_{0}) − (D̄′_{0} − Q_{0})Π. Clearly, D̄_{0}Π = 0 and σ(D̄_{0}) = σ(D̄′_{0}) = ∑_{j=1}^{m} ζ_{j}ϑ̄_{j}.

The characteristic variety of D̄_{0} is then equal to Σ. Since p_{θ} is the symbol of \frac{∂}{∂θ} and since the system \{σD̄_{0}, p_{θ}\} has no zeros in T^{∗}X − 0 it follows that \{D̄_{0}, \frac{∂}{∂θ}\} is an elliptic system.

One can then construct the higher D̄_{j} recursively so that D̄_{j}D̄_{j−1} = 0. We refer to [BoGu], Appendix §5, for further details.

We refer to the kernel H^{2}(X) = ker D̄_{0} ∩ L^{2}(X) as the Hardy space of square-integrable ‘almost CR functions’ on X. The L^{2} norm is with respect to the inner product

⟨F_{1}, F_{2}⟩ = \frac{1}{2π} ∫_{X} F_{1}\overline{F_{2}}dV_{X} , F_{1}, F_{2} ∈ L^{2}(X) ,  (53)

where

dV_{X} = \frac{1}{m!}α ∧ (dα)^{m} = α ∧ π^{∗}dV_{M} .  (54)

The S^{1} action on X commutes with D̄_{0}; hence H^{2}(X) = ⨁_{N=0}^{∞} H_{N}^{2}(X) where H_{N}^{2}(X) = {F ∈ H^{2}(X) : F (r_{θ}x) = e^{iNθ}F (x)}. We denote by H_{J}^{0}(M, L^{N}) the space of sections which corresponds to H_{N}^{2}(X) under the map s ↦ ŝ. Elements of H_{J}^{0}(M, L^{N}) are the almost holomorphic sections of L^{N}. (Note that products of almost holomorphic sections are not necessarily almost holomorphic.) We henceforth write ŝ = s and identify H_{J}^{0}(M, L^{N}) with H_{N}^{2}(X). Since (D̄_{0}, \frac{∂}{∂θ}) is a jointly elliptic system, elements of H_{J}^{0}(M, L^{N}) and H_{N}^{2}(X) are smooth. In many other respects, H_{J}^{0}(M, L^{N}) is analogous to the space of holomorphic sections in the complex case. Subsequent results will bear this out.

We let Π_{N} : L^{2}(X) → H_{N}^{2}(X) denote the orthogonal projection. The level N Szegö kernel Π_{N}(x, y) is defined by

Π_{N}F (x) = ∫_{X} Π_{N}(x, y)F (y)dV_{X}(y) , F ∈ L^{2}(X) .  (55)

It can be given as

Π_{N}(x, y) = ∑_{j=1}^{d_{N}} S_{j}^{N}(x)\overline{S_{j}^{N}(y)} ,  (56)

where S_{1}^{N}, …, S_{d_{N}}^{N} form an orthonormal basis of H_{N}^{2}(X).

Remark: The results stated here use only the D̄_{0} term of the complex; its kernel consists of the spaces of almost holomorphic sections of the powers L^{N} of the line bundle L, as explained below. The complex D̄_{j} was used by Boutet de Monvel -Guillemin [BoGu, Lemma 14.14] to show that the dimension of H_{J}^{0}(M, L^{N}) or H_{N}^{2}(X) is given by the Riemann-Roch formula (for N sufficiently large). For our results, we need only the leading term of Riemann-Roch, which we obtain as a consequence of Theorem 2(a). (The reader should be warned that the symbol is described incorrectly in Lemma 14.11 of [BoGu]. However, it is correctly described in Theorem 5.9 of the Appendix to [BoGu] and also in [GuUr]).

3. Scaling asymptotics for Szegö kernels

In [BSZ1, Theorem 3.1], we showed that in the complex case, the scaled Szegö kernel Π_{N} near the diagonal is asymptotic to the Szegö kernel Π_{1}^{H} of level one for the reduced Heisenberg group, given by

Π_{1}^{H}(z, θ; w, φ) = \frac{1}{π^{m}} e^{i(θ−φ)+iℑ(z·w̄)− \frac{1}{2} |z−w|^{2}} = \frac{1}{π^{m}} e^{i(θ−φ)+z·w̄− \frac{1}{2} (|z|^{2}+|w|^{2})} .  (57)

The method was to apply the Boutet de Monvel-Sjöstrand oscillatory integral formula

Π(x, y) ∼ ∫_{0}^{∞} e^{itψ(x,y)}s(x, y, t)dt mod C^{∞}(X × X)

arising from a parametrix construction (recall (52)).

Our goal now is to show that the universal asymptotic formula of [BSZ1] for the neardiagonal scaled Szegö kernel holds for the symplectic case (Theorem 3.1). In fact, our description adds some quite useful details to the formula given in [BSZ1].

We have shown that Szegö kernels can be expressed in the form Π(x, y) = S(x, y)+E(x, y), where S is the Fourier integral kernel in (52) and where E ∈ C^{∞}(X × X) is the remainder. We denote the N^{th} Fourier coefficient of these operators relative to the S^{1} action by Π_{N} = S_{N} + E_{N}. Since E is smooth, we have E_{N}(x, y) = O(N^{−∞}), where O(N^{−∞}) denotes a quantity which is uniformly O(N^{−k}) on X × X for all positive k. The Fourier coefficients S_{N} are given by

S_{N}(x, y) = ∫_{0}^{2π} e^{−iNθ}S(r_{θ}x, y)dθ = ∫_{0}^{∞} ∫_{0}^{2π} e^{−iNθ}e^{itψ(r_{θ}x,y)}s(r_{θ}x, y, t)dθdt  (58)

where r_{θ} denotes the S^{1} action on X. Changing variables t ↦ Nt gives

S_{N}(x, y) = N ∫_{0}^{∞} ∫_{0}^{2π} e^{iN(−θ+tψ(r_{θ}x,y))}s(r_{θ}x, y, Nt)dθdt .  (59)

We now determine the scaling limit of the Szegö kernel by continuing the argument of [BSZ1], and adding some new details on homogeneities which will be useful in our applications. To describe the scaling limit at a point x_{0} ∈ X, we choose a Heisenberg chart ρ : U, 0 → X, x_{0} centered at P_{0} = π(x_{0}) ∈ M. Recall (§1.2) that choosing ρ is equivalent to choosing preferred coordinates centered at P_{0} and a preferred local frame e_{L} at P_{0}. We then write the Szegö kernel Π_{N} in terms of these coordinates:

Π_{N}^{P_{0}}(u, θ; v, φ) = Π_{N}(ρ(u, θ), ρ(v, φ)) ,

where the superscript P_{0} is a reminder that we are using coordinates centered at P_{0}. (We remark that the function Π_{N}^{P_{0}} depends also on the choice of preferred coordinates and preferred frame, which we omit from the notation.) The first term in our asymptotic formula below says that the N^{th} scaled Szegö kernel looks approximately like the Szegö kernel of level one for the reduced Heisenberg group (recall (57)):

Π_{N}^{P_{0}}(\frac{u}{\sqrt{N}} , \frac{θ}{N}; \frac{v}{\sqrt{N}} , \frac{φ}{N}) ≈ Π_{1}^{H}(u, θ; v, φ) = \frac{1}{π^{m}} e^{i(θ−φ)+iℑ(u·v̄)− \frac{1}{2} |u−v|^{2}} .

In the following, we shall denote the Taylor series of a C^{∞} function f defined in a neighborhood of 0 ∈ ℝ^{K} by f ∼ f_{0} + f_{1} + f_{2} + … where f_{j} is the homogeneous polynomial part of degree j. We also denote by R_{n}^{f} ∼ f_{n+1} + ⋯ the remainder term in the Taylor expansion.

The following is our main result on the scaling asymptotics of the Szegö kernels near the diagonal. Since the result is of independent interest, we state our asymptotic formula in a more precise form than is needed for the applications in this paper.

Theorem 3.1. Let P_{0} ∈ M and choose a Heisenberg coordinate chart about P_{0}. Then

\begin{matrix}N^{−m}Π_{N}^{P_{0}}(\frac{u}{\sqrt{N}} , \frac{θ}{N}; \frac{v}{\sqrt{N}} , \frac{φ}{N}) \\ = Π_{1}^{H}(u, θ; v, φ) [1 + ∑_{r=1}^{K} N^{−r/2}b_{r}(P_{0}, u, v) + N^{−(K+1)/2}R_{K}(P_{0}, u, v, N)] , \end{matrix}

where:

• b_{r} = ∑_{α=0}^{2[r/2]} ∑_{j=0}^{[3r/2]}(ψ_{2})^{α}Q_{r,α,3r−2j} , where Q_{r,α,d} is homogeneous of degree d and

ψ_{2}(u, v) = u · v̄ − \frac{1}{2}(|u|^{2} + |v|^{2}) ;

in particular, b_{r} has only even homogeneity if r is even, and only odd homogeneity if r is odd;

• ∥R_{K}(P_{0}, u, v, N)∥_{C^{j}({|u|≤ρ, |v|≤ρ}} ≤ C_{K,j,ρ} for j ≥ 0, ρ > 0 and C_{K,j,ρ} is independent of the point P_{0} and choice of coordinates.

Proof. Since Π_{N} = S_{N} + O(N^{−∞}), it suffices to give the asymptotics of S_{N}. Hence, we fix P_{0} and consider the asymptotics of

\begin{matrix}S_{N}^{P_{0}} (\frac{u}{\sqrt{N}} , 0; \frac{v}{\sqrt{N}} , 0) \\ = N ∫_{0}^{∞} ∫_{0}^{2π} e^{iN(−θ+tψ( \frac{u}{\sqrt{N}} ,θ; \frac{v}{\sqrt{N}} ,0))}s(\frac{u}{\sqrt{N}} , θ; \frac{v}{\sqrt{N}} , 0, Nt)dθdt , \end{matrix}  (60)

where ψ and s are the phase and symbol from (52) written in terms of the Heisenberg coordinates.

On X we have λ = a(z)^{− \frac{1}{2}} e^{iφ}. So for (x, y) = (z, φ, w, φ′) ∈ X × X, we have by Theorem 2.3,

ψ(z, φ, w, φ′) = i [1 − \frac{a(z, w)}{\sqrt{a(z)}\sqrt{a(w)}}e^{i(φ−φ′)}] .  (61)

It follows that

\begin{matrix}ψ(\frac{u}{\sqrt{N}} , θ; \frac{v}{\sqrt{N}} , 0) \\ = i [1 − \frac{a(\frac{u}{\sqrt{N}} , \frac{v}{\sqrt{N}})}{\sqrt{a(\frac{u}{\sqrt{N}} , \frac{u}{\sqrt{N}})}\sqrt{a(\frac{v}{\sqrt{N}} , \frac{v}{\sqrt{N}})}}e^{iθ}] . \end{matrix}  (62)

We observe that the asymptotic expansion of a function f(\frac{u}{\sqrt{N}} , \frac{v}{\sqrt{N}}) in powers of N^{− \frac{1}{2}} is just the Taylor expansion of f at u = v = 0. By Lemma 2.4 and the notational convention established above, we have

a(\frac{u}{\sqrt{N}} , \frac{v}{\sqrt{N}}) = 1 + \frac{1}{N} u · v̄ + R_{3}^{a}(\frac{u}{\sqrt{N}} , \frac{v}{\sqrt{N}}) ,  R_{3}^{a}(\frac{u}{\sqrt{N}} , \frac{v}{\sqrt{N}}) = O(N^{−3/2}) . (63)

The phase in (60)

Ψ̃ := tψ(\frac{u}{\sqrt{N}} , θ; \frac{v}{\sqrt{N}} , 0) − θ = it [1 − \frac{a(\frac{u}{\sqrt{N}} , \frac{v}{\sqrt{N}})}{a(\frac{u}{\sqrt{N}} , \frac{u}{\sqrt{N}})^{\frac{1}{2}} a(\frac{v}{\sqrt{N}} , \frac{v}{\sqrt{N}})^{\frac{1}{2}}} e^{iθ}] − θ  (64)

then has the asymptotic N-expansion

Ψ̃ = it[1 − e^{iθ}] − θ − \frac{it}{N} ψ_{2}(u, v)e^{iθ} + tR_{3}^{ψ}(\frac{u}{\sqrt{N}} , \frac{v}{\sqrt{N}})e^{iθ} .  (65)

We use a smooth partition of unity {ρ_{1}(t), ρ_{2}(t)} to decompose the integral (60) into one over 0 < t < 3 and one over t > 2:

S_{N}^{P_{0}}(\frac{u}{\sqrt{N}} , 0; \frac{v}{\sqrt{N}} , 0) = I_{1} + I_{2} ,
I_{1} = N ∫_{0}^{3} ∫_{0}^{2π} e^{iNΨ̃}ρ_{1}(t)s(\frac{u}{\sqrt{N}} , θ; \frac{v}{\sqrt{N}} , 0, Nt)dθdt ,  (66)
I_{2} = N ∫_{2}^{∞} ∫_{0}^{2π} e^{iNΨ̃}ρ_{2}(t)s(\frac{u}{\sqrt{N}} , θ; \frac{v}{\sqrt{N}} , 0, Nt)dθdt .  (67)

To evaluate I_{1}, we absorb (−iψ_{2} + NR_{3}^{ψ})te^{iθ} into the amplitude (as in [BSZ1]), so that we view I_{1} as an oscillatory integral with phase

Ψ(t, θ) := it(1 − e^{iθ}) − θ  (68)

and with amplitude

A(t, θ; P_{0}, u, v) := ρ_{1}(t)e^{te^{iθ}ψ_{2}(u,v)+ite^{iθ}NR_{3}^{ψ}( \frac{u}{\sqrt{N}} , \frac{v}{\sqrt{N}})}s(\frac{u}{\sqrt{N}} , θ; \frac{v}{\sqrt{N}} , 0, Nt);  (69)

i.e.,

I_{1} = N ∫_{0}^{3} ∫_{0}^{2π} e^{iNΨ(t,θ)}A(t, θ; P_{0}, u, v)dθdt  (70)

We evaluate (70) by the method of stationary phase as in [BSZ1]. The phase Ψ is independent of the parameters (u, v) and we have

\begin{matrix}\frac{∂}{∂t}Ψ = i(1 − e^{iθ}) \\ \frac{∂}{∂θ}Ψ = te^{iθ} − 1 \end{matrix}  (71)

so the critical set of the phase is the point {t = 1, θ = 0}. The Hessian Ψ′′ on the critical set equals

( \begin{matrix}0 & 1 \\ 1 & i \end{matrix} )

so the phase is non-degenerate and the Hessian operator L_{Ψ} is given by

L_{Ψ} = ⟨Ψ′′(1, 0)^{−1}D, D⟩ = 2\frac{∂^{2}}{∂t∂θ} − i\frac{∂^{2}}{∂t^{2}} .

By the stationary phase method for complex oscillatory integrals ([Hör], Theorem 7.7.5), we have

I_{1} = γ ∑_{j=0}^{J} N^{−j}L_{j}[A(t, θ; P_{0}, u, v)]|_{t=1,θ=0} + R̂_{J}(P_{0}, u, v, N),  (72)

where

γ = N \frac{1}{\sqrt{det(NΨ′′(1, 0)/2πi)}} = \sqrt{−2πi}

and L_{j} is the differential operator of order 2j in (t, θ) defined by

L_{j}φ(t, θ) = ∑_{ν−μ=j} ∑_{2ν≥3μ} \frac{1}{2^{ν}i^{j}μ!ν!}L_{Ψ}^{ν}[φ(t, θ)(R_{3}^{Ψ})^{μ}(t, θ)]  (73)

with R_{3}^{Ψ}(t, θ) the third order remainder in the Taylor expansion of Ψ at (t, θ) = (1, 0). Also, the remainder is estimated by

|R̂_{J}(P_{0}, u, v, N)| ≤ CN^{−J} ∑_{|α|≤2J+2} sup_{t,θ} |D_{t,θ}^{α}A(t, θ; P_{0}, u, v)| ≤ C_{J}N^{m−J} ,  (74)

where the last inequality follows by observing from (69) that

|A(t, θ; P_{0}, u, v)| ≤ C′|s(\frac{u}{\sqrt{N}} , θ; \frac{v}{\sqrt{N}} , 0, Nt)| ≤ C′′N^{m} (0 ≤ t ≤ 3) ,  (75)

and similarly for its derivatives (using the fact that s is a symbol of order m and hence D_{t,θ}^{α}s(r_{θ}x, y, Nt) = O(N^{m}) uniformly for t ≤ 3).

To evaluate (72), we first expand exp [ite^{iθ}NR_{3}^{ψ}(\frac{u}{\sqrt{N}} , \frac{v}{\sqrt{N}})] in powers of N^{− \frac{1}{2}} , keeping track of the homogeneity in (u, v) of the coefficients. We simplify the notation by writing g = te^{iθ}. By definition,

R_{3}^{ψ}(\frac{u}{\sqrt{N}} , \frac{v}{\sqrt{N}}) ∼ N^{−3/2}ψ_{3}(u, v) + N^{−2}ψ_{4}(u, v) + ⋯ + N^{−d/2}ψ_{d}(u, v) + · · · .

We then have

e^{iNgR_{3}^{ψ}( \frac{u}{\sqrt{N}} , \frac{v}{\sqrt{N}})} ∼ ∑_{r=0}^{∞} N^{−r/2}c_{r}(u, v; t, θ) .  (76)

We further expand

c_{r} = ∑_{λ=1}^{r} c_{r,r+2λ}(u, v; t, θ) , r ≥ 1 , c_{0} = c_{00} = 1 ,  (77)

with c_{rd} homogeneous of degree d in u, v. Note that c_{r} is a polynomial of degree r in g. (The explicit formula for c_{rd} is:

c_{rd} = ∑ \{\frac{1}{n!}(ig)^{n}Π_{j=1}^{n}ψ_{a_{j}}(u, v) : n ≥ 1, a_{j} ≥ 3, ∑_{j=1}^{n} a_{j} = d, ∑_{j=1}^{n}(a_{j} − 2) = r\} , r ≥ 1 .

The range of d is determined by the fact that d = ∑_{j=1}^{n} a_{j} = r + 2n with 0 ≤ n ≤ r.)

We similarly expand the symbol:

s(\frac{u}{\sqrt{N}} , θ; \frac{v}{\sqrt{N}} , 0, Nt) = ∑_{k=0}^{∞} N^{m−k}t^{m−k}s_{k}(\frac{u}{\sqrt{N}} , \frac{v}{\sqrt{N}} , θ) = ∑_{k,ℓ=0}^{∞} N^{m−k−ℓ/2}t^{m−k}s_{kℓ}(u, v, θ)

where s_{kℓ} is the homogeneous term of s_{k} of degree ℓ in (u, v). Hence, we have

A ∼ ρ_{1}(t)e^{gψ_{2}(u,v)+iNgR_{3}^{ψ}( \frac{u}{\sqrt{N}} , \frac{v}{\sqrt{N}} ,θ)} ∑_{k=0}^{∞} N^{m−k}t^{m−k}s_{k}(\frac{u}{\sqrt{N}} , \frac{v}{\sqrt{N}} , θ)

∼ ρ_{1}(t)e^{gψ_{2}(u,v)}N^{m} ∑_{n=0}^{∞} N^{−n/2}f_{n}(u, v; t, θ, P_{0}) ,  (78)

where the remainder in (78) after summing K terms is O(N^{m− \frac{K+1}{2}} ). (Note that f_{n} is a polynomial in t of degree n + m.) We further have

f_{n} = ∑_{r+ℓ+2k=n} c_{r}s_{kℓ} = ∑_{k=0}^{[n/2]} t^{m−k} (s_{k,n−2k} + ∑_{r=1}^{n−2k} ∑_{λ=1}^{r} c_{r,r+2λ}s_{k,n−2k−r}) = ∑_{j=0}^{[3n/2]} f_{n,3n−2j} ,

where f_{n,d} is homogeneous of degree d in (u, v).

Hence,

L_{j}[A]|_{t=1,θ=0} = ∑_{n=0}^{K} N^{m−n/2}L_{j}[e^{gψ_{2}}f_{n}]|_{t=1,θ=0} + O(N^{m− \frac{K+1}{2}} ) .  (79)

Since L_{Ψ} is a second order operator in (t, θ), we see that

L_{j}[e^{gψ_{2}}f_{n}]|_{t=1,θ=0} = e^{ψ_{2}} ∑_{α≤2j}(ψ_{2})^{α}F_{njα} ,  (80)

where the F_{njα} are polynomials in u, v, ū, v̄ of degree ≤ 3n. Therefore, by (72), (79) and (80), we have

N^{−m}I_{1} ∼ \sqrt{−2πi} e^{ψ_{2}} ∑_{n=0}^{∞} ∑_{j=0}^{∞} ∑_{α=0}^{2j}(ψ_{2})^{α}N^{− \frac{n}{2} −j}F_{njα}

∼ \sqrt{−2πi} e^{ψ_{2}} ∑_{r=0}^{∞} ∑_{j=0}^{[r/2]} ∑_{α=0}^{2j}(ψ_{2})^{α}N^{−r/2}F_{r−2j,j,α}

∼ e^{ψ_{2}} ∑_{r=0}^{∞} ∑_{α=0}^{2[r/2]}(ψ_{2})^{α}N^{−r/2}Q_{rα} .  (81)

As with f_{n} we have the homogeneous expansion:

Q_{rα} = ∑_{j=0}^{[3r/2]} Q_{r,α,3r−2j} .  (82)

Here, Q_{r,α,d} is homogeneous of degree d in (u, v). Thus we have the desired Taylor series for I_{1}.

To show that I_{2} = O(N^{−∞}), we observe from (65) that

|\frac{∂}{∂θ}Ψ̃| ≥ [1 − O(N^{−1})] t − 1 ≥ \frac{1}{3}t for t ≥ 2, N ≫ 1 .

Hence by [Hör], Theorem 7.7.1, for all integers k ≥ 1 we have

∫_{0}^{2π} e^{iNΨ̃}ρ_{2}(t)s(\frac{u}{\sqrt{N}} , θ; \frac{v}{\sqrt{N}} , 0, Nt)dθ

≤ C′N^{−k}t^{−k} ∑_{α≤k} sup |D_{θ}^{α} (ρ_{2}(t)s(\frac{u}{\sqrt{N}} , θ; \frac{v}{\sqrt{N}} , 0, Nt))|
≤ C′_{k}N^{m−k}t^{m−k} .

(To see that C′ is independent of t, we write iNΨ̃ = i(Nt)Φ, with Φ = Ψ̃/t and apply [Hör, Theorem 7.7.1] to the phase Φ.) Integrating over t, we conclude that I_{2} = O(N^{m+1−k}); since k ≥ 1 is arbitrary, I_{2} = O(N^{−∞}). The estimate for the remainder now follows from (74) and (79).

4. Kodaira embedding and Tian almost isometry theorem

Definition: By the Kodaira maps we mean the maps Φ_{N} : M → P H^{0}(M, L^{N})′ defined by Φ_{N}(z) = {s^{N} : s^{N}(z) = 0}. Equivalently, we can choose an orthonormal basis S_{1}^{N}, …, S_{d_{N}}^{N} of H^{0}(M, L^{N}) and write

Φ_{N} : M → ℂℙ^{d_{N} −1} ,  Φ_{N}(z) = (S_{1}^{N}(z) : ⋯: S_{d_{N}}^{N} (z)) .  (83)

We also define the lifts of the Kodaira maps:

Φ̃_{N} : X → ℂ^{d_{N}} ,  Φ̃_{N}(x) = (S_{1}^{N}(x), …, S_{d_{N}}^{N} (x)) .  (84)

Note that

Π_{N}(x, y) = Φ̃_{N}(x) · \overline{Φ̃_{N}(y)} ;  (85)

in particular,

Π_{N}(x, x) = ∥Φ̃_{N}(x)∥^{2} .  (86)

We now prove Theorem 2, which generalizes to the symplectic category the asymptotic expansion theorem of [Zel] (also proved independently by [Cat] using the Bergman kernel in place of the Szegö kernel) and Tian’s approximate isometry theorem [Tian]:

(a) Using the expansion of Theorem 3.1 with u = v = 0 and noting that b_{r}(z, 0, 0) = 0 for r odd, we obtain the above expansion of Π_{N}(z, 0; z, 0) with a_{r}(z) = b_{2r}(z, 0, 0). (The expansion also follows by precisely the same proof as in [Zel].)

(b) In the holomorphic case, (b) followed by differentiating (a), using that Φ_{N}^{∗}(∂ ¯∂log |ξ|^{2}) = ∂ ¯∂log |Φ_{N}|^{2}. In the almost complex case, Φ_{N}^{∗} does not commute with the complex derivatives, so we need to modify the proof. To do so, we use the following notation: the exterior derivative on a product manifold Y_{1} × Y_{2} can be decomposed as d = d^{1} + d^{2}, where d^{1} and d^{2} denote exterior differentiation on the first and second factors, respectively. (This is formally analogous to the decomposition d = ∂ + ¯∂; e.g., d^{1}d^{1} = d^{2}d^{2} = d^{1}d^{2} + d^{2}d^{1} = 0.)

Recall that the Fubini-Study form ω_{F S} on ℂℙ^{m−1} is induced by the 2-form ω̃_{m} = \frac{i}{2}∂ ¯∂log |ξ|^{2} on ℂ^{m} \ {0}. We consider the 2-form Ω on (ℂ^{m} \ {0}) × (ℂ^{m} \ {0}) given by

Ω = \frac{i}{2}∂ ¯∂log ζ · η̄ = \frac{i}{2}d^{1}d^{2} log ζ · η̄.

Note that Ω is smooth on a neighborhood of the diagonal {ζ = η}, and

Ω|_{ζ=η} = ω̃_{m}

(where the restriction to {ζ = η} means the pull-back under the map ζ ↦ (ζ, ζ)). It suffices to show that

\frac{1}{N} Φ̃_{N}^{∗}ω_{d_{N}} → π^{∗}ω, π : X → M.

To do this, we consider the maps

Ψ_{N} = Φ̃_{N} × Φ̃_{N} : X × X → ℂ^{d_{N}} × ℂ^{d_{N}} , Ψ_{N}(x, y) = (Φ̃_{N}(x), Φ̃_{N}(y)) .

It is elementary to check that Ψ_{N}^{∗} commutes with d^{1} and d^{2}. By (85), we have

Ψ_{N}^{∗}(log ζ · η̄) = (log ζ · η̄) ∘ Ψ_{N} = log Π_{N} .

Therefore,

\frac{1}{N} Ψ̃_{N}^{∗}Ω_{d_{N}} = \frac{i}{2N} Ψ_{N}^{∗}d^{1}d^{2} log ζ · η̄ = \frac{i}{2N} d^{1}d^{2}Ψ_{N}^{∗} log ζ · η̄ = \frac{i}{2N} d^{1}d^{2} log Π_{N} .  (87)

Restricting (87) to the diagonal, we then have

\frac{1}{N} Φ̃_{N}^{∗}ω_{d_{N}} = \frac{i}{2N} (d^{1}d^{2} log Π_{N})|_{x=y} = diag^{∗}(d^{1}d^{2} log Π_{N}) ,

where diag : X → X × X is the diagonal map diag(x) = (x, x).

Using Heisenberg coordinates as in Theorem 3.1, we have by the near-diagonal scaling asymptotics

\frac{1}{N} Φ̃_{N}^{∗}ω_{d_{N}} |_{P_{0}} = \frac{i}{2N} diag^{∗}d^{1}d^{2} log Π_{N}^{P_{0}}(\frac{u}{\sqrt{N}} , \frac{θ}{N}; \frac{v}{\sqrt{N}} , \frac{φ}{N})|_{0}
= \frac{i}{2N} diag^{∗}d^{1}d^{2} log Π_{1}^{H}(u, θ; v, φ)|_{0} + O(N^{− \frac{1}{2}} ) .

Finally,

\frac{i}{2N} diag^{∗}d^{1}d^{2} log Π_{1}^{H}(u, θ; v, φ)|_{0} = \frac{i}{2N} diag^{∗}d^{1}d^{2}[i(θ − φ) + u · v̄ − \frac{1}{2}(|u|^{2} + |v|^{2})]

= \frac{i}{2N} ∑_{q=1}^{m} du_{q} ∧ dū_{q} = \frac{i}{2} ∑_{q=1}^{m} dz_{q} ∧ dz̄_{q} = ω|_{P_{0}}.(88)

Remark: A more explicit way to show (b) is to expand the Fubini-Study form:

ω̃_{m} = \frac{i}{2}|ξ|^{−4} [|ξ|^{2} ∑_{j=1}^{m} dξ_{j} ∧ dξ̄_{j} − ∑_{j,k=1}^{m} ξ̄_{j}ξ_{k}dξ_{j} ∧ dξ̄_{k}] .

Then

\frac{1}{N} Φ̃_{N}^{∗}ω_{d_{N}} = \frac{i}{2}Π_{N}(x, x)^{−2}\{(Π_{N}(x, x)d^{1}d^{2}Π_{N}(x, y) − d^{1}Π_{N}(x, y) ∧ d^{2}Π_{N}(x, y)\}|_{x=y} ,

and (b) follows from a short computation using Theorem 3.1 as above.

It follows from Theorem 2(b) that Φ_{N} is an immersion for N ≫ 0. Using in part an idea of Bouche [Bch], we now give a simple proof that the Kodaira embedding theorem holds for symplectic manifolds (Theorem 3): Let {P_{N}, Q_{N}} be any sequence of distinct points such that Φ_{N}(P_{N}) = Φ_{N}(Q_{N}). By passing to a subsequence we may assume that one of the following two cases holds:

(i) The distance r_{N} := dist(P_{N}, Q_{N}) between P_{N}, Q_{N} satisfies r_{N} \sqrt{N} → ∞; (ii) There exists a constant C independent of N such that r_{N} ≤ C/\sqrt{N}.

To prove that case (i) cannot occur, we let Π_{N}^{P_{N}} (x) = Π_{N}(x, P_{N}) denote the ‘peak section’ at P_{N}. By Theorem 3.1, we have

N^{−m} ∫_{B(P_{N} ,r_{N})} |Π_{N}^{P_{N}} |^{2}dV ≥ 1 − o(1) .

The same inequality holds for Q_{N}. If Φ_{N}(P_{N}) = Φ_{N}(Q_{N}) then the total L^{2}-norm of Π_{N}(x, ·) would have to be ∼ 2N^{m/2}, contradicting the asymptotic ∼ N^{m/2} from Theorem 2(a).

To prove that case (ii) cannot occur, we assume on the contrary that Φ_{N}(P_{N}) = Φ_{N}(Q_{N}), where P_{N} = ρ_{N}(0) and Q_{N} = ρ_{N}(\frac{v_{N}}{\sqrt{N}}), 0 ≠ |v_{N}| ≤ C, using a Heisenberg coordinate chart ρ_{N} about P_{N}. We consider the function

f_{N}(t) = \frac{|Π_{N}^{P_{N}} (0, \frac{tv_{N}}{\sqrt{N}})|^{2}}{Π_{N}^{P_{N}} (0, 0)Π_{N}^{P_{N}} (\frac{tv_{N}}{\sqrt{N}} , \frac{tv_{N}}{\sqrt{N}})} .  (89)

Then f_{N}(0) = 1 and we see from (85) and the Cauchy-Schwartz inequality that f_{N} ≤ 1. Furthermore, since we are supposing that Φ_{N}(P_{N}) = Φ_{N}(Q_{N}), we also have f_{N}(1) = 1. Thus for some value of t_{N} in the open interval (0, 1), we have f′′_{N}(t_{N}) = 0. By Theorem 3.1,

f_{N}(t) = e^{−|v_{N} |^{2}t^{2}} [1 + N^{−1/2}R̃_{N}(tv_{N})] ,  (90)

where

R̃_{N}(v) = R_{1}(P_{N}; 0, v, N) + R_{1}(P_{N}; v, 0, N) − R_{1}(P_{N}; v, v, N) − R_{1}(P_{N}; 0, 0, N) + O(N^{−1/2}) .

The estimate for R_{1} yields:

∥R̃_{N}∥_{C^{2}{|v|≤C}} = O(1)  (91)

Since f_{N}(1) = 1, it follows from (90)–(91) that |v_{N}|^{2} = O(N^{−1/2}). (A more careful analysis shows that we can replace N^{−1/2} with N^{−1} in (90) and thus |v_{N}| = O(N^{−1/2}).)

Write e^{x} = 1 + x + x^{2}φ(x). We then have

f_{N}(t) = 1 − |v_{N}|^{2}t^{2} + |v_{N}|^{4}t^{4}φ(|v_{N}|^{2}t^{2}) + N^{−1/2}R̃_{N}(tv_{N}) [1 − |v_{N}|^{2}t^{2} + |v_{N}|^{4}t^{4}φ(|v_{N}|^{2}t^{2})] .

Thus by (91),

f′′_{N}(t) = −2|v_{N}|^{2} + O(|v_{N}|^{4}) + O(N^{−1/2}|v_{N}|^{2}) , |t| ≤ 1 .

Since |v_{N}| = o(1), it follows that

0 = f′′_{N}(t_{N}) = (−2 + o(1))|v_{N}|^{2} ,

which contradicts the assumption that v_{N} ≠ 0.

5. Transversal sections

5.1. Quantitative transversality. To illustrate the connection between our almost holomorphic sections and the asymptotically holomorphic sections of Donaldson and of Auroux, we now explain how to adapt Donaldson’s proof of the existence of quantitatively transverse asymptotically holomorphic sections to our setting of almost holomorphic sections. We shall show that the normalized coherent states in our spaces H_{J}^{0}(M, L^{N}) of almost holomorphic sections satisfy the estimates of Proposition 34 in [Don1], suitably modified. Using these modified estimates together with the proof of Proposition 15 and of Lemma 14 in [Don1], one obtains the following result on the existence of almost holomorphic transversal sections.

Proposition 5.1. There is a constant C < +∞ such that for all sufficiently large N, there exists a section s_{N} ∈ H_{J}^{0}(M, L^{N}) with ∥¯∂s_{N}∥_{∞} ≤ C and which satisfies |∂s_{N}| > C^{−1}\sqrt{N} on Z_{s_{N}} .

Here Z_{s_{N}} denotes the zero set of the section s_{N}. The existence of asymptotically holomorphic sections satisfying these transversality estimates was given by Donaldson [Don1, Theorem 5]. (Donaldson expresses transversality in terms of the derivatives ∂_{scaled} = \frac{1}{\sqrt{N}} ∂, ¯∂_{scaled} = \frac{1}{\sqrt{N}} ¯∂ with respect to the scaled coordinates.) Recall that transversality implies that Z_{s_{N}} is a symplectic submanifold of M (whose homology class is the Poincaré dual of N[ω]).

Before we begin the proof of Proposition 5.1, we recall that the starting point of Donaldson’s construction is the existence of concentrated asymptotically holomorphic sections σ_{p}^{N} supported on a ball of radius O(N^{−1/3}) about each point p ∈ M satisfying certain C^{2} estimates [Don1, Proposition 11]. In the integrable case, Donaldson constructs holomorphic sections σ_{p}^{N} satisfying the estimates ([Don1], Proposition 34)

\begin{matrix}e^{−bd_{N} (p,q)^{2}} ≤ |σ_{p}^{N}(q)| ≤ e^{−ad_{N} (q,p)^{2}}, & d_{N}(q, p) ≤ ϵN^{1/6} \\ |σ_{p}^{N}(q)| ≤ e^{−aN^{1/3}}, & d_{N}(q, p) ≥ ϵN^{1/6} \end{matrix}

where d_{N}(q, p) = \sqrt{N}d(q, p) denotes the scaled distance on M. Given such sections, one forms the \frac{1}{\sqrt{N}} -‘lattice’ of points Λ_{N} = \{p_{i}\} and the associated complex vector space of sections

s_{N} = ∑_{i} w_{i}σ_{p_{i}}^{N}, w_{i} ∈ ℂ .  (92)

Donaldson shows ([Don1], Proposition 32) that one can choose the coefficients {w_{i}} with |w_{i}| < 1 so that the holomorphic section s_{N} satisfies the quantitative transversality estimate

|∂s_{N}(z)| > C^{−1}\sqrt{N}, ∀z ∈ Z_{s_{N}} .  (93)

Our principal claim is that the Szegö kernel Π_{N}(•, p) itself, when normalized to have modulus one at z = p, satisfies estimates sufficient to prove the existence of transversal almost-holomorphic sections of the form (92). We define

σ_{p}^{N}(x) = \frac{Π_{N}(x, p)}{Π_{N}(p, p)} ,  (94)

so that σ_{p}^{N}(p) = 1. (Recall that Π_{N}(p, p) = π^{−m}N^{m} + O(N^{m−1}).)

Lemma 5.2. For all ε, D ∈ ℝ^{+}, we have for N sufficiently large:

(i) |σ_{p}^{N}(q)| ≥ [1 − \frac{C}{\sqrt{N}} d_{N}(q, p)]e^{− \frac{1+ε}{2} d_{N} (q,p)^{2}}, d_{N}(q, p) ≤ D ;
(ii) |σ_{p}^{N}(q)| < ∼ [1 + \frac{C}{\sqrt{N}} d_{N}(q, p)]e^{− \frac{1−ε}{2} d_{N} (q,p)^{2}}, d_{N}(q, p) ≤ N^{1/6} ;
(iii) N^{−k/2}|∇^{k}σ_{p}^{N}(q)| < ∼ C_{k}e^{− \frac{1−ε}{2} d_{N} (q,p)^{2}} ,  k ≥ 0;
(iv) N^{−k/2}|∇^{k} ¯∂σ_{p}^{N}(q)| < ∼ C_{k}e^{− \frac{1−ε}{2} d_{N} (q,p)^{2}} ,  k ≥ 0;

where α < ∼ β means that α − β ≤ O(N^{−∞}), uniformly for (q, p) ∈ M × M.

Since the length |σ_{N}(p, θ)| is independent of θ, we have dropped θ from the notation and regard |σ_{N}| as a function on M. Note that (iii) implies that

|∇^{k}σ_{p}^{N}(q)| = O(N^{−∞}) , for d_{N}(q, p) ≥ N^{1/6}, k = 0, 1, 2, ….  (95)

Proof. We let q = p + \frac{u}{\sqrt{N}} (in preferred coordinates), so that u ≈ d_{N}(q, p), and we write σ_{p}^{N}(q) = σ_{p}^{N}(\frac{u}{\sqrt{N}} , 0). The lower bound (i) is an immediate consequence of Theorem 3.1. To verify (ii)–(iv) for u ≤ N^{1/6} (i.e., on the balls of unscaled radii N^{−1/3}), we observe that the asymptotic expansion of Theorem 3.1 has the following extension to the scaled N^{1/6} balls:

\begin{matrix}N^{−m}Π_{N}^{P_{0}}(\frac{u}{\sqrt{N}} , \frac{θ}{N}; 0, 0) \\ = Π_{1}^{H}(u, θ; 0, 0) [1 + ∑_{r=1}^{K} N^{−r/2}b_{r}(P_{0}, u, 0)] + N^{−(K+1)/2}R_{K}(P_{0}, u, N) , \end{matrix}

where

∥R_{K}(P_{0}, u, N)∥_{C^{j}(\{|u|≤N^{1/6}\}} < ∼ C_{K,j}e^{− \frac{1−ε}{2} |u|^{2}} for j ≥ 0 .  (96)

To verify (96), we modify the proof of Theorem 3.1 as follows: We first note that

NR_{3}^{ψ}(\frac{u}{\sqrt{N}} , 0) ≤ C , for |u| ≤ N^{1/6} ,  (97)

and hence we obtain the large-ball version of (75):

\begin{matrix}|A(t, θ; P_{0}, u, 0)| ≤ C′e^{−ℜg|u|^{2}/2}|s(\frac{u}{\sqrt{N}} , θ; 0, 0, Nt)| ≤ C′′e^{− \frac{1−ε}{2} |u|^{2}}N^{m} , \\ 1 − δ ≤ t ≤ 3 , −δ ≤ θ ≤ δ , |u| ≤ 2N^{1/6} . \end{matrix}  (98)

We use a smooth partition of unity to decompose the integral (70):

I_{1} = I′_{1} + I′′_{1},  I′_{1} = N ∫_{1−δ}^{3} ∫_{−δ}^{δ} e^{iNΨ(t,θ)}A(t, θ; P_{0}, u, 0)dθdt .  (99)

The integral I′′_{1} is over a compact region outside the critical point (1, 0) of the phase Ψ. In fact |\frac{∂Ψ}{∂θ} | > α > 0 in this region, and the method of proof that I_{2} = O(N^{−∞}) shows that we also have I′′_{1} = O(N^{−∞}), uniformly for |u| ≤ N^{1/6} by (98).

Furthermore by (97), the asymptotic expansion (76) of the exponential e^{NgR_{3}( \frac{u}{\sqrt{N}} ,0)} holds for |u| ≤ N^{1/6}, and hence we have the large-ball version of (78):

A(t, θ; P_{0}, u, 0) = ρ_{1}(t)e^{−g|u|^{2}/2}N^{m} [∑_{n=0}^{K} N^{−n/2}f_{n}(u, 0; t, θ, P_{0}) + R_{K}(u, t, θ)] ,
|R_{K}| ≤ C_{K}O(N^{− \frac{K+1}{2}} ) .  (100)

The remainder estimate (96) now follows from (98) and (100) as in the proof of Theorem 3.1.

To verify (ii) for |u| ≤ N^{1/6}, we apply (96) (with K = 0) to conclude that

|σ_{p}^{N}(p + \frac{u}{\sqrt{N}})| = e^{− \frac{1}{2} |u|^{2}} + \frac{1}{\sqrt{N}} e^{− \frac{1−ε}{2} |u|^{2}}R(p, u, N) , |D_{u}^{α}R(p, u, N)| ≤ C_{|α|} .  (101)

Since |σ_{p}^{N}(p)| = 1, it follows that R(p, 0, N) = 0 and hence

R(p, u, N) ≤ C|u| , for |u| < N^{1/6} .  (102)

Since |u| ≈ d_{N}(p, p + \frac{u}{\sqrt{N}}) for |u| < ε\sqrt{N}, the inequality (ii) follows from (101)–(102).

To show that (iii)–(iv) holds for d_{N}(q, p) ≤ N^{1/6}, we lift a local frame {Z̄_{j}^{M}} of the form (1) to obtain the local frame {Z̄_{1}, …, Z̄_{m}} for H^{0,1}X given by

Z̄_{j}(p + \frac{u}{\sqrt{N}}) = \frac{∂^{h}}{∂z̄_{j}} + ∑_{r=1}^{m} B_{jk}(\frac{u}{\sqrt{N}})\frac{∂^{h}}{∂z_{k}} ,  B_{jk}(\frac{u}{\sqrt{N}}) = O(\frac{1}{\sqrt{N}} |u|) ≤ O(N^{−1/3}) . (103)

Recalling (6) and (14), we further have

\frac{∂^{h}}{∂z̄_{j}} = \frac{∂}{∂z̄_{j}} + [−\frac{i}{2} \frac{u_{j}}{\sqrt{N}} − RĀ_{j} ; 1(\frac{u}{\sqrt{N}})] \frac{∂}{∂θ} = \frac{∂}{∂z̄_{j}} + i\sqrt{N} u_{j} − iNRĀ_{j} ; 1(\frac{u}{\sqrt{N}}) at p + \frac{u}{\sqrt{N}} , (104)

where RĀ_{j} ; 1(\frac{u}{\sqrt{N}}) = O(\frac{1}{N} |u|^{2}). The inequalities (iii)–(iv) for |u| ≤ N^{1/6} follow as before, using (103)–(104).

It remains to show (95). For this it suffices to show that ∇^{k}Π_{N}^{p}(\frac{u}{\sqrt{N}} , 0; 0, 0) = O(N^{−∞}) for |u| ≥ N^{1/6}. Since Π_{N} = S_{N} + E_{N}, where E_{N}(x, y) = O(N^{−∞}), we can neglect the remainder E_{N}. Furthermore, since S_{N} is rapidly decaying away from the diagonal, it suffices to consider

N^{1/6} ≤ |u| ≤ εN^{1/2} .

As in the proof of Theorem 3.1, we use a smooth partition of unity to write (60) as the sum of two integrals I_{1}, I_{2}. Note that in the proof that I_{2} = O(N^{−∞}), we use only that |u| ≤ εN^{1/2}. So it remains to consider the integral I_{1}. Recalling (65), we have

Ψ̃ = α_{N}(θ)t − θ , |α_{N}(θ)| ≥ \frac{|u|^{2}}{3N} ≥ \frac{1}{3}N^{−2/3} for N^{1/6} ≤ |u| ≤ εN^{1/2} .

Integrating by parts, we then obtain

|∫_{0}^{3} e^{iNΨ̃}ρ_{1}(t)s(\frac{u}{\sqrt{N}} , θ; \frac{v}{\sqrt{N}} , 0, Nt)dt|

≤ |Nα_{N}|^{−k} sup_{t≤3} |\frac{∂^{k}}{∂t^{k}} [ρ_{1}(t)s(\frac{u}{\sqrt{N}} , θ; \frac{v}{\sqrt{N}} , 0, Nt)]| ≤ C_{k}N^{−k/3+m} ,

Hence by (66),

|I_{1}| ≤ C′_{k}N^{−k/3+m+1} .  (105)

and thus S_{N}^{p}(\frac{u}{\sqrt{N}} , 0; 0, 0) = O(N^{−∞}). The estimate for the derivatives is similar.

Remark: It is of some interest to know how precise these off-diagonal estimates actually are. In the holomorphic case (with a C^{∞} but not necessarily analytic metric), M. Christ has shown (cf. [Chr]) that one has the global off-diagonal bounds

|Π_{N}(x, y)| ≤ CN^{m}e^{−\sqrt{N}d(x,y)}, ((x, y) ∈ X × X)  (106)

for a constant C which is uniform in (N, x, y). He also shows that this estimate cannot be improved in general. Thus, the decay estimates of Lemma 5.2 do not hold in general without the O(N^{−∞}) term. It is quite possible that (106) extends to the almost complex context of this paper, since it is based on weighted L^{2} estimates for an associated Green’s kernel which seem to have natural analogues for the pseudodifferential D̄_{j}-complex of [BoGu]. However, we do not need such a precise estimate as (106) and hence will not investigate it further.

We now outline how Proposition 5.1 follows from Lemma 5.2. The main steps in adapting the proof of Donaldson’s transversality theorem [Don1, Theorem 5] are to show the following: Let

G_{p_{i}}^{N}(q) = |σ_{p_{i}}^{N}(q) + |¯∂σ_{p_{i}}^{N}(q)| + \frac{1}{\sqrt{N}} |∇σ_{p_{i}}^{N}(q)| + \frac{1}{\sqrt{N}} |∇¯∂σ_{p_{i}}^{N}(q)| + \frac{1}{N} |∇^{2}σ_{p_{i}}^{N}(q)| ;

then • ∑_{i} G_{p_{i}}^{N}(q) is bounded independently of N;

• for all D > 0 there exists N_{0}(D) such that G_{p_{i}}^{N}(q) ≤ Ce^{−D^{2}/3} for d_{N}(q, p_{i}) ≥ D, N > N_{0}(D).

These statements follow immediately from parts (iii) and (iv) of Lemma 5.2; to obtain the first statement we also note that the number of points of the ‘lattice’ Λ_{N} = {p_{i}} is ∼ N^{m} and that we can choose D > 0 so that the balls \{B(p_{i}, \frac{D}{\sqrt{N}})\} cover M and at most (2D+2)^{2m} of them pass through any point. From there (as in the holomorphic case described in [Don1, p. 697]), we obtain Proposition 5.1 from the argument in [Don1, §§3–4].

5.2. The Adjunction Formula. As explained in [Don1], the zero set Z(s_{N}) of a section s_{N} ∈ C^{∞}(M, L^{N}) satisfying the transversality condition (93) is a symplectically embedded submanifold. In particular, when (M, ω) is a 4-manifold and Z(s_{N}) is a symplectically embedded Riemann surface, then it is well known that the genus g_{N} of Z(s_{N}) is given by the Adjunction Formula as in the holomorphic case:

g_{N} = \frac{c_{1}(L)^{2}}{2} N^{2} − \frac{c_{1}(M) · c_{1}(L)}{2}N + 1 .  (107)

To verify (107), we let C denote the complex curve Z(s_{N}) with complex structure J_{0} induced by ω. Then J_{0} can be extended to an almost complex structure J′ on M, compatible with ω, and it follows that C is a J-holomorphic curve in (M, ω, J′). By our assumptions, the homomorphism ∇s_{N} : T_{M} → L^{N} is surjective along C. Hence, for all z_{0} ∈ C, we have

ker_{z_{0}} (∂s_{N} : T_{M}^{1,0} → L_{N}) = (ker_{z_{0}} ∇s_{N}) ∩ T_{M,z_{0}}^{1,0} = T_{C,z_{0}} ∩ T_{M,z_{0}}^{1,0} = T_{C,z_{0}}^{1,0}

(where we use the almost complex structures J′, J_{0} on M, C, respectively). Proceeding as in the holomorphic case, from the resulting exact sequence of vector bundles

0 → T_{C}^{1,0} → T_{M}^{1,0}|_{C} ⟶^{∂s_{N}} L^{N}|_{C} → 0 ,

we obtain the Adjunction Formula:

(K_{M} ⊗ L^{N})|_{C} = K_{C} .  (108)

Here K_{M} = ⋀^{2} T_{M}^{1,0}^{∗} and K_{C} = T_{C}^{1,0}^{∗}. Note that c_{1}(K_{M}) is independent of the choice of J′, since the space of compatible almost complex structures on (M, ω) is connected (and in fact is contractible; see [McSa, 4.1]). Thus we have c_{1}(K_{M}) = −c_{1}(M). It follows further from Donaldson’s Lefschetz hyperplane theorem for asymptotically holomorphic sections satisfying (93) ([Don1, Prop. 39]) that C is connected. We then conclude from (108) that

2g − 2 = c_{1}(K_{C}) = ( − c_{1}(M) + Nc_{1}(L)) · Nc_{1}(L) ,

which yields (107).

Auroux [Aur1] generalized Donaldson’s construction to obtain asymptotically holomorphic, ‘quantitatively transversal’ sections of E ⊗ L^{N}, where E is an arbitrary complex vector bundle of rank k ≤ m over M. A section s_{N} ∈ C^{∞}(M, E ⊗ L^{N}) is quantitatively transversal ([Aur1], Definition 2) if ∇s_{N} : T_{M} → E ⊗ L^{N} is surjective and has a right inverse of norm < \frac{1}{η\sqrt{N}} at all points where |s_{N}| < η. The zero sets Z(s_{N}) are then embedded symplectic submanifolds of M ([Aur1], Prop. 1). In particular, for the case where E is the trivial vector bundle M × ℂ^{k}, sections of E ⊗ L^{N} are k-tuples of sections of L^{N}, and we can write s_{N} = (s_{N}^{1}, …, s_{N}^{k}). Then Z(s_{N}) is the set of simultaneous zeros of the sections s_{N}^{j} ∈ H_{J}^{0}(M, L^{N}) and s_{N} is asymptotically holomorphic if the s_{N}^{j} are asymptotically holomorphic (but quantitative transversality of the s_{N}^{j} is not sufficient to guarantee transversality of s_{N}).

By the previous comments, Auroux’s argument can be used to give asymptotically holomorphic, quantitatively transverse sections s_{N} ∈ H_{J}^{0}(M, L^{N})^{k}. In the case where k = m − 1, Z(s_{N}) is a complex curve, which is connected by Auroux’s generalization [Aur1, Prop. 2] of Donaldson’s Lefschetz hyperplane section theorem cited above, and the genus formula (107) extends to this case as follows:

Proposition 5.3. (see [Aur1, Proposition 5]) Let s_{N}^{1}, …, s_{N}^{m−1} ∈ C^{∞}(M, L^{N}) such that s_{N} := (s_{N}^{1}, …, s_{N}^{m−1}) is quantitatively transverse and asymptotically holomorphic, for N = 1, 2, 3, .... Then for N sufficiently large, Z(s_{N}) is connected and its genus g_{N} is given by

g_{N} = \frac{m − 1}{2}c_{1}(L)^{m}N^{m} − \frac{1}{2}c_{1}(M) · c_{1}(L)^{m−1}N^{m−1} + 1 .

The proposition follows by the argument in [Aur1], which is a generalization of the proof of (107). To summarize this argument, the hypotheses imply that Z(s_{N}) is a symplectic submanifold (for N ≫ 0) and one obtains as before an exact sequence

0 → T_{C}^{1,0} → T_{M}^{1,0}|_{C} ⟶^{∂s_{N}} L^{N} ⊕ ⋯ ⊕ L^{N}|_{C} → 0 ,  (109)

which yields

2 − 2g_{N} = c_{1}(C) = (c_{1}(M) − (m − 1)Nc_{1}(L), [C])
= [c_{1}(M) − (m − 1)Nc_{1}(L)] · c_{1}(L)^{m−1}N^{m−1} .

Remark: The results of this paper can be generalized to the case of almost holomorphic sections in H_{J}^{0}(M, E ⊗ L^{N}), for an arbitrary complex vector bundle E → M. Auroux’s argument can then be used to find quantitatively transversal, asymptotically holomorphic sections s_{N} ∈ H_{J}^{0}(M, E⊗L^{N}). The zero sets Z(s_{N}) are then (m−k−1)-connected symplectic submanifolds, for N ≫ 0; if k = rank E = m − 1, they are connected Riemann surfaces. In this case, the genus formula becomes:

2g_{N} − 2 = [(m − 1)Nc_{1}(L) + c_{1}(E) − c_{1}(M)] ∑_{j=0}^{m−1} c_{m−1−j}(E)c_{1}(L)^{j}N^{j} .  (110)

To verify (110), we proceed as in [Aur1], noting first that we have an exact sequence

0 → T_{C}^{1,0} → T_{M}^{1,0}|_{C} ⟶^{∂s_{N}} E ⊗ L^{N}|_{C} → 0 .  (111)

The genus formula (110) then follows from (111) by a routine Chern class calculation.

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Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA

  E-mail address: shiffman@math.jhu.edu, zelditch@math.jhu.edu