ASYMPTOTICS OF HOLOMORPHIC SECTIONS OF POWERS OF A POSITIVE LINE BUNDLE

STEVE ZELDITCH

1. Introduction

In this lecture we will describe some recent results [Z.1][Sh.Z] (obtained partly in collaboration with B.Shiffman) on the asymptotic properties of holomorphic sections s_{N} ∈ H^{0}(M, L^{N}) of the Nth tensor power L^{N} of a positive hermitian holomorphic line bundle (L, h) → (M, ω) over a compact Kahler manifold of dimension m. Here, ω = c_{1}(h) is the curvature form of h. Using the Boutet de Monvel-Sjostrand parametrix for the Szego kernel [BS] we first obtain a rather simple and direct proof of a result due to G. Tian on the asymptotic isometry of the Kodaira embeddings [T]. We then apply this result to prove the asymptotic uniform distribution of zeros of certain sequences of sections s_{N} ∈ H^{0}(M, L^{N}) as N → ∞, namely

• to ‘random sequences’ of sections
• to eigenfunctions of an ‘ergodic quantum map’.

In the simplest case where M = ℂℙ^{k}, where L = O(1) (the hyperplane line bundle) and where ω = ω_{F S} (the Fubini-Study form), H^{0}(ℙℂ^{k}, O(N)) consists of homogeneous holomorphic polynomials p(z_{0}, …, z_{k}) of degree N on ℂ^{k+1}. Our subject then specializes to the asymptotics of polynomials as the degree N → ∞. The distribution of zeros of random polynomials of large degree was first studied by Bloch-Polya, Littlewood, Kac and others (see [BlD] for references). More recently, mathematical physicists [BlD][BBL][Ha][NV][Z2][V] have been interested in random polynomials (and more general sections) as a model for eigenfunctions of quantum chaotic maps. The large N limit of powers L^{N} of a positive line bundle arises in physics as the semi-classical limit in the geometric quantization of a compact Kahler manifold.

Our first result concerns the Kodaira embeddings φ_{N} : M → ℙH^{0}(M, L^{⊗N})^{∗} defined by φ_{N}(z) = {s : s(z) = 0}. In a standard way [GH], one may express the Kodaira embeddings in terms of an orthonormal basis of sections. Namely, for each N ∈ N, h induces a hermitian metric h_{N} on L^{⊗N}. Let {S_{0}^{N}, …, S_{d_{N}}^{N} } be any orthonormal basis of H^{0}(M, L^{⊗N}), with respect to the inner product ⟨s_{1}, s_{2}⟩_{h_{N}} = ∫_{M} h_{N}(s_{1}(z), s_{2}(z))dV_{g} where dV_{g} is the volume form of g of volume one. For large N, there are no common zeroes of the sections {S_{0}^{N}, …, S_{d_{N}}^{N} } and the Kodaira map may be expressed as:

Φ_{N}(z) = [S_{0}^{N}(z), …, S_{d_{N}}^{N} (z)]  (1)

where [S_{0}^{N}(z), …, S_{d_{N}}^{N} (z)] denotes the line through (S_{0}^{N}(z), …, S_{d_{N}}^{N} (z)) as defined in a local holomorphic frame.

Date: June 12, 1998.

Research partially supported by NSF grant #DMS-9703775.

Theorem 1.1. ([Z.1]) There exists a complete asymptotic expansion:

∑_{i=0}^{d_{N}} ||S_{i}^{N}(z)||_{h_{N}}^{2} = a_{0}N^{m} + a_{1}(z)N^{m−1} + a_{2}(z)N^{m−2} + …

with a_{0} = \frac{c_{1}(L)^{m}}{m!} in the sense that, for any k, R,

|| ∑_{i=0}^{d_{N}} ||S_{i}^{N}(z)||_{h_{N}}^{2} − ∑_{j<R} a_{j}(z)N^{m−j}||_{C^{k}} ≤ C_{R,k}N^{n−R}.

It follows easily that the Kodaira embeddings are asymptotically isometric:

Corollary 1.2. Let ω_{F S} denote the Fubini-Study form on ℂP^{d_{N}}. Then:

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

for any k.

This result was first proved (simeltaneously and independently) by G.Kempf [K] and S.Ji [Ji] in the case of abelian varieties (with C_{0} convergence) and for general projective varieties with C^{4} convergence by G.Tian [T, Lemma 3.2(i)]. Heat kernel proofs of the C^{0}-convergence result were later found by T. Bouche [Bou.1] [Bou.2] and J.P.Demailly [D]. The author has also recently learned that D.Catlin independently proved the theorem and its corollary by a method similar to ours [C].

Corollary (1.2) is the starting point for some results on the distribution of zeros of random sequences of holomorphic sections. Namely, we consider the probability space (S, dμ), where S equals the product ∏_{N=1}^{∞} SH^{0}(M, L^{N}) of the unit spheres SH^{0}(M, L^{N}) in H^{0}(M, L^{N}) and μ is the product of Haar measures on these spheres. Given a sequence s = {s_{N}} ∈ S, we associate the currents of integration [Z_{s_{N}}] over the zero divisors Z_{s_{N}} of the sections s_{N}. In complex dimension 1, [Z_{s_{N}}] is the sum of delta functions at the zeros of s_{N}. Our first result states that for a random (i.e., for almost all) s ∈ S, the sequence of zeros of the sections s_{N} are asymptotically uniformly distributed:

Theorem 1.3. ([Sh.Z]) For μ-almost all s = \{s_{N}\} ∈ S, \frac{1}{N} [Z_{s_{N}}] → ω weakly in the sense of measures; in other words,

lim_{N→∞} (\frac{1}{N} [Z_{s_{N}}], φ) = ∫_{M} ω ∧ φ

for all continuous (m − 1, m − 1) forms φ. In particular,

lim_{N→∞} \frac{1}{N} Vol_{2m−2}\{z ∈ U : s_{N}(z) = 0\} = mVol_{2n}U ,

for U open in M (where Vol_{k} denotes the Riemannian k-volume in (M, ω) ).

Tian’s theorem combined with some plurisubharmonic analysis also gives a uniform distribution theorem for zeros of eigenfunctions of ergodic quantum maps. Recall that a quantum map is a sequence U_{χ,N} of unitary operators on H^{0}(M, L^{N}) which ‘quantizes’ a symplectic map χ : (M, ω) → (M, ω). For the precise definitions, we refer to [Z3]. We call U_{χ,N} a ‘quantum ergodic map’ if χ is an ergodic transformation of (M, ω). Since the zero set of a wave function represents the locus of points where the quantum particle is least likely to be, it is natural to try to connect the distribution of zeros to dynamical properties of the map. In the ergodic case we have:

Theorem 1.4. ([Sh.Z]; see also [NV]) Let (L, h) → (M, ω) be a positive Hermitian line bundle over a Kähler manifold with c_{1}(h) = ω and let U_{χ,N} : H^{0}(M, L^{N}) ↦ H^{0}(M, L^{N}) be a ergodic quantum map. Further, let {S_{0}^{N}, …, S_{d_{N}}^{N} } be an orthonormal basis of eigensections of U_{χ,N}. Then there exists a subsequence Λ ⊂ {(N, j) : N = 1, 2, 3, …, j ∈ {0, …, d_{N}}} of density one such that

lim_{N→∞,(N,j)∈Λ} \frac{1}{N} [Z_{S_{j}^{N}}] → ω

weakly in the sense of measures.

This result was conjectured by LeBoeuf-Voros and was proved independently (and prior to us) by Nonnenmacher-Voros [NV] in the case of the theta bundle over an elliptic curve ℂ/ℤ^{2}. We would also like to mention a result in a work in progress [NZ] at the opposite extreme where the quantum map U_{χ} is completely integrable in the strong sense of commuting with a Hamiltonian torus action. Let J : M → IR^{n} denote the moment map of the action and let Δ = J(M) denote the moment polytope. Then for each eigenfunction S_{N,j} of U_{χ,N}, we have J(Z_{S_{N,j}}) ⊂ ∂Δ, i.e. the zeros are contained in the inverse image of the boundary of the moment polytope.

2. Preliminaries

2.1. Notation. We begin by recalling some basic notions and establishing notation. A holomorphic line bundle L → M over an m-dimensional compact complex (projective) manifold M is called positive if it has a hermitian metric h of positive curvature. The curvature form c_{1}(h) is the (1, 1) form given locally by

c_{1}(h) = −\frac{\sqrt{−1}}{π}∂ ¯∂log ∥e_{L}∥_{h} ,

where e_{L} is a nonvanishing local holomorphic section of L, and ∥e_{L}∥_{h} = h(e_{L}, e_{L})^{1/2} denotes the h-norm of e_{L}. The form c_{1}(h) is a de Rham representative of the first Chern class c_{1}(L). Positivity means that ω = c_{1}(h) is a Kähler form, i.e. if ω = i ∑_{jk} ω_{jk̄}dz_{j} ∧ dz̄_{k} then the hermitian form ∑_{jk} ω_{jk̄}dz_{j}dz̄_{k} is positive definite. We give M the volume form dV = \frac{1}{c_{1}(L)^{m}} ω^{m} , so that M has unit volume: ∫_{M} dV = 1.

As above, we denote the space of global holomorphic sections of L by H^{0}(M, L) and that of its powers by H^{0}(M, L^{N}) where L^{N} = L ⊗ ⋯ ⊗ L. The dimension of H^{0}(M, L^{N}) is denoted by d_{N} + 1. It is well known that for N sufficiently large, d_{N} + 1 is given by the Hilbert polynomial of L, whose leading term is \frac{c_{1}(L)^{m}}{m!} N^{m} [G.H]. The metric h induces Hermitian metrics h_{N} on L^{N} given by ∥s^{⊗N}∥_{h_{N}} = ∥s∥_{h}^{N}. We give H^{0}(M, L^{N}) the inner product

⟨s_{1}, s_{2}⟩ = ∫_{M} h_{N}(s_{1}, s_{2})dV  (s_{1}, s_{2} ∈ H^{0}(M, L^{N}) ) ,  (2)

and we write |s| = ⟨s, s⟩^{1/2}.

2.2. Kodaira maps Φ_{N}. We now fix an orthonormal basis {S_{0}^{N}, …, S_{d_{N}}^{N} } of H^{0}(M, L^{N}). We also fix a local holomorphic section e_{L} of L over U ⊂ M. It induces sections e_{L}^{N} of L^{⊗N}|_{U} and we may write S_{i}^{N}(z) = f_{i}^{N}(z)e_{L}^{N}(z) for certain holomorphic functions f_{i}^{N} on U. The Kodaira map is defined by

Φ_{N}(z)M → ℙℂ^{d_{N}} , Φ_{N}(z) = [f_{0}^{N}(z), …, f_{d_{N}}^{N} (z)]  (3)

where [⋯] denotes the line thru the vector in ℂ^{d_{N} +1}. Recall that the Kahler form ω_{g_{F S}} of the Fubini-Study metric g_{F S} on ℂP^{m} is given in homogeneous coordinates [w_{0}, …, w_{m}] by ω_{F S} = \frac{\sqrt{−1}}{2π}∂ ¯∂log(∑_{j=0}^{m} |w_{i}|^{2}). Hence

Φ_{N}^{∗}ω_{F S} = \frac{\sqrt{−1}}{2π}∂ ¯∂log(∑_{j=0}^{m} |f_{j}|^{2}).  (4)

It is easy to see that this form is independent of the choice of orthonormal basis.

2.3. Currents of integration [Z_{s}]. For a holomorphic section s ∈ H^{0}(M, L^{N}), we let [Z_{s}] denote the current of integration over the zero divisor of s. Recall here the notion of current: We let D^{p,q}(M) denote the space of C^{∞} (p, q)-forms on M, and we let D′^{p,q}(M) = D^{m−p,m−q}(M)′ denote the space of (p, q)-currents on M; (T, φ) = T (φ) denotes the pairing of T ∈ D′^{p,q}(M) and φ ∈ D^{m−p,m−q}(M). Then for φ ∈ D^{m−1,m−1}(M), ([Z_{s}], φ) = ∫_{Z_{s}} φ. In the local frame e_{L}^{N} for L^{N}, we write s = fe_{L}^{N}, where f is a local holomorphic function. The zero current is then given by the Poincaré-Lelong formula

[Z_{s}] = \frac{\sqrt{−1}}{π}∂ ¯∂log |f| = \frac{\sqrt{−1}}{π}∂ ¯∂log ∥s∥_{h_{n}} + Nω .  (5)

We also consider the normalized zero divisor

[Z̃_{s}^{N}] = \frac{1}{N} [Z_{s}] ,

so that the currents [Z̃_{s}^{N}] are de Rham representatives of c_{1}(L), and thus

([Z̃_{s}^{N}], ω^{m−1}) = \frac{c_{1}(L)^{m}}{m!} .  (6)

Equation (6) says that the currents [Z̃_{s}^{N}]s all have the same mass. For background on currents of integration see [GH][LG].

2.4. Holomorphic sections and CR holomorphic functions. Following [B.G], we study asymptotics of high powers of line bundles by lifting the problem to the unit circle bundle X associated to L. That is, let L^{∗} be the dual line bundle to L and let D = {v ∈ L^{∗} : h(v, v) < 1} be its unit disc bundle relative to the metric induced by h and let X = ∂D = {v ∈ L^{∗} : h(v, v) = 1}. The positivity of c_{1}(h) is equivalent to the strict pseudoconvexity of the disc bundle D in L^{∗} (see [Gr]). We will denote the S^{1} action on X by r_{θ}x and its infinitesimal generator by \frac{∂}{∂θ}.

Let us also denote by T′D, T′′D ⊂ T D ⊗ ℂ the holomorphic, resp. anti-holomorphic subspaces and define d′f = df|_{T′}, d′′f = df|_{T′′} for f ∈ C^{∞}(D). Then X inherits the CR structure T X ⊗ ℂ = T′ ⊕ T′′ ⊕ ℂ\frac{∂}{∂θ}. Here T′X (resp. T′′X) denotes the holomorphic (resp. anti-holomorphic vectors) of D which are tangent to X. They are given in local coordinates by vector fields ∑ a_{j} \frac{∂}{∂z̄_{j}} such that ∑ a_{j} \frac{∂}{∂z̄_{j}} ρ = 0. A local basis is given by the vector fields Z_{j}^{k} = \frac{∂}{∂z̄_{j}} − (\frac{∂ρ}{∂z̄_{k}})^{−1}(\frac{∂ρ}{∂z̄_{j}})\frac{∂}{∂z̄_{k}} (j ≠ k.)

The Cauchy-Riemann operator on X is defined by

¯∂_{b} : C^{∞}(X) → C^{∞}(X, (T′′)^{∗}), ¯∂_{b}f = df|_{T′′}.  (7)

In terms of the local basis above, it is given by

¯∂_{b}f = ∑_{j≠k} Z_{j}^{k}fdz̄_{j}|_{T′′}.  (8)

Also associated to X are

• the contact form α = \frac{1}{i}d′ρ|_{X} = −\frac{1}{i}d′′ρ|_{X}
• the volume form dμ = α ∧ (dα)^{n}
• the Levi form L_{ρ}(z) = ∑ \frac{∂^{2}ρ}{∂z_{j}∂z̄_{k}} z_{j}z̄_{k}.
• the Levi form onX L_{X} = L_{ρ}|_{T′⊕T′′∩TX}

(9)

which are independent of the choice of ρ. The Levi form on X is related to dα = π^{∗}ω_{g} by: L_{X}(V, W) = dα(V, W̄).

The Hardy space H^{2}(X) is the space of boundary values of holomorphic functions on D which are in L^{2}(X), or equivalently H^{2} = (ker ¯∂_{b}) ∩ L^{2}(X). The S^{1} action commutes with ¯∂_{b}, hence H^{2}(X) = ⊕_{N=1}^{∞}H_{N}^{2}(X) where H_{N}^{2}(X) = {f ∈ H^{2}(X) : f(r_{θ}x) = e^{iNθ}f(x)}.

A section s of L determines an equivariant function ŝ on L^{∗} − 0 by the rule: ŝ(z, λ) = ⟨λ, s(z)⟩. Here, z ∈ M, λ ∈ L_{z}^{∗}. It is clear that if τ ∈ ℂ^{∗} then ŝ(z, τλ) = τŝ. We will usually 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 L^{∗}−0: put ŝ_{N}(z, λ) = ⟨λ^{⊗N}, s_{N}(z)⟩ where λ^{⊗N} = λ ⊗ λ ⊗ ⋯ ⊗ λ. The following proposition is well-known and easy to prove:

Proposition 2.1. The map s ↦ ŝ is a unitary equivalence between H^{0}(M, L^{⊗N}) and H_{N}^{2}(X).

2.5. Parametrix for the Cauchy-Szego kernel. The key analytic objects in this paper are the orthogonal (Szego) projection Π : L^{2}(X) → H^{2}(X) and its Fourier components the finite dimensional projections Π_{N} : L^{2}(X) → H_{N}^{2}(X). We define their kernels by

Π_{N}f(x) = ∫_{X} Π_{N}(x, y)f(y)dμ(y),  (10)

which differs from the definition of [B.S] in using dμ as the reference density.

The main result on the Szego kernel Π(x, y) for a strictly pseudoconvex domain is the following:

Theorem 2.2 (BS, Theorem 1.5 and §2.c). Let Π(x, y) be the Szego kernel of the boundary X of a bounded strictly pseudo-convex domain Ω in a complex manifold L. Then: there exists a symbol s ∈ S^{n}(X × X × IR^{+}) of the type

s(x, y, t) ∼ ∑_{k=0}^{∞} t^{m−k}s_{k}(x, y)

so that

Π(x, y) = ∫_{0}^{∞} e^{itψ(x,y)}s(x, y, t)dt

where the phase ψ ∈ C^{∞}(D × D) is determined by the following properties:

• ψ(x, x) = \frac{1}{i}ρ(x) where ρ is the defining function of X.

• d′′_{x}ψ and d′_{y}ψ vanish to infinite order along the diagonal.

• ψ(x, y) = −ψ̄(y, x).

The integrals are regularized by taking the principal value (see [BS]). The second condition states that ψ(x, y) is almost analytic. Roughly speaking, ψ is obtained by Taylor expanding ρ(z, z̄) and replacing all the z̄’s by w̄’s, i.e. the Taylor expansion of ψ near the diagonal is given by

ψ(x + h, x + k) = \frac{1}{i} ∑ \frac{∂^{α+β}ρ}{∂z^{α}∂z̄^{β}} (x)\frac{h^{α}}{α!} \frac{k̄^{β}}{β!} .

For example, in the case of the unit ball B_{n+1} ⊂ ℂ^{n+1}, the Szego kernel is given by

K_{∂B}(z, w) = \frac{1}{(1 − ⟨z, w⟩)^{n+1}} = ∫_{0}^{∞} e^{itψ_{∂B}(z,w)}t^{n}dt

with ψ_{∂B}(z, w) = 1 − ⟨z, w⟩.

The principal term s_{0}(x, x) on the diagonal was determined in [B.S (4.10)], using that Π is a projection:

s_{0}(x, x)dμ(x) = \frac{1}{4π^{m}} (det L_{X})||dρ||dx  (11)

where L_{X} = L_{ρ}|_{T′⊕T′′∩T X} is the restriction of the Levi form to the maximal complex subspace of T X.

3. Proof of Theorem (1.1) and Corollary (1.2)

The proof of Theorem 1.2 begins by lifting everything to X. As above, we fix an orthonormal basis {S_{i}^{N}} of H^{0}(M, L^{N}) and lift it to abasis {Ŝ_{i}^{N}} of H_{N}^{2}(X). The following proposition is straightforward and we refer to [Z1] for the proof:

Proposition 3.1. {Ŝ_{i}^{N}} is an orthonormal basis of H_{N}^{2}(X). Moreover, ||S_{j}^{N}(z)||_{h_{N}}^{2} = |Ŝ_{i}^{N}(x)|^{2} for any x with π(x) = z.

The following lemma links our problem to the Szego kernels:

Lemma 3.2. \frac{\sqrt{−1}}{2πN}Φ_{N}^{∗}ω_{F S} = ω_{g} + \frac{\sqrt{−1}}{2πN} ¯∂_{b}∂_{b} log Π_{N}(x, x).

Proof We first observe that

\frac{\sqrt{−1}}{2πN}Φ_{N}^{∗}ω_{F S} = ω_{g} + \frac{\sqrt{−1}}{2πN} ¯∂∂ log(∑_{j=0}^{d_{N}} ||S_{j}^{N}(z)||_{h_{N}}^{2} )  (12)

which follows by writing ||S_{j}^{N}(z)||_{h_{N}}^{2} = a^{N}|f_{j}^{N}| and using that ¯∂∂ log a^{N} = −Nω_{g}. We lift this statement to X in the form:

π^{∗} \frac{\sqrt{−1}}{2πN}Φ_{N}^{∗}ω_{F S} = \frac{\sqrt{−1}}{2πN} ¯∂_{b}∂_{b} log(∑_{j=0}^{d_{N}} |Ŝ_{j}^{N}|^{2}).  (13)

Next we observe that

Π_{N}(x, y) = ∑_{i=0}^{d_{N}} Ŝ_{i}^{N}(x)Ŝ^{N∗}(y)  (14)

or, in local coordinates,

Π_{N}(z, θ, w, θ′) = a(z)^{\frac{N}{2}} a(w)^{\frac{N}{2}} e^{iN(θ−θ′)} ∑_{i=0}^{d_{N}} f_{i}^{N}(z)f̄_{i}^{N}(w).  (15)

Hence we have

(a) ∑_{j=0}^{d_{N}} ||S_{j}^{N}(z)||_{h_{N}}^{2} = Π_{N}(z, 0, z, 0)
(b) \frac{\sqrt{−1}}{2πN}∂ ¯∂log(∑_{j=0}^{d_{N}} ||S_{j}^{N}||_{h_{N}}^{2} ) = \frac{\sqrt{−1}}{2πN} ¯∂_{b}∂_{b} log Π_{N}(z, 0, w, 0).

(16)

This completes the proof.

The projections Π_{N} are Fourier coefficients of Π and hence may be expressed as:

Π_{N}(x, y) = ∫_{0}^{∞} ∫_{S^{1}} e^{−iNθ}e^{itψ(r_{θ}x,y)}s(r_{θ}x, y, t)dtdθ  (17)

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

Π_{N}(x, y) = N ∫_{0}^{∞} ∫_{S^{1}} e^{iN(−θ+tψ(r_{θ}x,y)}s(r_{θ}x, y, tN)dtdθ.  (18)

From the fact that Imψ(x, y) ≥ C(d(x, X) + d(y, X) + |x − y|^{2} + O(|x − y|^{3}) (see [B.S]) it follows that the phase

Ψ(t, θ; x, y) = tψ(r_{θ}x, y) − θ.  (19)

has positive imaginary part. Hence the integral is a complex oscillatory integral. Before analysing its asymptotics we simplify the phase. As above, we choose a local holomorphic co-frame e_{L}^{∗}, put a(z) = |e_{L}^{∗}|_{z}^{2}, and write ν ∈ L_{z}^{∗} as ν = λe_{L}^{∗}. In the associated coordinates (x, y) = (z, λ, w, μ) on X × X we have:

ρ(z, λ) = a(z)|λ|^{2}, ψ(z, λ, w, μ) = \frac{1}{i}a(z, w)λμ̄  (20)

where a(z, w) is the almost analytic function on M × M satisfying a(z, z) = a(z). On X we have a(z)|λ|^{2} = 1 so we may write λ = a(z)^{− \frac{1}{2}} e^{iφ}. Similarly for μ. So for (x, y) = (z, φ, w, φ′) ∈ X × X we have

ψ(z, φ, w, φ′) = \frac{1}{i}(1 − \frac{a(z, w)}{i\sqrt{a(z)}\sqrt{a(w)}})e^{i(φ−φ′)}.  (21)

On the diagonal x = y we have ψ(r_{θ}x, x) = \frac{1}{i}(1 − \frac{a(z,z)}{a(z)}e^{iθ}) = \frac{1}{i}(1 − e^{iθ}). So

Ψ(t, θ; x, x) = \frac{t}{i}(1 − e^{iθ}) − θ.  (22)

We have

\begin{matrix}d_{t}Ψ = \frac{1}{i}(1 − e^{iθ}) \\ d_{θ}Ψ = te^{iθ} − 1 \end{matrix}  (23)

so the critical set is C = {(x, t, θ) : θ = 0, t = 1}. 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 is given by L_{Ψ} = ⟨(Ψ′′(1, 0)^{−1}D, D⟩ = 2\frac{∂^{2}}{∂t∂θ} − i\frac{∂^{2}}{∂t^{2}} . It follows by the stationary phase method that

Π_{N}(x, x) ∼ N \frac{1}{\sqrt{det(NΨ′′(1, 0)/2πi)}} ∑_{j,k=0}^{∞} N^{m−k−j}L_{j}s_{k}(x, x)  (24)

for various differential operators L_{j} of order 2j More precisely, for any R ≥ 0, one has (cf. [Ho I, Theorem 7.7.5])

|Π_{N}(x, x) − N \frac{1}{det(NΨ′′(1, 0)/2πi)} ∑_{j+k<R}^{∞} N^{m−k−j}L_{j}s_{k}(x, x)| ≤ CN^{m−R} ∑_{k<R,|α|≤2R−2k} ||D^{α}s_{k}||_{∞}. (25)

The expansion can be differentiated any number of times. After some rearrangement, the series has the form

Π_{N}(x, x) = N^{m}s_{0}(x, x) + N^{m−1}a_{1}(x, x) + …  (26)

where the coefficients s_{0}(x, x), a_{1}(x, x) … depend only on the jets of the terms s_{k} along the diagonal. From the description above of the leading coefficient s_{0}(x, x) we have:

Π_{N}(x, x)dμ(x) = \frac{1}{(2π^{m})}N^{m}α ∧ ω^{n} + O(N^{m−1}).  (27)

Relative to the canonical volume measure, the coefficient is a (non-zero) constant times N^{m}; it is determined by comparison to the leading coefficient of the Hilbert polynomial, completing the proof of the theorem.

3.1. Proof of Corollary (1.2). Because Φ_{N} is a CR map, the asymptotics of the derivatives follow immediately from the asymptotics of Π_{N}(x, x). Indeed, ¯∂_{b}∂_{b} log Π_{N}(x, x) = ¯∂_{b}∂_{b} log Π_{N}(x, y)|_{y=x}.

By (a) we have

log Π_{N}(x, x) = log(N^{m}s_{0}(x, x)[1 + N^{−1} \frac{s_{1}}{s_{0}} + …])

= m log N + log s_{0}(x, x) + log[1 + N^{−1} \frac{s_{1}}{s_{0}} + …]) = m log N + log s_{0}(x, x) + O(\frac{1}{N}). (28)

By differentiating the expansion we get

¯∂_{b}∂_{b} log Π_{N}(x, x) = ¯∂_{b}∂_{b} log s_{0}(x, x) + O(\frac{1}{N}) = O(1).  (29)

4. Zeros of random sequences of sections

We now turn to the distribution of zeros of random sequences of sections. Our purpose is to show that if we choose a sequence {s_{1}, s_{2}, … } at random from the space (S, μ), then the zero sets Z_{s_{N}} become uniformly distributed relative to ω.

A few remarks on the theorem before we begin the proof. First, it may seem to be rather tautologous that the limit distribution reproduces ω since the measure μ was defined in terms of ω in the first place. However, the result is not circular and in related settings the problem of determining the almost certain distribution of zeros is open. For instance, let us choose a (real-valued) spherical harmonic p_{N} of degree N at random from the unit sphere SV_{N} of the space of Nth order real spherical harmonics V_{N} ⊂ L^{2}(S^{n}, IR) and look at its zero set Z_{p_{N}}, a real hypersurface in S^{n}. Does the zero (nodal) set of almost any sequence {p_{N}} become uniformaly distributed relative to the volume measure as N → ∞? The answer is probably ‘yes’ but has not (at this time) been proved; the difficulty than is that the formula for the δ function on Z_{p_{N}} is more complicated than in the holomorphic case.

Now let us turn to the proof. What we do is calculate the expected values and variances of the random variables ([Z_{s}], φ). Corollary (1.2) determines the expected value and it turns out that the variances are so small that one can immediately deduce almost everywhere convergence. Somewhat surprisingly it is not even necessary to use the strong law of large numbers.

4.1. Expected distribution of zeros. We first determine the expected value of the normalized zero divisor Z̃_{s} as s is chosen at random from the unit sphere

SH^{0}(M, L^{N}) := {s ∈ H^{0}(M, L^{N}) : |s| = 1}

(or equivalently as [s] ∈ ℙH^{0}(M, L^{N}) is chosen at random with respect to the Fubini-Study volume). As above, we fix one orthonormal basis {S_{j}^{N}} of H^{0}(M, L^{N}) and write S_{j}^{N} = f_{j}e_{L}^{N} relative to a holomorphic frame (= nonvanishing section) e_{L}^{N} over an open set U ⊂ M. Any section in SH^{0}(M, L^{N}) may then be written as s = ∑_{j=1}^{d_{N}} a_{j}f_{j}e_{L}^{N} with ∑_{j=1}^{d_{N}} |a_{j}|^{2} = 1. To simplify the notation we let f = (f_{1}, …, f_{d_{N}}) : U → ℂ^{d_{N}} (which is a local representation of Φ_{N}) and we put

∑_{j=1}^{d_{N}} a_{j}f_{j} = ⟨a, f⟩.

Hence

Z̃_{s}^{N} = \frac{\sqrt{−1}}{Nπ}∂ ¯∂log |⟨a, f⟩| .  (30)

We shall frequently use the notation E(Y) for the expected value of a random variable Y on a probability space (Ω, dμ), i.e. E(Y) = ∫_{Ω} Y dμ.

We view Z̃_{s}^{N} as a D′^{1,1}(M)-valued random variable (which we call simply a ‘random current’) as s varies over SH^{0}(M, L^{N}) regarded as a probability space with the standard measure, which we denote by μ_{N}. The expected distribution of zeros of the random section s is the current E(Z̃_{s}^{N}) ∈ D′^{1,1}(M) given by

(E(Z̃_{s}^{N}), φ) = ∫_{S^{2d_{N} −1}} (Z̃_{s}^{N}, φ) dμ_{N}, φ ∈ D^{m−1,m−1}(M),  (31)

where we identify SH^{0}(M, L^{N}) with the unit (2d_{N} + 1)-sphere S^{2d_{N} −1} ⊂ ℂ^{d_{N}}. In fact, we have the following simple formula for the expected zero-distribution in terms of the map Φ_{N}:

Lemma 4.1 ([Sh.Z], Lemma 4.1). For N sufficiently large so that Φ_{N} is defined, we have:

E(Z̃_{s}^{N}) = \frac{1}{N} Φ_{N}^{∗}ω_{FS}

Proof For simplicity, write ω_{N} = \frac{1}{N} Φ_{N}^{∗}ω_{FS}. Then we have:

ω_{N} = \frac{\sqrt{−1}}{2πN}∂ ¯∂log ∑_{j=1}^{d_{N}} |f_{j}^{N}|^{2} = \frac{\sqrt{−1}}{2πN}∂ ¯∂log |f|^{2},  (32)

where f = (f_{0}, …, f_{d_{N}}). Let φ be a smooth (m − 1, m − 1) form, which we shall refer to as a ‘test form’. We may assume that we have a coordinate frame for L on Support φ. By (30), we must show that

\frac{\sqrt{−1}}{πN} ∫_{S^{2d_{N} −1}} ∫_{M} ∂ ¯∂log |⟨a, f⟩| ∧ φdμ_{N}(a) = (ω_{N}, φ) .  (33)

To compute the integral, we write f = |f|u where |u| ≡ 1. Evidently, log |⟨a, f⟩| = log |f| + log |⟨a, u⟩|. The first term gives

\frac{\sqrt{−1}}{πN} ∫_{M} ∂ ¯∂log |f| ∧ φ = ∫_{M} ω_{N} ∧ φ.  (34)

To complete the proof it suffices to show that the second term equals zero. But it is given by

\frac{\sqrt{−1}}{π} ∫_{S^{2d_{N} −1}} ∫_{M} ∂ ¯∂log |⟨a, u⟩| ∧ φdμ_{N}(a) = \frac{\sqrt{−1}}{π} ∫_{M} ∂ ¯∂ [∫_{S^{2d_{N} −1}} log |⟨a, u⟩|dμ_{N}(a)] ∧ φ = 0, (35)

since the average ∫ log |⟨a, ω⟩|dμ_{N}(a) is a constant independent of u for |u| = 1, and thus the operator ∂ ¯∂ kills it.

Combining Corollary 1.2 and Lemma 4.1, we obtain:

Corollary 4.2. E(Z̃_{s}^{N}) = ω + O(\frac{1}{N}); i.e., for each smooth test form φ, we have

E(Z̃_{s}^{N}, φ) = ∫_{M} ω ∧ φ + O(\frac{1}{N}) .

4.2. Variance estimate. The following estimate of the variance proved in [[Sh.Z], Lemma 4.3] is the key step in the proof of Theorem 1.3. The method of proof is somewhat similar to the calculation of the expected value.

Lemma 4.3. Let φ be any smooth test form. Then

E (|(Z̃_{s}^{N}, φ) − (ω_{N}, φ)|^{2}) = O(\frac{1}{N^{2}} ).

4.3. Almost everywhere convergence. We can now complete the proof of Theorem 1.3. An element in S will be denoted s = {s_{N}}. Since

|(Z̃_{s_{N}} , φ)| ≤ (Z̃_{s_{N}} , ω^{m−1})∥φ∥_{C^{0}} = c_{1}(L)^{m}∥φ∥_{C^{0}} ,

by considering a countable C^{0}-dense family of test forms, we need only consider one test form φ. By Lemma 1.2, it suffices to show that

(Z̃_{s_{N}} − ω_{N}, φ) → 0 almost surely .

Consider the random variables

Y_{N}(s) = (Z̃_{s_{N}} − ω_{N}, φ)^{2} ≥ 0 .  (36)

By Lemma 4.3,

∫_{S} Y_{N}(s)dμ(s) = O(\frac{1}{N^{2}} ) .

Therefore

∫_{S} ∑_{N=1}^{∞} Y_{N}dμ = ∑_{N=1}^{∞} ∫_{S} Y_{N}dμ < +∞,

and hence Y_{N} → 0 almost surely.

5. Ergodic eigenfunctions

Now we consider eigenfunctions of ergodic quantum maps. We recall that a quantum map is the quantization U_{χ,N} = Π_{N}σT_{χ}Π_{N} of a symplectic map χ_{o} of (M, ω). To be quantizable, χ_{o} must lift to a contact transformation χ of (X, α). We can then define the the unitary translation operator T_{χ} by χ on L^{2}(X). It commutes with the S^{1} action since it is lifted from the base. Unless χ_{o} is a holomorphic map, T_{χ} will not preserve H^{2}(X) so to get an operator on H_{N}^{2}(X) we need to compress it by Π_{N}. This will not usually be unitary so we need to put in a symbol σ ∈ C^{∞}(M) to make it so (at least modulo smoothing operators). The quantum map U_{χ,N} is then a sequence of unitary operators which forms a semiclassical Toeplitz-Fourier integral operator. We refer to [Z3] for further details.

We call the quantum map U_{χ,N} ergodic if χ_{o} is an ergodic transformation of (M, ω). The following result, proved in [Z3], belongs to a long line of results originating in the work of A. Shnirelman [Shn1] in 1974 on eigenfunctions of the Laplacian on compact Riemannian manifolds with ergodic geodesic flow.

Theorem 5.1. [Z3] Let {S_{j}^{N}} be an orthonormal basis of eigenfunctions of an ergodic quantum map U_{χ,N} on H^{0}(M, L^{N}). Then there is a subsequence {S_{j_{k}}^{N}} of density one such that ||S_{j}^{N}(z)||_{h_{N}}^{2} → 1 in the weak sense that for any open set U whose boundary has measure zero, ∫_{U} ||S_{j}^{N}(z)||_{h_{N}}^{2} dV → vol(U).

We now prove Theorem 1.4 by showing that if ||S_{j}^{N}(z)||_{h_{N}}^{2} → 1 then [Z_{S_{j}^{N}}] → ω. We write

u_{N} = \frac{1}{N} log ∥s_{N}(z)∥_{h_{N}} .

The main point of the proof is:

Lemma 5.2 ([Sh.Z], §5). If ||S_{j}^{N}(z)||_{h_{N}}^{2} → 1 then u_{N} → 0 in L^{1}(M).

Granted the lemma, it follows by the Poincaré-Lelong formula (5) that for any smooth test form φ ∈ D^{m−1,m−1}(M),

(\frac{1}{N} Z_{N} − ω, φ) = (u_{N}, \frac{\sqrt{−1}}{π}∂ ¯∂φ) → 0 ,

proving the theorem for smooth forms. Since by(6),

(\frac{1}{N} Z_{N}, φ) ≤ \frac{c_{1}(L)^{m}}{m!} sup |φ| ,

the conclusion extends to all C^{0} test forms φ.

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

  E-mail address: zel@math.jhu.edu